Stochastic block model and exploratory analysis in signed networks

1.6 MB·English

by  Jonathan Q. Jiang

Stochastic block model and exploratory analysis in signed networks
Jonathan Q. Jiang∗
Department of Computer Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
We propose a generalized stochastic block model to explore the mesoscopic structures in signed
networksbygroupingverticesthatexhibitsimilarpositiveandnegativeconnectionprofilesintothe
samecluster. Inthismodel,thegroupmembershipsareviewedashiddenorunobservedquantities,
andtheconnectionpatternsbetweengroupsareexplicitlycharacterizedbytwoblockmatrices,one
forpositivelinksandtheotherfornegativelinks. Byfittingthemodeltotheobservednetwork,we
can not only extract various structural patterns existing in the network without prior knowledge,
butalsorecognizewhatspecificstructuresweobtained. Furthermore,themodelparametersprovide
vitalcluesabouttheprobabilitiesthateachvertexbelongstodifferentgroupsandthecentralityof
eachvertexinitscorrespondinggroup. Thisinformationshedslightonthediscoveryofthenetworks
overlapping structures and the identification of two types of important vertices, which serve as the
cores of each group and the bridges between different groups, respectively. Experiments on a series
5 of synthetic and real-life networks show the effectiveness as well as the superiority of our model.
1
0 PACSnumbers: 89.75.Fb,05.10.-a
2

n
u I. INTRODUCTION toenablean“exploratory”analysisofnetworks,allowing
J ustoextractunspecifiedstructuralpatternsevenifsome
6 Thestudyofnetworkshasreceivedconsiderableatten- edges in the networks are missing [28, 29]. By fitting the
1 tion in recent literature [1–3]. This is mainly attributed model to the observed network structure, vertices with
to the fact that a network provides a concise mathemat- the same connection profiles are categorized into a pre-
] ical representation for social [4, 5], technological [6], bio- defined number of groups. The philosophy of these ap-
I
S logical [7–9] and other complex systems [1–3] in the real proachesisquitesimilartothatofthe“rolemodel”inso-
s. world,whichpavesthewayforexecutingproperanalysis ciology [30]—individuals having locally or globally anal-
c of such systems’ organizations, functions and dynamics. ogous relationships with others play the same “role” or
[ Many networks are found to possess a multitude of take up the same “position” [31]. It is clear to see that

2 mesoscopic structural patterns, which can be coarsely the possible topologies of the groups include community
v divided into “assortative” or “community” structure structure and multipartite structure, but they can be
4 and “disassortative” or “bipartitie/multipartite” struc- much, much wider.
9 ture [10, 11]. In addition, other types of mesoscopic One common assumption shared by these models is
5 structures,suchasthe“core-periphery”motif,havebeen thatthetargetnetworkscontainpositivelinksonly. How-
0 observed in real-life networks as well. Along with these ever,wefrequentlyencounterthesignednetworks,which
0
discoveries, a large number of techniques have been pro- havebothpositiveandnegativeedges,inbiology[19,32],
.
1 posed for mesoscopic structure extraction, in particular computer science [33], and last but definitely not least,
0 for community detection (see, e.g. [8, 10–14] and recent social science [34–37]. The negative connections usually
5
reviews [2, 3, 15]). Most, if not all, existing techniques represent hostility, conflict, opposition, disagreement,
1
: requireustoknowwhichspecificstructurewearelooking anddistrustbetweenindividualsororganizations,aswell
v for before we study it. Unfortunately, we often know lit- as the anticorrelation among objectives, whose coupled
Xi tleaboutagivennetworkandhavenoideawhatspecific relation with positive links has been empirically shown
structurescanbeexpectedandsubsequentlydetectedby toplayacrucialroleinthefunctionandevolutionofthe
r
a what specific methods. Biased results will be obtained whole network [32, 37].
if an inappropriate method is chosen. Even if we know Several works have been conducted to detect commu-
something beforehand, it is still difficult for a method nitystructureinthesekindsofnetworks. Yanget al.[34]
that is exclusively designed for a certain type of meso- proposed an agent-based method that performs a ran-
scopic structure to uncover the aforementioned miscel- domwalkfromonespecificvertexforafewstepstomine
laneous structures that may simultaneously coexist in a thecommunitiesinpositiveandsignednetworks. G´omez
network or may even overlap with each other [8, 16–20]. et al. [35] presented a generalization of the widely-used
To overcome these difficulties, a mixture model [11], a modularity [10, 14] to allow for negative links. Traag
stochastic block model [21] and their various extensions and Bruggeman [36] extended the Potts model to incor-
and combinations [22–27] have been recently introduced porate negative edges, resulting in a method similar to
the clustering of signed graphs. These approaches focus
on the problem of community detection and thus they
inevitably suffer a devastating failure if the signed net-
∗Current address: Department of Mathematics, Hong Kong Bap- works comprise other structural patterns, for example
tistUniversity,KowloonTong,HongKong the disassortative structure, as shown in Sec. IVA. To
2
make matters worse, they simply give a “hard” partition different from and much more complicated than that in
of signed networks in which a specific vertex could be- a positive network since both the density and the sign
long to one and only one cluster. Similar to the positive of the links should be taken into account at the same
networks,wehavegoodreasontobelievethatthesigned time. The intuitive descriptions of the assortative struc-
networks also simultaneously include all kinds of meso- tureanddisassortativestructuregiveninRef.[10,11]are
scopic structures that might overlap with each other. nolongersuitable. Anaturalquestionarises: Howcanwe
Inthispaper,weaimtocaptureandextracttheintrin- characterize the mesoscopic structures in a network that
sic mesoscopic structure of networks with both positive has both positive and negative edges? Guidance can be
and negative links. This goal is achieved by dividing the provided by the social balance theory [38], which states
vertices into groups such that the vertices within each that the attitudes of two individuals toward a third per-
group have similar positive and negative connection pat- son should match if they are positively related. In this
ternstoothergroups. Weproposeageneralizedstochas- situation, the triad is said to be socially balanced. A
tic block model, referred to as signed stochastic block networkiscalledbalancedprovidedthatallitstriadsare
model (SSBM), in which the group memberships of each balanced. This concept can be further generalized to k-
vertex are represented by unobserved or hidden quan- balance [39, 40] when the network can be divided into k
tities, and the relationship among groups is explicitly clusters, each having only positive links within itself and
characterized by two block matrices, one for the positive negative links with others.
links and the other for the negative links. By using the Following the principle, we can reasonably describe
expectation-maximizationalgorithm, wefitthemodelto the community structure in a signed network as a set
the observed network structure and reveal the structural ofgroupsofverticeswithinwhichpositivelinksarecom-
patterns without prior knowledge of what specific struc- paratively dense and negative links are sparser, and on
tures existing in the network. As a result, not only can thecontrarybetweenwhichpositivelinksaremuchlooser
various unspecific structures be successfully found, but andnegativelinksarethicker[34–36]. Obviously,itisan
also their types can be immediately elucidated by the extension of the standard community structure in net-
blockmatrices. Inaddition,themodelparameterstellus workswithpositiveedges. Incontrast,thedisassortative
the fuzzy group memberships and the centrality of each structure can be defined as a collection of vertices that
vertex, which enable us to discover the networks’ over- have most of their negative links within the group to
lapping structures and to identify two kind of important which they belong while have majority of their positive
vertices, i.e., group core and bridge. Experiments on a connections outside their group.
numberofsyntheticandrealworldnetworksvalidatethe
effectiveness and the advantage of our model.
The rest of this paper is organized as follows. We be- III. METHODS
gin with the depictions of the mesoscopic structures, es-
pecially the definitions of the community structure and A. The SSBM Model
disassortative structure, in signed networks in Sec. II.
Then we introduce an extension of the stochastic block GivenadirectednetworkG=(V,E),wecanrepresent
model in Sec. III, and show how to employ it to perform it by an adjacency matrix A. The entries of the matrix
anexploratoryanalysisofagivennetworkwithbothpos- are defined as: A = 1 if a positive link is present from
ij
itive and negative links. Experimental results on a series vertexitovertexj,A =−1ifanegativelinkispresent
ij
of synthetic networks with various designed structures from vertex i to vertex j, and A = 0 otherwise. For
ij
and three social networks are given in Sec. IV, followed weighted networks, A can be generalized to represent
ij
by the conclusions in Sec. V. the weight of the link. We further separate the positive
component from the negative one by setting A+ =A if
ij ij
A >0 and 0 otherwise, and A− =−A if A <0 and
ij ij ij ij
II. MESOSCOPIC STRUCTURES IN SIGNED 0 otherwise, so A=A+−A−.
NETWORKS Supposethattheverticesfallintocgroupswhosemem-
berships are “hidden” or “missing” for the moment and
It is well known that the mesoscopic structural pat- willbeinferredfromtheobservednetworkstructure. The
terns in positive networks can be roughly classified into number of groups c can also be inferred from the data,
the following two different types: “Assortative struc- which will be discussed in Sec. IIIC, but we take it as a
ture”, usually called “community structure” in most given here. The standard solution for such an inference
cases, refers to groups of vertices within which connec- problem is to give a generative model for the observed
tions are relatively dense and between which they are network structure and then to determine the parameters
sparser [10–12]. In contrast, “disassortative structure”, of the model by finding its best fit [11, 22–25].
alsonamed“bipartitestructure”ormoregenerally“mul- The model we use is a kind of stochastic block model
tipartite structure”, means that network vertices have that parameterizes the probability of each possible con-
mostoftheirconnectionsoutsidetheirgroup[10,11,13]. figurationofgroupassignmentsandedgesasfollows(see
For a signed network, its mesoscopic structure is quite Fig. 1 for a schematic illustration). Given an edge e ,
ij
3
A+ A- observe a positive edge e+ can be written as
ij ij ij
Node:1, , n i j i j
(cid:88)
(cid:537) (cid:307) (cid:537) (cid:307) Pr(e+|ω+,θ,φ)= ω+θ φ , (1)
ri sj ri sj ij rs ri sj
rs
Group:1, , c r s r s
(cid:550)+rs (cid:550)-rs and the probability of observing a negative edge e−ij is
(cid:88)
Pr(e−|ω−,θ,φ)= ω−θ φ . (2)
FIG. 1: Stochastic block model for signed networks. Unfilled ij rs ri sj
circles represent observed network structure and filled ones rs
correspond to hidden memberships. The solid line between
Themarginallikelihoodofthesignednetwork,therefore,
vertexiandj indicatestheexistenceofonepositiveornega-
can be represented by
tiveedgeconnectingthem. Thedashedlineindicatesthatthe
relation between the corresponding quantities is unobserved
and requires being learned from the observed network data. Pr(A|ω+,ω−,θ,φ)
(cid:18) (cid:19)A+(cid:18) (cid:19)A−
(cid:89) (cid:88) ij (cid:88) ij
= ω+θ φ ω−θ φ . (3)
rs ri sj rs ri sj
we choose a pair of group r and s for its tail and head
ij rs rs
with probability ω+ if e is positive, or with probability
rs ij
ωr−s if eij is negative. The two scalars ωr+s and ωr−s giv- Note that the self-loop links are allowed and the weight
ingtheprobabilitythatarandomlyselectedpositiveand A+ andA− arerespectivelyviewedasthenumberofpos-
ij ij
negative edge from group r to s respectively, explicitly itive and negative multiple links from vertex i to vertex
characterize various types of connecting patterns among j as done in many existing models [23–25].
groups, as we will see later. Then, we draw the tail ver- Toinferthemissinggroupmemberships←−g and→−g, we
tex i from group r with probability θri and the head needtomaximizethelikelihoodinEq.(3)withrespectto
vertex j from group s with probability φsj. Intuitively, themodelparametersω+,ω−,θ andφ. Forconvenience,
the parameter θri captures the centrality of vertex i in one usually works not directly with the likelihood itself
thegroup r fromthe perspectiveofoutgoing edgeswhile but with its logarithm
φ describes the centrality of vertex j in the group s
sj
from the perspective of incoming edges. The parameters L = lnPr(A|ω+,ω−,θ,φ)
ωr+s, ωr−s, θri and φsj satisfy the normalization condition (cid:32) (cid:33)
(cid:88) (cid:88)
= A+ln ω+ θ φ
c c c c ij r,s ri sj
(cid:88)(cid:88) (cid:88)(cid:88)
ω+ =1, ω− =1, ij rs
rs rs (cid:32) (cid:33)
r=1s=1 r=1s=1 (cid:88) (cid:88)
(cid:88)n (cid:88)n + A−ijln ωr−,sθriφsj . (4)
θri =1, φsj =1. ij rs
i=1 j=1
Themaximumofthelikelihoodanditslogarithmoccurin
←− →−
Let g and g to be respectively the group member- the same place because the logarithm is a monotonically
ij ij
ship of the tail and head of the edge e . So far, we increasing function.
ij
←− →−
have introduced all the quantities in our model: ob- Consideringthatthegroupmemberships g and g are
→− ←−
served quantities {A }, hidden quantities {g , g } unknown, it is intractable to optimize the log-likelihood
ij ij ij
and model parameters {ω+,ω−,θ ,φ }. To simplify L directly again. We can, however, give a good guess
rs rs ri sj ←− →−
the notations, we shall henceforth denote by ω+ the en- of the hidden variables g and g according to the net-
←− →−
tireset{ω+}andsimilarlyω−,θ,φ, g and g for{ω−}, work structure and the model parameters, and seek the
rs ←− →− rs
{θ }, {φ }, {g } and {g }. The probability that we maximization of the following expected log-likelihood
ri sj ij ij
L = (cid:88) Pr(←−g,→−g|A+,ω+,θ,φ)lnPr(A+|←−g,→−g,ω+,θ,φ)+ (cid:88) Pr(←−g,→−g|A−,ω−,θ,φ)lnPr(A−|←−g,→−g,ω−,θ,φ)
←−→− ←−→−
g,g g,g
(cid:20) (cid:21) (cid:20) (cid:21)
= (cid:88)Pr(r,s|e+,ω+,θ,φ) A+(cid:0)lnω+ +lnθ +lnφ (cid:1) +(cid:88)Pr(r,s|e−,ω−,θ,φ) A−(cid:0)lnω− +lnθ +lnφ (cid:1)
ij ij rs ri sj ij ij rs ri sj
ijrs ijrs
= (cid:88)q+ A+(cid:0)lnω+ +lnθ +lnφ (cid:1)+(cid:88)q− A−(cid:0)lnω− +lnθ +lnφ (cid:1), (5)
ijrs ij rs ri sj ijrs ij rs ri sj
ijrs ijrs
4
where q+ = Pr(←−g = r,→−g = s|e+,ω+,θ,φ) is the the expected log-likelihood appears at the places where
ijrs ij ij ij
probability that one find a positive edge e+ij with its tail  (cid:80) A+q+
vgisn|eievtre−ietjrne,pxωtrhei−tef,arθnto,iemoφtnw)gcotraoroknou.pbanerdmatanhddeeimtfosorhdqeei−aljrdpsav=rearmPterex(t←e−gjrisfjr.o=Amnrga,r→−lgooguijopu=ss ωωrr+−ss == (cid:80)(cid:80)(cid:80)ijiirjjsAAAi−ijj+i−jqqqii−jji+−rrjrsss,,
With the expected log-likelihood, we can get the best ijrs ij ijrs
(8)
teirhthasostmewimmeevavtaaexetlrruiesm,e.ossuFfimnoitcnfhegdeωiitvnv+heages,ltutcωhheae−elocm,mfuaθlLoaxstatiimtonloindukgmeeoφltfyh,sqtveia+irajlnlrlwusdpeiratsevhnsoidecftnehtqtheis−vjeepramrsospsaorrie.dotqibeoullOneiprmneaoes-f, φθsrji == (cid:80)(cid:80)(cid:80)(cid:80)ijjirssAAAA+i+i+i+jjjqqqqiii++++jjjrrrsss++++(cid:80)(cid:80)(cid:80)(cid:80)ijijrssAAAA−i−ij−ij−jqqqqi−i−jij−−jrrrsss,.
ijr ij ijrs ijr ij ijrs
possible solution is to adopt an iterative self-consistent
approachthatevaluatesbothsimultaneously. Likemany Eq. (6) and (8) constitute our EM algorithm for ex-
previous works [11, 23–25], we utilize the expectation- ploratory analysis of signed networks. When the algo-
maximization (EM) algorithm, which first computes the rithm converges, we obtain a set of values for hidden
posterior probabilities of hidden variables using esti- quantities q+ , q− and model parameters ω+, ω−, θ
ijrs ijrs
matedmodelparametersandobserveddata(theE-step), and φ.
and then re-estimates the model parameters (the M- It is worthwhile to note that the EM algorithm are
step). known to converge to local maxima of the likelihood but
In the E-step, we calculate the expected probabilities not always to global maxima. With different starting
q+ and q− given the observed network A and param- values, thealgorithmmaygiverisetodifferentsolutions.
ijrs ijrs
eters ω+, ω−, θ and φ Toobtainasatisfactorysolution,weperformseveralruns
with different initial conditions and return the solution
giving the highest log-likelihood over all the runs.
Now we consider the computational complexity of the
 Pr(←−g =r,→−g =s,e+|ω+,θ,φ) EMalgorithm. Foreachiteration,thecostconsistsoftwo
qi+jrs == (cid:80)ωrsr+sωθir+jrsiθφrPsijrφ(sej+i,ji|jω+,θ,φij) (6) qHpoi−fajerrgrtserso.umusTpinishs.getTEhfihqeres.et(sd6peg)cae,orswntidhnisoptsfahreroettminmisettehfwrecooomcrmkaltpcahluneelxdaeittsciytoiinmsisatoOhtfie(oqmi+nnjru×osmfacbt2nhe)der.
qi−jrs == P(cid:80)r(ω←−gr−sωiθj−ri=θφPsrrjφ,(→−eg−ij.i|jω=−,sθ,,eφ−ij)|ω−,θ,φ) miitmstheoear×daltsetocolit2oap)nOl.asc(rmoabIstmet×fieoostrfceedt2rhis)tffi.ehuceEsWuiMinlttegeartuEalosgtqeoigo.riTni(vt8ehp)tm,oraowfcdtohheersoenssoooeurtceerottinmimtcvhaoeeelrdgceenoelsusmtim.simpObTlaee(htxrTieiotno×nyf,
rs rs ri sj
to the number T of iterations. Generally speaking, T
is determined by the network structure and the initial
condition.
In the M-step, we use the values of q+ and q− es-
ijrs ijrs
timated in the E-step, to evaluate the expected log-
likelihood and to find the values of the parameters that B. Soft partition and overlapping structures
maximize it. Introducing the Lagrange multipliers ρ+,
ρ−, γ and λ to incorporate the normalization condi- The parameters, obtained by fitting the model to the
r s
tions, the expected log-likelihood expression to be maxi- observednetworkstructurewiththeE-Malgorithm,pro-
mized becomes videususefulinformationforthemesoscopicstructurein
agivennetwork. Specifically,thematricesω+andω−,an
analogywiththeimagegraphintherolemodel[41],char-
acterize the connecting patterns among different groups,
(cid:18) (cid:19) (cid:18) (cid:19)
L˜ = L +ρ+ 1−(cid:88)ω+ +ρ− 1−(cid:88)ω− + which determine the type of structural patterns. Fur-
rs rs
thermore, θ and φ indicate the centrality of a vertex in
rs rs
(cid:18) (cid:19) (cid:18) (cid:19) its groups from the perspective of outgoing edges and
(cid:88) (cid:88) (cid:88) (cid:88)
γr 1− θri + λs 1− φsj . (7) incoming edges, respectively. Consequently, the proba-
r i s j bility of vertex i drawn from group r when it is the tail
of edges can be defined as
(cid:80) (ω+ +ω−)θ
α = s rs rs ri , (9)
By letting the derivative of L˜ to be 0, the maximum of ir (cid:80) (ω+ +ω−)θ
rs rs rs ri
5
and vertex i can be simply assigned to the group r∗ to utilized in the previous generative models for network
which it most likely belongs, i.e., r∗ =argmax {α ,r = structure exploration [25].
r ir
1,2,...,c}. The result gives a hard partition of the According to MDL principle, the required length to
signed network. describe the network data comprises two components.
In fact, the set of scalars {α }c supply us with the The first one describes the coding length of the net-
ir r=1
probabilities that vertex i belongs to different groups, work, which is −L for directed network and −L/2
which can be referred to as the soft or fuzzy member- for undirected network. The other gives the length
ships. Assigning vertices to more than one group have for coding model parameters that is −(cid:80) lnω+ −
attracted by far the most interest, particularly in over- (cid:80) lnω− −(cid:80) lnθ −(cid:80) lnφ for directerdsnetwrsork
rs rs ri ri sj sj
lappingcommunitydetection[8,16–18]. Theverticesbe- and −(cid:80) lnω+ −(cid:80) lnω− −(cid:80) lnθ for undirected
rs rs rs rs ri ri
longingtoseveralgroups,arefoundtotakeaspecialrole network. The optimal c is the one which minimizes the
in networks, for example, signal transduction in biologi- total description length.
cal networks. Furthermore, some vertices, considered as
“instable” [16], locate on the border between two groups
and thus are difficult to classify into any group. It is of IV. EXPERIMENTAL RESULTS
great importance to reveal the global organization of a
signednetworkintermsofoverlappingmesoscopicstruc-
Inthissection,weextensivelytestourSSBMmodelon
tures and to find the instable vertices. We employ here
a series of synthetic signed networks with various known
the bridgeness [17] and group entropy [20] to capture
structure, including community structure and disassor-
the vertices’ instabilities and to extract the overlapping
tative structure. After that, the method is also applied
mesoscopicstructure. Thesetwomeasuresofvertexiare
to three real-life social networks.
computed as
(cid:118)
(cid:117)(cid:117) c (cid:88)c (cid:18) 1(cid:19)2 A. Synthetic networks
bi =1−(cid:116)c−1 αir− c , (10)
r=1
The ad hoc networks, designed by Girvan and New-
man [12], have been broadly used to validate and com-
c pare community detection algorithms [14–16, 20]. By
(cid:88)
ξ =− α log α . (11) contrast, there exists no such benchmark for community
i ir c ir
r=1 detection in networks with both positive and negative
links. We generate the signed ad hoc networks with con-
Notethatvertexihasalargebridgenessb andentropyξ
i i trolledcommunitystructurebythemethoddevelopedin
when it most likely participates in more than one group
Refs.[34,42]. Thenetworkshave128vertices, whichare
simultaneously and vice versa. From the perspective of
divided into four groups with 32 vertices each. Edges
incoming edges, we can represent the probability of ver-
are placed randomly such that they are positive within
tex j belonging to group s by
groupsandnegativebetweengroups,andtheaveragede-
(cid:80) (ω+ +ω−)φ gree of a vertex to be 16. The community structure is
βjs = (cid:80)rrs(ωrr+ss+ωrr−ss)φssjj. (12) caobniltirtyolloefdebacyhthvererteexpacroanmneetcetrisn,gptino ointdhiecratvienrgtictehseipnrtohbe-
same group, p the probability of positive links appear-
+
These statements for α also apply to β . So we don’t
ir js ing between groups, and p the probability of negative
−
need to repeat again.
linksarisingwithingroups. Thus, theparameterp reg-
in
Themodeldescribedabovefocusondirectednetworks.
ulates the cohesiveness of the communities and the re-
Actually, the model could be easily generalized to undi-
maining parameters p and p add noise to the commu-
+ −
rected networks by letting the parameter θ be identical
nity structure when p is fixed.
in
toφ. Thederivationfollowsthecaseofdirectednetworks
For the synthetic networks, we simply consider their
and the results are the same to Eq. (6) and (8).
hard partition as defined in Sec. IIIB. The results
are evaluated by the normalized mutual information
(NMI) [43], which can be formulated as
C. Model selection
c c
(cid:80) (cid:80) n ln nijn
So far, our model assumes that the number of groups ij n(1)n(2)
i=1j=1 i j
c is known as a prior. This information, however, is un- NMI(C1,C2)= (cid:115)
available for many cases. It is necessary to provide a ((cid:80)c n(1)lnn(i1))((cid:80)c n(2)lnn(i2))
criterion to determine an appropriate group number for i=1 i n i=1 i n
agivennetwork. Severalmethodshavebeenproposedto
deal with this model selection issue. We adopt the min- where C and C are the true group assignment and the
1 2
imum description length (MDL) principle, which is also assignmentfoundbythealgorithms,respectively,nisthe
6
(a) andsmaller. Forexample,theNMIoftheFECalgorithm
1 begins to drop once p exceeds 0.8, and then quickly re-
in
ducestolessthan0.2whenp =0.5andeventoapprox-
0.8 in
imately 0 when p is smaller than 0.3. Similar perfor-
FEC in
MI0.6 MMax mances can be observed for the MMax and PMMax ap-
N0.4 PMMax proaches as well. These results are quite understandable
GMMax since both the SSBM model and the GMMax method
0.2
SSBM consider the contribution made by the negative links in
0 signed networks, which is either neglected or removed in
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p the remaining three approaches. This highlights the im-
in
portance of the negative edges for community detection
(b)
in the signed networks. Moreover, the PMMax method
1
always outshines the MMax method, especially when p
in
0.8 in the range 0 ≤ pin ≤ 0.5, which is in agreement with
theresultsreportedinRef.[42],indicatingthattheposi-
MMin
MI0.6 PMMin tivelinksinsignednetworkshaveasignificantimpacton
N
0.4 GMMin community detection.
Then, we fix the parameter p = 0.8 and gradually
0.2 in
change other two parameters p and p from 0 to 0.5,
+ −
0 respectively. Clearly, all the synthetic networks are not
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p balanced in this setting. The results obtained by our
in
model and two updated algorithms are give in the up-
per row of Fig 3. As we can see, the SSBM model con-
FIG. 2: (Color online) NMI of our method and other ap-
proaches on balanced ad-hoc networks with controlled com- sistently, and sometimes significantly, outperforms the
munity structure (a) and disassortative structure (b). Each other two approaches. More specifically, its NMF is al-
point is an average over 50 realizations of the networks. ways1expectforafewnegligibleperturbations. Bycon-
trast, the FEC algorithm cannot offer a satisfactory par-
tition of the signed networks when 0 ≤ p ≤ 0.3 and
+
0 ≤ p ≤ 0.5, whose NMI is less than 0.4 at all times.
number of vertices, n is the number of vertices in the −
ij When0.3≤p ≤0.5and0≤p ≤0.5,theGMMaxap-
known group i that are assigned to the inferred group j, + −
proach exhibits a competitive performance, but its NMI
n(1) is the number of vertices in the true group i, n(2)
i j suddenlycollapsesandcontinuouslydecreasesoncep is
is the number of vertices in the inferred group j. The +
larger than 0.3.
larger the NMI value, the better the partition obtained
We turn now to the second experiment in which the
by the algorithms.
synthetic networks have the controlled disassortative
We conduct two different experiments. First, we set
structure. The signed networks are generated in the
the two parameters p and p to be zero and gradually
+ − same way, expect that we randomly place negative links
change p from 1 to 0. In this situation, all the gener-
in within groups and positive links between groups. Simi-
atedsyntheticnetworksare4-balanced. Fig.2(a)reports
larly, the disassortative structure in these networks are
theexperimentalresultsobtainedbyourmethodandtwo
controlled by three parameters again. p indicates the
state-of-the-artapproaches,namelygeneralizedmodular- in
probability of each vertex connecting to other vertices in
ity maximization through simulated annealing (denoted
the same group, p the probability of positive links ap-
byGMMax)[35,36]andthefindingandextractingcom- +
pearingwithingroups,andp theprobabilityofnegative
munity (FEC) method [34]. In addition, we also imple- −
links arising between groups.
mentthesimulatedannealingalgorithmtomaximizethe
We first study the balanced networks by setting p
standard modularity by ignoring the sign of the links +
and p to be zero and changing p from 1 to 0 once
(denoted by MMax) and removing the negative edges − in
again. As shown in Fig. 2(b), the FEC algorithm, the
(denoted by PMMax), respectively. Each point in the
MMax method and our model achieve the performances
curves is an average over 50 realization of the synthetic
thatisverysimilartothoseinthefirstexperiment. That
random networks. Bear in mind that the community
is, our model always successfully find the clusters in the
structure becomes less cohesive as the parameter p de-
in synthetic networks for all the cases, while the FEC algo-
creases from 1 to 0. We can see that both the SSBM
rithmandtheMMaxmethodperformfairlywellwhenp
model and the GMMax method perform fairly well and in
islargeenough, butquicklydegradeasp approaches0.
are almost able to perfectly recover the communities in in
The PMMax and the GMMax methods, however, per-
thesyntheticnetworksforallcases. When0≤p ≤0.1,
in form rather badly. The NMI of the PMMax method
our model is even slightly superior to the GMMax ap-
seems no greater than 0.5 even if p = 1, while the
proach. Theremainingthreemethods,however,canonly in
NMI of the GMMax approach nearly vanishes for all the
achieve promising results when p is sufficiently large.
in cases. Thisisbecausethetwomethods,whichseekstan-
Theyallshowafastdeteriorationasp becomessmaller
in
7
(a) (b) (c)

1
1 1
0.9
0.8 0.8 1
NMI 000...246 NMI 00..46 NMI 0.99 0.8
000.10.20.p30−.40.5 0.50.40p.3+0.20.10 0.200.10.20.p30−.40.5 0.50.40p.3+0.20.10 0.9800 .10.20.p30−.40.5 0.50.40p.3+0.20.10 000...567
(d) (e) (f)
0.4
0.3
1 1
0.8 0.8 1
MI 0.6 MI 0.6 MI 0.2
N 0.4 N N 0.99
0.2 0.4 0.1
0 0 0.2 0 0.98 0
0 0.1 0 0.1 0 0.1
0.10.20.3 0.30.2 0.10.20.3 0.30.2 0.10.20.3 0.30.2 0
0.4 0.4 0.4 0.4 0.4 0.4
p 0.5 0.5 p− p 0.5 0.5 p− p 0.5 0.5 p−
+ + +
FIG. 3: NMI on unbalanced ad-hoc networks with controlled community structure (first row) for (a) FEC, (b) GMMax and
(c)SSBM,andwithcontrolleddisassortativestructure(secondrow)for(d)FEC,(e)GMMinand(f)SSBM.Eachpointisan
average over 50 realizations of the networks.
dard and generalized modularity maximization, respec- all cases.
tively, are suitable only for community detection. As Finally, we focus on a synthetic network containing
a consequence, they deserve to suffer a serious failure a multitude of mesoscopic structures, whose adjacency
in this experiment. Instead, one should minimize the matrix is given in Fig. 4(a). Intuitively, according to
modularity to uncover the multipartite structure in net- the outgoing edges in this network, the second group is
works, as indicated in Ref. [10]. Therefore, we apply the community structure and the third group belongs to
the simulated annealing algorithm to minimize the gen- the disassortative structure. The first group with posi-
eralized modularity (denoted by GMMin) and the stan- tive outgoing links only, can be viewed as an example of
dard modularity by ignoring the sing of links (denoted the standard community structure in positive networks,
by MMin) and excluding the negative connections (de- whilethelastgroup,whichincludesonlynegativeoutgo-
noted by PMMin), respectively. We see from Fig. 2(b) inglinks,canbereferredtoasanextremeexampleofthe
that the GMMin method can obtain competitive perfor- disassortative structure in signed networks. Meanwhile,
mance with our SSBM model expect for a slight inferior from the perspective of incoming edges, the four groups
when0≤p ≤0.1. However,theMMinandthePMMin exhibit different types of structural patterns, which can-
in
approaches perform unsatisfactorily due to the fact that notbecategorizedsimplyascommunitystructureordis-
they do not consider the contributions derived from the assortative structure. We apply the FEC algorithm, the
negative links. GMMax method, the GMMin method and our model to
We investigate next the disassortative structure in un- this signed network. Limited by their intrinsic assump-
balanced synthetic networks by fixing p = 0.8 and tions, the FEC algorithm, the GMMax method and the
in
changing p and p from 0 to 0.5 step by step. The GMMin method fail to uncover the structural patterns,
+ −
lowerrowofFig.3givestheresultsobtainedbytheFEC as shown in Fig. 4(b)-(d). In particular, the general-
method, the GMMin approach and our SSBM model, ized modularity proposed in Refs. [35, 36], regardless of
which are quite similar to those in the first experiment. whether it is maximum or minimum, misleads us into
In particular, although the SSBM does not perform per- receiving an improper partition of the network in which
fectly in some cases, its NMF is still rather high, say, the four groups merge with each other. But by dividing
more than 0.98. When 0 ≤ p ≤ 0.3, the GMMin ap- vertices with the same connection profiles into groups,
−
proach yields sufficiently good results, but its NMF re- ourmodelcouldaccuratelydetectalltypesofmesoscopic
duces at a very fast speed along with p toward 0.5. structures, both from the perspective of outgoing links
−
The FEC algorithm achieves the worst performance in (Fig. 4(e)) and from the perspective of incoming edges
8
20
40
60
80
100
120
20 40 60 80 100 120
(a) (b) (c)
(d) (e) (f)
FIG. 4: (Color online) Detecting the mesoscopic structure of a synthetic network. (a) The adjacency matrix of the signed
networkwheretheblackdotsdenotethepositivelinksandthegraydotsrepresentthenegativeedges. Thepartitioningresults
for different methods (b) EFC, (c) GMMax, (d) GMMin and SSBM from the perspective of outgoing edges (e) and incoming
edges(f),wherethesolidedgesdenotethepositivelinksandthedashededgesrepresentnegativelinks. Thesizesofthevertices
in (e) and (f) indicate their centrality degree in the corresponding groups according to the parameters θ and φ, respectively.
x 104 (a) (b) B. Real-life networks
2.75
h h
ngt 2.7 ngt340
n le2.65 n le330 We further test our method by applying it to several
ptio 2.6 ptio realnetworkscontainingbothpositiveandnegativelinks.
scri2.55 scri320 The first network is a relation graph of 10 parties of
e e
D 2.5 D310 the Slovene Parliamentary in 1994 [44]. The weights of
2 3 4 5 6 2 3 4 5 6
linksinthenetworkwereestimatedby72questionnaires
Group number c Group number c
x 104 (c) among 90 members of the Slovene National Parliament.
9.3
ngth9.2 The questionnaires were designed to estimate the dis-
n le9.1 tance of the ten parties on a scale from -3 to 3, and the
ptio 9 final weights were the averaged values multiplied by 100.
cri We further test our method by applying it to several
s8.9
e
D8.8 realnetworkscontainingbothpositiveandnegativelinks.
3 4 5 6 7 8 9 10
The first network is a relation graph of 10 parties of
Group number c
the Slovene Parliamentary in 1994 [44]. The weights of
linksinthenetworkwereestimatedby72questionnaires
FIG.5: Modelselectionresultsfor(a)theSloveneParliamen-
tary network, (b) the Gahuku-Gama Subtribes network and among 90 members of the Slovene National Parliament.
(c) the international conflict and alliance network. The questionnaires were designed to estimate the dis-
tance of the ten parties on a scale from -3 to 3, and the
final weights were the averaged values multiplied by 100.
Applying our model to this signed network, we find
that the MDL achieves its minima when c=2, as shown
in Fig. 5(a), indicating that there are exactly two com-
munities in the network. Fig. 6(a) gives the partition
(Fig. 4(f)). Furthermore, the obtained parameters θ and obtained by our method, which divides the network into
φrevealthecentralityofeachvertexinitscorresponding two groups of equal size and produces a completely con-
group from the two perspectives.
9
TABLEI:Thesoftgroupmembershipα,bridgenessb [17]andgroupentropyξ [20]ofeachvertexintheSloveneParliamentary
i i
network [42]. Larger bridgeness or entropy means that the corresponding node are more “instable”.
Vertex SKD ZLSD SDSS LDS ZS-ESS ZS DS SLS SPS-SNS SNS
α 1.000 0 1.000 0 0 1.000 0 1.000 1.000 0.0186
i1
α 0 1.000 0 1.000 1.000 0 1.000 0 0 0.9814
i2
b 0 0 0 0 0 0 0 0 0 0.0372
i
ξ 0 0 0 0 0 0 0 0 0 0.1334
i
TABLE II: The soft group membership α, bridgeness b [17] and group entropy ξ [20] of each vertex in the Gahuku-Gama
i i
Subtribes network [45]. Larger bridgeness or entropy means that the corresponding node are more “instable”.
Vertex GAVEV KOTUN OVE ALIKA NAGAM GAHUK MASIL UKUDZ NOTOH KOHIK
α 1.000 1.000 0 0 0 0 0 0 0 0
i1
α 0 0 1.000 1.000 0 1.000 0.7143 1.000 0 0
i2
α 0 0 0 0 1.000 0 0.2857 0 1.000 1.000
i3
b 0 0 0 0 0 0 0.3773 0 0 0
i
ξ 0 0 0 0 0 0 0.5446 0 0 0
i
Vertex GEHAM ASARO UHETO SEUVE NAGAD GAMA
α 0 0 0 0 1.000 1.000
i1
α 1.000 1.000 0 0 0 0
i2
α 0 0 1.000 1.000 0 0
i3
b 0 0 0 0 0 0
i
ξ 0 0 0 0 0 0
i
ent groups.1 From Table I, we see that all the vertices
(a) LDS SDSS can be exclusively separated into two communities, ex-
pect for the vertex “SNS” which belongs to the circle
group with probability 0.0186 and to the square group
ZS-ESS SKD
withprobability0.9814. Inotherwords,thetwocommu-
ZLSD ZS nities overlap with each other at this vertex, resulting in
itshighbridgenessof0.0372andgroupentropyof0.1334.
This is validated by the observation that the vertex has
two negative links with vertices “ZS-ESS” and “DS” in
SNS SPS-SNS
the same community. We also visualize the learned pa-
DS
SLS rameters ω+ and ω− in Fig. 6(b), which indeed provide
a coarse-grained description of the signed network and
0 -1 0
(b) .3 1 2 6. reveal that this network actually has two communities.
6 4
The second network is the Gahuku-Gama Subtribes
network, which was created based on Read’s study
FIG. 6: Exploratory analysis of the Slovene Parliamentary on the cultures of Eastern Central Highlands of New
network [44]. The solid edges denote the positive links and
Guinea [45]. This network describes the political al-
the dashed edges represent negative links. The true commu-
liance and enmities among the 16 Gahuku-Gama sub-
nity structure in this network is represented by two different
tribes, which were distributed in a particular area and
shapes, circle and square. The shades of nodes indicate the
were engaged in warfare with one another in 1954. The
membershipαobtainedbyfittingourmodeltothisnetwork.
positive and negative links of the network correspond to
The sizes of the vertices, proportional to θ, indicates their
centrality degree with respect to their corresponding group. political arrangements with positive and negative ties,
respectively. Fig. 5(b) tells us that this signed network
consists of three groups because the MDL of the SSBM
model is minimum when c = 3. The three groups cat-
sistent split with the true communities in the network. egorized by our model are given in Fig. 7(a), and they
As expected, vertices within the same community are
mostly connected by positive links while vertices from
different communities are mainly connected by negative
links. We shade each vertex proportional to the pa- 1 ThisnetworkaswellastheGahuku-GamaSubtribesnetworkare
rameters {α }c , the magnitude of which supplies us bothundirectedgraph,andthereforetheparameterαisidentical
ir r=1
toβ,andθ isidenticaltoφ.
with the probabilities of each vertex belonging to differ-
10
(a) KOTUN OVE in several existing studies. These studies indicated that
GAVEV there are six main power blocs, each consisting of a set
GAMA of countries with similar actions of alliances or disputes.
NOTOH GAHUK ALIKA In Ref. [36], the authors labeled these power blocs as (i)
NAGAD
The West, (ii) Latin America, (iii) Muslim World, (iv)
Asia, (v) West Africa, and (vi) Central Africa. Apply-
KOHIK GEHAM
NAGAM ing the SSBM model to this network, we find that the
ASARO
MDL arrives its minimum when c = 6, as illustrated in
UHETO Fig. 5(c). By partitioning the network into six groups,
SEUVE MASIL
we summarize the results in Fig. 8. From the rearranged
UKUDZ
adjacency matrix [Fig. 8(c)], we can conclude that the
0.21
first, second, third and fifth groups, from bottom left to
(b)
top right, distinctly belong to the community structure,
1 -0.38 whilethesixthgroupcanbeviewedasthedisassortative
0 83.0- 42.0- 2 25.0 sctartuecgtourrizee.dHaosweeivtehre,rthcoemfomuurtnhitygrosturpucctaunrneootrbdeissaimsspolry-
.28 3 tative structure. In agreement with the assumption of
the SSBM model, vertices in the six groups exhibit the
FIG. 7: (Color online) Exploratory analysis of the Gahuku- similar connection profiles, although the miscellaneous
Gama Subtribes network [45]. The solid edges denote the structural patterns coexist in this network.
positive links and the dashed edges represent negative links. From the perspective of the outgoing edges, we ob-
The true community structure in this network is represented
tain a split of the network that is similar to the one
bythreedifferentshapeswhiletheinferredgroupsaredenoted
got in Ref. [36], as shown in Fig. 8(a). However, sev-
by different colors. The sizes of the vertices are proportional
eral notable difference exists between the two results.
to the parameters θ.
Specifically, “Pakistan” is grouped with the West and
“South Korea” is grouped with the Muslim World in
Ref. [36]. These false categorizations can be correctly
match perfectly with the true communities in the signed
amended, which is consistent with the configuration de-
network. AsshowninTableII,thevertex“MASIL”par-
pictedinHuntington’srenownedbookThe Clash of Civ-
ticipates in the circle group with probability 0.7143 and
ilizations [46]. In addition, we categorized “Australia”,
in the square group with probability 0.2857. As a result,
which is grouped with West in Ref. [36], into the group
it has a large value of bridgeness 0.3773 and group en-
Asia for understandable reasons. Fig. 8(b) gives a quite
tropy0.5446. Thisimpliesthatthese twogroupsoverlap
differentstructureofthisnetworkfromtheperspectiveof
with each other at this vertex, which is approved by the
incoming edges. Three groups, namely the West, Latin
fact that the vertex “MASIL”has two positive links con-
America and Muslim World, stay almost the same. But
nected to “NAGAM” and “UHETO”, respectively. The
“Russia”, togetherwithsomecountriesoftheformerSo-
learned parameters ω+ and ω− supply us with a thumb-
vietUnion,areisolatedfromtheAsiagroupandforman-
nail of the signed network again in Fig. 7(b).
otherindependentpowerbloc. Meanwhile,theremaining
Finally we test our model on the network of interna-
countries in Asia group join with the West Africa coun-
tional relation taken from the Correlates of War data
tries to constitute a bigger cluster. It is not difficult to
set over the period 1993—2001 [36]. In this network,
see that all the changes appear to be in accordance with
positive links represent military alliances and negative
the history and evolution of the international relations.
links denote military disputes. The disputes are asso-
Recall that the parameters θ and φ provide us with
ciated with three hostility levels, from “no militarized
the centrality degrees of each vertex in its corresponding
action” to “interstate war”. For each pair of countries,
groupfromtheperspectiveofoutgoingedgesandincom-
we chose the mean level of hostility between them over
ing edges, respectively. In other words, the parameters
the given time interval as the weight of their negative
measure the importance of each vertex in its group. For
link. The positive links denote the alliances: 1 for en-
a better visualization, the sizes of vertices in Fig. 8(d)
tente, 2 for non-aggression pact and 3 for defence pact.
and (e) are proportional to the magnitude of the scalars
Finally, we normalized both the negative links and pos-
θ and φ. Coincidentally, we discover that the big ver-
itive links into the interval [0, 1] and the final weight of
tices, marked by the red bold border, usually stand for
the link among each pair of countries is the remainder
the dominant countries in their corresponding groups.
oftheweightofthenormalizedpositivelinkssubtracting
For example, the largest vertex of the West is “USA” in
the weight of the normalized negative links. The ob-
Fig. 8(d). In fact, this state often serves as a leader in
tained network contains a giant component consisting of
its power bloc. A similar interpretation can be given for
161 vertices (countries) and 2517 links (conflicts or al-
the vertex “Russia” in Asia group. We further check the
liances). Here, we only investigate the structure of the
bridgeness and group entropy for each vertex in the net-
giant component.
work (data not shown), and we mark the vertices, which
The structure of this network has been investigated