Prominence and control: The weighted rich-club effect
Tore Opsahl,1Vittoria Colizza,2Pietro Panzarasa,1and Jos´ e J. Ramasco2
1School of Business and Management, Queen Mary College, University of London, UK
2Complex Systems Lagrange Laboratory, Complex Networks, ISI Foundation, Turin, Italy
Complex systems are often characterized by large-scale hierarchical organizations. Whether the
prominent elements, at the top of the hierarchy, share and control resources or avoid one another lies
at the heart of a system’s global organization and functioning. Inspired by network perspectives,
we propose a new general framework for studying the tendency of prominent elements to form
clubs with exclusive control over the majority of a system’s resources. We explore associations
between prominence and control in the fields of transportation, scientific collaboration, and online
PACS numbers: 89.75.Hc,89.65.-s,89.40.Dd
Research has long documented the abundance of sys-
tems characterized by heterogeneous distribution of re-
sources among their elements [1, 2]. Back in 1897, Pareto
noticed the social and economic disparity among people
in different societies and countries . This empirical reg-
ularity inspired the 80−20 rule of thumb stating that only
a select minority (20%) of elements in many real-world
settings are responsible for the vast majority (80%) of the
observed outcome. While recent studies have examined
the tendency of prominent elements to establish connec-
tions among themselves , how they leverage on their
connections to gain and maintain control over resources
circulating in a system still remains largely unexplored.
In particular, do they collude and choose to exchange
a disproportionately large amount of resources among
themselves rather than with others? Or does competi-
tion prevent them from deepening the connections they
have with one another? To answer these questions, we
need to test for the tendency of prominent elements to
engage in stronger or weaker interactions among them-
selves than expected by chance. We call this tendency
the weighted rich-club effect. In this Letter, we adopt
the framework of network theory – where the elements of
the system are seen as nodes and the links among the el-
ements represent interactions [4, 5, 6, 7, 8] – and provide
a novel method to properly assess this tendency.
Previous work has focused on highly connected nodes
and the degree to which they preferentially interact
among themselves . This feature is known as the rich-
club phenomenon [3, 9], a metaphor that alludes to the
tendency of the highly connected nodes (i.e., the rich
nodes) to establish more links among themselves than
randomly expected. Evidence of the phenomenon has
been reported for scientific collaboration networks ,
transportation networks  and inter-bank networks .
Conversely, research has shown that highly connected
routers on the Internet tend not to be connected with one
another , whereas the pattern of interactions among
proteins has been found to depend on the particular or-
ganism under consideration [3, 11]. Although uncovering
interesting structural aspects of the systems, these stud-
ies are limited in that they only detect whether or not
links among prominent nodes are present. In so doing,
they neglect a crucial piece of information encoded in the
weight of links, which is a measure of their intensity, ca-
pacity, duration, intimacy or exchange of services [12, 13].
A full understanding of how systems are organized re-
quires a shift towards a new paradigm that allows us to
evaluate whether nodes that rise to network prominence
also tend to exchange among themselves the majority of
the resources flowing within the network.
To this end, we rank all nodes of a system in terms
of a richness parameter r. For each value of r, we select
the group (the club) of all nodes whose richness is larger
than r. We thus obtain a series of increasingly selective
clubs. For each of these clubs, we count the number E>r
of links connecting the members, and measure the sum
W>rof the weights attached to these links (Fig. 1A). We
then measure the ratio φw(r) between W>rand the sum
of the weights attached to the E>rstrongest links within
the whole network (Fig. 1B). Formally, we have:
weights on the links of the network, and E is the total
number of links. Eq. (1) thus measures the fraction of
weights shared by the rich nodes compared with the total
amount they could share if they were connected through
the strongest links of the network. Other measures can
be introduced that depend on the local network structure
surrounding the rich nodes [3, 14, 15, 16]. Here we aim
at investigating the extent to which the prominent nodes
control the flow of resources over the whole system.
In analogy with the topological rich-club measure [3,
17], a high value of φw(r), however, is not in itself suffi-
cient to account for an actual tendency of the rich nodes
to preside over the strongest links. This is due to the fact
that even networks where links are randomly established
could display a non-zero value of φw(r). To assess the
actual presence of the weighted rich-club phenomenon,
l+1with l = 1, 2, ..., E are the ranked
arXiv:0804.0417v2 [physics.soc-ph] 20 Oct 2008
discounted of random expectations, φw(r) must be com-
pared with an appropriate benchmark. To this end, we
introduce a null model that is random but at the same
time comparable to the real network. In particular, this
model should break the associations between weights and
links while preserving some crucial features of the net-
work encoded in its degree distribution P(k) (i.e., the
probability that a given node is connected to k neigh-
bors) and weight distribution P(w) (i.e., the probability
that a given link has weight w). In addition, the nodes
in the rich club must be the same as in the real network,
which also preserves the richness distribution P(r) (i.e.,
the probability that a given node has richness r).
In what follows, we introduce three procedures for con-
structing null models (see Fig. 1C) that correspond to dif-
ferent ways of preserving P(r), depending on the choice
of the richness parameter r. In this Letter, we explore
three possible definitions of r: the degree k, the strength
s (i.e., the sum of the weights attached to the links orig-
inating from a node)  and the average weight ¯ w (i.e.,
the ratio between s and k) . If the richness of a node is
given by its degree, we adopt the following two random-
ization procedures. First, the Weight reshuffle procedure
consists simply in reshuffling the weights globally in the
network, while keeping the topology intact. Second, the
Weight & Link reshuffle procedure, which introduces a
higher degree of randomization, consists in reshuffling
also the topology, while preserving the original degree
distribution P(k) [6, 19]. Weights are automatically re-
distributed by remaining attached to the reshuffled links.
Both randomization procedures can be easily generalized
to directed networks. The Weight & Link reshuffle pro-
cedure, mixing the signal coming from the topology with
that generated by the location of weights, is considered
here to assess the effects of higher degrees of randomiza-
tion on the results, as well as for the sake of comparison
with the topological rich-club coefficient .
Inevitably, since weights are reshuffled globally, both
procedures produce null models in which the nodes do
not maintain the same strength s as in the real network.
When node richness is defined in terms of s, we need to
introduce a third procedure that preserves this quantity.
We construct a null model based on the randomization of
directed networks  that preserves not only the topol-
ogy and P(w), but also the strength distribution P(s)
(i.e., the probability that a given node has strength s)
of the real network. To this end, we reshuffle weights
locally for each node across its outgoing links (Directed
Weight reshuffle procedure). In so doing, we also obtain
null models where the average weight ¯ w of outgoing links
is kept invariant. We extend this procedure to the undi-
rected case by duplicating an undirected link into two
directed links, one in each direction.
For a given definition of the richness r, the weighted
Weight & Link
(C) Null models
FIG. 1: (A-B) Schematic representation of a weighted net-
work, with size of nodes proportional to their richness, and
width of links to their weight indicated by the corresponding
numbers. Several definitions of richness can be considered.
(A) The nodes and links in the rich club are highlighted, giv-
ing E>r = 6 links and W>r = 4+2+2+3+1+2 = 14. (B)
The strongest E>r = 6 links of the network are highlighted,
yielding the following value for the denominator of Eq. (1):
reshuffled; the numbers to their weight.
φw(r) = 14/21. (C) Null models. Solid lines refer to the links
= 4 + 4 + 4 + 3 + 3 + 3 = 21. We thus obtain:
rich-club effect can be detected by measuring the ratio:
sessed on the appropriate null model. When ρwis larger
than one, the original network has a positive weighted
rich-club effect, with rich nodes concentrating most of
their efforts towards other rich nodes compared with
what happens in the random null model. Conversely,
if it is smaller than one, the links among club members
are weaker than randomly expected.
In order to examine the applicability of our method,
we study three real-world networks drawn from different
domains: (i) The US Airport Network, obtained from
the US Department of Transportation , composed
of 676 commercial airports and 3,523 routes connect-
ing them. Each weight corresponds to the average num-
ber of seats per day available on the flights connecting
two airports [12, 22]. (ii) The Scientific Collaboration
Network , extracted from the arXiv  electronic
database in the area of Condensed Matter Physics, from
1995 to 1999. Nodes represent scientists and a link exists
between two scientists if they have co-authored at least
one paper.Link weight reflects the authors contribu-
tion in their collaboration  – the larger the number of
authors collaborating on a paper, the weaker their inter-
action. (iii) The Online Social Network  comprising
59,835 directed online messages exchanged among 1,899
college students at the University of California, Irvine,
from April to October 2004. Link weight is the number
null(r) refers to the weighted rich-club effect as-
of messages sent from one student to another.
We begin by defining network prominence in terms of
node degree. In this case, r = k. We examine whether
the highly connected nodes control the exchange of re-
sources. For the three networks, Fig. 2 (left column)
reports the weighted rich-club ratio and its topological
counterpart (inset).With only a mild topological ef-
fect , the airport network shows a strong weighted
rich-club effect, as can be identified from the remarkable
growth of ρwas a function of the degree of the airports.
This finding agrees with previous studies that reported
the presence of non-trivial correlations between weight of
the links and degrees of the nodes [12, 22, 26]. Connec-
tions among hub airports, with flights to many destina-
tions, are characterized by large travel fluxes. Different
results are found for the scientific collaboration network:
while there is evidence of a strong positive topological
rich-club effect, the network does not display a weighted
one. As shown in Fig. 2, ρwremains flat around 1 for al-
most the whole range of k. The authors that have many
collaborators tend to work together. However, the in-
tensity of their collaboration does not differ from what
is randomly expected, thus providing additional support
to the observed lack of correlations between collabora-
tion intensity and number of collaborators [27, 28]. Fi-
nally, the weighted and topological rich-club effects dis-
play strikingly different trends for the online social net-
work. Very gregarious individuals, with a large number
of contacts, poorly communicate with one another. How-
ever, when they do, they choose to forge stronger links
than randomly expected.
To investigate how different definitions of prominence
might affect the results, we restricted our attention to
a subset of the arXiv collaboration network based on
the publications on Network Science . In Fig. 3A-
B, we mapped the interaction patterns within the clubs
obtained by defining r in terms of the degree k (num-
ber of co-authors) and the strength s (number of pub-
lished papers), respectively. In this network, each paper
corresponds to a fully connected group of collaborators.
When a paper is co-written by a large number of authors,
these authors take on a high degree and thus increase
their chances to become members of the club based on
k. Large collaborations tend to secure club membership,
yet generate weaker links than smaller ones . Exper-
imental papers on biological networks are authored by a
large number of scientists, and therefore only few such
papers may suffice to substantially increase the topologi-
cal rich-club effect (see the very large clique in Fig. 3A).
By contrast, they bring about weaker links than small-
scale collaborations, thereby reducing their contribution
to the weighted rich-club effect. However, when network
prominence is defined in terms of s, club members as well
as their interaction patterns substantially change with re-
spect to the case in which r = k (Fig. 3B).
The next step is thus to define network prominence in
Weight & Link reshuffle
US Airport Network
Scientific Collaboration Network
Online Social Network
FIG. 2: Weighted rich-club effect in: the US Airport Network
(top); the Scientific Collaboration Network (center); and the
Online Social Network (bottom). Left column: r = k. The
insets refer to the topological rich-club coefficient ρ(k) ,
defined as the ratio between φ(k) (i.e., the fraction of links
connecting rich nodes, out of the maximum possible number
of links among them)  and φnull(k) (i.e., φ(k) measured on
the corresponding Weight & Link reshuffle null model). Right
column: r = s (diamonds) and r = ¯ w (circles).
terms of node strength s and shift our attention from the
most connected to the most involved nodes in the sys-
tem’s activity. Our findings show that active nodes pref-
erentially direct their efforts towards one another, and
this tendency becomes more pronounced as the involve-
ment of nodes in network activity increases (see Fig. 2,
right column). Not only do busy airports direct routes
to one another, but they also secure control over travel
fluxes by channeling on those routes a larger proportion
of their passengers than randomly expected. This behav-
ior is in sharp contrast with what was found using a differ-
ent null model , a pointed reminder of the crucial role
played by such models in assessing the rich-club effect.
When scientists are chosen on the basis of their scientific
productivity s, exclusive clubs emerge in which scientists
tend to collaborate with one another to a greater extent
than randomly expected, unlike what was found within
the club of the most connected scientists. In the online
social network, highly active users tend to communicate
with one another more frequently than would be the case
if contacts were chosen at random.
While node strength gives a general indication of how
involved a node is in the activity of a network, it does
not allow us to discriminate between nodes with a large
number of weak links and nodes with a small number of
4 Download full-text
r = k
r = s
collaboration network  based on degree (A) and strength
(B). Only links among the rich nodes are shown. The size of
the nodes is proportional to their richness; the width of the
links to their weight.
Subsets of the rich nodes in the Network Science
strong links. To address this issue, we define the rich-
ness parameter in terms of the average weight ¯ w. We
find positive effects in all networks (see Fig. 2, right col-
umn). Airports that optimize the traffic per link tend to
direct their busy routes to one another. Scientists that
show the ability to maximize their resources per collab-
oration tend to intensively collaborate with one another.
Online communication tends to occur among users that
maximize the attention directed to each contacts.
By shifting focus from the network topology to the
weight of links, we have proposed a new general frame-
work for studying the tendency of prominent nodes to
attract and exchange among themselves the majority of
the resources available in a system. Unlike a merely topo-
logical assessment of the network, our method allows us
to uncover organizing principles that would otherwise re-
main undetected. In addition, by varying the definition
of prominence, we found evidence of different organizing
principles, and paved the way towards a deeper under-
standing of the multiple layers of a system’s organization.
Our method is widely applicable, in that it enables us
to study the control benefits of prominent elements in a
variety of empirical settings, by decoupling prominence
from strictly network properties. To the extent that the
components of a system can be sorted into a hierarchy
according to a given property, our framework suggests
several new ideas for future research, including the im-
pact of performance, centrality, status, age, size on the
ability of elements to control flows of resources. In this
respect, our study helps shed a new light on the global
organization of complex systems.
The authors would like to thank Alessandro Vespignani
and Alain Barrat for useful discussions and suggestions.
 V. Pareto, Cours d’economie politique, Macmillan, Paris,
 A.-L. Barab´ asi and R. Albert, Science 286, 509 (1999).
 V. Colizza, A. Flammini, M.A. Serrano, and A. Vespig-
nani, Nature Phys. 2, 110 (2006).
 R. Albert and A.-L. Barab´ asi, Rev. Mod. Phys. 74, 47
 S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks:
From Biological Nets to the Internet and WWW, Oxford
Univ. Press, (2003).
 M.E.J. Newman, SIAM Rev. 45, 167 (2003).
 R. Pastor-Satorras, A. Vespignani, Evolution and Struc-
ture of the Internet:A Statistical Physics Approach,
Cambridge Univ. Press, (2004).
 G. Caldarelli, Scale-Free Networks: Complex Webs in
Nature and Technology, Oxford Univ. Press, (2007).
 S. Zhou and R.J. Mondragon, IEEE Commun. Lett. 8,
 G. De Masi, G. Iori and G. Caldarelli, Phys. Rev. E 74,
 S. Wuchty, PLoS ONE 2, e355 (2007).
 A. Barrat, M. Barth´ elemy, R. Pastor-Satorras and A.
Vespignani, Proc. Natl. Acad. Sci. (USA) 101, 3747
 M. Granovetter, Am. J. Sociology 78, 1360 (1973).
 M.A. Serrano, arXiv:0802.3122 (2008).
 S. Valverde and R.V. Sol´ e, Phys. Rev. E 76, 046118
 V. Zlatic et al., arXiv:0807.0793 (2008).
 L.A.N. Amaral and R. Guimer` a, Nature Phys. 2, 75
 J.J. Ramasco and S. Morris, Phys. Rev. E 73, 016122
 M. Molloy and B. Reed, Random Struct. & Alg. 6, 161
 M.A. Serrano, M. Bogu˜ n´ a and A. Vespignani, J. Econ.
Interac. Coor. 2, 111 (2007).
 R. Guimer` a, S. Mossa, A. Turtschi and L.A.N. Amaral,
Proc. Natl. Acad. Sci. (USA) 102, 7794 (2005).
 M.E.J. Newman, Phys. Rev. E 64, 016132 (2001).
 P. Panzarasa, T. Opsahl, Proc. of the XXVII Int. Sunbelt
Social Network Conf. (2007).
 Z. Wu et al., Phys. Rev. E 74, 056104 (2006).
 J.J. Ramasco and B. Gon¸ calves, Phys. Rev. E 76, 066106
 J.J. Ramasco, Eur. J. Phys. St. 143, 47 (2007).
 M.E.J. Newman, Phys. Rev. E 74, 036104 (2006).