Network Science (2019), 1–20
doi:10.1017/nws.2019.62
ORIGINAL ARTICLE
Identifying key players in bipartite networks
Scott W. Duxbury
Department of Sociology, University of North Carolina Chapel Hill, 155 Hamilton Hall, 102 Emerson Drive, Chapel Hill,
NC 27514, USA (email: duxbury.5@osu.edu)
Action Editor: Stanley Wasserman
Abstract
Measures of bipartite network structure have recently gained attention from network scholars. However,
there is currently no measure for identifying key players in two-mode networks. This article proposes
measures for identifying key players in bipartite networks. It focuses on two measures: fragmentation and
cohesion centrality. It extends the centrality measures to bipartite networks by considering (1) cohesion
and fragmentation centrality within a one-mode projection, (2) cross-modal cohesion and fragmentation
centrality, where a node in one mode is influential in the one-mode projection of the other mode, and
(3) cohesion and fragmentation centrality across the entire bipartite structure. Empirical examples are
provided for the Southern Women’s data and on the Ndrangheta mafia data.
Keywords: two-mode network, vulnerability, influence, centrality, disruption
Network scholars have recently shown interest in developing structural measures for bipartite networks (Jasny & Lubell, 2015; Latapy et al., 2008; Opsahl et al., 2010; Opsahl, 2013). First, advances
in statistical network models have been extended to affiliation data (Fujimoto et al., 2011; Wang
et al., 2009, 2013), opening avenues for inferential analyses of bipartite networks. Second, onemode projections of bipartite networks (e.g. Breiger, 1974) typically overestimate local structural
effects, such as brokerage and clustering (Opsahl et al., 2010; Opsahl, 2013), and sacrifice information contained in the two-mode structure. With these issues and developments comes a need
for explicit measures of network structure and node position in bipartite networks.
Notable developments in measurement for bipartite network structure includes global and
local clustering coefficients (Opsahl, 2013), geodesic distances (Borgatti & Everett, 1997; Latapy
et al., 2008; Newman, 2001), brokerage (Jasny & Lubell, 2015), and determining central cliques
(Larsen & Ellersgaard, 2017). However, key player identification has yet to be translated to bipartite networks. Key player identification is of concern for scholars interested in the influence of
certain actors over the entire network. For instance, criminologists use key player analysis to locate
vulnerabilities in gangs and other organized criminal networks (Duijn et al., 2014; Duxbury &
Haynie, 2019). Epidemiologists have applied key player measures to identify actors who are positioned to quickly diffuse a disease or health behavior through a community or population (Chen
et al., 2008; Cobb et al., 2010; Valente, 2012), and ecologists have suggested using key player centralities to identify nature reserves that are integral to a functional landscapes (Urban et al., 2009).
Extending key player measures to bipartite graphs thus offers a useful scientific tool for research
in these fields and related ones.1
Borgatti (2006) proposed two measures for key player identification in one-mode networks:
fragmentation and cohesion centrality.2 The former refers to vulnerable positions within a
© Cambridge University Press 2019
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S. W. Duxbury
network, where the latter refers to positions of influence. Unlike brokerage or degree centrality, these measures identify actors who are positioned to yield maximal network disruption when
removed (fragmentation centrality)3 or who are positioned to quickly transmit a signal to the
greatest number of actors (cohesion centrality). Such identification strategies are of great interest,
for example, when measuring the vulnerability of networks, or when evaluating the potential for
a signal to diffuse throughout a network (e.g., disease, new ideas, innovations, or resources).
The goal of this paper is to extend key-player identification to bipartite graphs. It first
reviews Borgatti’s (2006) measures in depth. It then extends the measures to bipartite networks,
extrapolating on the one-mode measure to construct unique metrics for (1) intramodal fragmentation and cohesion centrality, where a node wields influence within a one-mode projection of
the bipartite network; (2) cross-modal fragmentation and cohesion centrality, where a node in
one mode of the network may be influential in the one-mode projection of the other mode in
the two-mode network, and (3) total fragmentation and cohesion centrality, where a node in the
network is influential across the entire bipartite structure. It uses simulations to illustrate that the
proposed measures of fragmentation and cohesion centrality identify distinct sets of nodes compared to alternative measures of influence when working with bipartite networks. An empirical
example is provided on the Ndrangheta mafia network for fragmentation centrality and another
on Southern Women data (Davis et al., 1941) for cohesion centrality.4 R code to implement the
measures is available at the author’s GitHub.
1. The key player problem
Borgatti (2006) proposed fragmentation and cohesion centrality to isolate actors who are either in
positions of great vulnerability or in great influence in one-mode networks. The measures focus
on the relative distance between actors.
1.1 Fragmentation
Fragmentation centrality calculates disruption potential. The measure is based on the shape of
the subgraph that results from a focal actor’s removal. Let F be the fragmentation centrality of an
actor i in a one-mode network with n actors. The fragmentation centrality Fi is proportional to
the average distance between actors when i is removed from the network,
2
Fi = 1 −
1
dhj
(1)
n(n − 1)
where d is the geodesic distance between two actors h and j when a focal actor i is removed from
the network. The equation utilizes the reciprocal of d so that d1 is equal to 0 when actors are
hj
completely disjointed (when d is infinite). The value is subtracted from 1 to ensure that increasing
values indicate increasing disruption potential. Like most centrality measures, the value ranges
between 0 and 1, where higher values of Fi reflect more vulnerable positions in a network (e.g.,
their removal results in greater path lengths, on average, between any two randomly chosen actors
in the network). By identifying the maximum Fi , a researcher can isolate the actor who is the
greatest vulnerability in the network—that is, the actor who, when deleted, yields the greatest
possible increase in global mean geodesic distance.
1.2 Cohesion
Like fragmentation centrality, cohesion centrality is also measured by relative distances between
two actors. Cohesion centrality measures influence in a one-mode network as indicated by the
mean path length between a focal actor and all other actors. Let C be the cohesion centrality of an
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actor i in one-mode network with n actors. Ci is computed by averaging over the distance between
a focal actor and all alters in the network,
1
dij
(2)
Ci =
n
as above, Ci is constrained to range between 0 and 1. Higher values indicate shorter path lengths,
on average, from the focal actor to a randomly chosen alter, and therefore greater diffusion potential. By identifying the maximum Ci in a one-mode network, a research can isolate the actor who
is positioned to most quickly diffuse a signal throughout the network.
1.3 Key player sets
One of Borgatti’s (2006) innovations was to identify key player sets. The utility of these measures
is such that they can be updated to identify groups of key players in a network when the influence
of a group of actors may be greater than the sum of each actor’s influence. This is ideal when
diffusion or fragmentation potential are nonlinear—that is, for example, when the removal of the
three leading key players may have a weaker effect than the removal of the leading three-player set.
The measures can be adjusted by replacing i in each equation with a predetermined set of size k.
This results in:
1
2
dhj
Fk = 1 −
(3)
n(n − 1)
for fragmentation centrality, and
1
dkj
(4)
n
for cohesion centrality. A researcher interested in key sets of players merely determines the size of
the desired set. A greedy optimization algorithm can be used to iterate across possible k set combinations and identify the leading k set (Borgatti, 2006; An & Liu, 2016). Since the basic approach
to key player set identification does not change in the bipartite context, this issue will not be discussed at length. However, the measurements proposed for identifying key players in two-mode
networks can easily be adapted for key player sets in the same fashion as those above.
Ck =
2. Key players in bipartite networks
A core component of key player identification is the relative distance of a focal actor to alters. Yet,
there is little consensus on the best way to measure distance in a two-mode network. To derive
key player metrics in the bipartite context, the paper proceeds by first discussing how to measure
distance in a bipartite network before developing measurements for fragmentation and cohesion
centrality.
2.1 Distance
The conventional strategy for assessing distance in a bipartite network is to first project each mode
of the network and then compute distance between actors in each simple one-mode projection
(e.g. Newman, 2001, 2010). This approach offers the benefit of parsimony and allows researchers
to consider the implicit relationships between actors who belong to the same mode of the network (Breiger, 1974). However, one-mode projections often overestimate some structural features,
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S. W. Duxbury
such as local clustering (Opsahl, 2013). More pertinent to the detection of key players in bipartite
networks is that simple one-mode projections sacrifice information contained in two-mode networks. Such information loss has obvious implications for key player identification, where over
or underestimating distance in the one-mode projections may affect which actors are identified
as influential. Given a two-mode network which can be represented by an affiliation matrix with
m actors in the first mode and p actors in the second, one possible approach is to reconfigure the
m × p affiliation matrix into a square (m + p) × (m + p) matrix, such that both modes appear in
both the columns and the rows of the reconfigured matrix (Borgatti & Everett, 1997). However,
this approach yields almost identical results as the one-mode projection strategy.
An alternative strategy to account for distances in projections of bipartite networks is to weight
the edges in the projected networks by mutual affiliations in the bipartite network (Newman, 2001;
Larsen & Ellersgaard, 2017). Newman (2001) defines this measure as:
1
wij =
(5)
nq − 1
q
where, given a projection of ij edges, the weight w for an ij edge is proportional to the degree
centrality of q (nq ), who is connected to both i and j in the unprojected two-mode network. Since
an ij edge can only exist if nq ≥ 2, there is no risk of dividing by infinite values. The distance
between two actors can then be computed by summing over the weights of each edge on the
shortest path between i and j in the weighted one-mode projection.
This approach is appealing for measuring distance in bipartite networks because the weighting
procedure preserves information from the two-mode network. In some instances, however, the
strategy of summing over weighted edges to calculate distance raises the possibility that smallest
value or “shortest” path between two actors may not actually be the path with the smallest number
of intermediaries. Considering this issue in the case of weighted one-mode networks, Opsahl et al.
(2010) propose a solution to considering both edge-weights and the number of intermediaries
between actors in a weighted one-mode network. Here, the researcher specifies a tuning parameter
which assigns greater or lesser priority to edge weights. When applied to a weighted one-mode
projection, the shortest path length d between i and j can be expressed as
dijb =
1
1
+···+ a
a
wiz
wzj
(6)
where superscript b distinguishes the measure of distance in bipartite structures from simple
distance d, z represents the intermediary nodes on the path between i and j, and α is a tuning
parameter. Values of α larger than 1 place greater importance on the weights in the projected network, while values below 1 assign greater weight to the number of intermediaries in a path. Since
a case can be made that both the strength of the weighted ties and the number of intermediaries in
a one-mode projection are important when evaluating distance in a bipartite network, it is often
appropriate to assign a value of 1 to α. When a value of 1 is used, the distance measure reduces to
the summation approach discussed by Newman (2001), and can be represented as:
dijb =
1
1
+···+
wiz
wzj
(7)
Equation (7) will be used for all empirical analyses in this study. Opsahl et al. (2010) and
Agneessens et al. (2017) provide guidance on selecting a substantively meaningful value for α.
To summarize, distance can be measured in bipartite networks by first weighting the edges
in one-mode projections by the number of mutual affiliations in the bipartite network and then
summing over the edge weights for all intermediary links in the path between two actors. The
result is a distance measure which accounts for proximity in bipartite network structure as well as
the absolute path lengths between actors in the one-mode projections.
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2.2 Fragmentation
In a two-mode network, the projection between actors in the first mode, the projection between
actors in the second mode, and the affiliations between modes are distinct relationships. Separate
key player measures should be created to summarize the relationships in both projections and the
overall bipartite structure (Latapy et al., 2008). There are three ways in which actors’ influence
can be conceptualized in a bipartite network. First, actors may be influential within the one-mode
projection to which they belong (intramodal influence). Second, actors may be influential in the
one-mode projection to which they do not belong (cross-modal influence). Third, actors may be
influential in the entire bipartite network structure—that is, across both projected networks (total
influence).
2.2.1 Intramodal fragmentation
Intramodal fragmentation centrality considers the vulnerability of actors within a one-mode projection of a bipartite network. A simple way to extend fragmentation centrality in one-mode
networks to intramodal fragmentation centrality is to replace the simple distance measure in
Equation (1) with the distances obtained from Equation (7). In this approach, an actor is deleted
in one mode of the network. Next, the network is projected into a weighted one-mode network,
with edge weights assigned using Equation (7). The researcher then computes Equation (1) and
repeats the procedure for all actors in the network.
2
1
db
hj
While intuitively appealing, this approach raises some interpretive issues. Namely, n(n−1) will
not be constrained to achieve a maximum value of 1. As a consequence,
subtracting the value from
2
1 will yield negative values in some circumstances (when
2
1
db
hj
1
db
hj
n(n−1)
> 1), and positive values in others
(when n(n−1) < 1). In applied research, this may sometimes yield negative intramodal fragmentation centralities for actors who are in vulnerable locations, and positive intramodal fragmentation
centralities for actors who are not vulnerable.
One way to resolve
this issue is to construct intramodal fragmentation centrality by taking the
2
1
db
hj
n(n−1) . This provides an alternate approach to ensuring that increasing values reflect
greater fragmentation potential. While values will exceed 1, they will not dip below 0. Let Fib be the
reciprocal of
intramodal fragmentation centrality for an actor in a one-mode projection of a bipartite network.
Fib can be defined as,
nb nb − 1
(8)
Fib =
1
2
b
dhj
where nb is the number of actors in the intramodal projection and the numerator of the equation
indicates the number of possible ties in the projected network. As in the case of Equation (1), 1
b to avoid dividing infinite values, and to ensure that disconnected components of
is divided by dhj
the network yield
1
b
dhj
=
1
Inf
= 0. Thus, infinite values of Fib are only obtainable in completely dis-
connected graphs where all nodes are isolates. Higher values reflect greater disruption potential,
where removing an actor with the largest Fib will yield the greatest obtainable increase in distances
in the projected network to which i belongs. The benefit of this approach is that it standardizes
the distance measure by the size of the network. Thus, intramodal fragmentation centralities in
networks of different sizes can be compared. The actor with the greatest intramodal vulnerability
can be identified by maximizing over Fib .
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S. W. Duxbury
2.2.2 Cross-modal fragmentation
An interesting property of two-mode networks is that removing an actor in one mode of the
network will have a ripple effect on the projection of the alternate mode. Thus, fragmentation
measures can be constructed to evaluate how node deletion results in cross-modal disruption.
Following the approach above, cross-modal fragmentation centrality can be computed by first
deleting a node in one mode of the bipartite network. The alternate mode of the network is
then projected and weights computed. The distances between actors in the projected network are
1
summed as
c , where superscript c denotes that distances are being computed in the crossdhj
modal projection—that is, the projected mode of the network of which i is not a member. The
score is then standardized by the number of actors in the cross-modal projection. Let Fic indicate
the cross-modal fragmentation centrality of an actor i,
Fic =
nc (nc − 1)
1
2
c
dhj
(9)
c is the distance score for ties connecting alternate mode actors h and j in the cross-modal
where dhj
projection, and nc is the number of actors in the cross-modal projection. The interpretation of
this measure is that removing an actor with higher cross-modal fragmentation centrality will yield
larger increases on the distance between actors in the cross-modal projection. The actor with the
greatest cross-modal vulnerability can be identified by maximizing over Fic .
2.2.3 Total fragmentation
Latapy et al. (2008) note that there is not a good measure of average distance for the two-mode
structure of a bipartite network; only measures for one-mode projections. Consequently, creating
a key player statistic for the affiliation structure poses a problem. A possible means to translate the
key player statistic to bipartite structure is to create an aggregate measure of fragmentation based
on the affiliations between first and second mode actors. Such a measure would estimate how an
actor i disrupts the p × p projection as well as the m × m projection. The strategy of projecting
both matrices and combining their measures is similar in concept (though not process) to the
dual projection approach of Everett & Borgatti (2013); see also Everett (2016).
The concern with this measure is to gauge the effect an actor’s removal has on both modes
of the network. Let Fi∗ be the total fragmentation centrality for an actor i. Since Fib and Fic are
standardized by both the size of the projected networks and the distance between nodes in the
affiliation structure, these measures can simply be averaged to create a composite fragmentation
centrality statistic. The resulting value indicates the average disruption potential of an actor across
both projections,
Fib + Fic
(10)
2
where higher values reflect greater average disruption across both projected networks. The actor
with the greatest average vulnerability across both modes of the bipartite network can be identified
by maximizing over Fi∗ .
Fi∗ =
2.3 Cohesion
2.3.1 Intramodal cohesion
As in the case of intramodal fragmentation centrality, the simplest way to calculate intramodal
cohesion centrality is to replace dij in Equation (1) with dijb (Equation (7)). Here, the intramodal
cohesion centrality of an actor is the sum of the distances in a projected network divided by the
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number of actors in the projection. Increasing values indicate greater cohesion, such that actors
with greater cohesion centrality are positioned to diffuse a signal in the shortest amount of time.
As above, the value is not constrained to sum 1.
1
Cib =
dijb
(11)
nb
Like fragmentation centrality, the cross-modal cohesion score for actors can also be constructed.
Cross-modal cohesion represents the reachability of an actor i to actors in the cross-modal projection. As the actors’ cross-modal cohesion centrality increases, their ability to quickly transmit a
signal to actors in the cross-modal projection also increases.
2.3.2 Cross-modal cohesion
The approach to calculating cross-modal cohesion is somewhat different from cross-modal fragmentation. For cross-modal cohesion, the concern is to summarize the reachability of an actor i
to all actors in the cross-modal projection. One way to do so would be to calculate cross-modal
cohesion as the average intramodal cohesion centrality of alters in the cross-modal projection. The
logic of this strategy is that the actor who is most able to diffuse a signal to alters in the cross-modal
projection will be the actor who is highly connected to alters who are influential within their own
respective intramodal projection. In this approach, the intramodal cohesion of alters is standardized by the number of alters nominated. Let Cic be the cross-modal cohesion centrality of i,
b
Cj
c
(12)
Ci =
nj
where the numerator is the summation of the intramodal cohesion centralities for all of i’s alters
in the alternate mode of the two-mode network. The denominator is the number of alters in the
numerator. The benefit of this measure is that it summarizes the diffusion potential of a focal
actor’s alters. The problem, however, is that actors with more connections will often be penalized
for their connectivity because a large number of lower-scoring alters will offset the influence of
a handful of high-scoring alters. A simple way to fix this issue is to standardize the measure by
the total number of actors in the cross-modal projection. Doing so ensures that highly connected
actors are not penalized for their connectivity. This offers the final measure of cross-modal
cohesion centrality,
b
Cj
c
Ci =
(13)
nc
where the denominator represents the total number of actors in the cross-modal projection.
Maximizing the measure identifies the actor who is most able to diffuse a signal in the cross-modal
projection.
2.3.3 Total cohesion
As in the case of fragmentation centrality, a composite measure for two-mode network cohesion
can be constructed. The question is how to identify nodes that are influential in the two-mode
structure and across both projections. Let Ci∗ be the total cohesion centrality for the two-mode
structure. Since Cib and Cic are both standardized by the number of actors in the respective onemode projections, Ci∗ can be computed by averaging the cross-modal and intramodal cohesion
centrality of i. The resulting value measures both an actor’s influence in its one-mode projection
and the influence of the actor’s alters in the cross-modal projection. Higher values indicate greater
total cohesion centrality:
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S. W. Duxbury
Ci∗ =
Cib + Cic
2
(14)
Maximizing the measure identifies the actor who is able to most quickly diffuse a signal across
both projections of the two-mode network.
2.4 Key player sets
The proposed measures can also be used to identify key player sets with little adjustment. The
approach would mirror that described in the first section, where a set of size k is assigned and
the measures are computed for each unique combination of k actors. Of course, this will be
computationally intensive if the researcher is interested in F ∗ or C∗ or the network is large.
3. Proof of concept
A simple way to demonstrate the utility of the measure is to compare the proposed measures
to simple calculations of key player statistics which do not account for distance in two-mode
networks. To do so, I first simulate a random bipartite network with density varying randomly
between 0.05 and 0.35. Each network has 65 total actors, 50 in the first mode and 15 in the second. I then calculate the fragmentation and cohesion centrality for each network, using both
the equations proposed above as well as “simple” projections which do not adjust for distance
in bipartite networks. In the simple projections, cross-modal fragmentation and centrality statistics are calculated using Equations (9) and (13), and intramodal key player statistics are calculated
using Equations (1) and (2). The total fragmentation and cohesion centralities are computed using
Equations (14) and (10).
Each simulation identified the actors with the largest intramodal, cross-modal, and total fragmentation and cohesion centralities. Identifiers for these actors were then stored for analysis. I
computed simple matching coefficients between the vectors of identified actors to assess how frequently the bipartite and one-mode measures identify the same actors. To contextualize these
matches against random chance, I also compute correlation coefficients between the vectors of
identified actors for each strategy. With 500 iterations, the results reported in Table 1 are the
matching coefficients and correlations for key player identification in 500 random bipartite networks. In the analyses below, subscript i denotes actors in the first mode of the network with 50
nodes, while subscript q denotes actors in the second mode of the network with 15 nodes.
Beginning with fragmentation centrality, the two measures are most consistent for intramodal
fragmentation centrality in the first mode with 50 actors. Here, both approaches identified the
same actors in 15.8% of cases and are correlated at 0.319 (p < 0.001). The measures are less consistent when calculating intramodal fragmentation centrality in the second mode with 15 actors,
where the two approaches only identified the same actors in 5% of cases (r = −0.052, p < 0.001).
Turning to cross-modal fragmentation, the proposed measure and simple one-mode projection
almost never identified the same actors. Fic had a 0% similarity between the simple measures
and the proposed approach (r = −0.024, p > 0.1). Similarly, there was only 6% similarity for
Fqc (r = −0.05, p > 0.1). Aggregating the intramodal and cross-modal measures to F ∗ , the onemode projection and proposed approach are correlated at 0.126 (p < 0.001) with 16.4% similarity.
This indicates that while there is some overlap between the proposed approach and the one-mode
projections for fragmentation centrality, the two measures typically differ.
Results for cohesion centrality are less divergent but tell a similar story. One-mode projections and the proposed method for evaluating intramodal cohesion centrality in the first mode
identify the same actors in 28.2% of cases (r = 0.292, p < 0.001). However, much like second
mode intramodal fragmentation, the % similarity intramodal cohesion centrality in the second
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Table 1. Comparison of key player measures
% similarity
Correlation
Intramodal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Large mode (Fib )
15.8
0.319†
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Small mode (Fqb )
5.00
−0.052
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Large mode (Cib )
28.2
0.292†
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
0.200
0.069
Small mode (Cqb )
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Cross-modal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Large mode (Fic )
0.000
−0.024
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Small mode (Fqc )
0.600
−0.050
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Large mode (Cic )
71.8
0.743†
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
84.6
0.815†
Small mode (Cqc )
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Total
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Fragmentation (F ∗ )
16.4
0.126†
Cohesion (C∗ )
60.4
0.700†
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
∗ , † p < 0.001. % similarity is a simple matching coefficient multiplied by 100
mode is very low (0.2) and the two approaches are correlated at 0.069 (p > 0.1). Simple projections perform best when identifying cross-modal cohesion scores. Here, the % similarity for
cross-modal cohesion centrality in the first mode is 71.8, and the two approaches are correlated
at 0.743 (p < 0.001). Similarly, when considering cross-modal cohesion centrality in the second
mode, the two measures identify the same actor in 84.6% of simulations (r = 0.815, p < 0.001).
Aggregating the intramodal and cross-modal measures, total cohesion centrality is correlated at
0.7 (p < 0.001) for the two approaches with 60.4% similarity. Taken in sum, these results demonstrate that the proposed measures routinely identify distinct sets of key players. While the two
strategies show greater overlap when considering cohesion centrality, there is still substantial
variation between the two methods.
Next, I conducted a second simulation varying the properties of the simulated bipartite networks across a range of global characteristics. The networks were simulated with random density
between 0.01 and 0.3 and with a random number of nodes in each mode (between 10 and 100). I
then computed the simple and weighted fragmentation and cohesion centralities for each network.
I also computed the degree centrality for the entire network and the betweenness and degree centrality for each projection.5 With 500 replications each, the second simulation reports the results
from multiple measures of influence across 500 simulated bipartite networks. To assess the performance of the proposed measures, I recorded the number of cases where the actor with maximum
bipartite key player centrality also maximized an alternative centrality measure. Unmatched cases
indicate that simpler measures identified different actors than the bipartite key player statistics.
Figure 1 provides conditional density plots comparing the weighted fragmentation centralities to betweenness centrality, degree centrality, and unweighted fragmentation centrality. For
intramodal fragmentation centrality, there were no matches in 77% of networks, meaning that
the intramodal fragmentation centrality uniquely identified actors in four out of five simulated
networks. For total fragmentation centrality, the maximum total fragmentation centrality was
not identified by any other measure in 86% of cases, and cross-modal fragmentation centrality
uniquely identified leading actors in 95% of cases. The panels in Figure 1 provide insight into the
network properties where the differences between intramodal fragmentation centrality and alternative measures are greatest. Across the panels, the probability of identifying the same leading
actor increases as networks increase in either size or density. The relationship between the difference in mode size and the probability of identifying the same actor with another measure is
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Figure 1. Conditional density plots comparing fragmentation centrality to alternative measures of influence. Y axis is the
proportion of networks where the actor with the maximum weighted fragmentation centrality was equivalent to an alternative measure of influence. The difference in mode size is the absolute difference. The unweighted measure is the simple
fragmentation centrality.
negative, indicating that the measures become more balanced as the absolute difference in mode
size decreases.
As in the prior simulation, there is greater overlap between the cohesion centrality measures
and simple measures of influence. Although the maximum intramodal cohesion centrality identifies unique actors in 95% of networks, the maximum cross-modal cohesion centrality only
identifies unique actors in 23% of networks, and the maximum total cohesion centrality identifies unique actors in 26% of networks. Figure 2 plots the conditional densities for each measure.
Trends in Figure 2 for intramodal cohesion centrality are similar to Figure 1: As network density
and network size increase, so does the probability that alternative measures of influence will be
unable to identify the same leading actor as the intramodal cohesion centrality. However, there
is no clear relationship between network properties and cross-modal cohesion centrality. Total
cohesion centrality tends to uniquely identify leading actors in sparse networks, and only rarely
identifies unique leading actors in dense networks. There is no clear relationship between differences in mode size and cohesion centrality across measures. These results suggest that the
researchers may be able to identify actors who are able to quickly diffuse a signal across the
bipartite structure (total cohesion centrality) using simple degree centrality, and that cross-modal
cohesion centrality can often be approximated with simple cohesion centrality. That said, alternative measures of influence yielded different results than weighted cross-modal and total cohesion
centrality measures in roughly one out of every four simulated networks, which is a non-negligible
margin.
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Figure 2. Conditional density plots comparing cohesion centrality to alternative measures of influence. Y axis is the proportion of networks where the actor with the maximum weighted cohesion centrality was equivalent to an alternative
measure of influence. The difference in mode size is the absolute difference. The unweighted measure is the simple cohesion
centrality.
Figure 3 correlates the weighted key player measures with the unweighted measures. Panel A
reveals that there is little association between the weighted and unweighted fragmentation centrality measures, regardless of network structure. Consistent with the results for leading actor
identification, Panel B shows that weighted and unweighted intramodal cohesion centrality tend
to be weakly-to-moderately correlated associated. However, both total and cross-modal cohesion centrality measures are highly correlated with their unweighted counterparts. These results
suggest that, if a researcher is simply interested in aggregating network statistics and is not interested in leading actor identification, unweighted cross-modal and total cohesion centrality may be
sufficient.
In sum, results illustrate that fragmentation centrality and intramodal cohesion centrality overwhelmingly identify unique actors when compared to alternative measures, while degree centrality
and unweighted cohesion centrality identify similar actors to cross-modal and total cohesion
centrality in many cases. Since cross-modal cohesion centrality is an average measure of alters’
intramodal cohesion centrality, the comparability of weighted and unweighted cross-modal cohesion centralities indicates that the ranking of average intramodal cohesion centralities across
groups of nodes tend to be similar for the weighted and unweighted measures. This similarity,
in turn, drives the similarity between the weighted and unweighted total cohesion centralities.
Indeed, this is reflected in the high correlations between unweighted and weighted cross-modal
and total cohesion centralities. Results also suggest that the divergence between the weighted
key player measures metrics increases as network density and size increase for most key player
measures, but declines as the absolute difference in mode size increases. Researchers should
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Figure 3. Correlation between simple and unweighted key player measures. The difference in mode size is the absolute
difference.
note that the various measures of influence will usually yield divergent results for leading actor
identification in networks larger than 50 nodes or with densities above 0.2.
4. Empirical applications
The measures are illustrated using two datasets: the Ndrangheta mafia dataset and the Southern
Women dataset (Davis et al., 1941). The former is a two-mode network of mafia members and
mafia summits; the latter is a two-mode network of women’s attendance at community events. In
both sets of empirical demonstrations, I calculate both the bipartite key player measures and the
simple key player statistics (as discussed in the Proof of Concept).
4.1 Ndrangheta mafia: fragmentation centrality
The Ndrangheta data describe an Italian mafia network compiled from a large investigation,
Operazione Infinito (Coutinho, 2016). The data consists of 156 mafia members attending 48 separate summits. Oftentimes, criminologists seek to identify which actors in a criminal network can
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Table 2. Fragmentation centrality for Ndrangheta mafia data
Mean
Std. Dev.
Range
Intramodal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Members (Fib )
0.327
0.002
0.320–0.335
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Summits (Fqb )
0.508
0.077
0.485–1.026
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Cross-modal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Members (Fic )
0.474
0.003
0.469–0.494
Summits (Fqc )
0.350
0.218
0.313–1.834
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Total
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Members (Fi∗ )
0.397
0.002
0.391–0.405
Summits (Fq∗ )
0.429
0.148
0.401–1.430
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
be arrested to maximally disrupt the behaviors of a criminal organization. Fragmentation centrality provides an explicit measure of such vulnerability. In the case of the Ndrangheta data, each
measure gives insight into which mafia members or mafia summits should be targeted to maximally disrupt the Ndrangheta criminal organization. The intramodal fragmentation centrality for
mafia members gives insight into which members are most vulnerable in the member by member
projection, while the intramodal fragmentation centrality for summits gives insight into which
summit is most vulnerable in the summit by summit projection. The cross-modal fragmentation centrality for mafia members measures which member, when deleted, will yield the most
damage to the summit by summit projection, while the cross-modal fragmentation centrality for
summits measures which summit, when deleted, will yield the most damage to the summit by
summit projection. The total fragmentation centralities for both members and summits evaluates
which summits and which members would yield the most damage to the entire two-mode network
when deleted. As in the simulation studies above, subscript i denotes actors in the more populous
mode of the network (mafia members), while subscript q denotes actors in the less populous mode
(summits).
Table 2 presents summary statistics for the bipartite fragmentation centrality measures. There
is little variation in the intramodal fragmentation centrality for mafia members, where the standard deviation of Fib = 0.002. Alternatively, there is much more variation in the distribution of
Fqb , where the standard deviation is 0.077. While the mean Fqb (0.508) sits near its minimal value
(0.485), the maximum Fqb is 1.026. The relatively large standard deviation for Fqb when compared
to Fib indicates that there is more variation in how deleting a summit would damage the summit by summit projection network than there is how deleting a mafia member would damage
the member by member projection. Moreover, the largest Fqb (1.026) is larger than the largest Fib
(0.335), indicating that, respective to the size of each projection, removing the leading summit
will yield more damage to the summit by summit projected network—as indicated by increases in
average distance—than removing the leading mafia member will yield to the member by member
projection.
Turning to cross-modal effects, the mean Fic is 0.474 with a standard deviation of 0.003 and a
range of 0.469–0.494. The standard deviation for Fic is slightly larger than the standard deviation
for Fib (0.002), indicating that there is slightly more variation in how deleting a random mafia
member would affect the member by member projection as compared to the summit by summit projection. The cross-modal fragmentation centrality measures for summits also show much
variation. The mean Fqc is 0.350 with a standard deviation of 0.218 and a range of 0.313–1.834, indicating substantial right skew. This result indicates that while removing many of the low-scoring
summits will not yield much damage to the mafia by mafia projection, removing the summit with
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Figure 4. Intramodal fragmentation centrality. Panel A identifies the actor with the maximum Fib (0.335) in the member by
member projection of the Nradhengta network, while Panel B identifies the actor with the leading Fqb (1.03). Simple fragmentation centrality statistics were also computed on unweighted variations of the one-mode projections. In the member by
member projection, the simple key player statistics were unable to prioritize between the members, yielding a value of 0 for
all members. In the summit by summit projection, the simple measures identified the same summit as the weighted metric
(maximum simple fragmentation centrality = 0.299).
the largest fragmentation centrality will yield substantial damage. The standard deviation of Fqc
is also roughly three times larger than the standard deviation of Fqb , indicating that there is more
variation in how deleting a random summit would damage the member by member projection
than there is in how deleting a random summit would damage the summit by summit projection.
Figure 4 plots the member by member projected network (Panel A) as well as the summit by
summit projected network (Panel B). In both panels, the simple and weighted fragmentation centrality measures identified the same actors (blue nodes)—a result which was uncommon in the
simulated networks above. This indicates that, in some applied cases, simple and weighted measures of intramodal fragmentation centrality will identify the same actors. Examining the location
of actors within the structure of both projected networks yields intuitive insight into each actor’s
influence: both appear to be centrally located within the projected networks (though this is clearer
in Panel B).
One benefit of the bipartite key player measures is that the scores are standardized by the size
of the projected networks. Thus, it is possible to compare cross-modal and intramodal fragmentation centrality to evaluate whether, for instance, deleting a high-profile mafia member will yield
more or less damage to the member by member projection than deleting a high-profile summit
would yield to the member by member projection. A comparison of Fqc to Fib reveals that the
smallest Fqc (0.313) is smaller than the smallest Fib (0.320). This indicates that deleting the summit
with the smallest cross-modal fragmentation centrality would yield less damage to the member by
member projection than deleting the mafia member with the smallest intramodal fragmentation
centrality, though this difference is small and likely not substantively meaningful in this network.
Alternatively, deleting the summit with the largest Fqc (1.834) will have a larger disruptive effect
on the member by member projection than deleting the mafia member with the largest Fib (0.335).
This indicates that although there are some mafia members who are in more vulnerable locations than some summits, the actor with the most potential to fragment the member by member
projection is a summit, rather than a mafia member.
A similar analysis can be conducted for the summit by summit projection. Here, the mafia
member with the smallest cross-modal fragmentation centrality Fic (0.469) is smaller than the
smallest Fqb (0.485), indicating that deleting the mafia member with the smallest cross-modal fragmentation centrality will yield less damage to the summit by summit projection than deleting the
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Figure 5. Total and cross-modal fragmentation scores for
Ndrangheta mafia network. Blue squares are events; red circles are actors. The yellow circle is the mafia member with the
largest Fic (0.494) and Fi∗ (0.405). The green square is the summit with the largest Fq∗ (1.830) and Fq∗ (1.430). The black circle is
the mafia member with the largest total (0.468) and cross-modal
(0.935) simple fragmentation centrality, when calculated from
unweighted projections of the bipartite network. The simple calculation of total (0.673) and cross-modal (1.047) fragmentation
centrality for summits also identified the green-square as the
summit with the maximum fragmentation potential.
summit with the smallest intramodal fragmentation centrality. Likewise, deleting the summit with
the largest Fqb (1.026) will yield more damage to the summit by summit projection than deleting
the mafia member with the largest Fic (0.494). These results indicate that summits tend to occupy
more vulnerable positions with respect to the summit by summit projection than mafia members.
Given that deleting certain summits tends to yield more damage to both the summit by summit
projection and the member by member projection, it is likely that the most vulnerable actor in the
entire two-mode structure is a summit. This can be assessed by calculating Fi∗ and Fq∗ . Consistent
with the results above, Fq∗ tends to be greater in value than Fi∗ , reflecting that summits tend to
hold greater fragmentation potential (i.e., tend to be more vulnerable) than mafia members: the
mean Fq∗ is 0.429 with a standard deviation of 0.148 and a range of 0.401–1.430, while the mean
Fi∗ is 0.397 with a standard deviation of 0.002 and a range of 0.391–0.405. Figure 5 highlights the
Ndrangheta mafia members and summits with the leading cross-modal and total fragmentation
centralities. Red nodes represent mafia members, while blue squares are summits. The green circle
is the summit which is the largest Fq∗ and the largest Fqc . It is centrally located within the network
structure, reflecting its high position of influence. This summit was also identified by simple calculations of cross-modal and total fragmentation centrality, further corroborating its high influence.
The yellow node is the mafia member with the largest Fic (0.494) and Fi∗ (0.405). The black node
is the mafia member with the largest simple calculations of cross-modal (1.047) and total (0.673)
fragmentation centrality. These results reflect that when there are events or actors that wield overwhelming influence (as in the case of summits), the two measures will sometimes identify the
same actors. However, when there is less variation among actors, as in the case of the mafia members, the simple and weighted measures will likely not concur and may identify distinct actors. In
these instances, the weighted bipartite measures will often be preferable to the simple measures of
fragmentation centrality because the weighted measures preserve more information available in
the bipartite network structure.
In sum, analyses of the Ndrangheta mafia network illustrate how measures of fragmentation
centrality can be used to identify vulnerable locations in a two-mode network. The application
illustrates how fragmentation centrality measures can be compared to assess how deleting actors
in each respective mode of the network may damage the projections of each mode in the two-mode
network. The analyses also show that while simple measures sometimes identify the same actors
as weighted measures, they also sometimes diverge. In cases of divergence, it is recommended to
use the weighted measures, which preserve more information from the bipartite network when
computing one-mode projections.
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Table 3. Cohesion centrality for southern women data
Mean
Std. Dev.
Range
Intramodal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Women (Cib )
0.745
0.253
0.326–1.098
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Events (Cqb )
0.919
0.361
0.486–1.612
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Cross-modal
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Women (Cic )
0.388
0.129
0.175–0.607
Events (Cqc )
0.299
0.140
0.154–0.592
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Total
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Women (Ci∗ )
0.567
0.187
0.272–0.842
Events (Cq∗ )
0.609
0.246
0.331–1.104
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4.2 Southern women: cohesion centrality
The Southern Women data were collected by Davis et al. (1941) in the 1930s. It consists of 18
women who attended 14 social events. The purpose of cohesion centrality is to determine who is
the most influential or reachable actor in the network. A signal emanating from this actor (e.g., an
idea, a disease) should have the shortest average path in the network structure to all other actors
in the intramodal projection, cross-modal projection, and overall bipartite network.
Table 3 shows the cohesion centrality scores for the southern women. Women in the social
event network show considerable variation in intramodal cohesion centrality. The average Cib is
0.745, with a standard deviation of 0.253 and a range of values stretching between 0.326 and 1.098.
This indicates that the woman with the leading Cib can disseminate a signal much more quickly
to the other women in the women by women projection than the woman with the smallest Cib .
Turning to the intramodal cohesion centrality of events, the mean Cqb is 0.919 with a standard
deviation of 0.361 and a range of values stretching between 0.486 and 1.612.
As in the case of intramodal cohesion centrality among women, a signal emanating from the
event with the leading Cqb will spread much more quickly than a signal emanating from the event
with the smallest Cqb . In some cases, such as the Davis network, the interpretation of cohesion
centrality will have unclear meaning for certain modes of a two-mode network. Events do not
diffuse signals without womens’ participation. Interpretation of events’ cohesion centrality therefore requires reference to the womens’ social network. The intramodal cohesion centrality of the
event by event projection indicates which event is positioned to most quickly diffuse a signal to
the women who are most likely to spread the signal through other events. In less abstract terms,
if a disease were to appear at a random event in the Southern Women data, Event 8—the event
with the largest intramodal cohesion centrality—would have the greatest potential to spread the
disease to the women who attend other events.
Figure 6 demonstrates these measures graphically and compares them to the simple measures
of intramodal cohesion centrality. Panel A plots the woman by woman projection. The blue node
is Evelyn, who has the largest Cib at a value of 1.098. An interesting finding in Panel A is that
simple calculations of intramodal cohesion centrality are unable to discriminate between women
in the woman by woman projection. While only one woman obtained the highest value of Cib ,
seven women obtained the maximum possible value of 1 when computing simple measures of
intramodal cohesion centrality. Six of these women are identified in Panel A as black nodes, while
the seventh woman is Evelyn (identified as a blue node because she also maximizes Cib ). This
result indicates that, in some circumstances, simple measures of intramodal cohesion centrality
are unable to prioritize actors when examining bipartite networks, while the weighted measures
do not encounter this problem.
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Figure 6. Panel A plots cohesion centrality in the woman by woman one-mode projection of the Davis network, while
Panel B plots cohesion centrality in the event by event projection. Blue nodes demark the maximum Cib in Panel A (Evelyn,
max(Cib ) = 1.098), and the maximum Cqb in Panel B (Event 8, max(Cqb ) = 1.61. Black nodes identify those nodes in each projection which obtain the maximum value of 1 when using simple (e.g. Borgatti, 2006) cohesion centrality measures on the
one-mode projections in Panels A and B.
This result is replicated in Panel B, which plots the intramodal cohesion centrality of events in
the event by event projection. Here, the leading event (Event 8), as measured by Cqb , is identified as
a blue node, with a value of 1.61. This is the event expected to be able to most quickly disseminate
a signal to other events in the event by event projection. Simple measures of intramodal cohesion
centrality again identify numerous events which obtain the maximum possible value of 1. Here,
Event 8 as well as three other events (black nodes) are identified as sharing equal potential to
spread a signal through the event by event network. In sum, results in Figure 5 indicate that, unlike
weighted measures of cohesion centrality in bipartite networks, simple measures are sometimes
unable to prioritize actors when examining intramodal cohesion in two-mode networks.
Turning to cross-modal cohesion centrality, the mean Cic is 0.388 with a standard deviation of
0.129 and a range of 0.175–0.607. There is considerable variation in this distribution, indicating
that while some women can quickly diffuse a signal throughout the event by event projection,
others cannot. As in the case of fragmentation centrality, Cic can be compared to Cqb to evaluate
whether signals emanating from women or events more quickly diffuse throughout the event by
event projection. The mean Cqb (0.919) is over twice as large as the mean Cic (0.388), indicating
that the event with the highest intramodal cohesion centrality can more quickly send a signal
throughout the event projection compared to the woman with the highest cross-modal cohesion
centrality. However, the smallest Cqb (0.486) is also smaller than the largest Cic (0.607), indicating
that some women can diffuse a signal throughout the event by event projection more quickly than
some events. Turning to the cross-modal influence of events on the women by women projection,
the mean Cqc is 0.299 (SD = 0.140) with a range of 0.154–0.592. The maximum Cqc (.592) is smaller
than the maximum Cbc (1.098), indicating that a signal emanating from the most influential event
would diffuse less quickly throughout the woman by woman projection than a signal emanating
from the women with the highest intramodal centrality. Where Cqc and Cbc are Tex notation for
the equations.
It is also possible to assess which woman and which event wield the most influence over the
entire bipartite network structure. The mean Ci∗ is 0.567 with a standard deviation of 0.187 and a
range of 0.272–843. The total cohesion centrality scores are comparable for events, with a mean
Cq∗ of 0.609 (SD = 0.246) and a range of 0.331–1.104. The similar means for the two indicates that
both women and events, on average, yield comparable levels of influence over the entire network.
Indeed, a t-test fails to identify a statistically significant difference in values between the two sets
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Figure 7. Total cross-modal cohesion for Southern Women
Data. Red circles are first mode actors, blue squares are events.
The green node is Theresa, an integral woman who maximizes
Ci∗ (0.842) and Cic (0.607), as well as the simple calculation of
total cohesion among women (0.692). The black node is Nora,
who maximizes the simple calculation of cross-modal cohesion
among women. The yellow square is Event 8, which maximizes
Cq∗ (1.104) and Cqc (0.592), as well as the simple measures of cross
(0.758) and total (0.879) cohesion among events.
of cohesion centralities (p > 0.1). That said, the largest total cohesion centrality is for Event 8
(Cq∗ = 1.104), indicating that the most influential actor across both modes of the Southern Women
data is Event 8.
Figure 7 compares simple calculations of cross-modal and total cohesion centrality to weighted
bipartite measures. The blue squares are events, while the red nodes are women. The green node is
Theresa, the woman who maximizes both Ci∗ (0.842) and Cic (0.607), as well as the simple calculation of total cohesion among women (0.692). The black node is Nora, a woman who maximizes the
simple calculation of cross-modal cohesion among women. The yellow square is Event 8, which
maximizes Cq∗ (1.104) and Cqc (0.592), as well as the simple measures of cross (0.758) and total
(0.879) cohesion among events. Consistent with the results discussed for fragmentation centrality
among summits in the Ndrangheta mafia network, the simple and weighted metrics of crossmodal and total cohesion centrality reach consensus when there is clearly one influential event
(Event 8). However, the measures also sometimes identify distinct actors when distances in the
projected network are weighted to capture information in the bipartite structure. In the case of
cross-modal cohesion, the simple measure identifies a different woman as most influential (Nora)
than the weighted measure (Theresa).
In sum, results in empirical applications illustrate how the weighted bipartite key player measures can be used to assess influence in two-mode networks. Key player measures can be used to
identify actors which are influential within an intramodal projection, a cross-modal projection,
or across the entire bipartite structure. Empirical applications also demonstrate how cross-modal
and intramodal key player statistics can be compared to identify whether actors in the first or
second mode of a two-mode network yield more or less influence over each projected network.
Finally, empirical applications also consider the performance of these measures against simple calculations of key player statistics which do not account for distance in bipartite networks. Results
indicate that while the two strategies often identify similar actors, they also often diverge. In some
cases, such as intramodal cohesion centrality in the Southern Women projections, simple key
player measures were unable to prioritize between sets of four or six vertices, while the weighted
measures did not encounter this problem.
5. Discussion
This article proposed a method to identify key players in bipartite networks. The method adapts
Borgatti’s (2006) cohesion and fragmentation centrality by weighting one-mode projections by
mutual affiliations in a two-mode network (Newman, 2001). Geodesic distances in the one-mode
Network Science
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projection are determined using a shortest path algorithm, which can be tuned to assign greater
or lesser priority to edge weights and the number of intermediaries on a path (Opsahl et al., 2010).
With the adjusted geodesic distances, key player statistics can be calculated. Distinct measures
were proposed to assess intramodal, total, and cross-modal influence in bipartite networks. The
measures are standardized so that centrality scores from actors in different modes of a network or
different networks can be compared.
The method contributes to a growing interest in examining two-mode network structure
(Everett & Borgatti, 2013; Jasny & Lubell, 2015; Latapy et al., 2008; Opsahl et al., 2010; Opsahl,
2013). Like Latapy et al. (2008) and Everett & Borgatti (2013), it is oriented towards summarizing the three-embedded network structures in affiliation data: m × m, p × p, and m× p. As such,
it offers a useful tool to analyze bipartite networks (e.g Fujimoto et al., 2011; Wang et al., 2009,
2013).
While not discussed at length in this paper, the method can be updated to identifying key
player sets. This approach may be useful when an investigator is interested in a subset or clique
of actors who yield influence beyond what the linear combination of their key player statistics
would suggest. However, the method proposed in this paper may be computationally intensive
in large networks, particularly if a researcher is interested in fragmentation centrality. Researchers
should use caution and careful theoretical reasoning when specifying the size of the k set to reduce
computational burden.
It is also worth discussing here the role of the number of actors in each mode of a bipartite network. Oftentimes, as the absolute difference in the number of actors in each mode of the network
increases, the mode with fewer actors will yield more influence. This should not be surprising.
In a one-mode network, path lengths will tend to be shorter in smaller networks because there
are fewer possible intermediaries between any two actors. Likewise, in a two-mode network, as
the number of actors in one mode decreases, the number of shared affiliations between actors
in the other mode will also tend to increase, reducing the average distance between two actors
in the larger mode. Thus, researchers should bear in mind that when the absolute difference in
the sizes of each mode increases, the likelihood that the smaller mode will be more influential
increases. Even in these cases, intramodal, cross-modal, and total key player centrality measures
in the mode with fewer actors may still be of interest to researchers, depending on the nature of
the research question and the goals of key player identification.
When Borgatti (2006) first outlined the key player problem, he proposed examining key players
in relation to certain attributes. The method proposed here pushes towards this goal by examining
players who are connected through affiliations. In the case of public health, a disease may spread
quickly because of an organization which is particularly reachable by both other actors and other
organizations. In the case of criminal networks, drug markets may be particularly resilient because
of the properties of certain buyers who traffic drugs between sellers. The measures proposed in this
article identify such multifaceted positions of influence and vulnerability.
Acknowledgments. I would like to thank David Melamed, Stanley Wasserman, Ron Breiger, and three anonymous reviewers
at Network Science for their helpful suggestions and feedback.
Conflict of interest. Scott W. Duxbury has nothing to disclose.
Notes
1 Examples of bipartite networks in each of these contexts are drug markets (buyer–vendor ties), memberships in smoking
cessation communities ( person–community ties), and interaction of organism with nature reserves (organism–reserve ties).
2 Borgatti (2006) used the terms KPP-Pos (cohesion) and KPP-Neg (fragmentation).
3 Node removal refers to deleting the node and all incident edges from the network.
4 Both datasets were obtained from the UCINET network database.
5 Degree centrality and betweenness centrality have unclear cross-modal analogues, so they are not measured here.
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Cite this article: Duxbury SW. Identifying key players in bipartite networks. Network Science https://doi.org/10.1017/
nws.2019.62
Scott Duxbury