Detecting global bridges in networks

TLDR: In this article, a decomposition of betweenness centrality into two terms, one highlighting the local contributions and the other the global ones, is proposed to identify global bridges in air transportation and scientific collaboration networks.

Abstract: The identification of nodes occupying important positions in a network structure is crucial for the understanding of the associated real-world system. Usually, betweenness centrality is used to evaluate a node capacity to connect different graph regions. However, we argue here that this measure is not adapted for that task, as it gives equal weight to "local" centers (i.e. nodes of high degree central to a single region) and to "global" bridges, which connect different communities. This distinction is important as the roles of such nodes are different in terms of the local and global organisation of the network structure. In this paper we propose a decomposition of betweenness centrality into two terms, one highlighting the local contributions and the other the global ones. We call the latter bridgeness centrality and show that it is capable to specifically spot out global bridges. In addition, we introduce an effective algorithmic implementation of this measure and demonstrate its capability to identify global bridges in air transportation and scientific collaboration networks.

Figures

FIG. 5. Co-citation and co-author network of articles published by scientists at ENS de Lyon. Nodes represent the authors or references appearing in the articles, while links represent co-appearances of these features in the same article. The color of the nodes corresponds to the modularity partition and their size is proportional to their BC (left) or to their bridgeness (right), which clearly leads to different rankings (references cited are used in the computations of the centrality measures but appear as dots to simplify the picture). We only keep nodes that appear on at least four articles and links that correspond to at least 2 co-appearances in the same paper. After applying these thresholds, the 8000 articles lead to 8883 nodes (author or references cited in the 8000 articles) and 347,644 links. The average degree is 78, the density 0.009 and the average clustering coefficient is 0.633. Special care was paid to avoid artifacts due to homonyms. Weights are attributed to the links depending on the frequency of co-appearances (cosine distance, see [15].

FIG. 2. Artificial network with a clear community structure using Lancichinetti et al [18] method. For clarity, we show here a smaller network containing 1000 nodes, 30 communities, 7539 links (20% inter-and 80% intra-community links). Each color corresponds to a community as detected by modularity optimization [2, 25].

FIG. 3. (a) Ability of BC or bridgeness to reproduce the ranking of bridging nodes, taking as reference the global indicator (Eq 2). For each of the three networks, we first compute the cumulative sums for the global measure G, according to three sorting options: the G measure itself and the two centrality metrics, namely BC and bridgeness. By construction, sorting by G leads to the highest possible sum, since we rank the nodes starting by the highest G score and ending by the lowest. Then we test the ability of BC or bridgeness to reproduce the ranking of bridging nodes by computing the respective ratios of their cumulative sum, ranking by the respective metric (BC or Bri), to the cumulative obtained by the G ranking. A perfect match would therefore lead to a ratio equal to 1. Since we observe that the ratio for bridgeness is higher than for BC, this means that ordering nodes by their decreasing bridgeness leads to a better ranking of the ’global’ scores as measured by G. To smooth the curves, we have averaged over 200 points. Curves corresponding to different networks are colorised as LFR (red), Airports (blue), ENS (green). (b, c, d): average relative local terms as function of node degree for the three investigated networks (for definition see text).

FIG. 4. Example of the two largest Argentinean airports, Ezeiza (EZE) and Aeroparque (AEP). Both have a similar degree (54 and 45 respectively), but while the first connects Argentina to the rest of the world (85% of international connections, average distance 2.848 miles, G=2327.2), Aeroparque is only a local center (18% of international connections, average distance 570 miles, G=9.0). However, as in the simple graph (Figure 1), BC gives the same score to both (BCEZE=79,000 and BCAEP = 82,000), while bridgeness clearly distinguishes the local center and the bridge to the rest of the world, by attributing to the global bridge a score 250 times higher (BriEZE=46,000 and BriAEP = 174). Red nodes represent international airports while blue nodes are domestic.

FIG. 1. The figures show the betweenness (a) and bridgeness (b) scores for a simple graph. Betweenness does not distinguish centers from bridges, as it attributes a slightly higher score (Figure a, scores = 27) to high-degree nodes, which are local centers, than to the global bridge (Figure a, score = 25). In contrast, bridgeness rightly spots out the node (Figure b, score = 16) that plays the role of a global bridge.

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Copyright
Detecting global bridges in networks
Pablo Jensen, Matteo Morini, Márton Karsai, Tommaso Venturini, Alessandro
Vespignani, Mathieu Jacomy, Jean-Philippe Cointet, Pierre Mercklé, Eric
Fleury
To cite this version:
Pablo Jensen, Matteo Morini, Márton Karsai, Tommaso Venturini, Alessandro Vespignani, et al..
Detecting global bridges in networks. Journal of Complex Networks, Oxford University Press, 2015,
4 (3), pp.319-329. �10.1093/comnet/cnv022�. �hal-01206166�

IMA Journal of Complex Networks (2015) Page 1 of 15
doi:10.1093/comnet/xxxxxx
Detecting global bridges in networks
PABLO JENSEN
∗
IXXI, Institut Rhonalpin des Syst
`
emes Complexes, ENS Lyon; Laboratoire de Physique, UMR
5672, ENS Lyon 693 64 Lyon, France
∗
Correspon ding author: pablo.jensen@ens-lyon.fr
MATTEO MORINI
IXXI, I nstitut Rhonalpin des Sy st
`
emes Complexes, ENS Lyon; LIP, INRIA, UMR 5668, ENS de
Lyon 6 9364 Lyon, France
M
´
ARTON KARSAI
IXXI, I nstitut Rhonalpin des Sy st
`
emes Complexes, ENS Lyon; LIP, INRIA, UMR 5668, ENS de
Lyon 6 9364 Lyon, France
TOMMASO VENTURINI
M
´
edialab, Sc iences Po, Paris
ALESSANDRO VESPIGNANI
MoBS, Northeastern U niversity, Boston MA 02115 USA; ISI Foundation, Turin 1013 3, Italy
MATHIEU JACOMY
M
´
edialab, Sc iences Po, Paris
JEAN-PHILIPPE COINTET
Universit
´
e Paris-Est, SenS-IFRIS
PIERRE MERCKL
´
E
Centre Max Weber, UMR 5283, ENS Lyon 69364 Lyon, France
ERIC FLEURY
IXXI, I nstitut Rhonalpin des Sy st
`
emes Complexes, ENS Lyon; LIP, INRIA, UMR 5668, ENS de
Lyon 6 9364 Lyon, France
[Received on XX XX XXXX; revised on XX XX XXXX; accep te d on XX XX XXXX]
c
 The author 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 of 15 PABLO JENSEN ET AL.
The identification of nodes occupying important positions in a network structure is crucial for the under-
standing of the associated real-world system. Usually, betweenness centrality is used to evaluate a node
capacity to connect different graph regions. However, we argue here that this measure is not adapted for
that task, as it gives equal weight to “local” centers (i.e. nodes of high degree central to a single region)
and to “global” bridges, which connect different communities. This distinction is important as the roles
of such nodes are different in terms of the local and global organisation of the network structure. I n this
paper we propose a decomposition of betweenness centrality into two terms, one highlighting the local
contributions and the other the global ones. We call the latter bridgeness centrality and show that it is
capable to specifically spot out global bridges. In addition, we introduce an effective algorithmic imple-
mentation of this measure and demonstrate its capability t o identify global bridges in air tr ansportation
and scientifi c collaboration networks.
Keywords: Centrality Measures, Bet w eenness Centrality, Bridgeness Centrality
JXX, JYY
1. Introduction
Although the h isto ry of graphs as scientific objects begins with Euler’s [ 10] famous walk across K¨onigsberg
bridges, th e notion of ’bridge’ has rarely been tackled by network theorists
1
. Among the few articles
that took br idges seriously, the most famous is probably Mark Granovetter’s paper on The Streng th of
Weak Ties [14]. Despite the hug e influence of this pa per, few works have remarked th a t its most orig-
inal insig hts concern prec isely the n otion of ’bridge’ in social networks. Granovetter sugg e sted that
there might be a fundamental functional difference be twe en strong and weak ties. While strong ties
promote homogeneous and isolated communities, weak ties foster heter ogeneity and cro ssbreeding. Or,
to use the o ld t¨onnesian clich´e, strong ties generate Gemeinshaft, while weak ties generates Gesellshaft
[8]. Although Granovetter does realize that bridging is the phenomenon he is looking after, two major
difficulties prevented him from a direct ope rationalization of such concep t: “We have had neither the
theory nor the measurement and sampling techniques to move sociometry from the usual small-group
level to that of larger structures” (ibidem, p. 1360). Let’s start from “the measurement and sam pling
techniques”. In order to compute th e br idging force of a given node or link, one needs to be able to
draw a sufficiently comprehensive graph of the system under investigation. Networks constructed with
traditional ego-centered and sampling techniques are too biased to c ompute bridging forces. Exhaustive
graphs of small so c ia l groups will not work either, since such groups are, by definition, dominated by
bounding relations. Since the essence of bridges is to connect individuals across distant social regions,
they can only be computed in large and complete social graphs. Hopeless until a few years ago, such
endeavor seems more and more reasonable as digital media spread through society. Thanks to digital
traceability it is now possible to draw large and even huge social networks [20, 30, 31].
Let’s discuss now the second point, the “theory” needed to measure the bridging force of different
edges or nodes
2
. Being able to identify bounding and bridging nodes has a clear interest for any type
of network. In social networks, bounding and bridging measures (or ”closure” an d “brokerage”, to use
Burt’s terms [6]) tell us w hich nodes build social territories an d which allow items (ideas, pieces of
informa tion, opinions, money... ) to travel through them. In scientometrics’ networks, these notions tell
us which authors define disciplines and paradigms and which breed interdisciplinarity. In ecological
1
We refer to the common use of the word ’bridge’, and not to the technical meaning in graph theory as ’an edge whose deletion
increases its number of connected components’
2
In this paper, we will focus on defining the bridgeness of nodes, but our definition can straightforwardly be extended to edges,
just as the betweenness of edges is derived from that of nodes.

DETECTING GLOBAL BRIDGES IN NETWORKS 3 of 15
networks, they identify relations, which create specific ecological communities and the ones c onnecting
them to larger habitats.
In all these contexts, it is the very same question tha t we wish to a sk: do nodes or edges reinforce
the density of a cluster of nodes (bounding) or do they connect two separated clusters (bridging)? For-
mulated in this way, the bridging/bounding question seems easy to answer. After having identified the
clusters of a network, one should simply observe if a node connects nodes of the same cluster (bound -
ing) or of different clusters (bridging) . However, the intra-cluster/inter-cluster approach is both too
dependent on the method used to detect communities an d flawed by its inherent circular logic: it uses
clustering to define bridging and bounding ties when it is precisely the balance of bridges an d bounds
that determines clusters. Remark that, far from be ing a mathematical subtlety, this question is a key
problem in social theory. D e fining internal (gemeinschaf t) and external (gesellschaft) re la tions by pre-
supposing the existence and the composition of social groups is absurd as groups are themselves defined
by social relations.
In this paper, we introduc e a mea sure of bridgeness of nodes that is independent on the community
structure and thus e scapes this vicious circle, contra ry to other proposals [7, 24]. Moreover, sin ce
the computation of bridg eness is straightforwardly relate d to that of the usual betweenness, Brandes’
algorithm [5] can be u sed to compute it efficiently
3
. To demonstrate the power of our method and
identify nodes acting as local or global bridges, we apply it on a synthetic network and two real ones:
the world airport network and a scientometric n etwork.
Measuring bridgeness
Identify ing impo rtant nod es in a network structur e is cr ucial for the understanding of the associated
real-world system [3, 4, 9], for a review see [25]. The most common mea sure of c e ntrality of a node
for network connections on a global scale is betweenness centrality (BC), which “measures the extent
to which a vertex lies on paths between other vertices” [11, 12]. We show in the following that, when
trying to identif y spe cifically glo bal bridges, BC has some limitations as it assigns the same importance
to paths between the immediate neighbo urs of a node as to paths between further nodes in the network.
In other words BC is built to ca pture the overall cen trality of a node, and is n ot specific enough to
distinguish between two types of centralities: local (center of a co mmunity) and global (brid ge between
communities). Instead, our measure of bridging is more specific, as it gives a higher score to global
bridges. The fact that BC may attribute a higher score to local centers than to global bridges is easy
to see in a simple network (Figure 1). The logics is that a “star” node with degree k, i.e. a node
without links between all its first neighbors (clustering coefficient 0) rece ives automatically a BC =
k(k − 1)/2 arising from paths of len gth 2 connecting the node’s first neighbors and crossing the central
node. More generally, if there exist nodes with high degree but connected only locally (to nodes of the
same community), the ir betweenness ma y be of the order of that measured for more globally connected
nodes. Consistent with this observation, it is w e ll-known that for many networks, BC is highly correlated
with degree [13, 23, 26]. A recent scientometric s study tried to use between ness centrality as “a n
indicator of the interdiscip linarity of jo urnals” but noted that this idea on ly worked “in local citation
environments and after normalization because otherwise the influence of degree centrality dominated
the betweenness centrality measure [21].
To avoid th is problem and specifically spot out global centers, we decompose BC into a local and a
3
We have written a plug-in for Gephi [1] that computes this measure on large graphs. See Supplementary Informations for a
pseudo-algorithm for both node and edge bridgeness.

4 of 15 PABLO JENSEN ET AL.
0
0
0
0
0
9
27
27
25
0
0
5
16
0
0
0
0
0
0
0
0
(a) Betweenness centrality
(b) Bridgeness centrality
5
FI G. 1. The figures show the betweenness (a) and bridgeness (b) scores for a simple graph. Betweenness does not distinguish
centers from bridges, as it attributes a slightly higher score (Figure a, scores = 27) to high-degree nodes, which are local centers,
than to the global bridge (Figure a, score = 25). In contrast, bridgeness rightly spots out the node (Figure b, score = 16) that plays
the role of a global bridge.
global term, the latter being called ’bridgene ss’ centrality. Since we want to distinguish global bridges
from local ones, the simplest approach is to discar d shortest paths, whic h either start or end at a node’s
first neighbors from the summatio n to compute BC (Eq. 1.1). This completely removes the paths that
connect two non connected neighbors for ’star nodes’ (see Figure 1) and greatly diminishes the effect of
high d egrees, while keeping those paths that connect more distant regions of the network.
More formally in a graph G = (V, E), where V assigns the set of nodes and E the set of links the
definition of the betweenness centrality for a node j ∈ V stands as:
BC( j) = Bri(j) + local( j), (1.1)
where
BC( j) =
∑
i6= j6=k
σ
ik
( j )
σ
ik
Bri( j) =
∑
i6∈N
G
( j)∧k6∈N
G
( j)
σ
ik
( j )
σ
ik
local( j) =
∑
i∈N
G
( j)∨k∈N
G
( j)
σ
ik
( j )
σ
ik
.
(1.2)
Here the summation r uns over any distinct node p airs i and k;
σ
ik
represents the number of shortest paths
between i and k; while
σ
ik
( j ) is the number of such shortest paths r unning through j. Decomposing
BC into two parts (right hand side) the first term defines actually the global term, bridgeness centrality,
where we consider shortest paths between nodes not in the neighbourhood of j (N
G
( j )), while the second
local term considers the sh ortest paths starting or ending in the neighbourhood of j. This definition
also demonstrates that the bridgeness centrality value of a node j is always smaller or equal to the
correspo nding BC value and they only differ by the local contr ibution o f the first neighbours. Fig. 1
illustrates the ability of bridgeness to specifically highlight nodes that connect different regions of a
graph. Here the BC (Fig. 1a) and bridgeness centrality values (Fig. 1b) calculated for nodes of the
same network demonstrate that bridgeness centr ality gives the highest score to the no de which is central
globally (green), while BC does not distinguish among local or g lobal centers, and actually assigns the
highest score to nodes with high degrees (red).

Frequently asked questions

Q1. What is the advantage of the measure of bridgeness over former propositions?

One crucial advantage of their measure of bridgeness over former propositions is that it is independent of the definition of communities. 

Q2. What is the problem with intra-cluster/inter-cluster approach?

the intra-cluster/inter-cluster approach is both too dependent on the method used to detect communities and flawed by its inherent circular logic: it uses clustering to define bridging and bounding ties when it is precisely the balance of bridges and bounds that determines clusters. 

Q3. How do the authors compute the bridging force of a given node or link?

In order to compute the bridging force of a given node or link, one needs to be able to draw a sufficiently comprehensive graph of the system under investigation. 

Q4. What is the simplest way to detect global bridges in networks?

The identification of nodes occupying important positions in a network structure is crucial for the understanding of the associated real-world system. 

Q5. What is the meaning of the word bridge?

In ecological1The authors refer to the common use of the word ’bridge’, and not to the technical meaning in graph theory as ’an edge whose deletion increases its number of connected components’ 

Q6. How does the bridgeness of the network compare to the global score?

ordering nodes by their decreasing bridgeness leads to a ranking that is closer to the ranking obtained by the global score than the ranking by decreasing BC. 

Q7. Why does BC assign a higher score to nodes that are not connected to other communities?

Bridgeness never ranks higher local centers than global bridges: here, it correctly assigns a 5 times higher score to node 758 than to node 515. 

Q8. Why does BC give a high score to nodes that are not connected to other communities?

Second problem with BC, it gives a high score to nodes that are not connected to other communities, merely because they are local centers, i.e. they have a high degree.