A Model of Robust Positions in Social Networks

Forthcoming: American Journal of Sociology

Matthew S. Bothner, University of Chicago

Edward Bishop Smith, University of Michigan

Harrison C. White, Columbia University

Word Count (including notes): 15,838

We have benefited from the advice of Peter Bearman, Ron Breiger, Ron Burt, Tim Conley, James Evans, John Levi

Martin, Mark Mizruchi, and John Padgett, as well as seminar participants at the University of Chicago, Columbia
University, the Tuck School of Business, and the Yale School of Management. Financial support from the University
of Chicago Booth School of Business and from the Ewing Marion Kauffman Foundation is gratefully acknowledged.
We thank Val Burris for generously making data available on the sociology PhD exchange network. Correspondence
may be directed to Matt Bothner at mbothner@chicagobooth.edu

A Model of Robust Positions in Social Networks

Abstract:

What are features of robust rather than fragile social positions? This article introduces a network

model that pictures occupants of robust positions as recipients of diversified support from durably

located others, and portrays occupants of fragile positions as dependents on tenuously situated

others. The model extends Herfindahl’s index of concentration by bringing in the recursivity of

Bonacich’s (1987) method. Using Newcomb’s panel study of status-conferring flows among

members of a college fraternity, we find empirical support for the contention that fragility reduces

future growth in status. Extensions of the model both to input-output networks among industries

in the U.S. economy and to hiring networks among academic departments are presented.

Implications for future research are discussed.

What are features of robust rather than fragile social positions? Various approaches have

long equated robust positions with diversification, and fragile, vulnerable ones with

concentration. For instance, the early treatments of Emerson (1962) and Blau (1964) pictured

occupants of fragile positions as those who depend primarily on just one other entity in a social

network. Since then, many investigations have discussed both the hazards of relying on a single

source and the advantages that result from ties to a wide range of others. Examples of this theme

appear in the core tenets of resource dependency theory (Pfeffer and Salancik 1978), ecological

conceptions of niche width as a shelter from environmental change (Hannan and Freeman 1989;

Peli 1997), and network-theoretic accounts of the social structures that sustain malleability and

preserve options in complex games (Padgett and Ansell 1993). It is thus well understood that the

range of an actor’s ties in a network is an important determinant of the robustness of its position.

We develop a conception of robustness and fragility that builds on these earlier lines of

research but also departs from them in an important respect. While harnessing notions of

diversification and concentration, we also draw on the concepts of “decoupling” and “coupling”

developed in White’s (2008) theory of identity and control to allow for mutual influence among

social actors’ identities and networks. We use the term decoupling to refer to clean breaks and

sharp distinctions between social actors, as when consumers’ perceptions of an industry are

unaffected by the practices of its overseas suppliers. Conversely, by coupling we refer to close

connections that yield consequences beyond those of mere contact, as when an executive’s style

of leadership spills over and colors others’ opinions of her direct reports (White 2008; cf. White

2002, pp. 200-220). In our view, to be coupled with another actor is not simply to have contact,

exchange, or a social tie with that actor. It is also to be intertwined with that actor’s identity and

network.

Goffman’s (1968, pp. 30, 47) account of stigmatization and its diffusion offers acute

examples of this intertwining. According to his view, “the loyal spouse of the mental patient, the

daughter of the ex-con, the parent of the cripple, the friend of the blind, the family of the

hangman, are all obliged to share some of the discredit of the stigmatized person to whom they

are related [and] persons who acquire a degree of stigma in this way can themselves have

connections who acquire a little of the disease twice-removed. … The issue is that in certain

circumstances the social identity of those an individual is with can be used as a source of

information concerning his own social identity, the assumption being that he is what the others

are.” Thus, as long as social ties are detectable by third parties, in networks of intangible flows—

like the ones Goffman (1968) described, or the more familiar flows of recognition and esteem

among scientists described by Merton (1968)—coupling equates to the influence of another’s

identity on one’s own. And in networks of tangible flows, such as goods and services moving

among production markets, coupling occurs when the position of a given market is affected by

the exchange patterns of contiguous markets.

We contend that a conceptualization of robustness—or fragility, its theoretical

complement—must take into account concentration and coupling. We define the occupant of a

fragile position as one who is both highly dependent on and closely coupled with others who are

themselves tenuously located in social structure. An example of a fragilely positioned nonprofit

is one whose support comes exclusively from an unpopular politician (Hannan 1998:149-50). An

example of a fragilely positioned company is one whose footing resides solely in a precariously

undiversified market (cf. White 2002:246-7). We view fragility as dependency on dependents,

and robustness as diversification across the diversified.

Consistent with this view, in what follows, we develop a network model of fragility and

robustness that jointly takes concentration and coupling into account. We then illustrate its

applicability in three empirical settings: our primary setting is Newcomb’s (1961) study of a

college fraternity, where we examine the effect of fragility on growth and decline in status; next,

in an extension that permits us to examine our model’s implications in a network of flows

unrelated to social status, we assess the effect of fragility on the performance of industries tracked

by the U.S. Department of Commerce’s Bureau of Economic Analysis (Moyer et al. 2004); we

then return to status-related themes by investigating the effect of fragility on academic

departments’ prestige, using Burris’s (2004) data on the sociology PhD exchange network.

Our aims, which are theoretical as well as methodological, integrate several streams of

earlier research. We move toward our primary theoretical aim—to bring into focus the status-

eroding effects of fragility in social systems—by drawing from Abbott’s (2001) etiology of

status, Weber’s (1978) discussion of types of charisma, and Whyte’s (1993) ethnography of a

North Boston street gang. We work toward our methodological objective—to construct a general

measure of fragility of positions in networks—by drawing on Herfindahl’s measure of

concentration and melding it with the underlying mathematics of Bonacich’s (1987) measure of

status.

Theoretical Antecedents

Our starting point is Abbott’s (2001) observation that an individual’s status may turn

more significantly on the range of his or her current options than on a history of affiliations with

high-status institutions. According to this view, “it should not be assumed that the ultimate elite

career is one that hooks together all the perfect elite trajectories... The most elite individual may

be that person who maintains the largest number of possible future trajectories that s/he could

jump onto” (Abbott 2001, p. 247).

This insight carries several interrelated implications. One is that, although many

sociologists have long viewed status largely as the product of stable traits (Berger et al. 1977),

status may instead rise and fall in response to shifts in the durability of an actor’s location in

social structure. Another implication is that much of the growth and decline in status we observe

in social life may well have their most fundamental antecedents in audience members’ taste for

robustness and distaste for fragility. Abbott’s claim points directly to the appealing theoretical

possibility that possession of durable social support invites prestige-conferring reactions, whereas

occupants of fragile social positions are seen as illegitimate and therefore attract responses from

others that attenuate their social standing. We believe that deference accruing to durably situated

actors, as well as disdain for those in fragile roles (evident in diatribes like “he’s only hanging on

by a thread—cut him loose”) should figure centrally in our portrayals of status-based processes.

Cases of fragility’s corrosive effects also appear as far back as Weber’s (1978)

comparison of two distinct types of charismatic leaders. The first (most familiar) type of

charismatic leader is the extraordinary figure whose military exploits, intellectual powers, or

miraculous works elicit the deference of a broad community of followers. This individual,

standing on a wide base of support, inhabits what for us is a robust position. Conversely, for

Weber, the second type is the person whom the first type eventually (with age) consecrates as his

or her successor. The second type’s social standing therefore rests perniciously on the

sponsorship of the original charismatic leader. Unlike the “pure” charisma of the predecessor,

mere “hereditary charisma” or “lineage charisma” (Weber 1978, pp. 248, 1135-41) marks the

successor’s role—a role not nearly as legitimate as that of the first type. This second type is

fragilely situated, having been chosen by the original, now deceased leader, and frequently

evokes suspicion and conflict among prospective followers—those whose allegiance the first type

garnered with relative ease. Weber’s distinction is important because it brings into focus the

unfavorable perceptions that encircle and damage occupants of fragile social roles. With fragility

often come reactions that erode legitimacy, such as Quakers’ epithetical descriptions of

successors as “hirelings” (Weber 1978, p. 249).

Whyte’s ([1943] 1993) celebrated ethnography of Boston’s North End, especially his

analysis of the Nortons street gang, offers further examples. In particular, two components of

Whyte’s qualitative account—his description of the Nortons’ bowling match and its competitive

aftermath, and his portrayal of the realignment of the gang after its leader fell from grace—neatly

illustrate two main pieces of our argument: that fragility invites status-reducing penalties, and that

an actor is increasingly fragile when the sources of that actor’s social support are themselves

vulnerable. We draw on this case in detail because of its depth and relevance.

We begin by turning in figure 1 to Whyte’s (1993, p. 13) original hand-drawn sociogram

of the Nortons. Of the thirteen depicted, three are most relevant for our theory: Doc, the gang’s

leader, Alec at the bottom, and, most significantly, Long John, who while enjoying a moderate

level of prestige nonetheless stood on fragile social ground. With weak ties to most Nortons,

Long John’s social support was strongly concentrated in his relations with Doc, Danny, and

Mike. According to Whyte (1993, p. 12), “Long John was in an anomalous position … his

friendship with the three top men gave him a superior standing. As Doc explained: ‘It’s because

we’ve always catered to Long John. When we go somewhere, we ask Long John to go with us.

We come up to him and slap him on the back. We give him so much attention that the rest of the

fellows have to respect him.’ Nevertheless, he had little authority over the followers.”

The social consequences of Long John’s fragility within the structure of the gang were

most visible as Alec (positioned at the bottom of the sociogram in figure 1) singled out Long John

for one-on-one competition after a high-stakes intragang bowling match (Whyte 1993, pp. 20-24).

In the intragang match, individuals’ bowling scores ended up being virtually identical to their

locations in the status ordering. According to Whyte, it was normative for high-status gang

members to heckle their lower-status rivals, but not vice versa, and so there was a tendency for

each contestant to bowl in accordance with his status.

Nevertheless, the apparent equilibration of status and athletic performance was not left

unchallenged, particularly by Alec. In an effort to recover, Alec targeted Long John, daring him

to one-on-one bowling contests and beating him repeatedly. Additionally, “when bowling was

resumed in the fall, Long John became Alec’s favorite opponent, and for some time Alec nearly

always came out ahead. He gloated. Long John explained: ‘He seems to have the Indian sign on

me’” (p. 22). Summarizing Alec’s attack, Whyte drew the following conclusion, which nicely

illustrates the status-corrosive reactions occupants of fragile roles tend to elicit: “It is significant

that, in making his challenge, Alec selected Long John instead of Doc, Danny, or Mike. It was

not that Long John’s bowling ability was uncertain. His average was about the same as that of

Doc or Danny and better than that of Mike. As a member of the top group but not a leader in his

own right, it was his social position that was vulnerable” (p. 22). Alec persistently trounced Long

John in the bowling alleys, before Doc intervened and shored up Long John’s confidence.

In the future, Doc was not nearly as able to prop up Long John, however. And this fact

takes us to our second point—namely, that an individual’s position becomes more fragile if that

individual’s source of social support becomes less robust. Doc’s fall—and, due to their coupling

in the network, Long John’s heightened fragility—resulted from his failed foray into the North

End’s political ambit.

In his summary of Long John’s dilemma, Whyte (1993, p. 45) highlighted a central

component of the model of fragility we develop, specifically that the fragility of an actor’s

position is exacerbated when that actor’s source of social support—in this case, Danny, Mike, and

especially Doc—grows more vulnerable: “Long John divided his time between Spongi’s and the

Norton Street corner. The realignment left him in a vulnerable position. There were two groups

that hung around Spongi’s “joint”: the inner circle and the hangers-on. Spongi included his

brother, Danny, Doc, and two others in the inner circle. When he went for “coffee-ands,” for a

drive, or to the movies, he would invite them to accompany him. He did not include Long John in

his invitations, so Long John was excluded from the inner circle. Without the support of Doc,

Danny, and Mike, he had no standing among the boys who remained on Norton Street, and he did

The Nortons grew disillusioned with Doc and underwent sharp social-

structural changes after the summer of 1938 (Whyte 1993, pp. 42-51). Although there were

several exits, entries, and shifts in stayers’ positions, the most jugular realignments, particularly

for Long John, followed Doc’s repositioning. When Doc’s position became less robust—Doc

had lost much of the support of the Nortons and had grown dependent on Spongi, a new player,

not included in figure 1—Long John became nearly an isolate.

Although this second point concerns our conceptualization, rather than the expected consequences, of fragility, it does

imply a corollary prediction—namely, that Alec would have been even more likely to challenge Long John in the
bowling alleys had the social positions occupied by Long John’s supporters grown less robust.

not know where to go.”

Measurement

The distinctiveness of social positions resembling the one occupied by Long John brings

us to a more formal description of our measurement of fragility and robustness. We start with the

Herfindahl index to account for concentration and then bring in the recursivity of Bonacich’s

(1987) method to account for coupling.

The Herfindahl index has long been used to assess industry concentration, where it is

written as the sum of firms’ squared market shares (Schmalensee 1989, pp. 966-7; McLean and

Padgett 1997). It has also been used to quantify religious homogeneity within a geographical area

(Ellison et al. 1997). We take it as our point of departure for measuring fragility because it aptly

captures the degree to which one node in the network depends on a narrow base of alters rather

than facing diversification across others. More specifically, if we consider a hypothetical

relational matrix

, where

 

=  

, and

denotes the flow of respect

(endorsements, recognition, esteem, deference, liking, support, precedence, or honor) directed

from individual

, then we can express the Herfindahl index for members of the social

network summarized by

very simply as:

−

∑

(1)

where the important transformation is from

as follows:

−





= 







∑

(2)

When applied to relational matrices like

, the Herfindahl index thus involves these

straightforward steps: divide each entry through by its row sum, square the resulting proportions,

and then collect the row sums for the new matrix of squared proportions. If

equals one,

then receives all of his or her respect or social support from just one other actor in the network.

Under this scenario,

is completely dependent. If, by contrast, the index for a given actor equals

(

)

−

, that actor is then minimally dependent, and his or her position is consequently far less

fragile. The Herfindahl index captures the concentration of sources of incoming flows and

therefore serves as a constructive baseline for measuring the fragility of social positions.

Used by itself, however, the Herfindahl index has limitations. When applied to a given

social network, it considers only the distribution of an actor’s immediate ties, making no

allowance for the identities and networks of that actor’s contacts to exert influence. The use of

Herfindahl’s measure thus assumes complete decoupling by construction because it contains no

provision for network spillovers. Although full decoupling—that is, the complete absence of

network spillovers—may accurately describe some social structures, it likely mischaracterizes

most networks of interest to sociologists. Consequently, to build coupling into our measure of

fragility, we turn to a salient feature of Bonacich’s (1987) status measure, whose structure we

now summarize as the backcloth of our approach.

Using again a hypothetical relational matrix

as our reference point, Bonacich

(1987:1172-75) recursively defines the status of

as:

( )

(

)

α,β

α+β

S x

∑

(3)

where

again denotes the respect

directs to

is the status of

is a scaling constant,

and

is a parameter allowing for the incorporation of White’s (1992) notion of coupling in

networks (cf. Bonacich 1987: 1174). The intuition underlying the measure hinges on

. When

equals zero,

i s

status is just the sum of the flows of respect directed to

from the other actors

in the network—

i s

row sum in

adjusted by the scaling constant

, in other words. As

rises, flows of respect carry greater force when they emanate from higher-status actors. To the

extent that

exceeds zero, the status of

affects the status of

through

j s

respect for

Used widely in empirical research, and increasingly in formal models (Hannan et al. 2003;

Ballester et al. 2004; Bothner et al. 2010), Bonacich’s (1987) measure thus has the particularly

appealing feature of permitting the researcher to operationalize status as a state enjoyed by those

who are highly regarded by highly-regarded others. That is, it folds into the computation of an

actor’s status score an added boost insofar as that actor receives respect (endorsements,

recognition, esteem, deference, liking, support, precedence, or honor) from those who themselves

receive respect (see also Abbott 1981). It is therefore sensitive to the identity or source—not just

the count or intensity—of the endorsements an actor receives.

Using this feature of Bonacich’s (1987) approach in the context of measuring fragility is

straightforward. To do so, we shift attention from totals (Bonacich’s row sums, collected from

) to dispersions (Herfindahl’s summed squared proportions, taken from

), while allowing for

coupling in the network. More specifically, we express our measure of fragility as follows, where

represents the fragility of actor

( )

(

)

a b

a bF

∑

(4)

The core elements in equation (4) are

and

. Recall that, in discussing Herfindahl’s index,

we set

−





= 







∑

in equation (2), so that

was the squared proportion of

j s

respect

for

. Summarizing Bonacich’s measure, we also stressed the salience of the

parameter,

underscoring its affinity with the notion of coupling. The parameter

in equation (4) for

fragility is the analog of

for status in (3), and

is a scaling constant. Since

denotes the fragility of actor

determines the extent to which the fragility of

shapes the

fragility of

A high value of

, which corresponds to coupling in the social structure, is

consistent with connected actors’ identities and networks mattering discernibly. According to

equation (4), an actor is fragile to the extent that a large proportion of his or her respect comes

from those who are themselves dependent.

To view coupling among fragility levels from another vantage point, we rewrite equation

(4) as an infinite sum, where

is an entry in the matrix

( )

k=0

a b

∞

∑

(5)

For equation (5) to converge,

−

must hold, where

equals the largest normed eigenvalue

Given this constraint, we write the relationship between and

as follows, where

denotes a coupling coefficient.

−

= ⋅

(6)

Thus, for

, full decoupling is taken to characterize the relevant network, and fragility

therefore correlates perfectly with the Herfindahl index. Under

, in other words, fragility

reduces to concentration. Conversely, as

→

, each actor’s fragility score

is increasingly

affected by the fragility score

of those on whom

depends.

The substantive interpretation of

will of course vary across different kinds of networks.

When analyzing a network through which intangible resources circulate, such as the Nortons

street gang, a researcher would likely select a high value of

when the members of the network

attach significance to instances in which individuals are “propped up” by others who are in turn

The solution (readily computed with standard packages) to equation (5) in matrix form is

( )

(

)

a b

−

, where

is an identity matrix (cf. Bonacich 1987, eq. 4).

weakly positioned. Stated more generally,



accommodates settings where actors’

identities seep out and affect one another. A promotion tournament—for instance, one

comprising junior executives racing one another for a senior position—also fits this image.

There, gossip likely mars contestants whose strongest endorsements issue from the most weakly

positioned vice chairmen. Consequently, just as the parameter

for status “can be thought of as

a radius” inside of which the analyst chooses to evaluate the status formation process (Bonacich

1987, p. 1174), the coupling coefficient

from equation (6) could correspond to the weight

applied to contiguous actors’ identities when quantifying fragility.

Conversely, in a network through which tangible resources flow, such as the network of

loans and business opportunities tying together the Florentine families trenchantly analyzed by

Padgett and Ansell (1993), one would choose a large value of

when measuring fragility insofar

as economic shocks reverberate over long distances through the chains of the network. Under

such conditions, assuming that the goal for families is economic and political survival, it would

be especially risky for a given family to depend appreciably on a small set of counterparts whose

own alliances are sparse. Therefore, for networks of tangible flows, the coupling coefficient

from (6) could correspond to the weight given to adjacent actors’ economic networks.

With these descriptions in place, we can now express robustness as the complement of

fragility. That is, the occupant of a robust position is one who is not dependent on dependents,

but is anchored solidly in multiple nodes that are in turn durable constituents of the network. We

define robustness by linearly transforming (5) as follows.

Although it may be more accurate to refer to our measure as “structural robustness” or “relational robustness” to

distinguish it from the separate ideational component of robustness (see Moody and White’s [2003, esp. p. 104]
discussion), we have opted for the simplest label. In addition, we choose the scaling constant

so that the sum of

squared fragility scores equals the number of actors in the matrix (cf. Bonacich 1987, p. 1173). We could also of
course substitute in place of 1 , as the minuend, a column vector of constants equal to the largest fragility score plus
the smallest fragility score, in order to linearly convert fragility scores into robustness scores. Using the max plus the
min has the benefit of making the robustness distribution a mirror image of the fragility distribution, with the same
endpoints and variance.

( )

k=0

a b

∞

= −

∑

(7)

Positions and Dynamics in Newcomb’s Fraternity

We turn to data from Newcomb’s (1961) panel study of a college fraternity to pursue two

main objectives: Our first aim is to depict simply and visually the distinctive features of our

measure of fragility introduced in equation (4); our second aim is to empirically test our

theoretical proposition that fragility is status-eroding. Newcomb’s study tracked 17 male college

students who were recruited to live expense-free for a semester-long period in a fraternity-style

house. Each week, for fifteen weeks, the students were required to rank one another from 1 to 16

according to likeability or “favorableness of feeling.” Newcomb and his colleagues utilized the

resulting data to draw inferences about friendship formation in small groups. The data, which can

be represented as a series of interpersonal networks, have since been used in several studies

(White, Boorman, and Breiger 1976; Doreian et al. 1996; Gould 2002; Moody, McFarland, and

Bender-deMoll 2005) and are useful for the present study for at least four reasons.

First, as a small-scale social structure, Newcomb’s fraternity serves as a particularly

transparent site for bringing relief to differences between fragile and robust social positions in

cross-section (consistent with our first aim). Second, it allows us to compute time-varying levels

of fragility and status (keeping with our second aim). The regularly collected “favorabless of

feeling” appraisals convert easily into asymmetric matrices suitable for the construction of time-

changing status and fragility scores. With them, we can assess the impact of fragility on future

adjustments in status, net of the current level of status, allowing us to assess our expectation that

fragility induces status decline. Third, Newcomb’s data allow us to disentangle the effects of

fragility on status growth and decline from those of individuals’ time-constant characteristics.

Fourth, participants in the experiment did not know one another before its inception, because they

were transfer students. This feature of the research design permits us to focus exclusively on

dynamics that occur within, rather than starting before, the semester-long window. Before

turning to dynamic models, we begin by visualizing differences in the social positions of

fraternity members in the first week of the study.

Cross-sectional differences in social positions . Our starting point is an examination of

the first of fifteen 17-by-17 weekly relational matrices

that summarize the social structure of

the fraternity. Cell

ijt

equals 16 if, in week

, individual

gave

a ranking of 1 in

terms of likeability, and equals 1 if

gave

a ranking of 16. Reverse-coding thus transforms

the initial rankings into positive flows from columns to the rows of

. For example, according

to the first weekly relational matrix

depicted in table 1, individual 2 gave individual 3 the

least favorable possible ranking, while individual 4 dispensed to individual 2 the most favorable

ranking attainable. To each of the fifteen weekly matrices, we applied Bonacich’s (1987) status

measure, setting

equal to ¾ of the inverse of the largest normed eigenvalue of

(Podolny

2005). Using this approach, a high-status individual is the target of favorable sentiments from

others who are themselves favorably appraised. We also used

( )

a b

from equation (5) to

compute individuals’ time-varying fragility scores. Because peer monitoring in close living

quarters likely made fraternity members intensely aware of the nature of one another’s social ties,

we chose to assign to the coupling coefficient

its maximum possible value of

.99

Using the relational matrix

in table 1, the plot of status against fragility in figure 2

offers a portrayal of individuals’ social positions in week 1 of the study. The most apparent

feature of figure 2 is of course that more fragilely positioned individuals filled subordinate status

However, in panel models that supplement those we subsequently discuss, we also explored results using five distinct

levels of c —in particular,

{

}

0, .25, .5, .75, .99

∈

. We took this approach to evaluate our assertion that coupling is a

central component of fragility, while also assessing our anticipation of a negative effect of fraternity members’ fragility
levels.

positions in the network. In figure 2, individual 17 inhabits an especially high-status and robust

position, unlike individual 8, who is low in status and highly fragile.

More interesting is the fact

that some fraternity members, such as 13 and 16, are near status-equivalents, but differ

considerably in the fragility of their positions. That is, individuals 13 and 16 offer an example of

how actors may enjoy equivalent amounts of recognition from esteemed others, while one

depends disproportionately on just a few sponsors (as did Long John, going back to figure 1),

while another stands on a much wider base of support (as did Nutsy—Long John’s close

counterpart in figure 1). More generally, differences in fragility among status-equals is not unlike

a case of two towers of equal height that nevertheless stand on markedly dissimilar foundations.

We further explore differences in social foundations in figures 3 and 4—where we depict the

positions of individuals 13 and 16, respectively—to highlight distinctive features of our

approach.

The raw data in

offer a valuable initial vantage point. An inspection of rows 13 and

16 of table 1 gets quickly to the main analytical difference between status and fragility. Very

simply, the measurement of status begins with computing row sums, whereas the measurement of

fragility starts with calculating dispersions. It is for this reason that status and fragility are

analytically distinct summary statistics conveying unique information about actors’ social

positions. Corresponding to the comparable heights of individuals 13 and 16 on the vertical axis

Although our panel models predict the growth (rather than level) of future status as a function of current fragility net

of current status, we also simulated versions of Newcomb’s week 1 matrix to ensure that far weaker cross-sectional
correlations between status and fragility are plausible. Using these alternative matrices (again with seventeen nodes,
each ranking each other from 1 to 16 followed by reverse-coding), we not only observed correlations that were virtually
at zero, we also viewed individuals who were much more sharply positioned on the off-diagonal than 13 and 16 in
figure 2—and who offer leads for future research on fragility and status. More specifically, our simulations (not
reported, but available on request) brought relief to “drones” (that is, individuals receiving lukewarm support from
many, while lighting up no one) as inhabitants of low-status, low-fragility roles, while also highlighting “polarizers” as
occupants of high-status, high-fragility roles. We believe that a promising line of future research would involve
investigating contextual conditions that give rise to particularly low correlations between fragility and status. One such
context may be deeply contested settings where factions emerge and where polarizers deploy drones as their trusted
between-group emissaries.

An alternative, though broadly related, theoretical approach is brought forward in Mizruchi et al.’s (1986)

disaggregation of status scores into derived (generated by alters) and reflected (generated by ego) components.
Although their concept of derived status bears some resemblance to our concept of fragility, our approach differs from
theirs in that we construct a separate measure rather than present a decomposition of status.

in figure 2, 13’s row sum equals 120 and 16’s row sum is nearly identical—equaling 127. In

contrast, we see a discernible difference in the extent to which 13 and 16 face concentrated social

support: Individual 13’s incoming flows are noticeably lumpy—given 13’s relatively high

dependence on individuals 1 and 6—whereas flows directed at individual 16 are more “even.”

The greater fragility of individual 13’s position relative to that of individual 16 is also

apparent in a comparison of pictures of ego-networks we present in figures 3 and 4. Summarized

very briefly, figures 3 and 4 jointly convey that individual 13 is heavily dependent on particularly

fragile alters—specifically, individuals 8, 1, and 6—while individual 16 faces comparatively

distributed social support and is far less dependent on the most fragilely positioned alters in the

network. We chose line-widths to reflect relative levels of dependence and selected smaller

(larger) circles for the fragilely (robustly) situated alters in the focal individual’s ego-network.

Summarized in greater detail, figures 3 and 4 result from applying the two-step

transformation introduced in equation (2) to the relational matrix

in table 1. That is, we made

row-stochastic and then squared the resulting proportions, thereby converting

. We

collected fragility scores by applying

( )

a b

from equation (5) to

for all members of the

fraternity in week 1—precisely the scores arrayed along the horizontal axis in figure 2. Placing

individual 13 at the center of figure 3, we then arranged all other fraternity members clockwise

along its outer edge—in increasing order of 13’s level of dependence on each alter. More

specifically, we extracted and sorted individual 13’s row in

, arrayed 13’s alters accordingly,

and made the thickness of the connecting lines proportional to entries in row 13 in

. We then

embedded individual 13’s alters in circles whose radii are a decreasing function of fragility.

Code for generating figures 3 and 4 is available on request. We made line thickness a linear function of entries in

Consistent with figure 2, individual 17 resides in the largest circle, while individual 8’s circle is

and for added emphasis nonlinearly transformed alters’ fragility levels when selecting the radii of their surrounding
circles.

the smallest. We took exactly the same steps when constructing figure 4. Viewed together,

figures 3 and 4 are important because they highlight salient structural differences in the positions

of individuals occupying nearly the same level of status computed by Bonacich’s (1987) method.

In addition, figures 3 and 4 reflect our conception of fragility as a function of dependency

(captured by line-thickness) on dependents (summarized by alters’ circle-sizes).

Status growth and decline. Consistent with the observations of Abbott (2001), Weber

(1978), and Whyte (1993), we expect an individual to face status decline as that individual’s

social position becomes increasingly fragile. In particular, the mechanism we envision is one

where fraternity members develop negative impressions about—and thus rank less favorably—

those whose social footings are narrow and slippery. Although the precise mechanisms linking

fragility to outcomes will of course vary across empirical sites, we believe the one just proposed

is accurate for a network of young college students who are groping for a place in their world and

likely to esteem those who have secured durable standing.

We began our investigation of status dynamics by taking a sequence-analytic approach

(Abbott 1995; Abbott and Tsay 2000), which allows us to examine typical paths through the

status distribution. We proceeded by collecting the time series of status scores, across weeks 1

through 15, for all seventeen individuals in the study, and created a 17-by-17 matrix of pairwise

distances in these time series. In this matrix, cell

( )

i j

was large to the extent that

i s

fifteen-

week movement through the status distribution differed substantially from

j s

. We then grouped

individual trajectories into clusters using standard model-based clustering techniques (Banfield

and Raftery 1993). Three clusters emerged with roughly equal numbers of individuals allocated to

each. By averaging the fragility and status scores of all individuals in a cluster, we were able to

generate a characteristic trajectory for each cluster through the status and fragility state spaces

over time.

Figure 5 highlights these movements. The left-hand plot illustrates how a typical

individual’s level of fragility might vary across time for each cluster. The right-hand plot, by

contrast, shows week-to-week movements in status. Clusters are represented in each plot by a

marker: closed boxes (individuals 4, 5, 9, 14, 17), open boxes (3, 10, 15, 16), and circles (1, 2, 6,

7, 8, 11, 12, 13). Viewed together the plots indicate an association between fragility and status

decline. For example, the cluster showing the lowest fragility (closed boxes) maintains the

highest average level of status across the 15-week semester. Conversely, the cluster marked by

highest fragility in the left-hand plot (open squares) also drops most precipitously in status in the

right-hand plot.

We also estimated panel models predicting growth in status that take the form:

i t

−

+ +

(8)

where

and

denote the status and fragility scores,

1,...,17

and

1,...,14

. With lagged

status included on the right-hand side, equation (8) models dynamics of adjustment as a function

of time-changing levels of fragility.

We enter fixed effects represented by

to absorb invariant, subject-specific propensities

to fill fragile positions. These sweep out fraternity members’ levels of prior educational

attainment, charisma, intrinsic likeability, social skill (Leifer 1988), and all other stable traits that

plausibly channel sorting into fragile social positions (cf. Gould 2002, pp. 1143-4). Using a

fixed-effects specification thus eliminates all between-person variation, so that the coefficients

reflect within-person effects of fragility on growth in status. Additionally, as denoted by the

indicators

, we adjust for all types of overall temporal heterogeneity, such as aggregate levels

of reciprocity and transitivity, the eigenvalue from which fragility is computed, and all other

global properties of the network (Doreian et al. 1996; Moody, McFarland, and Bender-deMoll

2005). Since

enter jointly with

, the effect of age is also fully accounted for. Correlations

and descriptive statistics for variables in our analyses are included in table 2.

Table 3 presents six models predicting future status growth in Newcomb’s fraternity. We

present one-tailed tests for all continuous covariates given the clarity of our expectations about

the directions of the effects. We also present standardized regression coefficients because of their

usefulness in conveying effect magnitudes in models containing network-related covariates.

Model 1 contains just lagged status and fixed effects for individuals and periods as a baseline.

Inspection of the individual fixed effects (not reported to conserve space) confirms that variation

in fraternity members’ average levels of status growth is clearly pronounced, reinforcing the

merits of the within-estimator. Consistent with our expectations about the importance of

coupling, we find in model 2 that the effect of fragility (as Herfindahl’s measure, where complete

decoupling is assumed) is indiscernibly different from zero.

In model 3 we enter

it c

and

.99

it c

jointly. Whereas

it c

reduces to the

Herfindahl index,

.99

it c

assumes coupling, with adjacent individuals’ positions influencing the

fragility of a given individual’s position in social structure. In model 3, the coefficient on

it c

stays insignificant, whereas

.99

it c

exerts a discernibly negative effect. Through regressing

status growth on

it c

and

.99

it c

jointly, the estimate on

.99

it c

reflects the variation that is

orthogonal to

it c

. Thus, the persistence of the effect of

.99

it c

, where coupling is taken into

account, offers preliminary evidence to suggest that, at least in the present empirical setting,

network spillovers may matter more than immediate dependencies. In particular, this result is

consistent with the notion that identities of actors’ contacts matter more than concentration levels

of ego-networks. When estimating other models (available upon request), with

equal to .25, .5,

and .75, we also found that fragility is significant only at the highest possible level of coupling,

We report within-person correlations because our estimator focuses only on within-person variance.

.99

. This set of outcomes is important, because it suggests that taking coupling into account

is a necessary part of understanding the link between positional fragility and dynamics of

prestige. Model 4 confirms that the negative effect of fragility measured with coupling stays

significant when its counterpart is excluded from the set of covariates.

In models 5 and 6, we take two additional steps to assess further our main finding that

fragility negatively affects status growth, before turning to an application of our fragility measure

in a network of economic flows unrelated to status. First, we guard against the possibility that

behavioral correlates of fragility fuel the effect we observe. It is easy to imagine that (at least

some) occupants of fragile roles tend to act sycophantically, whereas occupants of robust

positions are more prone to feather displays and, in general, carry themselves with considerable

aplomb—much to their advantage, especially in a fraternity-style setting. We have argued,

building on prior sociological theory, that fragility undermines legitimacy and is therefore status-

eroding, whereas robustness solidifies legitimacy and is consequently status-enhancing. We

nonetheless address the counterpossibility that locations on an axis marked by fragility and

robustness are in fact undetectable by others in a network, making them by definition

inconsequential, in a direct sense, for processes of growth and decline in status. If this alternative

view is descriptive, then an easily tracked, behavioral correlate of fragility, such as sycophantic

conduct (cf. Burt 1976:104-9) may well fuel the effect we observe. Although the fixed effects

absorb time-constant intrinsic propensities to behave subserviently, conduct of this type may have

a time-changing component as well.

We constructed a weighted asymmetry measure to proxy individuals’ shifting

propensities to act obsequiously by working from the premise that a focal actor is sycophantic to

the degree that he sends out more positive affect in a given exchange than he receives. Our

measure thus captures imbalances in the flows that mark a chosen actor’s set of implicit trades, so

that the individual meting out lots of favorable rankings to those who send back unfavorable

rankings gets a large value. Using our reverse-coded weekly relational matrices

, a fraternity

member’s time-varying propensity for sycophantic behavior is measured as follows:

(

)

ijt

jit

ijt

−

⋅

∑

(9)

where the indicator

ijt

equals 1 if

gave

a less favorable ranking than

gave

, -1

otherwise. We expect

to diminish future growth in status. Correspondingly, when we enter

in model 5 we observe a strong negative effect, and also find that fragility remains discernibly

negative. Having accounted fraternity members’ time-varying levels of asymmetric exchange,

we can conclude with greater confidence that fragilely situated individuals face status-related

penalties because of the unique features of the positions they occupy.

Second, we evaluate our primary result further by ensuring that the pattern of results does

not hinge on any particular subject’s series. We do so both because of the relatively small size of

Newcomb’s panel, and due to the fact that one member of the fraternity persistently garnered

poor ratings. White, Boorman, and Breiger (1976, p. 759) drew attention to “a scapegoat …

(man 10), who received one of the bottom three choices of each of the other 16 persons.”

Correspondingly, the intercept for the tenth individual is strongly and conspicuously negative

across all models in table 4. We therefore estimated a version of model 5 omitting the tenth

individual.

We also estimated a version of our final model with status scores computed with

Results of this approach appear in model 6. Our effect of interest stays strongly

negative (-2.43 t-test for fragility), leading us to conclude that the scapegoat does not

disproportionately configure the effects we observe. To assess the findings more generally, we

β = 0 , rather than the standard ¾

multiplier, to ensure that our results did not hinge on a particular choice of beta. Although this is in our view a
theoretically less attractive way to proceed (here, status is measured just as counts of reverse-coded rankings, rather
than weighting by evaluators’ status), the effect of fragility measured at c

.99

was still discernible with a coefficient

of -.209 and -2.03 t-test.

also estimated sixteen other versions of model 5 in which we omitted each of the other subjects of

the study in turn. Without exception, across these sixteen alternative specifications, we found

evidence of a negative effect of fragility, whose impact was always significant at the .05 level of

confidence.

Fragility and Robustness of Industries in the U.S. Economy

We have thus far primarily discussed the fragility and robustness of positions in networks

of intangible flows, such as flows of esteem, recognition, and respect among individuals.

Although we have framed our measurement strategy as one that is valuable mainly for the

analysis of purely social systems, particularly status-based systems, we believe it to be reasonably

context-invariant (cf. White 1992, pp. 207-209) in the sense that it also applies (in connection

with different substantive mechanisms) to systems of tangible flows, such as biological,

electronic, and economic networks. For example, in a predator-prey network, a fragile population

of animals is one that preys on a vulnerable input (see Clark 1987, for the classic case of black-

footed ferrets’ near extinction because of their dependence on at-risk prairie dogs), whereas a

robust population’s food web is diversified at several layers. Or, a fragilely situated website

(Kleinberg 2007) sits at the end of a chain, getting the majority of its traffic from a site that itself

depends for traffic on one or just a few other sites. Similarly, organizations and industries are

fragile or robust as a function of their transaction patterns (cf. Hirschman 1962, pp. 98-119 on

satellite versus master industries and on linkage effects). A robustly situated industry is one

whose incumbents sell to (or buy from) a range of industries that have a diversified mix of

transaction partners.

We turn to standard input-output data for industries defined by the North American

Industrial Classification System (NAICS) to illustrate the applicability of our approach in a

These supplementary models are available upon request. Significance tests for fragility all exceed in absolute value

the critical value of 1.645 for a one-tailed test, ranging from -1.71 when the eighth individual is deleted
to -2.43 when the tenth individual is omitted, as discussed above.

network of economic flows. Like its predecessor, the U.S. Standard Industrial Classification

(SIC) system, the NAICS classifies establishments by the kind of economic activity in which they

engage, but unlike the SIC, the NAICS offers more accurate categorizations of emerging and

service industries (Yuskavage 2007). Transactions among 67 industrial categories tracked in the

input-output accounts of the Bureau of Economic Analysis (BEA) allowed us to compute annual

levels of fragility for industries from 2000 through 2005, and to assess their effects on industries’

total value added, where total value added is defined as total industry output minus total industry

input (Smith 2005). These industry categories, together with their average levels of total input,

total output, and total value added are shown in table 4. The industry ID refers to row number in

the BEA’s annual Use of Commodities by Industries matrices (Moyer et al. 2004).

To explore the association of total value added with measures of fragility in input-output

networks, we began by assembling six yearly 67-by-67 matrices

, where

ijt

 

=  

and

for all

ijt

Cell

records for year

the sales of industry

to industry

—or, stated

differently,

j s

input from

. We then used

( )

a b

from equation (5) to calculate industries’

time-changing fragility scores.

We did so in two ways in order to exploit the opportunity our data provide to detect

fragility levels in terms of both selling and buying. First, we used the matrices

to calculate sell

fragility,

SELL it

. When

SELL it

for a chosen industry is high, it is revenue-dependent on other

industries that are dependent in their selling patterns. To compute

SELL it

, we proceeded as

before, normalizing each entry in

by its row sum and then squaring each proportion in the

resulting row-stochastic matrix. That is, consistent with equation (2), we let

We address the possible effects of large diagonals (the amount of intracategory sales, as when automakers sell to

automakers) with a separate adjustment. Whereas our measure of fragility deals with dependency on other nodes in the
network, the adjustment we introduce captures dependency on oneself at an aggregate level.

I /

ijt





Θ = 







∑

, where

ijt





= Θ





, and then applied equation (5) to

to calculate

SELL it

. Second, to compute buy fragility,

BUY it

, we worked with the transpose of

. We

again used the same two-step conversion process—make the matrix row-stochastic and then

square proportions—on

, and applied our measure

( )

a b

to the twice-transformed matrix. If

BUY it

for a given industry is high, that industry is dependent, in the procurement of its goods or

services, on other industries that are dependent in their buying patterns.

To convey a sense of the chainlike underpinnings of fragile positions in the input-output

network, in figures 6 and 7 we depict patterns for two industries that, on average, have high

fragility scores. In figure 6, we map out selling patterns contributing to a high sell fragility score

for Chemical Products, and in figure 7 we trace buying patterns leading to a high buy fragility

score for Truck Transportation.

To identify the sell-chain for Chemical Products in figure 6, we constructed the matrix

as the average of six annual matrices

2000 through 2005, and then computed sell fragility

scores for industries, with the coupling parameter

equal to

.99.

Two of the 67 categories—Federal General Government and State and Local General Government—necessarily have

undefined sell fragility scores because their row sums in

Chemical Products neared

the top of the distribution, with just ten industries ahead of it—for example, Farms as well as

Support Activities for Mining both had higher scores. Chemical Products was therefore ranked

considerably ahead of robustly positioned sectors—for instance, Computer and Electronic

Products and Food Services and Drinking Places—that offer inputs to a wide array of other

industries. Then, with Chemical Products as our point of departure, and using entries in the row-

I are zero. These sectors are of course important buyers in

the columns of

I , however, and so are included in the computation of fragility scores. After making the original

matrix

I row-stochastic and squaring each entry, the undefined entries in their rows of the resulting matrix were reset

to zeroes. We exclude both governmental categories from our panel models because we are interested primarily in
within-effects of fragility. When between or overall-effects of fragility are of interest, we suggest assigning a fragility
score of zero to actors with zero row sums in the relational matrix, and capturing the consequence of their isolate
position with a separate dummy variable.

stochastic version of

, we moved from first to third-order connections in the network, at each

remove identifying the adjacent industry on which the focal industry was most relatively

dependent for its sales. As shown by the pattern of sell dependencies in figure 6, Chemical

Products sells primarily to Plastics and Rubber Products, which sells mainly to Construction,

which in turn sells mainly to Real Estate.

Turning to the buy-chain for Truck Transportation in figure 7, we followed the same

steps just outlined, except that we worked from the transpose of

. Using

to calculate buy

fragility scores for industries,

we found that Truck Transportation ranked near the top of the

distribution, close to other input-dependent industries, such as Food Services and Drinking

places, the lion share of whose inputs come from Food and Beverage and Tobacco Products.

Using a column-stochastic version of , we then tracked the buy-dependencies evident in the

chain in figure 7. Truck Transportation buys mainly from Petroleum and Coal Products, which

buys disproportionately from Oil and Gas Extraction, which then relies principally on Rental

Leasing Services and Lessors of Intangible Assets.

To investigate associations between industries’ time-varying levels of fragility and total

value added, we turn to panel models of the form:

(

)

(

)

i t

TVA

TII

θ X γ

(10)

where

, 1

i t

TVA

denotes the total value added by industry

in year

, 1

i t

TVA

equals gross

industry output in the focal year, minus total industry input in that year, which we represent by

Two additional categories of the original 67—Noncomparable Imports as well as Scrap, Used and Secondhand

Goods—did not receive inputs from any of the other categories and therefore had row sums equal to zero in

. We

also exclude them from our panel models, given that they allow no within-category variation. We include them,

however, in the columns of

when calculating fragility scores because of the inputs they provide.

, 1

i t

TII

. We focus on

, 1

i t

TVA

as an outcome because it is a transparent measure of industry

performance, capturing the extent of a given industry’s contributions to overall U.S. economic

growth (Smith 2005, p. 2). We enter

, 1

i t

TII

as an adjustment for industry size, given the

possibility of scale-based efficiencies. We enter logs of both measures because of the skewness

of their original distributions.

The matrix

whose coefficients are in

contains four versions of our fragility

measure from equation (5). Two of the four covariates in

are the fragility scores just

discussed—

SELL it

and

BUY it

—each of which was computed with the coupling parameter

set to its maximum value of

.99.

The other two are versions of sell fragility and buy fragility

with

c = 0

, and are therefore perfectly correlated with Herfindahl indices of concentration in

selling and buying relations, respectively. Thus, using more precise notation, the full set of

measures in

are:

, |

.99

SELL it c

, |

.99

BUY it c

, |

SELL it c

, and

, |

BUY it c

. Unlike our analysis

of Newcomb’s fraternity, where we had strong, theoretically informed expectations about both

the appropriate level of

in measuring fragility and the direction of its effect on status, there is

less guidance available in the existing literature for anticipating effects of these fragility scores on

industries’ annual levels of value creation.

Yet, it is possible informally to sketch reasons why we might observe a negative effect of

fragility on total value added, especially for

SELL it

when coupling in chains of sales is

accounted for. As an industry is increasingly fragile with respect to its customer base—that is, as

its customer base grows more specialized—that industry will be more susceptible to the vagaries

of its buyers. Additionally, if we consider demand for an industry’s outputs to be at least partly

random, quantity demanded will certainly at times fall beneath the threshold at which the focal

industry’s establishments can sell at a positive profit. As an industry grows more fragile, the

value it adds may shrink because its establishments are dying, laying off workers, or making

other costly adjustments. Just as important, it is also not hard to imagine that a long chain of

output-related dependencies (from Chemical Products to Real Estate in figure 6) is worse than a

short chain (from Chemical Products just to Plastic and Rubber Products). In a chain of

considerable length, there are multiple sites, not just a single site, at which problems (perhaps

related to being held up by powerful suppliers or other events) can surface. We believe that a

measure sensitive to such contingencies is better suited to characterizing fragility than one that

ignores them. Consequently, we anticipate that

, |

.99

SELL it c

will negatively affect future total

value added.

Moving to the matrix

in equation (10), we also enter as adjustments two realizations

of Bonacich’s measure from equation (3). As with our measurement of fragility in this empirical

setting, we again consider industries’ positions in terms of both sales and procurement. Although

for Whyte’s street gang and Newcomb’s fraternity we drew on equation (3) to measure status, in

the context of inputs and outputs tying together industries, we view it as a measure of centrality.

Although there is much evidence pointing to the positive effects of organizational status on

economic performance (Podolny 2005), the industry categories on which we rely are defined at a

sufficiently high level of aggregation that we suspect any meaningful prestige-related distinctions

among establishments are primarily within, rather than between, categories.

Consequently, we used equation (3) to compute sell centrality,

SELL it

, from

and to

compute buy centrality,

BUY it

, from

SELL it

is large for industries, such as Broadcasting

and Telecommunications, that send substantial outputs into several other sell-central industries.

Correspondingly,

BUY it

is large for industries, such as Construction, that take in substantial

inputs from a host of other buy-central industries. Thus, although we can view high sell

centrality as being analogous to the epicenter of a ripple effect, high buy centrality is not unlike a

vortex suctioning in other industries’ goods and services. We view

SELL it

and

BUY it

proxies for an industry’s importance in the economy—having to do with the scale of transaction

partners—and we therefore expect them to affect total value added positively, adjusting for total

industry input.

Shifting to

in equation (10), we include additional covariates that address two

particular features of input-output tables. First, we devised the dummy variable

negativeflow in

light of the small number of cases in which negative flows of dollars appeared in an industry’s

row or column of the input-output matrices. For example, in year 2000, the dollar value of the

input from Insurance to Farms was -$181.7 million. This is because the insurance industry

weathered losses as the farming industry’s claims exceeded benefits. Since we recoded negative

entries to zeros in

in order to calculate fragility and centrality scores, we include the indicator

negativeflow as a cautionary adjustment.

Second, because our measure of fragility concerns the nature of an actor’s ties to others in

a network, as noted previously we set the diagonal equal to zero in the matrices

. Zeros along

the diagonal is the rule rather than the exception in most analyses of social networks. Conversely,

given the broad level of aggregation at which industries are defined for input-output tables,

diagonals measuring intraindustry sales (as when computer manufacturers sell components to one

another) are frequently nonzero. We therefore enter two further adjustments:

intrarow , the ratio

of industry

i s

diagonal to its row sum in

, and

intracolumn , its diagonal over its column

sum in

. We use these measures to capture a focal industry’s time-varying levels of internal

exchange.

Finally, as denoted by

and

, we enter fixed effects for industries and for years,

respectively. Using industry-specific fixed effects

accounts for a number of factors, including

cross-industry heterogeneity in demand, average distance from the end consumer, and other time-

constant traits. With year indicators

, we adjust for various aggregate macroeconomic

fluctuations likely to influence industry performance. Correlations and descriptive statistics for

variables in our analyses are shown in table 5.

Table 6 presents results of five models predicting total value added. We begin with very

simple specifications and then proceed to models that more stringently adjust for consequential

sources of variation. Model 1 contains only total industry input and sell fragility measured with

maximal coupling. We find, keeping with our expectations, that industries that are less robustly

positioned in their selling patterns add less value. Model 1 thus offers preliminary evidence in

support of the supposition that positional robustness in an economic network elevates

performance. Turning to model 2, we observe a negative effect as well for our measure of buy

fragility.

In model 3, we move to a two-way fixed effects specification that conservatively absorbs

between-industry, as well as temporal, heterogeneity. Given this approach, the available variation

is within-industry and unrelated to aggregate-level macroeconomic factors. We observe first that

the effect of sell centrality,

SELL it

, is strongly positive, indicating that as a chosen industry sells

increasingly to other important industries, its future performance rises. Also, although the effect

of sell fragility without coupling (the equivalent of the Herfindahl index) is indiscernibly distinct

from zero, the estimate of sell fragility measured with coupling,

, |

.99

SELL it c

, retains its

significance in the presence of various adjustments. Using the estimates of model 3 together with

the descriptive statistics presented in table 5, we find that a one within-industry standard

deviation increase in sell fragility with coupling lowers future total value added by about 2%

(exp[-.2299599*.078491] = 0.98). This result is important because it suggests that occupancy of

a position implicated in chains of dependencies has performance-eroding consequences.

Although we do not at this juncture take this effect as definitive evidence of a causal relationship

because we do not exogenously vary fragility, given the conservative nature of our panel model,

we do believe it is an important association for consideration in future research.

We arrive at a different pattern of effects in model 4, where we turn from measures keyed

to downstream ties to those that build from ties a chosen industry has to others situated upstream

in the supply chain. Unlike model 2, which omitted industry-specific fixed effects, model 4

presents us with a pattern in which buy fragility is no longer consequential. Similarly, whereas

sell centrality mattered appreciably in model 3, its counterpart in model 4 loads poorly, given its

strong association with unobserved stable traits.

We also see in model 4 that the effect of

, |

BUY it c

is both strongly significant and

positive (3.03 t-test). Although this effect lacks the attractive stability of sell fragility measured

with coupling—which holds up across models, with and without industry-specific indicators—it

is still a suggestive result. In particular, it points to the importance of a discussion of the scope

conditions bracketing the generality of our approach—a discussion we pursue in our concluding

section. We underscore for now the fact that, although we have portrayed our fragility measure

as a factor conducive to declines in coveted outcomes, it is not difficult to envision settings where

just the opposite is true.

Given the distinctive features of our current empirical domain, it is not entirely surprising

that

, |

BUY it c

is positively associated with future value added. One plausible account follows

from the high level of aggregation in input-output data and from the fact that input markets differ,

in terms of availability of substitutes, from output markets. More precisely, there are no

substitutes in terms of suppliers, at least in the short-run, at this level of aggregation: although it

is reasonable to anticipate relatively fast changes in the mix of industries to which a focal industry

can sell—for example, telecommunications can shift from selling mainly to one market to another

quickly—it is unlikely that any given industry can significantly and speedily alter its mix of

suppliers. As an example, an instantaneous transition from rail transportation to truck

transportation as an input is implausible. Consequently, at this level of aggregation, it may be

desirable to draw from just one supplier industry, within which substitutes may be found, rather

than a wide range of supplier industries—all of whom, in turn, could increase their input prices.

We examine finally model 5, where all previously discussed covariates enter jointly,

together with

negativeflow

intrarow

, and

intracolumn

. Including these three additional

measures is important because input-output data differ in material ways from most sociometric

data: Input-output data present both the numerical equivalent of negative ties and nonzero

diagonal elements. When these three covariates enter in model 5, they fall just shy of

significance at a standard level and our coefficient of interest—on sell fragility with coupling—

retains its significance. Thus, in a setting that is substantively very distant from the status-based

systems of Whyte (1993) and Newcomb (1961), we find evidence to suggest that taking chains of

dependency into account is important for understanding outcomes in networks.

Prestige and Fragility in the PhD Exchange Network

Our third empirical application is a reanalysis of the data used in Burris’s (2004)

investigation of the determinants of sociology departments’ prestige and thus brings us back to

status-related themes. An advantage of revisiting Burris’s data is the opportunity they offer to

assess the effect of fragility on an important outcome now known to vary discernibly with

network centrality. Using a sample of 94 departments listed in the ASA’s 1995 Guide to

Graduate Departments of Sociology, Burris (2004) documented that, net of established measures

of research productivity, centrality in the PhD exchange network explains a disproportionate

share of the variation in departmental prestige scores.

These prestige scores were drawn from a National Research Council survey (Goldberger,

Maher, and Flattau 1995) and were “based on ratings by disciplinary peers of the ‘scholarly

quality of program faculty’ on a five-point scale ranging from ‘distinguished’ to ‘marginal’”

(Burris 2004, p. 245). The key independent variable—social capital—was measured as logged

We are grateful to an anonymous AJS reviewer for suggesting that we apply our measure to Burris’s (2004) data.

network centrality using Bonacich’s (1972) eigenvector measure on a symmetric matrix recording

the trading of scholars between departments.

( )

i j

Cell

of the relational matrix used to

compute social capital (and from which we calculate fragility) tallies the number of full-time

faculty members who had been hired from department

and from department

during the chosen observation window. The data for this matrix came from the ASA’s

1995 Guide. Burris’s (2004) results are interesting and important because they counter

conventional, purely productivity-based etiologies of departmental prestige by underscoring the

salience of the status-related consequences of departments’ positions in the hiring network.

Centrally located departments appear to enjoy a number of interrelated advantages, such as

recognition from other well-situated departments and a large base of well-placed researchers

poised to preserve the status of their intellectual roots (pp. 243-5).

Adjusting for the predictors entered in Burris (2004), we anticipate a negative effect of

fragility in the PhD exchange network on departmental prestige. A fragilely located department

is one that trades scholars with a limited set of departments that are similarly restricted in their

sets of exchange partners. We think that occupancy of a fragile position will negatively affect

departmental prestige for at least two reason: One plausible mechanism is that fragility operates

as a market signal of intellectual myopia; another is that occupancy of a fragile position

unwittingly elicits indifference or antagonism among those departments falling outside the

exchange-set and thus (perhaps unconsciously) negatively affects peers’ appraisals.

Consequently, we begin with model 1 in table 7, which replicates model 4 in table 4 of

Burris (2004). Detailed summaries of measures of scholarly productivity entered in model 1

appear on pp. 254-5 of the original article. Very briefly, both Article Publications and Citations

The correlation between Burris’s logged measure of social capital based on Bonacich (1972) and the measure

centrality based on Bonacich (1987) we use throughout the current article is .90. In addition, results are virtually
identical if we replace the former measure with the latter in our subsequent regression models. We use the former
measure in our reanalysis for sake of consistency with Burris’s original study.

are averages of counts across a focal department’s full-time faculty over a prior five-year

window, and Research Grants is the percentage of faculty members garnering external research

grants in the seven-year period before the NRC survey. In addition, consistent with Keith and

Babchuk (1998), Weighted Article Publications is a measure that takes into account the

heightened impact that results from publishing in the most influential journals, and Book

Publications is the mean count of books per faculty member earning reviews in Contemporary

Sociology from 1980 through 1989. Model 1 thus re-presents results showing that, on top of

influential article and book production, centrality in the hiring network is strongly associated with

favorable peer-appraisals.

Model 2 adds to model 1 our measure fragility with the coupling coefficient

.99

Consistent with our expectation, net of the strong positive effect of social capital, fragility in the

PhD exchange network is discernibly associated with lower ratings on the NRC scale.

Furthermore, the effect magnitude of fragility is comparable in absolute value to those of the two

consequential productivity-based predictors in the model—weighted articles and book

publications. This is most apparent in model 3, where we present standardized regression

coefficients.

According to model 3, a one standard deviation increase in a department’s fragility

effectively counterbalances an equivalent increase in either (weighted) journal or book

publishing. This result is important substantively because it indicates that a standard shift in

fragility within the doctoral hiring network matters as much as a standard shift in production. It

also suggests that, net of centrality, robustness in the network may convey openness and generate

goodwill among peers. More broadly, the effect of interest in model 3 is suggestive of a social

(rather than merit-based) process by which changes in departmental status occur over time.

Table A in Burris (2004, p. 262) provides descriptive statistics and correlations for all covariates included in model 1.

An extended version of this table that includes our measures of fragility is available on request.

Although these data are not dynamic, it is not hard to imagine departments shifting in the extent

to which they exchange provincially and consequently altering views held by their peers.

Finally, whereas in models 2 and 3 we entered only fragility with coupling set to its

maximum value, we conclude in model 4 by adding fragility with coupling set to zero. This

allows us to assess the value of our measure against that of Herfindahl’s concentration index.

Similar to the pattern of results from our analyses of Newcomb’s fraternity and of input-output

networks, the significance of

.99

i c

taken together with the insignificance of

i c

reinforces

the importance, on a methodological level, of incorporating coupling in the measurement of

fragility. Finally, on a substantive level, the results in model 4 are important because they point

directly to the relevance of exchange partners’ positions in the network for a given department’s

level of prestige.

Discussion and Conclusion

Our aims in this article have been both to advance our understanding of status dynamics

by bringing into focus the negative consequences of fragility and to develop a method for

arraying network positions along a continuum defined by fragility and robustness. To summarize

briefly: We framed our approach on the backdrop of prior analyses that harnessed concentration

versus diversification of social ties as salient constructs. We then brought forward the importance

of taking into account coupling among the nodes that comprise a social system. Melding

disparate approaches, we developed a measure of fragility that is attentive to the extent to which a

focal actor is propped up by those who are themselves precariously situated. Using Whyte’s

(1993) ethnography of the Nortons as an initial illustrative case, we turned to an application of

our measure in three distinct empirical settings and found similar results. In Newcomb’s panel,

we found that more fragilely situated fraternity members faced declines in status over the course

of a semester-long period. In our panel of industries, adjusting for centrality, we found in both

pooled cross-sectional and within-models that our measure of fragility in selling patterns lowered

industry value added. Our reanalysis of Burris’s (2004) data demonstrated that fragility in the

PhD exchange network is negatively associated with departmental prestige.

Thus, in addition to the empirical results we have presented, one of our primary

contributions has been to demonstrate that an actor’s ranking in an ordering of status (or of

centrality, in the case of industrial sectors) may be insufficient for accurately characterizing its

network position. Using the first week of Newcomb’s panel as an example, we illustrated that,

although metrics like Bonacich’s (1987) approach serve as powerful methods for detecting

prestige orderings from relational data, our measure of fragility illuminates important, and

otherwise occluded, distinctions between actors that appear status-equivalent and yet rely on very

different social footings. We believe that future analyses of the antecedents and consequences of

status will profit from looking beyond the aggregation of prestige-conferring flows to the paths

by which those flows converge on positions in social structure. When virtually all respect,

esteem, or social support emanates from a single vulnerable source, the receiver’s position is

considerably less robust.

We have also drawn attention to the importance of constructing network models that

account for extralocal processes. We found in particular that taking coupling into account in the

measurement of fragility was important for explaining variance in the outcomes we examined.

Our results indicated that attention to ego-network composition through Herfindahl’s measure

alone was insufficient. In this sense, our approach is consonant with other portrayals of processes

traveling through chains of considerable length. Examples of work in this vein include

Rapoport’s (1963) models of interaction, White’s (1970) analyses of vacancy chains, Tilly’s

(1990) discussion of immigration patterns, Abbott and Hrycak’s (1990) analyses of careers, and

Burt’s (2007) models of reputational stability as a function of closure among indirect contacts.

As Burt (2007) argues, whether or not extralocal processes matter deserves careful scrutiny and

carries central implications for research design. If causality resides entirely in the immediate

network—that is, if decoupling is a characteristic feature of the population—then it is

unnecessary to capture the network as a whole. Data on ego-networks alone are sufficient.

Conversely, attention to the full pattern of ties within a population becomes increasingly

necessary insofar as identity-related spillovers occur. Our approach indicates that collecting

complete network data is important, at least in some empirical contexts, for moving toward richer

portrayals of status-related processes.

Limits on generality. Before sketching implications for other lines of sociological

theory, we emphasize three conditions that restrict the generality of the perspective we have

presented. First, there are clearly institutional settings in which extralocal processes do not

matter, and where decoupling therefore defines the network. In these settings, fragility is best

measured just as a Herfindahl measure of concentration, without attention to contiguous actors’

identities and networks. Consider in this connection the two-by-two table in figure 8, whose

vertical axis extends from fragile to robust and whose horizontal axis goes from decoupling to

coupling. If in fact decoupling marks the network, then fragility should be measured in way that

is consistent with the bottom-left cell in figure 8, where A relies completely on D, but where D’s

identity and network are inconsequential for A. This quadrant is therefore quite different from its

counterpart on the bottom right, where A is fragile because of its position at the end of a chain,

and from the upper-right cell, where A is robust by virtue of its wide range of connections to

others who are diversified in their sources of social support.

The extent to which a social structure is “readable” (White 1992, pp. 106-11) contributes

significantly to the degree of coupling depicted on the horizontal axis, and thus is a factor

determining the best measurement strategy. Although many social systems allow incumbents (as

well as observers) to map out who is connected to whom and in what way—consider, as

extremes, a rank-order tournament or, less commonly, a matrix organization whose publicly

displayed organizational chart accurately reflects its informal structure—other systems are far

more opaque, even for insiders. There, the concrete patterning of social ties is fuzzy and

uncertain for all involved—incumbents and observers have difficulty discerning, or even

inchoately sensing, nonlocal patterns—and network spillovers are therefore rare. Consequently,

an important condition for our approach is a reasonable level of perception within the network.

Coupled identities, where spillovers occur, necessarily require opportunities for detecting or

discerning social ties.

We therefore expect extralocal ties to matter appreciably in the estimation of fragility

effects within transparent settings like investment banking, where firms’ locations in public

pecking orders are widely monitored (Podolny 1993). In such settings, we recommend

computing fragility with a large

weight. Conversely, for less penetrable settings that would

tend toward the left column in figure 8—such as semiconductors (Podolny, Stuart, and Hannan

1996), where consumers likely have difficulty discerning the circuitous patterns of interfirm

patenting ties—we anticipate little in the way of network spillovers and thus suspect that firms’

levels of fragility will be best calculated more restrictively—that is, with a smaller

weight, or

even with

set equal to zero.

Second, our depiction of A’s position as fragile due to chaining in the bottom right of

figure 8 leads us to reinforce the importance of examining global-network structure when

deciding whether to implement our measure. As we discuss in more technical detail in our

appendix, a central consideration in determining if our measure will yield information over and

above Herfindahl’s index is whether the structure of the global network permits chains of

reasonable length among its nodes, or is instead a tightly connected clique. Simulations

presented in our appendix show that social structures marked by longer mean path lengths allow

for greater expression of differences between our measure of fragility and one agnostic with

respect to context. We believe these simulations are important because they shed additional light

on the conditions in which the

weight is large in the practical sense. Although in a closely knit

cult our measure differs little from Herfindahl’s, it offers much additional information in more

differentiated networks that are typically the subject of sociological investigation.

Third, in empirical settings different from the status systems of Whyte (1993) and

Newcomb (1961), it is entirely plausible to expect that what we have measured as fragility in

equation (4) will exert positive effects on coveted outcomes. Consequently, in such settings,

although its algebraic form will remain unchanged, our measure will require substantive

recasting. Consistent with this possibility, Zuckerman’s (1999, 2000) work clearly demonstrates

the benefits of constructing a narrow, concentrated identity. When individuals or organizations

winnow their focus to an existing node in a way that renders their performance easier to appraise,

corresponding advantages result. There are also the possibilities that sharp focus on another node

in a network enhances trust and reduces learning costs (Uzzi 1996) and that specialized firms

both more readily respond to distinctive customer preferences (Carroll and Hannan 2000; Bothner

2003) and more effectively convey future work experiences to prospective employees (Baron

2004). More generally, there are certainly settings in which fragility is ideologically construed as

positive.

To adjudicate among ways of discussing the likely effects of our measure, we believe it is

essential as a starting point to consider differences in where concentration comes from—from ego

or from alter. Consider a case from the computer industry as an example. Dell’s choice

historically to focus exclusively on the direct sales channel is markedly different from a fraternity

brother who gets a disproportionate share of his social support from one other member of the

house—although Herfindahl’s index will be high for both. Dell’s strategy has largely proven

status-enhancing, but in Newcomb’s fraternity we found that concentration is status-eroding.

Where these cases differ importantly is not in levels of analysis, but in the provenance of

concentration and therefore in terms of control. Whereas a computer maker can deliberately

decide to specialize within a specific segment, the fraternity brother has far less discretion over

whether he receives the lion’s share of his social support from a particular peer. When this does

occur, those other than the focal individual are consequently the ones imputing (unfavorable)

meaning to his position.

Thus, in keeping with lines of research differing from our own, the measure of fragility

we have introduced might be recast as a measure of focus in other contexts. In such contexts,

actors receive rewards for establishing exclusive ties to those with narrow allegiances, and the

fragile-robust axis labels for the ordinate in figure 8 could therefore be replaced respectively with

focused-disordered. When the cost of forging such ties varies inversely with actors’ quality

levels, as at times they do in apprenticeship programs, for instance, equation (4) will measure

focus, rather than unwanted complexity or disorder, as a viable market signal in Spence’s (1974)

sense.

To mitigate concerns about generality, in addition to describing how to substantively

reframe our measure to accommodate different settings, we stress the pervasiveness of the social

structural conditions that are consistent with the viewpoint we have advanced: our approach

posits that actors sort into positions in a status ordering as a function of the recognition they

receive, and that the network is sufficiently transparent that they are aware of the means by which

positions in this ordering are determined—whether these positions are durably built through

diversification or exist vulnerably because of concentration. Although the conditions necessary

for our perspective do not mark all social settings, they certainly characterize many. We believe

our approach is generally applicable for theoretical and methodological reasons: both because of

the frequency with which peer monitoring and gossip make the underpinnings of social positions

transparent and because of the frequency with which network-analytic study designs collect data

on ties for whole populations.

Although we believe that attention to the origins of fragility—with ego or with alter—serves as a useful starting

point, it is insufficient. Like a laser steadily aimed at the wrong target, ego can of course deliberate narrow his or her
focus on the wrong audience.

This point also raises the issue of lineages, if actors in a system span cohorts or generations. Under this scenario,

equation (4) could measure purity of lineage, as when an artist or academic comes from the right (and narrow) line of
teachers. Purity is then a by-product of having collaborated exclusively with a master who was him or herself sharply
discriminating as a disciple. For instance, this is almost certainly the legitimating process the Apostle Paul had in mind
when, in his defense, he cited Gamaliel as his teacher (Acts 22).

Future directions. In addition to replicating our results in other settings, especially in those

where status in known to precede coveted outcomes, we see three particularly promising ways to

proceed further. First, although our measure of fragility operates at the level of individual nodes,

it is easy to imagine opportunities to link up with important group-level themes highlighted by

Moody, White, Harary, and colleagues in their work on structural cohesion (see, e.g., Moody and

White 2003; White and Harary 2001). Moody and White (2003) discuss robustness at the group-

level in terms of a focal group’s capacity to persist in spite of node removal and thus picture

cohesive groups as those that have “a status beyond any individual group member” (p. 112).

They also suggest that future research may benefit from focusing on the hazards of removal faced

by particular nodes. Along this line, we believe that our measure of fragility may be particularly

valuable.

Consider, as a potential empirical site, recent work in biology on metabolic and other

cellular networks (Jeong et al. 2000) that focus on individual substrates, connected by

biochemical pathways, constituting a metabolic network. In this domain, a highly fragile

substrate is one that is connected to one or a few narrowly focused alternate substrates. In

addition, not unlike many social networks, biological networks are often characterized by long-

tailed distributions where a few substrates account for much of the network connectivity.

Although this can result in stability for the network as a whole—in the sense that networks stay

largely intact in the face of random error—researchers are now interested in examining cross-

organism differences in the identities and durability of individual substrates (Jeong et al. 2000).

Arraying nodes in these and similar kinds of networks according to their fragility may be useful

for locating otherwise subaltern fault lines in global structures and thus for further understanding

group-level robustness as envisioned by Moody and White (2003).

Two additional next steps are to link notions of fragility and robustness more tightly to

etiologies of status and also to harness the approach we have developed to further pinpoint the

structural foundations of robust action—or action that preserves options (Leifer 1988; Padgett

and Ansell 1993). We conclude with very preliminary sketches of how theories of status and of

robust action might be extended in light of our approach.

We found in our analysis of Newcomb’s data that fraternity members enjoyed increments

in status as their positions within the social structure became more robust. This finding is

consistent with Abbott’s (2001) observation that eliteness is a by-product of possessing a wide

range of future options. We interpreted this finding on a purely structural level as evidence

consistent with the premise that audience members esteem those in robust positions. In arguing

that structure directly affects perceptions, we sought to adjust as stringently as possible for fixed

and time-changing behavioral differences in our empirical models. Nevertheless, we believe that

future work on the determinants of eliteness may benefit from examining some of the possible

behavioral concomitants of robustness. These are actions that emerge from the durability of

actors’ positions and in turn contour their levels of prestige. We refer here in particular to actions

that are permissible and sustainable exclusively for inhabitants of robust positions. Examples

include the “attractive arrogance” Merton (1968, p. 61) ascribed to elite scientists, or excessive

acts of generosity directed toward others who lack the resources to reciprocate. Through tracing

out the links among robustness, modes of conduct, and harvests of future deference, we anticipate

that our conceptions of where status comes from are likely to sharpen.

Turning in conclusion to further implications for theory, we suspect that robust positions,

as we have defined them, render robust action more sustainable. Work on robust action has

immediate precedents in Leifer’s (1991) research on chess masters. This study found that the

very best chess players are not those endowed with uncommon forecasting abilities, but rather

those who can sustain flexibility with respect to (or remain robust with respect to) opponents’

moves. More generally, robust action refers to strategic conduct that is difficult for a competitor

to interpret and that therefore keeps its practitioner from getting absorbed in an undesirable role.

For example, by acting robustly in a romantic relation, a suitor veils actual intentions and

therefore circumvents appearing overly committed and becoming the less powerful; or by acting

robustly in international relations, one nation wishing to conquer another nation adroitly provokes

its target to start a war and then successfully brands it as the instigator (Leifer 1988, pp. 867-9).

Correspondingly, for a political figure, this strategy equates to maneuvering in ways that are

indecipherable to opponents and therefore congenial to continued flexibility (Padgett and Ansell

1993).

We believe that attention to positional robustness may shed new light on the incidence of

robust action. Although we fully agree that social skills (Leifer 1988) and protean identities

(Padgett and Ansell 1993:1263-4) greatly facilitate robust action, we contend as well that this

strategy may be aided by durability of social position. This is because of the costs involved.

Notwithstanding its many advantages, robust action has a dark side and can be dangerous. The

cagey suitor invites accusations of insincerity and indecisiveness. Robust actors, though they

may be exquisitely patient and shrewd, open themselves to the accusation that they are slow

ciphers. Given these risks, we conclude with a disconfirmable prediction. We expect robust

action to be more likely among robustly positioned actors because of the capacity of their social

positions to absorb, much like an insurance policy, the costs of accidents as they behave

strategically.

Appendix: Linking Local Measurement to Global Network Structure

This appendix works toward the construction of a bridge between our egocentric

measurement of fragility and the global network structure on which the fragility of a focal node

depends. Since our model reaches beyond the ego-network through our weighting parameter

to reflect larger structure, we believe it is important to examine, at least preliminarily, how this

broader context conditions our measure. One way to bring into focus and better understand this

local-global link is to identify the global-level circumstances in which a large

weight induces a

vector of fragility scores that differs most significantly from a corresponding set of Herfindahl

indices. Consequently, returning to equation (3) in the main text, our question is the following:

Under what global-network conditions is fragility computed with

.99

least correlated with

fragility computed with

(i.e., with Herfindahl’s index)? Through identifying the extralocal

circumstances in which

.99

differs most from

, we can better appreciate when a chosen

coupling coefficient is indeed “large” in the practical sense that its application yields information

beyond a metric that is insensitive to alters’ positions in the network.

We examine the measurement-related consequences of varying global structure through

the lens of mean path length—a long-studied network-level metric. The path distance between

two nodes is the shortest number of steps between them: Two directly connected individuals have

a path distance of one; individuals capable of reaching one another by a single go-between have a

path distance of two; a pair linked most directly through five intermediaries has a path distance of

six; and so forth (see Burt 1982, pp. 25-29 for measurement; Burt 2005, pp. 32-24 for empirical

illustration). Moving from the dyadic to the global level, the mean path length—the average

shortest distance between all connected pairs—for a given social structure is an index of its

overall reachability or efficiency (Wasserman and Faust 1994). To the extent that mean path

length is low, members of the social structure can get to each other quickly. In contrast, a high

mean path length reflects the presence of long winding strands of social ties.

We investigate how mean path length conditions fragility scores in order to assess a

direct implication of our approach—namely, that fragility measured with a high coupling

coefficient is most likely to offer unique information in networks marked by extended chains of

dependence. We believe that pursuing this implication is important on theoretical and empirical

grounds. On a theoretical level, it begins to illuminate the contextual underpinnings of fragility

by bringing relief to global-level dynamics likely to increase variation along a fragility-robustness

axis. On an empirical level, it has the potential to alert researchers to the types of observable

social structures in which attending to context (over and above the immediate composition of the

ego-network) in the computation of fragility yields the greatest payoffs.

We proceeded by generating 1,000 simulated networks of various sizes and levels of

connectivity. Using these networks, we could then assess the effect of mean path length on our

correlation-based outcome of interest,

(

)

.99

cor

, which we refer to as cormeasures.

The square matrices from which we constructed these networks ranged in size

from 50 to 150

rows. To induce variation in the total number of non-zero entries across our various matrices, we

worked with an array of “cutpoints” in each matrix size-category.

More precisely, after generating a square matrix of a given size

from

random draws

from the standard normal distribution, we set all entries above the chosen cutpoint equal to 1,

brought other entries to zero, and constrained the main diagonal to equal zero regardless of the

value of the initial random draw. We used six cutpoints in total: 0, .5, 1, 1.5, 2, and 2.25. With

equal frequency, we applied the first four of these cutpoints to 160 matrices of size 50; we applied

all but the largest to a total of 600 matrices of sizes 75, 100, and 125; and we used all six

cutpoints on 240 matrices of size 150. Thus, for each size-cutpoint pair, we simulated 40 random

matrices, yielding a total of 1,000 directed non-valued networks.

We then computed cormeasures as the correlation between fragility for

.99

and

fragility for

, as well as mean path length, for each of these networks. Although our results

are virtually identical if we set isolates’ fragility scores equal to zero, we eliminated isolates from

our analysis both because fragility is undefined for completely disconnected nodes and because

the average path length of a network is undefined when no path exists between two nodes.

The plot shown in figure A.1 reveals a strong negative association between mean path

length and cormeasures. Thus, the greater the average shortest distance between connected pairs

in a network, the more significant the consequences of applying a large coupling coefficient. As

the non-parametric smoothing function (Friedman 1984) we have added suggests, this association

appears to stay strong until a mean path length of roughly six.

We also took steps to guard against the possibility that the association we observe is an

artifact of the number of nodes. In a simple model regressing cormeasures on mean path length

and network size, the coefficient on mean path length was -.155 (-94.56 t-test) and the coefficient

on network size was .000492 (7.19 t-test). In addition, in a second model in which our predictor

of interest enters reciprocally due to the nonlinear pattern evident in figure A.1, the coefficient on

1/(mean path length) was 1.25 (63.37 t-test) and the estimate on network size was .000421 (4.37

t-test).

We conclude by depicting specific networks created in the course of our simulations that

differ by mean path length and cormeasures. We focus on the extremes and a midpoint for sake

of illustration. In the successive sociograms shown in figure A.2, we move from simulated

networks in which our measure of fragility overlaps with standard concentration to those in which

Consistent with our earlier discussion of relational matrices as images of status-conferring flows among members of

a social structure, on a substantive level one might envision (in addition to any number of other interpretations of a
simulated tie) a value of 1 in a chosen cell of a given matrix as an indicator of whether the individual in that column
recognizes or acknowledges the individual in that row. Thus, a focal individual can “reach” a distant alter along a path
composed of third parties that convey recognition, drawing on one to get to another if the first intermediary dispenses
recognition to the focal individual.

it differs appreciably. Each network was generated from a matrix of size 100, with the full 100

included in panel 1, 92 nodes (due to the removal of 8 isolates) in panel 2, and 82 nodes in panel

3. We report for each graph its mean path length and the correlation between fragility scores with

minimal and maximal coupling. Panel 1 shows a highly connected network where nodes can

reach others almost immediately on average. For such networks, whose basic structure is perhaps

found most often in small cults, measuring fragility with a large

weight has little impact.

Conversely, the networks depicted in panels 2 and 3 correspond to higher mean path length and

are ones in which incorporating broader context in the measurement of fragility has discernible

consequences. These are also more similar in typology to the kinds of networks frequently of

interest to sociologists. It is also in such networks that nodes can depend highly on other nodes

that are themselves caught in chains of dependence, and thus occupy positions that differ

significantly from their more robustly positioned counterparts.

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Figure 1 A depiction of the social structure of the Nortons street
gang, as hand-illustrated by Whyte (1993 [1943], p. 13).

Figure 2 Status versus fragility in week 1 of Newcomb’s study.
Status and fragility scores were computed from the relational
matrix

shown in table 1, the first of fifteen weekly matrices

used to compute network measures in our panel models. Numbers of
individuals’ rows in Newcomb’s matrices label each observation.
Dashed lines correspond to the status and fragility scores of
individuals 13 and 16.

Fragility in Newcomb's Fraternity: Week 1

tat

n N

rat

: W

eek

0.8

0.9

1.0

1.1

1.2

1.3

Figure 3 A portrayal of the ego-network of individual 13 in week 1
of Newcomb’s study. Individual 13’s contacts are arrayed clockwise
in increasing order of 13’s relative dependence on them. Levels of
relative dependence on each alter are proportional to line-thickness.
Radii of circles surrounding alters decrease (increase) with alters’
levels of fragility (robustness) depicted along the horizontal axis of
figure 2.

Figure 4 A portrayal of the ego-network of individual 16 in week 1
of Newcomb’s study. Individual 16’s contacts are arrayed clockwise
in increasing order of 16’s relative dependence on them. Levels of
relative dependence on each alter are proportional to line-thickness.
Radii of circles surrounding alters decrease (increase) with alters’
levels of fragility (robustness) depicted along the horizontal axis of
figure 2.

T i m e

gilit

1 0 1 1 1 2 1 3 1 4 1 5

T i m e

tat

1 0 1 1 1 2 1 3 1 4 1 5

Figure 5 Times series plots of average fragility scores (left side) and
status scores (right side) for three groups of individuals in
Newcomb’s fraternity. Groups were identified from a sequence
analysis of status trajectories: closed boxes (individuals 4, 5, 9, 14,
17), open boxes (3, 10, 15, 16), and circles (1, 2, 6, 7, 8, 11, 12, 13).
Time is measured in weeks of the study. These series suggest that
increases (decreases) in fragility over time were associated with
declines (growth) in status.

Chemical Products

Plastics and Rubber Products

Construction

Real Estate

Figure 6 A chain of sales-based ties leading to a high sell fragility
score for Chemical Products. Chemical Products sells primarily to
Plastics and Rubber Products, which sells mainly to Construction,
which in turn sells mainly to Real Estate. Numbers of industries’
rows in the BEA’s use matrices label each node in the chain.

Truck Transportation

Petroleum and Coal Products

Oil and Gas Extraction

Rental Leasing Services and

Lessors of Intangible Assets

Figure 7 A chain of procurement-based ties leading to a high buy
fragility score for Truck Transportation. Truck Transportation buys
mainly from Petroleum and Coal Products, which buys
disproportionately from Oil and Gas Extraction, which then relies
mainly on Rental Leasing Services and Lessors of Intangible Assets.
Numbers of industries’ rows in the BEA’s use matrices label each
node in the chain.

Decoupling

Coupling

Fragile

Robust

Figure 8 Visual representations of fragility and robustness for
decoupled and coupled networks. Solid lines denote first-order
ties of social support. Dotted lines represent second-order ties
of social support. Diameters of circles around nodes reflect
levels of robustness. Nodes are circled if their identity and
network affect the robustness of the focal actor A.

Figure A.1 The negative association between cormeasures—the
correlation between fragility measured with

.99

and fragility

measured with

—and mean path length. Outcomes for 1,000

simulated networks are depicted. A curve provided by a bivariate
smoother is included.

mean path length

eas

cormeasures = .99

mean path length = 1.69

cormeasures = .52

mean path length = 4.44

cormeasures = .19

mean path length = 6.97

Figure A.2 Three networks differing in mean path length and in the
correlation between fragility measured with

.99

and fragility

measured with

Table 1. A relational matrix

recording likeability levels in week 1 of Newcomb’s study

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

Table 2. Correlations and Descriptive Statistics for Variables in the Analysis
of Newcomb’s Fraternity

[1]

[2]

[3]

[4]

[1] Status

[2] Fragility (c = 0)

-0.6131

[3] Fragility (c = .99)

-0.5670

0.9558

[4] Sycophant

-0.9013

0.5388

0.4617

Mean

0.9511

0.9813

0.9706

Standard Deviation

0.3095

0.1929

0.2411

12.4961

Min

0.1794

0.7822

0.7377

-27.2813

Max

1.4963

1.9030

2.1452

34.1563

Note.— Within-correlations

Table 3. Standardized Regression Coefficients for Fixed-Effects Models of Status Growth in Newcomb’s Fraternity

Status

-1.7583

-1.8841

-1.8630

-1.9317

-2.6064

-2.7008

(0.1406)***

(0.1798)***

(0.1790)***

(0.1722)***

(0.3641)***

(0.3941)***

Fragility (c = 0)

-0.1286

0.4359

(0.1147)

(0.3186)

Fragility (c = .99)

-0.5417

-0.1770

-0.2167

-0.3190

(0.2855)*

(0.1023)*

(0.1033)**

(0.1312)**

Sycophant

-0.6984

-0.6112

(0.3329)**

(0.3518)*

238

224

within

0.4321

0.4355

0.4453

0.4402

0.4520

0.4018

Standard errors in parentheses

* significant at 10%; ** significant at 5%; *** significant at 1%

Individual and time fixed effects included in each model

Table 4

Industry

Industry Input

Industry Output

Value Added

Farms

140653.30

226356.90

85703.56

Forestry, fishing, and related activities

26545.78

52722.33

26176.55

Oil and gas extraction

65476.18

163180.70

97704.47

Mining, except oil and gas

24639.00

52866.15

28227.15

Support activities for mining

27785.12

50811.48

23026.37

Utilities

133361.30

335695.80

202334.40

Construction

543079.20

1085010.00

541930.80

Food and beverage and tobacco products

441142.70

609195.10

168052.40

Textile mills and textile product mills

50982.08

74036.68

23054.60

Apparel and leather and allied products

26584.37

45694.48

19110.12

Wood products

60706.13

93724.62

33018.50

Paper products

101460.00

152891.80

51431.83

Printing and related support activities

48617.33

94093.69

45476.37

Petroleum and coal products

227630.40

268945.50

41315.13

Chemical products

293146.30

464042.70

170896.30

Plastics and rubber products

110445.40

175049.10

64603.68

Nonmetallic mineral products

51249.98

98002.30

46752.32

Primary metals

108063.60

155912.00

47848.32

Fabricated metal products

137038.30

251089.20

114050.90

Machinery

158110.90

257307.40

99196.48

Computer and electronic products

253429.70

384404.20

130974.50

Electrical equipment, appliances, and components

58029.42

105076.70

47047.34

Motor vehicles, bodies and trailers, and parts

356501.60

466462.00

109960.40

Other transportation equipment

102267.90

168526.90

66259.02

Furniture and related products

42478.95

74952.20

32473.23

Miscellaneous manufacturing

68293.34

128761.80

60468.45

Wholesale trade

253107.90

912496.80

659388.90

Retail trade

326095.50

1030896.00

704800.90

Air transportation

69189.37

117946.70

48757.37

Rail transportation

17717.27

43247.35

25530.08

Water transportation

23009.57

30959.83

7950.28

Truck transportation

117959.80

218202.50

100242.70

Transit and ground passenger transportation

10787.68

26521.35

15733.67

Pipeline transportation

20509.60

32113.18

11603.58

Other transportation and support activities

30587.42

107669.30

77081.88

Warehousing and storage

10725.12

40223.25

29498.13

Publishing industries (includes software)

117607.90

234023.60

116415.70

Motion picture and sound recording industries

45352.96

82347.10

36994.13

Broadcasting and telecommunications

315438.50

588061.90

272623.30

Information and data processing services

53465.05

101594.70

48129.63

Federal Reserve banks, credit intermediation, and related activities

187492.00

580050.10

392557.90

Securities, commodity contracts, and investments

123229.80

281169.80

157940.00

Insurance carriers and related activities

235777.30

490853.30

255075.90

Funds, trusts, and other financial vehicles

63316.13

81845.45

18529.28

Real estate

416120.80

1679320.00

1263199.00

Rental and leasing services and lessors of intangible assets

119552.40

257450.60

137898.20

Legal services

58146.05

213350.30

155204.30

Miscellaneous professional, scientific and technical services

334861.70

790237.80

455376.10

Computer systems design and related services

70193.47

240492.40

170298.90

Management of companies and enterprises

118088.00

309771.70

191683.70

Administrative and support services

172480.50

457265.90

284785.30

Waste management and remediation services

27604.10

55652.32

28048.23

Educational services

60413.46

150490.00

90076.53

Ambulatory health care services

174752.50

545440.70

370688.20

Hospitals and nursing and residential care facilities

226020.40

511579.50

285559.10

Social assistance

41515.53

105436.40

63920.84

Performing arts, spectator sports, museums, and related activities

27695.32

72705.16

45009.85

Amusements, gambling, and recreation industries

37699.79

101712.40

64012.57

Accommodation

36536.68

100576.60

64039.87

Food services and drinking places

237611.20

462741.80

225130.60

Other services, except government

274611.50

600479.10

325867.50

Federal government enterprises

22250.10

85822.66

63572.57

State and local government enterprises

101491.50

175110.70

73619.20

Note.-- In millions of dollars, averaged over years 2000 through 2005. Omitted categories include: Federal general government (62),

State and local general government (65), Noncomparable imports (66), Scrap, used and secondhand goods (67)

Table 5. Correlations and Descriptive Statistics for Variables in the Analysis of Industry Performance

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[1] ln(Total Value Added)

[2] ln(Total Industry Input)

0.6875

[3] Sell Fragility (c = .99)

-0.2625

-0.115

[4] Buy Fragility (c = .99)

0.1774

0.1063

0.0466

[5] Sell Centrality

0.1644

0.0961

-0.0028

-0.0661

[6] Buy Centrality

-0.0268

0.1121

0.1276

0.0147

0.389

[7] Sell Fragility (c = 0)

0.0512

0.0806

-0.0891

-0.2732

0.0148

-0.0268

[8] Buy Fragility (c = 0)

0.2092

0.081

0.0678

0.729

-0.0174

0.0061

-0.2988

[9] Intrarow

-0.0349

0.1066

0.0948

0.0776

0.149

0.4178

-0.1032

0.0113

[10] Intracolumn

0.1003

0.0085

-0.024

-0.0032

0.4356

0.0971

-0.1175

-0.0351

0.5209

[11] Negativeflow

-0.0768

-0.0107

0.027

0.047

-0.1613

-0.0482

0.09

-0.0082

0.0187

0.0386

Mean

11.3431

11.3445

0.31242

0.77544

0.66943

0.66276

0.68663

0.7888

0.1486

0.16088

0.04444

Standard Deviation

0.11939

0.12078

0.07849

0.12639

0.04804

0.05283

0.04811

0.09485

0.01328

0.011

0.11838

Min

8.83203

9.08469

0.0068

0.26115

0.00704

0.04558

0.13135

0.33719

Max

14.199

13.3905

5.86601

4.23476

4.39572

3.29411

3.57717

4.52637

0.70106

0.82611

Note.-- Within correlations and standard deviations

ln(Total Industry Input)

0.903

0.888

0.381

0.398

0.316

(0.035)**

(0.033)**

(0.057)**

(0.062)**

(0.064)**

Sell Fragility (c = .99)

-0.089

-0.230

-0.241

(0.035)*

(0.057)**

(0.055)**

Buy Fragility (c = .99)

-0.374

-0.011

0.007

(0.047)**

(0.053)

(0.049)

Sell Centrality

0.394

0.298

(0.094)**

(0.115)*

Buy Centrality

0.045

-0.010

(0.095)

(0.102)

Sell Fragility (c = .00)

-0.031

0.181

(0.092)

(0.094)

Buy Fragility (c = .00)

0.218

0.261

(0.072)**

(0.068)**

2002

0.023

0.024

0.016

(0.014)

(0.013)

2003

0.057

0.047

0.043

(0.014)**

(0.015)**

(0.014)**

2004

0.101

0.100

0.104

(0.015)**

(0.016)**

2005

0.120

0.130

(0.018)**

(0.020)**

Intrarow

0.084

(0.481)

Intracolumn

1.048

(0.544)

Negative Flow

-0.045

(0.036)

Constant

1.134

1.571

6.804

6.592

7.081

(0.405)**

(0.377)**

(0.631)**

(0.674)**

(0.676)**

Industry Fixed Effects

Yes

315

0.6836

0.7312

0.6139

0.5903

0.6648

Standard errors in parentheses

* significant at 5%; ** significant at 1% (two-tailed tests)

Note.-- All continuous covariates are lagged one year with the exception of industry input. Models 3-5 report within R2

Table 6. Models Predicting ln(Total Value Added)

Table 7. Models Predicting Departmental Prestige in Burris’s PhD Exchange Network

Social Capital

1.118

1.105

0.747

1.068

(0.069)**

(0.067)**

(0.045)**

(0.078)**

Fragility (c = 0)

-0.085

(0.093)

Fragility (c = .99)

-0.101

-0.093

-0.096

(0.039)*

(0.036)*

(0.039)*

Article Publications

0.072

0.067

0.070

0.064

(0.051)

(0.049)

(0.052)

(0.049)

Citations

0.005

0.026

0.005

(0.010)

(0.053)

(0.010)

Research Grants

-0.000

-0.001

-0.008

-0.000

(0.003)

(0.045)

(0.003)

Weighted Article Publications

0.180

0.176

0.112

0.165

(0.082)*

(0.079)*

(0.051)*

(0.080)*

Book Publications

0.245

0.217

0.114

0.232

(0.090)**

(0.088)*

(0.046)*

(0.089)*

Constant

-0.401

-0.294

-0.000

-0.138

(0.150)**

(0.151)

(0.034)

(0.227)

0.89

0.90

Standard errors in parentheses. Model 3 reports standardized regression coefficients and corresponding standard errors.

* significant at 5%; ** significant at 1%

Document Outline

A Model of Robust Positions in Social Networks0F*
Matthew S. Bothner, University of Chicago
Edward Bishop Smith, University of Michigan
Harrison C. White, Columbia University
Word Count (including notes): 15,838

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