Table Of ContentEnhance the Efficiency of Heuristic Algorithm
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for Maximizing Modularity Q
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2
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Yanqing Hu1, Jinshan Wu2, Zengru Di1∗
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J
3 1. Department of Systems Science, School of Management,
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Beijing Normal University, Beijing 100875, P.R.China
] 2. Department of Physics & Astronomy, University of British Columbia,
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p Vancouver, B.C. Canada, V6T 1Z1
-
c
o February 2, 2008
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.
s
c
i
s
y Abstract
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ModularityQisanimportantfunctionforidentifyingcommunitystructureincom-
p
plex networks. In this paper, we prove that the modularity maximization problem
[
maxS∈S Q¯ =Tr(STBS)isequivalenttoanonconvexquadraticprogrammingproblem
1 maxS∈S Qm =Tr(ST(B+D)mS). Thisresultprovideusasimplewaytoimprovethe
v efficiencyofheuristicalgorithmsformaximizingmodularityQ. Manynumericalresults
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demonstrate that it is very effective.
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Keyword: Complex Network, Community Structure, Modularity Q
1
1. PACS: 89.75.Hc, 05.40.-a,87.23.Kg
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8
0 1 Introduction
:
v
i
X Complex network has received an enormous amount of attention in recent years [1, 2, 3].
r Scientists havebecome interestedin the study of networksdescribing the topologiesof wide
a
variety of systems such as the world wide web, social and communication networks, bio-
chemicalnetworksandmanymore. Basedoncomplexnetworksmanyquantitativemethods
can be applied so as to extract the characteristics embedded in the system. One of the
important quantitative methods is to analysis the community structure [1, 2, 3]. Distinct
communities within networks can loosely be defined as subsets of nodes which are more
densely linked, when compared to the rest of the network. Nodes belonging to a tight-knit
community are more than likely to have other properties in common. In the world wide
web, community analysis has uncovered thematic clusters. In biochemical or neural net-
works, communities may be functional groups [1, 4, 5], and separating the network into
∗Authorforcorrespondence: [email protected]
1
such groups could simplify the functional analysis considerably. As a result, the problem of
identification of communities has been the focus of many recent efforts.
Maximizing modularity Q is the most widely accepted method for detecting community
structure among many algorithms [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22], although modularity index has been proved that it may fail to identify small
modules [23]. Modularity Q was presented as a index of community structure by Newman
and Grive, which was introduced as Q = (e −a2), where e are the fraction of links
r rr r rr
P
that connect two nodes inside the community r, a the fraction of links that have on or
r
both vertices in side the community r, and sum extends to all communities r in a given
network. Note that this index provides a way to determine if a certain description of the
graph in terms of communities is more or less accurate. Generally speaking, the larger the
value of Q, the more accurate is a partition into communities. So maximizing modularity
Q can detect community structures. There are many algorithms of maximizing Q directly
suchas extremaloptimization (EO)[21], greedy algorithm[9] and other optimal algorithm.
Infact,they areusuallyheuristic algorithmsfor modularitymaximizationproblemandthis
problem has been proved to be a NPC in the strong sense by Ulrik Brandes et al [24].
Canweimprovetheefficiencyofcorrespondingheuristicalgorithmsbydetailedinvestiga-
tionofmathematicstructureofmodularityQ? Accordingtoref[12],maxQ= (e −a2)
r rr r
canbesimplifiedasmaxQ¯ =Tr(STBS). Inthispaper,weprovedthatmaxQ¯P=Tr(STBS)
isanonconvexquadratic0−1programming. AssumeD =diag( ni=1|B1,i|, ni=1|B2,i|,···, ni=1|Bn,i|),
where B is the element of B. Then maxQ¯ = Tr(STBS)Pis equivalenPt to maxQ =P
i,j m
Tr(ST(B+D)mS) for allpositive integer number m. (B+D)m is a positive matrix, so the
modularity maximization problem can map to a continuous nonconvex quadratic program-
ming. These theorems will be detailed in Section 2. In this way, modularity maximization
problem is equivalent to maxQ = Tr(ST(B +D)mS). We have done many numerical
m
experiments onartificialand real-worldnetworkssuchas physics-economicsscientists coop-
erationnetwork,E.colinetworkandCollagefootballnetwork,andfoundthataproperlarge
m is very helpful for two basic neighborhood transformation algorithms and EO algorithm
for maximizing Q. It implies that our results has great possibility to enhance the efficiency
of many heuristic algorithms.
2 Theorems about modularity maximization problem
NewmanandGivanproposedthemodularityQindexbasedonthecommonexperiencethat
suchnetworksseemto havecommunitiesin them: subsetsofnodes withinwhichnode-node
connections are dense, but between which connections are less dense [5]. According to [12],
modularity Q can be simplified. Suppose we have a network N which has n nodes and can
be represented mathematically by an adjacency matrix A with elements A = 1 if there
i,j
is an edge from i to j and A = 0 otherwise. d denotes the degree of node i and P is
i,j i
a matrix, P = didj. Without losing any generality we assume that the network N has
i,j 2L
n communities (if the number of community is less than n we can use 0 to substitute).
Suppose S = (S1,S2,···,Sn) is the community structure matrix, Si ∈ {0,1}n denotes the
i community, i = 1,2,···,n. For example: assume S = (0,1,0,1,0,···,0)T, it denotes
i
that community i only contains two nodes which are node 2 and 4. Because a node only
belongs to one community, each row of S just has one 1. We use S to denotes the set of all
2
possible S. Let B =A−P, we easily have modularity maximization problem is equivalent
to maxS∈S Q¯ = Tr(STBS) [12], where Tr means trace which denotes the sum of diagonal
entries of a matrix.
Now we will map the maximization modularity Q problem to nonconvex quadratic 0-1
S1
S2
programming. Let S˜= .. , then maxS∈S Q¯ =Tr(STBS) can be write as:
.
Sn
B
B
maxS∈S Q¯ =S˜T .. S˜
.
B
s1,1+s1,2+···+s1,n =1
st. s··2,·1·+···s·2·,2+···+s2,n =1
sn,1+sn,2+···+sn,n =1
si,j ∈{0,1} i,j =1,2,···,n
From the subject conditions we can easily get that the set S contain nn elements, S =
{S1,S2,···,Snn}. According to the definition of S˜ we also have the corresponding set
S˜ ={S˜1,S˜2,···,S˜nn}.
Theorem1: LetD =diag( ni=1|B1,i|, ni=1|B2,i|,···, ni=1|Bn,i|),then,maxS∈S Q¯ =
Tr(STBS)problemis equivalenPtto the maxPimizationproblePmofmaxS∈S Q1 =Tr(ST(B+
D)S) which can be map to a nonconvex quadratic continuous programming.
Proof:
∵ Tr(ST(B+D)S)=Tr(STBS)+Tr(STDS)=Tr(STBS)+ n D .
i=1 i,i
P
∴ maxS∈S Q=Tr(STBS) problem is equivalent to the maximization problem of
maxS∈S Q=Tr(ST(B+D)S).
According to Gerschgorin Circle Theory [25], easily we have B +D is a symmetrical
positive matrix.
maxS∈S Q=Tr(ST(B+D)S) is a continuous nonconvex quadratic programming [26].
Theorem2: For all positive integer number m, maxS∈S Q¯ = Tr(STBS) problem is
equivalent to the maximization problem of maxS∈S Qm =Tr(ST(B+D)mS).
Proof: ∵ maxS∈S Q¯ =Tr(STBS)
is equivalent to maxS∈S Q1 =Tr(ST(B+D)S)
B+D
B+D
is equivalent to maxS˜∈˜S Q1 =S˜T ... S˜
B+D
and S˜TS˜=Tr(STS)=n
3
B+D
B+D
∴ maxS˜∈˜S Q1 =S˜T ... S˜
B+D
m
B+D
B+D
is equivalent to maxS˜∈˜S Qm =S˜T ... S˜
B+D
(B+D)m
(B+D)m
is equivalent to maxS˜∈˜S Qm =S˜T ... S˜
(B+D)m
is equivalent to maxS∈S Qm =Tr(ST(B+D)mS).
3 Application of the theorems
Based on the theorem 2, maximizing Q is equivalent to maxS∈S Qm =Tr(ST(B+D)mS).
CanweenhancetheefficiencyofheuristicalgorithmsformaximizingmodularityQbychang-
ingitintothisnewmaximizingproblemwithaproperlargem? Therearesomanyheuristic
algorithm for maximizing modularity Q, we cannot investigate all of them. If we could, we
also cannot promise our method satisfy the future heuristic algorithms. But it is well-know
that, for many heuristic algorithmssuch as EO,Potts [22] andso on, their key methods are
to find optimal neighborhood transformations, where neighborhood transformation means
moving a node for one community to another community at each optimizing step. So if
our method is effective on the basic neighborhood transformation algorithms, it will has
great possibility to be effective on many other heuristic algorithms. There are two basic
neighborhood transformation algorithms. One is random neighborhood transformation al-
gorithm. We randomly initiate the beginning partition (with sufficient number of groups),
then at each step, randomly choose a node form one community and move it into another
onethatcanmakeQbecomelarger,untilmovinganynodecannotmakeQlargeranymore.
The other algorithm is greedy neighborhood transformation algorithm. The corresponding
process is similar with the process of random one, but the difference is that at each step,
the node will be movedto a groupthat makes Q has the largestincrement. We choose four
different fields’ networks to test our method. One is the classical artificial random network
which has n = 128 nodes divided into 4 communities of 32 nodes each. Edges between
two nodes are introduced with different probabilities depending on whether the two nodes
belong to the same community or not: every node has < k = 8 > links on average to
intra
its fellows in the same community, and < k = 8 > links to the outer-world. Here we
inter
chose the artificial network with the diffuse community structures to test our method. It
is because when the network contains clear community structure, m has almost no effects
on the final partition. The rest 3 networks are scientists cooperation network [27], E.coli
network [28] and college football network [5]. The results show that for a proper large m,
our method is helpful for finding large value of Q (as shown in Fig. 1 and Fig. 2). But it is
hard to say it need more or less time in maximizing Q process.
4
We also use the extremal optimization algorithm(EO) [21] to test our method. EO was
proposed by Jordi Duch and Alex Arenas, which is heuristic algorithm. In their algorithm,
they define a fitness of each node. The fitness f of node i is defined as
i
q
f = i (1)
i
k
i
where, k denotes the degree of node i, and the q is the contribution of individual node i
i i
to the Q. Assume e denotes the n-dimensional vector in which the ith element is 1, others
i
0 then
q =eTBS (2)
i i i
For the maximization problem maxS∈S Q=Tr(ST(B+D)mS), the contribution is
qm =eT(B+D)mS (3)
i i i
Unfortunately, wecannotuse the function fm = qim (as Eq. 1) to define the fitness, forit is
i ki
notsatisfytheoriginalconditions(see[21]). SowedefinethenewfitnessfunctionastheEq.
3. Moreover,JordiDuchandAlex Arenasdidn’t define the ‘optimalstate’quantitativelyin
[21]. Inthis paper,wethink apartitionprocesshasarrivedthe optimalstateatstept ifthe
Q of t is equal or larger than each Q from step t+1 to t+n, where n is the node number
of a network.
We investigate extremal optimization with new fitness function (NEO) for different m
and compare the NEO algorithm with the EO algorithm in the above four networks. The
resultsshowthatthe properlargermisveryhelpfulbothformaximizingmodularityQand
reducing computing time, but sometimes the too large m is not helpful (as shown in Fig.
3). We guess one of the main reasons is that too large m will bring more computing errors.
4 Conclusion and discussion
Weprovethatthemodularitymaximizationproblemisequivalenttoanonconvexquadratic
programming problem. Based the characteristics of nonconvex quadratic programming, we
demonstrate that the modularity maximization problem is equivalent to the maximization
problemmaxS∈S Qm =Tr(ST(B+D)mS). Thisconclusionprovideasimplewaytoimprove
the efficiency of algorithms for maximizing modularityQ. Many numerical experiments are
done in different networks include artificial networks, scientists cooperation network, E.coli
network and Collage football network. The results show that new maximization problem
with proper large m can enhance the efficiency of the heuristic algorithms for maximizing
Q. Especially, it is helpful in both maximization Q and time complicity for EO algorithm.
But it is a real challenge problem to strictly give the most optimal m.
Acknowledgement
The authors want to thank M. E. J. Newman from providing the college football network
and Qiang Yuan for some useful discussion. This work is partially supported by 985 Projet
and NSFC under the grant No.70431002,No.70771011.
5
Random Network Scientists Cooperation Network
0.13 0.41
0.4
0.125
0.39
Q Q
0.38
0.12
0.37
0.115 0.36
2 4 m 6 8 10 2 4 m 6 8 10
0.38 E.coil Network 0.298 College football Network
0.37 0.297
0.296
0.36
Q
Q 0.295
0.35
0.294
0.34
0.293
0.33 0.292
2 4 m 6 8 10 2 4 m 6 8 10
Figure 1: Results of random neighborhood transformation algorithm .
Random Network Scientists Cooperation Network
0.39
0.125
0.38
0.12
Q
Q 0.37
0.115
0.36
0.11 0.35
2 4 m 6 8 10 2 4 m 6 8 10
E.coil Network College Football Network
0.36
0.289
0.35
0.288
Q0.34 0.287
Q
0.286
0.33
0.285
0.32 0.284
2 4 m 6 8 10 2 4 m 6 8 10
Figure 2: Results of greedy neighborhood transformation algorithm.
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Random Network Random Network
620
0.12
NEO 615
EO
0.115 me610
Q Ti
605
0.11 EO 600 NEO
595
2 4 m 6 8 10 2 4 m 6 8 10
Scientists Cooperation Network Scientists Cooperation Network
0.44 2050
EO
0.42 2000
Q0.4 NEO me1950
0.38 Ti1900
NEO
0.36 1850
EO
1800
0.34 2 4 m 6 8 10 2 4 6 8 10
m
E.coli Network E.coil Network
0.37
3400
0.36 NEO EO
3300
0.35
e 3200
Q m
Ti
0.34 3100 NEO
0.33 EO 3000
0.32 2 4 6 8 10 2900 2 4 m 6 8 10
m
College Football Network College Footabll Network
760
NEO 740 EO
720
0.295
Q EO Time678000 NEO
0.29
660
640
0.285 620
2 4 m 6 8 10 2 4 m 6 8 10
Figure 3: These plots show the results of NEO and EO algorithm in different networks.
The line means the Q value which is gotby EO (with originalfitness function). From these
plots we canconclude that large m is very helpful in both maximizing Q and reducing time
complicity. But sometimes too largem will bring overdoneeffects such as the results in the
collage football network.
7
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