Stochastic block model and exploratory analysis in signed networks

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
Abstract

We propose a generalized stochastic block model to explore the mesoscopic structures in signed networks by grouping vertices that exhibit similar positive and negative connection profiles into the same cluster. In this model, the group memberships are viewed as hidden or unobserved quantities, and the connection patterns between groups are explicitly characterized by two block matrices, one for positive links and the other for negative links. By fitting the model to the observed network, we can not only extract various structural patterns existing in the network without prior knowledge, but also recognize what specific structures we obtained. Furthermore, the model parameters provide vital clues about the probabilities that each vertex belongs to different groups and the centrality of each vertex in its corresponding group. This information sheds light on the discovery of the networks¡¯ 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 of synthetic and real-life networks show the effectiveness as well as the superiority of our model.

pacs:
89.75.Fb, 05.10.-a

Current address: ] Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

I Introduction

The study of networks has received considerable attention in recent literature Newman03 (); MJP09 (); Fortunato10 (). This is mainly attributed to the fact that a network provides a concise mathematical representation for social GAT07 (); AEPM08 (), technological GSCF02 (), biological RL05 (); GIIT05 (); MP07 () and other complex systems Newman03 (); MJP09 (); Fortunato10 () in the real world, which paves the way for executing proper analysis of such systems’ organizations, functions and dynamics.

Many networks are found to possess a multitude of mesoscopic structural patterns, which can be coarsely divided into “assortative” or “community” structure and “disassortative” or “bipartitie/multipartite” structure Newman06 (); ME07 (). In addition, other types of mesoscopic structures, such as the “core-periphery” motif, have been observed in real-life networks as well. Along with these discoveries, a large number of techniques have been proposed for mesoscopic structure extraction, in particular for community detection (see, e.g. GIIT05 (); Newman06 (); ME07 (); MM02 (); PFCB03 (); MM04 () and recent reviews Fortunato10 (); MJP09 (); LJAA05 ()). Most, if not all, existing techniques require us to know which specific structure we are looking for before we study it. Unfortunately, we often know little about a given network and have no idea what specific structures can be expected and subsequently detected by what specific methods. Biased results will be obtained if an inappropriate method is chosen. Even if we know something beforehand, it is still difficult for a method that is exclusively designed for a certain type of mesoscopic structure to uncover the aforementioned miscellaneous structures that may simultaneously coexist in a network or may even overlap with each other GIIT05 (); DJP05 (); TAF08 (); ISMB11 (); CSA11 (); JL12 ().

To overcome these difficulties, a mixture model ME07 (), a stochastic block model PKS83 () and their various extensions and combinations BM11 (); TYSYR11 (); EDSE08 (); HXJ11 (); Peixoto14 (); AFL11 () have been recently introduced to enable an “exploratory” analysis of networks, allowing us to extract unspecified structural patterns even if some edges in the networks are missing ACM08 (); RM09 (). By fitting the model to the observed network structure, vertices with the same connection profiles are categorized into a predefined number of groups. The philosophy of these approaches is quite similar to that of the “role model” in sociology FH71 ()—individuals having locally or globally analogous relationships with others play the same “role” or take up the same “position” JD07 (). It is clear to see that the possible topologies of the groups include community structure and multipartite structure, but they can be much, much wider.

One common assumption shared by these models is that the target networks contain positive links only. However, we frequently encounter the signed networks, which have both positive and negative edges, in biology CSA11 (); MGKQS09 (), computer science GGC12 (), and last but definitely not least, social science BWJ07 (); SPA09 (); VJ09 (); MRS10 (). The negative connections usually represent hostility, conflict, opposition, disagreement, and distrust between individuals or organizations, as well as the anticorrelation among objectives, whose coupled relation with positive links has been empirically shown to play a crucial role in the function and evolution of the whole network MGKQS09 (); MRS10 ().

Several works have been conducted to detect community structure in these kinds of networks. Yang et al. BWJ07 () proposed an agent-based method that performs a random walk from one specific vertex for a few steps to mine the communities in positive and signed networks. Gómez et al. SPA09 () presented a generalization of the widely-used modularity Newman06 (); MM04 () to allow for negative links. Traag and Bruggeman VJ09 () extended the Potts model to incorporate 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 networks comprise other structural patterns, for example the disassortative structure, as shown in Sec. IV.1. To make matters worse, they simply give a “hard” partition of signed networks in which a specific vertex could belong to one and only one cluster. Similar to the positive networks, we have good reason to believe that the signed networks also simultaneously include all kinds of mesoscopic structures that might overlap with each other.

In this paper, we aim to capture and extract the intrinsic mesoscopic structure of networks with both positive and negative links. This goal is achieved by dividing the vertices into groups such that the vertices within each group have similar positive and negative connection patterns to other groups. We propose a generalized stochastic block model, referred to as signed stochastic block model (SSBM), in which the group memberships of each vertex are represented by unobserved or hidden quantities, and the relationship among groups is explicitly characterized by two block matrices, one for the positive links and the other for the negative links. By using the expectation-maximization algorithm, we fit the model to the observed network structure and reveal the structural patterns without prior knowledge of what specific structures existing in the network. As a result, not only can various unspecific structures be successfully found, but also their types can be immediately elucidated by the block matrices. In addition, the model parameters tell us the fuzzy group memberships and the centrality of each vertex, which enable us to discover the networks’ overlapping structures and to identify two kind of important vertices, i.e., group core and bridge. Experiments on a number of synthetic and real world networks validate the effectiveness and the advantage of our model.

The rest of this paper is organized as follows. We begin with the depictions of the mesoscopic structures, especially the definitions of the community structure and disassortative structure, in signed networks in Sec. II. Then we introduce an extension of the stochastic block model in Sec. III, and show how to employ it to perform an exploratory analysis of a given network with both positive and negative links. Experimental results on a series of synthetic networks with various designed structures and three social networks are given in Sec. IV, followed by the conclusions in Sec. V.

Ii Mesoscopic structures in signed networks

It is well known that the mesoscopic structural patterns in positive networks can be roughly classified into the following two different types: “Assortative structure”, usually called “community structure” in most cases, refers to groups of vertices within which connections are relatively dense and between which they are sparser MM02 (); Newman06 (); ME07 (). In contrast, “disassortative structure”, also named “bipartite structure” or more generally “multipartite structure”, means that network vertices have most of their connections outside their group PFCB03 (); Newman06 (); ME07 ().

For a signed network, its mesoscopic structure is quite different from and much more complicated than that in a positive network since both the density and the sign of the links should be taken into account at the same time. The intuitive descriptions of the assortative structure and disassortative structure given in Ref. Newman06 (); ME07 () are no longer suitable. A natural question arises: How can we characterize the mesoscopic structures in a network that has both positive and negative edges? Guidance can be provided by the social balance theory Heider46 (), which states that the attitudes of two individuals toward a third person should match if they are positively related. In this situation, the triad is said to be socially balanced. A network is called balanced provided that all its triads are balanced. This concept can be further generalized to -balance Davis67 (); DF68 () when the network can be divided into clusters, each having only positive links within itself and negative links with others.

Following the principle, we can reasonably describe the community structure in a signed network as a set of groups of vertices within which positive links are comparatively dense and negative links are sparser, and on the contrary between which positive links are much looser and negative links are thicker BWJ07 (); VJ09 (); SPA09 (). Obviously, it is an extension of the standard community structure in networks with positive edges. In contrast, the disassortative structure can be defined as a collection of vertices that have most of their negative links within the group to which they belong while have majority of their positive connections outside their group.

Iii Methods

iii.1 The SSBM Model

Given a directed network , we can represent it by an adjacency matrix . The entries of the matrix are defined as: if a positive link is present from vertex to vertex , if a negative link is present from vertex to vertex , and otherwise. For weighted networks, can be generalized to represent the weight of the link. We further separate the positive component from the negative one by setting if and otherwise, and if and otherwise, so .

Suppose that the vertices fall into groups whose memberships are “hidden” or “missing” for the moment and will be inferred from the observed network structure. The number of groups can also be inferred from the data, which will be discussed in Sec. III.3, but we take it as a given here. The standard solution for such an inference problem is to give a generative model for the observed network structure and then to determine the parameters of the model by finding its best fit ME07 (); BM11 (); TYSYR11 (); EDSE08 (); HXJ11 ().

Figure 1: Stochastic block model for signed networks. Unfilled circles represent observed network structure and filled ones correspond to hidden memberships. The solid line between vertex and indicates the existence of one positive or negative edge connecting them. The dashed line indicates that the relation between the corresponding quantities is unobserved and requires being learned from the observed network data.

The model we use is a kind of stochastic block model that parameterizes the probability of each possible configuration of group assignments and edges as follows (see Fig. 1 for a schematic illustration). Given an edge , we choose a pair of group and for its tail and head with probability if is positive, or with probability if is negative. The two scalars and giving the probability that a randomly selected positive and negative edge from group to respectively, explicitly characterize various types of connecting patterns among groups, as we will see later. Then, we draw the tail vertex from group with probability and the head vertex from group with probability . Intuitively, the parameter captures the centrality of vertex in the group from the perspective of outgoing edges while describes the centrality of vertex in the group from the perspective of incoming edges. The parameters , , and satisfy the normalization condition

Let and to be respectively the group membership of the tail and head of the edge . So far, we have introduced all the quantities in our model: observed quantities , hidden quantities and model parameters . To simplify the notations, we shall henceforth denote by the entire set and similarly , , , and for , , , and . The probability that we observe a positive edge can be written as

(1)

and the probability of observing a negative edge is

(2)

The marginal likelihood of the signed network, therefore, can be represented by

(3)

Note that the self-loop links are allowed and the weight and are respectively viewed as the number of positive and negative multiple links from vertex to vertex as done in many existing models TYSYR11 (); EDSE08 (); HXJ11 ().

To infer the missing group memberships and , we need to maximize the likelihood in Eq. (3) with respect to the model parameters , , and . For convenience, one usually works not directly with the likelihood itself but with its logarithm

(4)

The maximum of the likelihood and its logarithm occur in the same place because the logarithm is a monotonically increasing function.

Considering that the group memberships and are unknown, it is intractable to optimize the log-likelihood directly again. We can, however, give a good guess of the hidden variables and according to the network structure and the model parameters, and seek the maximization of the following expected log-likelihood

(5)

where is the probability that one find a positive edge with its tail vertex from group and its head vertex from group given the network and the model parameters. Analogous interpretation can be made for too.

With the expected log-likelihood, we can get the best estimate of the value of together with the position of its maximum gives the most likely values of the model parameters. Finding the maximum still presents a problem, however, since the calculation of and requires the values of , , and , and vice versa. One possible solution is to adopt an iterative self-consistent approach that evaluates both simultaneously. Like many previous works ME07 (); TYSYR11 (); EDSE08 (); HXJ11 (), we utilize the expectation-maximization (EM) algorithm, which first computes the posterior probabilities of hidden variables using estimated model parameters and observed data (the E-step), and then re-estimates the model parameters (the M-step).

In the E-step, we calculate the expected probabilities and given the observed network and parameters , , and

(6)

In the M-step, we use the values of and estimated in the E-step, to evaluate the expected log-likelihood and to find the values of the parameters that maximize it. Introducing the Lagrange multipliers , , and to incorporate the normalization conditions, the expected log-likelihood expression to be maximized becomes

(7)

By letting the derivative of to be 0, the maximum of the expected log-likelihood appears at the places where

(8)

Eq. (6) and (8) constitute our EM algorithm for exploratory analysis of signed networks. When the algorithm converges, we obtain a set of values for hidden quantities , and model parameters , , and .

It is worthwhile to note that the EM algorithm are known to converge to local maxima of the likelihood but not always to global maxima. With different starting values, the algorithm may give rise to different solutions. To obtain a satisfactory solution, we perform several runs 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 EM algorithm. For each iteration, the cost consists of two parts. The first part is from the calculation of and using Eq. (6), whose time complexity is . Here is the edges in the network and is the number of groups. The second part is from the estimation of the model parameters using Eq. (8), whose time complexity is also . We use to denote the number of iterations before the iteration process converges. Then, the total cost of the EM algorithm for our model is . It is difficult to give a theoretical estimation to the number of iterations. Generally speaking, is determined by the network structure and the initial condition.

iii.2 Soft partition and overlapping structures

The parameters, obtained by fitting the model to the observed network structure with the E-M algorithm, provide us useful information for the mesoscopic structure in a given network. Specifically, the matrices and , an analogy with the image graph in the role model JS06 (), characterize the connecting patterns among different groups, which determine the type of structural patterns. Furthermore, and indicate the centrality of a vertex in its groups from the perspective of outgoing edges and incoming edges, respectively. Consequently, the probability of vertex drawn from group when it is the tail of edges can be defined as

(9)

and vertex can be simply assigned to the group to which it most likely belongs, i.e., . The result gives a hard partition of the signed network.

In fact, the set of scalars supply us with the probabilities that vertex belongs to different groups, which can be referred to as the soft or fuzzy memberships. Assigning vertices to more than one group have attracted by far the most interest, particularly in overlapping community detection GIIT05 (); DJP05 (); TAF08 (); ISMB11 (). The vertices belonging to several groups, are found to take a special role in networks, for example, signal transduction in biological networks. Furthermore, some vertices, considered as “instable” DJP05 (), locate on the border between two groups and thus are difficult to classify into any group. It is of great importance to reveal the global organization of a signed network in terms of overlapping mesoscopic structures and to find the instable vertices. We employ here the bridgeness TAF08 () and group entropy JL12 () to capture the vertices’ instabilities and to extract the overlapping mesoscopic structure. These two measures of vertex are computed as

(10)
(11)

Note that vertex has a large bridgeness and entropy when it most likely participates in more than one group simultaneously and vice versa. From the perspective of incoming edges, we can represent the probability of vertex belonging to group by

(12)

These statements for also apply to . So we don’t need to repeat again.

The model described above focus on directed networks. Actually, the model could be easily generalized to undirected networks by letting the parameter be identical to . The derivation follows the case of directed networks and the results are the same to Eq. (6) and (8).

iii.3 Model selection

So far, our model assumes that the number of groups is known as a prior. This information, however, is unavailable for many cases. It is necessary to provide a criterion to determine an appropriate group number for a given network. Several methods have been proposed to deal with this model selection issue. We adopt the minimum description length (MDL) principle, which is also utilized in the previous generative models for network structure exploration HXJ11 ().

According to MDL principle, the required length to describe the network data comprises two components. The first one describes the coding length of the network, which is for directed network and for undirected network. The other gives the length for coding model parameters that is for directed network and for undirected network. The optimal is the one which minimizes the total description length.

Iv Experimental results

In this section, we extensively test our SSBM model on a series of synthetic signed networks with various known structure, including community structure and disassortative structure. After that, the method is also applied to three real-life social networks.

iv.1 Synthetic networks

The ad hoc networks, designed by Girvan and Newman MM02 (), have been broadly used to validate and compare community detection algorithms LJAA05 (); MM04 (); DJP05 (); JL12 (). By contrast, there exists no such benchmark for community detection in networks with both positive and negative links. We generate the signed ad hoc networks with controlled community structure by the method developed in Refs. BWJ07 (); KF08 (). The networks have 128 vertices, which are divided into four groups with 32 vertices each. Edges are placed randomly such that they are positive within groups and negative between groups, and the average degree of a vertex to be 16. The community structure is controlled by three parameters, indicating the probability of each vertex connecting to other vertices in the same group, the probability of positive links appearing between groups, and the probability of negative links arising within groups. Thus, the parameter regulates the cohesiveness of the communities and the remaining parameters and add noise to the community structure when is fixed.

Figure 2: (Color online) NMI of our method and other approaches on balanced ad-hoc networks with controlled community structure (a) and disassortative structure (b). Each point is an average over 50 realizations of the networks.

For the synthetic networks, we simply consider their hard partition as defined in Sec. III.2. The results are evaluated by the normalized mutual information (NMI) SJ02 (), which can be formulated as

where and are the true group assignment and the assignment found by the algorithms, respectively, is the number of vertices, is the number of vertices in the known group that are assigned to the inferred group , is the number of vertices in the true group , is the number of vertices in the inferred group . The larger the NMI value, the better the partition obtained by the algorithms.

Figure 3: NMI on unbalanced ad-hoc networks with controlled community structure (first row) for (a) FEC, (b) GMMax and (c) SSBM, and with controlled disassortative structure (second row) for (d) FEC, (e) GMMin and (f) SSBM. Each point is an average over 50 realizations of the networks.

We conduct two different experiments. First, we set the two parameters and to be zero and gradually change from 1 to 0. In this situation, all the generated synthetic networks are 4-balanced. Fig. 2(a) reports the experimental results obtained by our method and two state-of-the-art approaches, namely generalized modularity maximization through simulated annealing (denoted by GMMax) VJ09 (); SPA09 () and the finding and extracting community (FEC) method BWJ07 (). In addition, we also implement the simulated annealing algorithm to maximize the standard modularity by ignoring the sign of the links (denoted by MMax) and removing the negative edges (denoted by PMMax), respectively. Each point in the curves is an average over 50 realization of the synthetic random networks. Bear in mind that the community structure becomes less cohesive as the parameter decreases from 1 to 0. We can see that both the SSBM model and the GMMax method perform fairly well and are almost able to perfectly recover the communities in the synthetic networks for all cases. When , our model is even slightly superior to the GMMax approach. The remaining three methods, however, can only achieve promising results when is sufficiently large. They all show a fast deterioration as becomes smaller and smaller. For example, the NMI of the FEC algorithm begins to drop once exceeds 0.8, and then quickly reduces to less than 0.2 when and even to approximately 0 when is smaller than 0.3. Similar performances can be observed for the MMax and PMMax approaches as well. These results are quite understandable since both the SSBM model and the GMMax method consider the contribution made by the negative links in signed networks, which is either neglected or removed in the remaining three approaches. This highlights the importance of the negative edges for community detection in the signed networks. Moreover, the PMMax method always outshines the MMax method, especially when in the range , which is in agreement with the results reported in Ref. KF08 (), indicating that the positive links in signed networks have a significant impact on community detection.

Then, we fix the parameter and gradually change other two parameters and from 0 to 0.5, respectively. Clearly, all the synthetic networks are not balanced in this setting. The results obtained by our model and two updated algorithms are give in the upper row of Fig 3. As we can see, the SSBM model consistently, and sometimes significantly, outperforms the other two approaches. More specifically, its NMF is always 1 expect for a few negligible perturbations. By contrast, the FEC algorithm cannot offer a satisfactory partition of the signed networks when and , whose NMI is less than 0.4 at all times. When and , the GMMax approach exhibits a competitive performance, but its NMI suddenly collapses and continuously decreases once is larger than 0.3.

We turn now to the second experiment in which the synthetic networks have the controlled disassortative structure. The signed networks are generated in the same way, expect that we randomly place negative links within groups and positive links between groups. Similarly, the disassortative structure in these networks are controlled by three parameters again. indicates the probability of each vertex connecting to other vertices in the same group, the probability of positive links appearing within groups, and the probability of negative links arising between groups.

We first study the balanced networks by setting and to be zero and changing from 1 to 0 once again. As shown in Fig. 2(b), the FEC algorithm, the MMax method and our model achieve the performances that is very similar to those in the first experiment. That is, our model always successfully find the clusters in the synthetic networks for all the cases, while the FEC algorithm and the MMax method perform fairly well when is large enough, but quickly degrade as approaches 0. The PMMax and the GMMax methods, however, perform rather badly. The NMI of the PMMax method seems no greater than 0.5 even if , while the NMI of the GMMax approach nearly vanishes for all the cases. This is because the two methods, which seek standard and generalized modularity maximization, respectively, are suitable only for community detection. As a consequence, they deserve to suffer a serious failure in this experiment. Instead, one should minimize the modularity to uncover the multipartite structure in networks, as indicated in Ref. Newman06 (). Therefore, we apply the simulated annealing algorithm to minimize the generalized modularity (denoted by GMMin) and the standard modularity by ignoring the sing of links (denoted by MMin) and excluding the negative connections (denoted by PMMin), respectively. We see from Fig. 2(b) that the GMMin method can obtain competitive performance with our SSBM model expect for a slight inferior when . However, the MMin and the PMMin approaches perform unsatisfactorily due to the fact that they do not consider the contributions derived from the negative links.

We investigate next the disassortative structure in unbalanced synthetic networks by fixing and changing and from 0 to 0.5 step by step. The lower row of Fig. 3 gives the results obtained by the FEC method, the GMMin approach and our SSBM model, which are quite similar to those in the first experiment. In particular, although the SSBM does not perform perfectly in some cases, its NMF is still rather high, say, more than 0.98. When , the GMMin approach yields sufficiently good results, but its NMF reduces at a very fast speed along with toward 0.5. The FEC algorithm achieves the worst performance in all cases.

Figure 4: (Color online) Detecting the mesoscopic structure of a synthetic network. (a) The adjacency matrix of the signed network where the black dots denote the positive links and the gray dots represent the negative edges. The partitioning results for different methods (b) EFC, (c) GMMax, (d) GMMin and SSBM from the perspective of outgoing edges (e) and incoming edges (f), where the solid edges denote the positive links and the dashed edges represent negative links. The sizes of the vertices in (e) and (f) indicate their centrality degree in the corresponding groups according to the parameters and , respectively.

Finally, we focus on a synthetic network containing a multitude of mesoscopic structures, whose adjacency matrix is given in Fig. 4(a). Intuitively, according to the outgoing edges in this network, the second group is the community structure and the third group belongs to the disassortative structure. The first group with positive outgoing links only, can be viewed as an example of the standard community structure in positive networks, while the last group, which includes only negative outgoing links, can be referred to as an extreme example of the disassortative structure in signed networks. Meanwhile, from the perspective of incoming edges, the four groups exhibit different types of structural patterns, which cannot be categorized simply as community structure or disassortative structure. We apply the FEC algorithm, the GMMax method, the GMMin method and our model to this signed network. Limited by their intrinsic assumptions, the FEC algorithm, the GMMax method and the GMMin method fail to uncover the structural patterns, as shown in Fig. 4(b)-(d). In particular, the generalized modularity proposed in Refs. VJ09 (); SPA09 (), regardless of whether it is maximum or minimum, misleads us into receiving an improper partition of the network in which the four groups merge with each other. But by dividing vertices with the same connection profiles into groups, our model could accurately detect all types of mesoscopic structures, both from the perspective of outgoing links (Fig. 4(e)) and from the perspective of incoming edges (Fig. 4(f)). Furthermore, the obtained parameters and reveal the centrality of each vertex in its corresponding group from the two perspectives.

Figure 5: Model selection results for (a) the Slovene Parliamentary network, (b) the Gahuku-Gama Subtribes network and (c) the international conflict and alliance network.

iv.2 Real-life networks

We further test our method by applying it to several real networks containing both positive and negative links. The first network is a relation graph of 10 parties of the Slovene Parliamentary in 1994 KM96 (). The weights of links in the network were estimated by 72 questionnaires among 90 members of the Slovene National Parliament. The questionnaires were designed to estimate the distance of the ten parties on a scale from -3 to 3, and the final weights were the averaged values multiplied by 100.

Vertex SKD ZLSD SDSS LDS ZS-ESS ZS DS SLS SPS-SNS SNS
1.000 0 1.000 0 0 1.000 0 1.000 1.000 0.0186
0 1.000 0 1.000 1.000 0 1.000 0 0 0.9814
0 0 0 0 0 0 0 0 0 0.0372
0 0 0 0 0 0 0 0 0 0.1334
Table 1: The soft group membership , bridgeness  TAF08 () and group entropy  JL12 () of each vertex in the Slovene Parliamentary network KF08 (). Larger bridgeness or entropy means that the corresponding node are more “instable”.
Figure 6: Exploratory analysis of the Slovene Parliamentary network KM96 (). The solid edges denote the positive links and the dashed edges represent negative links. The true community structure in this network is represented by two different shapes, circle and square. The shades of nodes indicate the membership obtained by fitting our model to this network. The sizes of the vertices, proportional to , indicates their centrality degree with respect to their corresponding group.

We further test our method by applying it to several real networks containing both positive and negative links. The first network is a relation graph of 10 parties of the Slovene Parliamentary in 1994 KM96 (). The weights of links in the network were estimated by 72 questionnaires among 90 members of the Slovene National Parliament. The questionnaires were designed to estimate the distance of the ten parties on a scale from -3 to 3, and the final weights were the averaged values multiplied by 100.

Vertex GAVEV KOTUN OVE ALIKA NAGAM GAHUK MASIL UKUDZ NOTOH KOHIK
1.000 1.000 0 0 0 0 0 0 0 0
0 0 1.000 1.000 0 1.000 0.7143 1.000 0 0
0 0 0 0 1.000 0 0.2857 0 1.000 1.000
0 0 0 0 0 0 0.3773 0 0 0
0 0 0 0 0 0 0.5446 0 0 0
Vertex GEHAM ASARO UHETO SEUVE NAGAD GAMA
0 0 0 0 1.000 1.000
1.000 1.000 0 0 0 0
0 0 1.000 1.000 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Table 2: The soft group membership , bridgeness  TAF08 () and group entropy  JL12 () of each vertex in the Gahuku-Gama Subtribes network Read54 (). Larger bridgeness or entropy means that the corresponding node are more “instable”.

Applying our model to this signed network, we find that the MDL achieves its minima when , as shown in Fig. 5(a), indicating that there are exactly two communities in the network. Fig. 6(a) gives the partition obtained by our method, which divides the network into two groups of equal size and produces a completely consistent split with the true communities in the network. 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 parameters , the magnitude of which supplies us with the probabilities of each vertex belonging to different groups.111This network as well as the Gahuku-Gama Subtribes network are both undirected graph, and therefore the parameter is identical to , and is identical to . From Table 1, we see that all the vertices can be exclusively separated into two communities, expect for the vertex “SNS” which belongs to the circle group with probability 0.0186 and to the square group with probability 0.9814. In other words, the two communities overlap with each other at this vertex, resulting in its high bridgeness of 0.0372 and group entropy of 0.1334. This is validated by the observation that the vertex has two negative links with vertices “ZS-ESS” and “DS” in the same community. We also visualize the learned parameters and in Fig. 6(b), which indeed provide a coarse-grained description of the signed network and reveal that this network actually has two communities.

Figure 7: (Color online) Exploratory analysis of the Gahuku-Gama Subtribes network Read54 (). The solid edges denote the positive links and the dashed edges represent negative links. The true community structure in this network is represented by three different shapes while the inferred groups are denoted by different colors. The sizes of the vertices are proportional to the parameters .
Figure 8: (Color online) Exploratory analysis of the international conflict and alliance network VJ09 (). Maps of the groups found using the SSBM model from the perspective of outgoing edges (a) and incoming edges (b). (c) The rearrange adjacency matrix, in which the black dots denote positive links and the gray dots represent negative edges, respectively. Six groups are separated by black solid lines. The partition of this network obtained by the SSBM model from the perspective of outgoing edges (d) and incoming edges (e), where the solid edges denote the positive links and the dashed edges represent negative links. The sizes of vertices are respectively proportional to their centrality degree and . The red bold border vertices have the large centrality degrees while the black bold border vertices have the large values of bridgeness and group entropy.

The second network is the Gahuku-Gama Subtribes network, which was created based on Read’s study on the cultures of Eastern Central Highlands of New Guinea Read54 (). This network describes the political alliance and enmities among the 16 Gahuku-Gama subtribes, which were distributed in a particular area and were engaged in warfare with one another in 1954. The positive and negative links of the network correspond to 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 . The three groups categorized by our model are given in Fig. 7(a), and they match perfectly with the true communities in the signed network. As shown in Table 2, the vertex “MASIL” participates in the circle group with probability 0.7143 and in the square group with probability 0.2857. As a result, it has a large value of bridgeness 0.3773 and group entropy 0.5446. This implies that these two groups overlap with each other at this vertex, which is approved by the fact that the vertex “MASIL”has two positive links connected to “NAGAM” and “UHETO”, respectively. The learned parameters and supply us with a thumbnail of the signed network again in Fig. 7(b).

Finally we test our model on the network of international relation taken from the Correlates of War data set over the period 1993—2001 VJ09 (). In this network, positive links represent military alliances and negative links denote military disputes. The disputes are associated with three hostility levels, from “no militarized action” to “interstate war”. For each pair of countries, we chose the mean level of hostility between them over the given time interval as the weight of their negative link. The positive links denote the alliances: 1 for entente, 2 for non-aggression pact and 3 for defence pact. Finally, we normalized both the negative links and positive links into the interval [0, 1] and the final weight of the link among each pair of countries is the remainder of the weight of the normalized positive links subtracting the weight of the normalized negative links. The obtained network contains a giant component consisting of 161 vertices (countries) and 2517 links (conflicts or alliances). Here, we only investigate the structure of the giant component.

The structure of this network has been investigated in several existing studies. These studies indicated that there are six main power blocs, each consisting of a set of countries with similar actions of alliances or disputes. In Ref. VJ09 (), the authors labeled these power blocs as (i) The West, (ii) Latin America, (iii) Muslim World, (iv) Asia, (v) West Africa, and (vi) Central Africa. Applying the SSBM model to this network, we find that the MDL arrives its minimum when , as illustrated in Fig. 5(c). By partitioning the network into six groups, we summarize the results in Fig. 8. From the rearranged adjacency matrix [Fig. 8(c)], we can conclude that the first, second, third and fifth groups, from bottom left to top right, distinctly belong to the community structure, while the sixth group can be viewed as the disassortative structure. However, the fourth group cannot be simply categorized as either community structure or disassortative structure. In agreement with the assumption of the SSBM model, vertices in the six groups exhibit the similar connection profiles, although the miscellaneous structural patterns coexist in this network.

From the perspective of the outgoing edges, we obtain a split of the network that is similar to the one got in Ref. VJ09 (), as shown in Fig. 8(a). However, several notable difference exists between the two results. Specifically, “Pakistan” is grouped with the West and “South Korea” is grouped with the Muslim World in Ref. VJ09 (). These false categorizations can be correctly amended, which is consistent with the configuration depicted in Huntington’s renowned book The Clash of Civilizations Huntington96 (). In addition, we categorized “Australia”, which is grouped with West in Ref. VJ09 (), into the group Asia for understandable reasons. Fig. 8(b) gives a quite different structure of this network from the perspective of incoming edges. Three groups, namely the West, Latin America and Muslim World, stay almost the same. But “Russia”, together with some countries of the former Soviet Union, are isolated from the Asia group and form another independent power bloc. Meanwhile, the remaining countries in Asia group join with the West Africa countries to constitute a bigger cluster. It is not difficult to see that all the changes appear to be in accordance with the history and evolution of the international relations.

Recall that the parameters and provide us with the centrality degrees of each vertex in its corresponding group from the perspective of outgoing edges and incoming edges, respectively. In other words, the parameters measure the importance of each vertex in its group. For a better visualization, the sizes of vertices in Fig. 8(d) and (e) are proportional to the magnitude of the scalars and . Coincidentally, we discover that the big vertices, marked by the red bold border, usually stand for the dominant countries in their corresponding groups. For example, the largest vertex of the West is “USA” in Fig. 8(d). In fact, this state often serves as a leader in its power bloc. A similar interpretation can be given for the vertex “Russia” in Asia group. We further check the bridgeness and group entropy for each vertex in the network (data not shown), and we mark the vertices, which have large values of these two measures, with the black bold border. As anticipated, these kinds of vertices are particularly prone to reside on the boundaries of different groups. That is to say, the vertices that are very difficult to divide into one group build a fuzzy watershed of the overlapping structures. In Fig. 8(b), three vertices “Janpan” , “Philippines” and “Australia”, with high values of bridgeness and group entropy, play a transitional role between the West and Asia groups. In reality, the above-mentioned Asian counties frequently collaborated with the counterparts in West group in many areas, from economics to military.

V Conclusions

We propose an extension of the stochastic block model to study the mesoscopic structural patterns in signed networks. Without prior knowledge what specific structure exists, our model can not only accurately detect broad types of intrinsic structures, but also can directly learn their types from the network data. Experiments on a number of synthetic and real world networks demonstrate that our model outperforms the state-of-the-art approaches at extracting various structural features in a given network. Due to the flexibility inherited from the stochastic model, our method is an effective way to reveal the global organization of the networks in terms of the structural regularities, which further helps us understand the relationship between networks’ structure and function. As future work, we will generalize our model by releasing the requirement that the block matrices are square matrices and investigate the possible applications of the more flexible models.

Acknowledgements.
The author would like to thank Vincent A. Traag for providing the international conflict and alliance network used in this paper. The author is also grateful to the anonymous reviewers for their valuable suggestions, which were very helpful for improving the manuscript.

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