DOI: 10.1126/science.1184819
, 876 (2010); 328Science
et al.Peter J. Mucha,
Multiscale, and Multiplex Networks
Community Structure in Time-Dependent,
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Community Structure in
Time-Dependent, Multiscale,
and Multiplex Networks
Peter J. Mucha,
1,2
* Thomas Richardson,
1,3
Kevin Macon,
1
Mason A. Porter,
4,5
Jukka-Pekka Onnela
6,7
Network science is an interdisciplinary endeavor, with methods and applications drawn from across
the natural, social, and information sciences. A prominent problem in network science is the
algorithmic detection of tightly connected groups of nodes known as communities. We developed a
generalized framework of network quality functions that allowed us to study the community
structure of arbitrary multislice networks, which are combinations of individual networks coupled
through links that connect each node in one network slice to itself in other slices. This framework
allows studies of community structure in a general setting encompassing networks that evolve over
time, have multiple types of links (multiplexity), and have multiple scales.
T
he study of graphs, or networks, has a long
tradition in fields such as sociology and
mathematics, and it is now ubiquitous in
academic and everyday settings. An important
tool in network analysis is the detection of
mesoscopic structures known as communities (or
cohesive groups), which are defined intuitively as
groups of nodes that are more tightly connected to
each other than they are to the rest of the network
(1–3). One way to quantify communities is by a
quality function that compares the number of
intracommunity edges to what one would expect
at random. Given the netwo rk adjacency matri x A,
where the element A
ij
details a direct connection
between nodes i and j, one can construct a qual-
ity function Q (4, 5) for the partitioning of nodes
into communities as Q = ∑
ij
(A
ij
− P
ij
)d(g
i
, g
j
),
where d(g
i
, g
j
) = 1 if the community assignments
g
i
and g
j
of nodes i and j are the same and 0
otherwise, and P
ij
is the expected weight of the
edge between i and j under a specified null model.
The choice of null model is a crucial con-
sideration in studying network community struc-
ture (2). After selecting a null model appropriate
to the network and application at hand, one can
use a variety of computational heuristics to assign
nodes to communities to optimize the quality Q
(2, 3). However, such null models have not been
available for time-dependent networks; analyses
have instead depended on ad hoc methods to
piece together the structures obtained at different
times (6–9) or have abandoned quality functions
in favor of such alternatives as the Minimum
Description Length principle (10). Although tensor
decompositions (11) have been used to cluster
network data with different types of connections,
no quality-function method has been developed
for such multiplex networks.
We developed a methodology to remove these
limits, generalizing the determination of commu-
nity structure via quality functions to multislice
networks that are defined by coupling multiple
adjacency matrices (Fig. 1). The connections
encoded by the network slices are flexible; they
can represent variations across time, variations
across different types of connections, or even
community detection of the same network at
differ ent scales. However , the usual procedure for
establishing a quality function as a direct count of
the intracommunity edge weight minus that
expected at random fails to provide any contribu-
tion from these interslice couplings. Because they
are specified by common identifications of nodes
across slices, interslice couplings are either present
or absent by definition, so when they do fall inside
communities, their contribution in the count of intra-
community edges exactly cancels that expected at
random. In contrast, by formulating a null model in
terms of stability of communities under Laplacian
dynamics, we have derived a principled generaliza-
tion of community detection to multislice networks,
REPORTS
1
Carolina Center for Interdisciplinary Applied Mathematics,
Department of Mathematics, University of North Carolina,
Chapel Hill, NC 27599, USA.
2
Institute for Advanced Materials,
Nanoscience and Technology, University of North Carolina,
Chapel Hill, NC 27599, USA.
3
Operations Research, North
Carolina State University, Raleigh, NC 27695, USA.
4
Oxford
Centre for Industrial and Applied Mathematics, Mathematical
Institute, University of Oxford, Oxford OX1 3LB, UK.
5
CABDyN
Complexity Centre, University of Oxford, Oxford OX1 1HP, UK.
6
Department of Health Care Policy, Harvard Medical School,
Boston, MA 02115, USA.
7
Harvard Kennedy School, Harvard
University, Cambridge, MA 02138, USA.
*To whom correspondence should be addressed. E-mail:
mucha@unc.edu
1
2
3
4
Fig. 1. Schematic of a multisli ce network. Four slices
s = {1, 2, 3, 4} represented by adjacencies A
ijs
encode
intraslice connections (solid lines). Interslice con-
nections (dashed lines) are encoded by C
jrs
,specifying
the coupling of node j to itself between slices r and s.
For clarity, interslice couplings are shown for only two
nodes and depict two different types of couplings: (i)
coupling between neighboring slices, appropriate for
ordered slices; and (ii) all-to-all interslice coupling,
appropriate for categorical slices.
nodes
resolution parameters
coupling = 0
1 2 3 4
5
10
15
20
25
30
nodes
resolution parameters
coupling = 0.1
1 2 3 4
5
10
15
20
25
30
nodes
resolution parameters
coupling = 1
1 2 3 4
5
10
15
20
25
30
Fig. 2. Multislice community detection of the
Zachary Karate Club network (22) across multiple
resolutions. Colors depict community assignments of
the 34 nodes (renumbered vertically to group
similarly assigned nodes) in each of the 16 slices
(with resolution parameters g
s
={0.25,0.5,…,4}),
for w = 0 (top), w = 0.1 (middle), and w =
1 (bottom). Dashed lines bound the communities
obtained using the default resolution (g =1).
14 MAY 2010 VOL 328 SCIENCE www.sciencemag.org
876
CORRECTED 16 JULY 2010; SEE LAST PAGE
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with a single parameter controlling the interslice
correspondence of communities.
Important to our method is the equivalence
between the modularity quality function (12) [with
a resolution parameter (5)] and stability of com-
munities under Laplacian dynamics (13), which
we have generalized to recover the null models for
bipartite, directed, and signed networks (14). First,
we obtained the resolution-parameter generaliza-
tion of Barber’s null model for bipartite networks
(15) by requiring the independent joint probability
contribution to stability in (13) to be conditional
on the type of connection necessary to step
between two nodes. Second, we recovered the
standard null model for directed networks (16, 17)
(again with a resolution parameter) by generaliz-
ing the Laplacian dynamics to include motion
along different kinds of connections—in this case,
both with and against the direction of a link. By
this generalization, we similarly recovered a null
model for signed networks (18). Third, we
interpreted the stability under Laplacian dynamics
flexibly to permit different spreading weights on
the different types of links, giving multiple reso-
lution parameters to recover a general null model
for signed networks (19).
We applied these generalizations to derive null
models for multislice networks that extend the
existing quality-function methodology, including
an additional parameter w to control the coupling
between slices. Representing each network slice s
by adjacencies A
ijs
between nodes i and j, with
intersli ce couplings C
jrs
that connect node j in slice
r to itself in slice s (Fig. 1), we have restricted our
attention to unipartite, undirected network slices
(A
ijs
= A
jis
) and couplings (C
jrs
= C
jsr
), but we can
incorporate additional structure in the slices and
couplings in the same manner as demonstrated for
single-slice null models. Notating the strengths of
each node individually in each slice by k
js
= ∑
i
A
ijs
and across slices by c
js
= ∑
r
C
jsr
,wedefinethe
multislice strength by k
js
= k
js
+ c
js
. The continuous-
time Laplacian dynamics given by
p
˙
is
¼ ∑
jr
ðA
ijs
d
sr
þ d
ij
C
jsr
Þp
jr
k
jr
− p
is
ð1Þ
respects the intraslice nature of A
ijs
and the
intersli ce couplings of C
jsr
. Using the steady-state
probability distribution p
∗
jr
¼ k
jr
=2m,where2m =
∑
jr
k
jr
, we obtained the multislice null model in
terms of the probability r
is| jr
of sampling node i in
slice s conditional on whether the multislice struc-
ture allows one to step from ( j, r)to(i, s), accounting
for intra- and interslice steps separately as
r
isj jr
p
∗
jr
¼
k
is
2m
s
k
jr
k
jr
d
sr
þ
C
jsr
c
jr
c
jr
k
jr
d
ij
k
jr
2m
ð2Þ
where m
s
= ∑
j
k
js
. The second term in parentheses,
which describes the conditional probability of
motion between two slices, leverages the definition
of the C
jsr
coupling. That is, the conditional
probability of stepping from ( j, r)to(i, s)along
an interslice coupling is nonzero if and only if i = j,
and it is proportional to the probability C
jsr
/k
jr
of
selecting the precise interslice link that connects to
slice s. Subtracting this conditional joint probability
from the linea r (in time) appro ximation of the
exponential describing the Laplacian dynamics, we
obtained a multislice gen eralization of modularity
(14):
Q
multislice
¼
1
2m
∑
ijsr
h
A
ijs
− g
s
k
is
k
js
2m
s
d
sr
þ
d
ij
C
jsr
i
dðg
is
,g
jr
Þð3Þ
where we have used reweighting of the conditional
probabilities, which allows a different resolution g
s
in each slice. We have absorbed the resolution pa-
rameter for the interslice couplings into the mag-
nitude of the elements of C
jsr
,which,forsimplicity,
we presume to take binary values {0,w} indicating
the absence (0) or presence (w) of interslice links.
1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000
40PA, 24F, 8AA
151DR, 30AA, 14PA, 5F
141F, 43DR
44D, 2R
1784R, 276D, 149DR, 162J, 53W, 84other
176W, 97AJ, 61DR, 49A,
24D, 19F, 13J, 37other
3168D, 252R, 73other
222D, 6W, 11other
1490R, 247D, 19other
Year
Senator
10 20 30 40 50 60 70 80 90 100 110
CT
ME
MA
NH
RI
VT
DE
NJ
NY
PA
IL
IN
MI
OH
WI
IA
KS
MN
MO
NE
ND
SD
VA
AL
AR
FL
GA
LA
MS
NC
SC
TX
KY
MD
OK
TN
WV
AZ
CO
ID
MT
NV
NM
UT
WY
CA
OR
WA
AK
HI
Con
g
ress #
A
B
Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23)withw = 0.5 coupling
of 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicate
assignments to nine communities of the 1884 unique senators (sorted vertically and connected across
Congressesbydashedlines)ineachCongressinwhich they appear. The dark blue and red communities
correspond closely to the modern Democratic and Republican parties, respectively. Horizontal bars
indicate the historical period of each community, with accompanying text enumerating nominal party
affiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration;
AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J,
Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communities
appeared simultaneously. (B) The same assignments according to state affiliations.
www.sciencemag.org SCIENCE VOL 328 14 MAY 2010
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Community detection in mul tislice networks
can then proceed using many of the same com-
putational heuristics that are currently available for
single-slice networks [although, as with the stan-
dard definition of modularity , one must be cautious
about the resolution of communities (20) and
the likelihood of complex quality landscapes that
necessitate caution in interpreting results on real
networks (21)]. We studied examples that have
multiple resolutions [Zachary Karate Club (22)],
vary over time [voting similarities in the U.S. Senate
(23)], or are multiplex [the “Tastes, Ties, and T ime”
cohort of university students (24)]. We provide
additional details for each example in (14).
We performed simultaneous community de-
tection across multiple resolutions (scales) in the
well-known Zachary Karate Club network, which
encodes the friendships between 34 members of a
1970s university karate club (22). Keeping the
same unweighted adjacency matrix acr oss slices
(A
ijs
= A
ij
for all s), the resolution associated with
each slice is dictated by a specified sequence of
g
s
parameters, which we chose to be the 16 values
g
s
= {0.25, 0.5, 0.75, …, 4}. In Fig. 2, we depict
the community assignments obtained for cou-
pling strengths w = {0, 0.1, 1} between each
neighboring pair of the 16 ordered slices. These
results simultaneously probe all scales, includ-
ing the partition of the Karate Club into four com-
munities at the default resolution of modularity
(3, 25). Additionally , we identified nodes that have
an especially strong tendency to break off from
largercommunities(e.g.,nodes24to29inFig.2).
We also considered roll call voting in the U.S.
Senate across time, from the 1st Congress to the
1 10th, covering the years 1789 to 2008 and includ-
ing 1884 distinct senator IDs (26). W e defined
weighted connection s between each pair of sen-
ators by a similarity between their voting, specified
independently for each 2-y ear Congre ss (23). W e
studied the multislice collection of these 1 10
networks, with each individual senator coupled to
himself or herself when appearing in consecutive
Congresses. Multislice community detection un-
covered interesting details about the continuity
of individual and group voting trends over time
that are not captured by the union of the 110 in-
dependent partitions of the separate Congresses.
Figure 3 depicts a partition into nine communities
that we obtained using coupling w =0.5.The
Congresses in which three communities ap pea r e d
simultaneously are each historically noteworthy:
The 4th and 5th Congresses were the first with
political parties; the 10th and 11th Congresses
occurred during the political drama of former V ice
President Aaron Burr’s indictment for treason ; the
14th and 15th Congresses witnessed the beginning
of changing group structures in the Democratic-
Republican party amidst the dying Federalist party
(23); the 31st Congress included the Compromise
of 1850; the 37th Congress occurred during the
beginning of the American Civil War; the 73rd and
74th Congresses followed the landslide 1932
election (during the Great Depression); and the
85th to 88th Congresses brought the major
American civil rights acts, including the congressio-
nal fights over the Civil Rights Acts of 1957, 1960,
and 1964.
Finally, we applied multislice community
detection to a multiplex network of 1640 college
students at a northeastern American university
(24), including symmetrized connections from the
first wave of this data representing (i) Facebook
friendships, (ii) picture friendships, (iii) roommates,
and (iv) student housing-group preferen ces. Be-
cause the diff erent connectio n types are categorical,
the natural interslice couplings connect an individ-
ual in a slice to himself or herself in each of the
other three network slices. This coupling between
categorical slices thus differs from that above,
which connected only neighboring (ordered) slices.
T able 1 indicates the numbers of communities and
the percentages of individuals assigned to one, two,
three, or four communities across the four types of
connections for dif ferent values of w,asafirst
investigation of the relative redundancy across the
connection types.
Our multislice framework makes it possible to
study community structure in a much broader class
of networks than was previously possible. Instead
of detecting communities in one static network at a
time, our formulation generalizing the Laplacian
dynamics approach of (13) permits the simulta-
neous quality-function study of community struc-
ture across multiple times, multiple resolution
parameter values, and multiple types of lin ks. W e
used this method to demon strate insights in real-
world networks that would have been difficult or
impossible to obtain without the simultaneous
consideration of multiple network slices. Although
our examples included only one kind of variation at
a time, our framework applies equally well to
networks that have multiple such features (e.g.,
time-dependent multiplex networks). W e expect
multislice community detection to become a
powerful tool for studying such systems.
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data; S. Reid and A. L. Traud for help developing code;
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Supporting Online Material
www.sciencemag.org/cgi/content/full/328/5980/876/DC1
SOM Text
References
17 November 2009; accepted 22 March 2010
10.1126/science.1184819
Table 1. Communities in the first wave of the multiplex “Tastes, Ties, and Time” network (24), using the
default resolution (g = 1) in each of the four slices of data (Facebook friendships, picture friendships,
roommates, and housing groups) under various couplings w across slices, which changed the number of
communities and percentages of individuals assigned on a per-sl ice basis to one, two, three, or four
communities.
w
Number of
communities
Communities per individual (%)
1234
0 1036 0 0 0 100
0.1 122 14.0 40.5 37.3 8.2
0.2 66 19.9 49.1 25.3 5.7
0.3 49 26.2 48.3 21.6 3.9
0.4 36 31.8 47.0 18.4 2.8
0.5 31 39.3 42.4 16.8 1.5
1 16 100 0 0 0
14 MAY 2010 VOL 328 SCIENCE www.sciencemag.org878
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1
CORRECTIONS & CLarifiCations
www.sciencemag.org sCiEnCE ERRATUM POST DATE 16 JULY 2010
Erratum
Repor ts: “Community structure in time-dependent, multiscale, and multiplex networks”
by P. J. Mucha et al. (14 May, p. 876). Equation 3 contained a typographical error that was
not caught during the editing process: The δ
sr
term should have been outside of the paren-
theses within the square brackets. The correct equation, which also appears in the support-
ing online material as equation 9, is as follows:
See the revised supporting online material (www.sciencemag.org/cgi/content/full/
sci;328/5980/876/DC2), which also includes a correction to equation 11. The computations
supporting the examples described in the Report were all performed with the correct for-
mula for Q
multislice
. The authors thank Giuseppe Mangioni for pointing out the error.
CORRECTIONS & CLarifiCations
Post date 16 July 2010
on August 30, 2010 www.sciencemag.orgDownloaded from
