Comput Econ (2008) 32:407–413
DOI 10.1007/s10614-008-9145-3

Network Formation Under Cumulative Advantage:
Evidence from The Cambridge High-Tech Cluster

Hinnerk Gnutzmann

Accepted: 2 May 2008 / Published online: 5 June 2008
© Springer Science+Business Media, LLC. 2008

Abstract

When joining a social network, the already well-connected often make

for especially attractive partners because they can facilitate links to other network
members. However, the effect is potentially weakened by increasing redundancy of
contacts with network size. We consider the trade-off between these two factors in a
dataset of the Cambridge High-Tech Cluster and compare results with the county of
Cambridgeshire as a whole. As expected, network effects are stronger in the former,
but in both datasets, redundancy does not offset the benefits of reach in attracting new
partners.

Keywords

Social networks

· Cumulative advantage · Monte Carlo simulation

1 Introduction

High-technology clusters generally consist of a multitude of small firms operating in
an ill-defined and highly risky upstream market for technology; they often rely on
finance from informal sources and employ highly specialised workers (

Arora et al.

2004

). In such an environment of uncertainty, asymmetric information and matching

problems, personal contacts appear to be especially valuable. As Castilla et al. (

2000

)

put it, “there is no proposition so universally agreed upon and so little studied” as the
importance of social networks in such settings. In this paper, we study the formation
of the network of joint board memberships. The network grows as entrants join and
seek established partners; individuals are more likely to be chosen as partners the more
links they already have within this network. This is known as a cumulative advantage
process in the social networks literature.

H. Gnutzmann (

B

)

European University Institute, Florence, Italy
e-mail: [e-mail]

123

408

H. Gnutzmann

The role of social networks in Clusters has been studied in the literature. In the

context of the Cambridge cluster, Myint et al. (

2005

) study the impact of social capital

on venture creation; they find that serial entrepreneurs primarily leverage their net-
works effectively. Garnsey and Heffernan (

2005

) also emphasise the importance of

networks on firm creation. Such findings are reflected in other fields as well. While
these studies—if they consider network data directly—typically use graph-theoretic
measures of the network at a given point in time, this paper complements the literature
by focussing on the impact of different formation processes on network structure.

We proceed by considering the relationship between social capital and cumulative

advantage in Sect.

2

. The next section reviews the Simon model of network growth,

which occurs under a special form of cumulative advantage. The simulation relates
the strength of cumulative advantage directly to the skew of distribution of the number
of link held. These results are confronted with two data sets in Sect.

5

: firstly, a pro-

prietary data set of firms in the Cluster—referred to as a the “High-Tech Sample”—is
considered. The results are then compared to the directorship network formed by all
firms the county of Cambridgeshire.

As expected, the High-Tech sample exhibits the strongest cumulative advantage

effects; the Cambridgeshire network still exhibits a relatively strong form of cumulative
advantage, hinting at the probable importance of social capital in a wide number of
contexts. Section

6

concludes in more detail.

2 Cumulative Advantage and Social Capital

Directors are partly chosen on the basis of their social capital, as discussed above. In our
framework, this is represented by a director receiving an appointment with a probability
depending on the number of contacts k already held. This can be summarised in an
attachment function a

(k):

a

(k) ∝ k

γ

(1)

Directors that have already accumulated social capital are generally in a more

favourable position to win even more contacts; this cumulative advantage effect is
consistent with the large inequality in the number of contacts observed in our data
sets. The marginal impact of an additional contact one’s social capital depends on two
factors: firstly, an additional contact increases the reach within the network (

Granovet-

ter 1973

). This benefit can to some extent be offset by redundancy: the new contact

may be very similar to those already held, which decreases the marginal impact on
social capital.

For positive values of

γ , the “cumulative advantage” effect discussed above is

present, and gets stronger as the parameter rises. Between zero and one, the marginal
effect of further contacts on the probability of appointment is decreasing. In this case,
which we refer to as “weak cumulative advantage”, the effect of redundancy puts a
check on the increased reach. On the other hand, when gamma is greater than one,
marginal effects are increasing, and reach is said to outweigh redundancy (cumula-
tive advantage is “strong”). In the knife-edge case of

γ = 1, the two effects exactly

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Network Formation Under Cumulative Advantage

409

Table 1 Forms of cumulative
advantage

γ

Description

γ < 0

Cumulative disadvantage

γ = 0

Random attachment

0

< γ < 1

Weak cumulative advantage

γ = 1

Proportional cumulative advantage

γ > 1

Strong cumulative advantage

balance and cumulative advantage is proportional to appointments; the negative is not
considered here. Table

1

presents this taxonomy.

This mechanism gives rise to a stochastic process which determines the distrib-

ution of contacts held. Each period, an entrant—who has no previous links—enters
the network and assembles a board with one existing member; the partner is chosen
according to the attachment function. Hence, two changes happen to the contact dis-
tribution: first, the new entrant joins the group of directors with a single appointment.
Secondly, the on existing member each receives an additional appointment.

Let N be the number of directors in the network, N

k

the number of directors with k

links, and p

k

=

N

k

N

be their proportion. A director with k links receives an appointment

with probability proportional to a

(k); the probability that anyone out of the directors

with said number of links receives an appointment is thus given by

A

(k) =

p

k

a

(k)

∀k

p

k

a

(k)

(2)

This set-up gives rise to the two master equations, which show the expected net

changes in the number of directors with a single link, and those with k

> 1 links

respectively:

(N + 1)p

N

+1

1

− N p

N

1

= 1 − A(1)

k

= 1

(N + 1)p

N

+1

k

− N p

N

k

= [A(k − 1) − A(k)] k > 1

(3)

After a large number of directors have joined the network, this distribution no longer

changes with new entrants (a steady state has been reached), which provides a natural
focal point for analysis.

3 Steady State Solution for

γ = 1

Models in this vein have been studied for some time now, e.g. by Simon (

1955

) and

Price (

1965

). Their approach is analytical and considers the case of

γ = 1; in this

case, a

(k) = k and Eq.

2

simplifies to:

A

(k) =

p

k

k

∀k

p

k

k

=

p

k

k

2

(4)

123

410

H. Gnutzmann

The second equality follows because the denominator equals the mean degree of

the network. Because in each time step one new director is added to the network and
connected to one more individual, the number of links in the network rises by two for
each new member. Hence the mean degree must be constant at 2 also.

By definition of the steady state, the degree distribution must be unchanging when

new joiners arrive, i.e.

(N + 1)p

∗

k

− N p

∗

k

= p

∗

k

. Substituting this into the master

equations yields:

p

∗

1

=

2
3

k

= 0

p

∗

k

=

kp

∗

k1

−(k+1)p

∗

k

2

k

> 1

(5)

Solving by repeated substitution gives

p

∗

k

= (2)B(k + 1, 3)

(6)

where B

() is the Beta-function. From the properties of this function, we obtain in the

limit—as k tends to infinity:

p

∗

k

∝ k

−3

(7)

This result establishes that the generating mechanism implies a Power Law degree

distribution for large values of k. Furthermore, the proportional cumulative attachment
model implies a power law exponent of

−3. For more details on this model, see

Newman (

2003

).

4 Simulation

Individuals with the same number of links have the same probability of being chosen
as partners—in the context of the model, they are interchangeable. Thus we store the
number of directors with k links as the kth element of the vector

bi ns. Secondly, we

create a vector

bi n Mass; here, the element k is proportional to the probability that

one of the directors with k links receives another one. It is obtained by element wise
multiplication of

bi ns with the attachment function,

1

:

bi n Mass

k

=

bi ns

k

∗ k

γ

(8)

The network is set up by creating an initial network with 2 incumbents who are

linked to one another. Thus each director has 1 link. If the desired network size is n,
a further n

− 2 members need to be created. For each new member,

• update the

bi n Mass vector and normalise the sum of elements to one; create a

new vector

bi nCumulati

veMass, where the kth element is the sum of elements

1

..k of

bi n Mass

• draw a random number on the unit interval. Find the k such that binCumulative-

Mass

k

−1

≤ rnd <

bi nCumulati

veMass

k

and move one director form this bin

to bin k

+ 1.

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Network Formation Under Cumulative Advantage

411

Fig. 1 Simulation results

• add member to the bin of directors with one link

Having created the network, the Power Law exponent is estimated using the pl f i t

routine developed by Clauset et al. (

2007

).

The simulation was run with n

= 50000 iterations. The former parameter was

chosen to be in line with both data sets, and the latter is close to the size of the larger
sample. The experiment was carried out for values of

γ in the range 0.8–1.4; this

appears to be the economically interesting range. The increment between steps was
0

.01.

Results are shown in Fig.

1

. It is apparent that the exponent of the stationary dis-

tribution changes very slowly as

γ departs from 1; as γ > 1.2, however, it rises

swiftly.

5 Empirical Results

Figure

2

shows the empirical degree distributions for the three data sets described

in the introduction. In the left hand panel, linear scales are used. It is evident that
the degree distribution is highly skewed in all data sets and approximately has the
rectangular hyperbola form expected from a power law distribution. This is at first
sight consistent with a form of cumulative advantage process.

On the right hand side, the same frequency-degree diagram is drawn on double-log

scales. Recall from above that, if the power law distribution held exactly, this plot
should produce a straight line. Here, the results are much harder to interpret. First note
that, in all three panels, there are fewer poorly connected directors than predicted by

123

412

H. Gnutzmann

Fig. 2 Frequency distributions on linear and double-log scales

Table 2 Maximum likelihood
estimation results for

γ

Dataset

Cumulative

Density

distribution

exponent

Cambridgeshire

1.48

2.48

High-Tech cluster

1.55

2.55

the power law. This is apparent in the Cluster data set. Interestingly, the distribution
of the Cambridgeshire data set—which is also by far the most comprehensive—is
considerably closer to the power law in the left hand region.

Table

2

presents the estimated power law exponent for the two data sets. Estimation

was done using the maximum likelihood method outlined above on the cumulative
distribution, from which exponents for the density function were calculated. Compar-
ing the point estimates, however, yields two insights: first, the coefficients are quite
close – between cumulative and cross-sectional networks, and the different sampling
definitions. Secondly, the coefficients are less than three, the value derived from the
analytical model below; the frequency of well-connected directors thus decays less
quickly than predicted. The maximum likelihood estimator (MLE) of the exponent,
of course, is only valid if the distribution does in fact, follow a power law distribution.
Since this assumption is unlikely to hold exactly, and over the entire distribution, this
attaches some extra uncertainty to the estimates.

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Network Formation Under Cumulative Advantage

413

The MLE results between 2.4 and 2.5, as estimated from the sample, are not as

predicted by the proportional cumulative advantage model above, but exhibit even
more skew. Such values of the exponent occur when gamma is slightly

>1, at around

1

.2 < γ < 1.3. These values were obtained through a grid search on the simulation

results.

6 Conclusion

Personal links between individuals can facilitate transactions of many kinds. In such
cases, individuals that are already well-connected are especially attractive partners
for those seeking to enter an industry; hence they are disproportionately likely to
receive new links, giving rise to a process of cumulative advantage. However, the
well-connected are in danger of having increasingly redundant links, weakening their
advantage. The overall cumulative advantage effect thus reflects a trade-off between
reach and redundancy.

Since cumulative advantage models are already difficult to solve analytically in

those cases relatively favourable to such treatment, and no known analytical solution
exists for the general case, this paper used numerical simulation to relate the strength
of the cumulative advantage process to the skew in the degree distribution In both
data sets, cumulative advantage effects were found to be stronger than proportional.
Cumulative advantage was strongest in the High-Tech data set, and slightly weaker in
the Cambridgeshire one. This is consistent with prior expectations.

The results presented here should be seen as indicative, for it is apparent that the

data sets do not literally meet the parametric assumption of a power law distribution,
and the functional form of the cumulative advantage process was restricted rather
narrowly. However, we believe cumulative advantage to be a potentially important
factor in explaining the structure of real-world directorship networks, and especially
the stark inequalities between individuals.

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