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Community Detection Using Modularity Approach in Social Network Analysis

ABSTRACT

We use complex network theory to study the differences between the international relationship concepts in Formal Interstate Alliance (FIA) and Dyadic Militarized Interstate Disputes (MID) of the countries. Seven IR networks were identified. Six of these networks are from Formal Interstate Alliance; one is from Militarized Interstate Disputes. Since community detection is computationally challenging, using modularity to calculate real world social network datasets, we provide a simple modularity algorithm and verify its performance with real social network datasets. The idea is to identify emerging trends besides using network techniques to examine the IR concept. Additionally, we are interested to identify the most influential, central, as well as active nodes using sociometric analyses. We analyzed the structure and the communities of these IR networks and found significant differences in Formal Interstate Alliance (FIA) compared with Dyadic Militarized Interstate Disputes (MID). In FIA, the countries make alliance because of the presence of various types of relationships such as defense pact, neutrality or non-aggression treaty, and entente agreement. These kinds of relationships are called positive relationships. These differences made up a topological structure quite different at different international levels. We also found differences in the network characteristics. Once these differences are understood, the topological structure of the IR network and the communities shaped in FIA could be predicted if we know the total number of nodes and the ties. However, this discovery implies that IR is a dynamic concept that produces several changes in the IR network structure and the way that countries make groups of relationships; it provides the opportunity to give analytic support to empirical studies.

INTRODUCTION

Social, technological and information systems can usually be defined in terms of complex networks which have a topology of interconnected nodes combining organization and randomness. Network theory provides a way to study the complex systems effectively, such as the metabolic networks, protein–protein interaction networks and transportation networks .And it has been found that these complex networks own many common topological properties .For example, many complex networks have been found to have community structure or modular structure. Social networks consist of different kinds of relationships. Two representative relationships are positive (friendship) and negative (hostility) relationships between agents. For example, nations may be formal allies like Japan and the United States or may have military conflicts like South Korea and North Korea. For these kinds of relationships, social balance theory in psychology and sociology states that when a social triad consisting of three agents is considered, the triad probably consists either of three positives or of one positive and two negatives [24, 25].

One of the key research in this area is community structures. Community is a bunch of nodes in a network having similar properties or having more connections between them. Detection of a group of nodes in a network is a goal to identify strongly connected components by separating sparse connections. A partitioning procedure that attempts to find relatively dense sets of nodes which are relatively sparsely connected to the other sets, is called community detection. Therefore numerous algorithms have been proposed to find reasonably good partitions in a reasonably fast way. In recent years this search has attracted much interest for fast algorithms due to the increasing availability of large network data sets and the influence of networks on everyday life.

There is a renewed interest in communities and their structure as well as in the algorithms to describe the structures [15, 22, 23]. Computational efficiency is one of the main issue that must be taken into account while dealing with the problems of finding communities in real world social networks [18-21]. Yet researchers have been trying to get reasonable computation for clustering using modularity.

In this paper we are interested in identifying how communities appear in the real world social network datasets such as formal interstate alliance and Dyadic Militarized Interstate Disputes datasets. We take into international relationships (IR) within the countries network. To do this we employ Louvain’s algorithm using modularity approach [13], for all seven networks. This algorithm is ideal for finding large scale communities because of its analytical base, which is really clear with respect to network division into communities. We consider this study important because it allows us to provide an analytic and computational basis to the empirical studies, thus provides the lines of research for applied physics.

The outline of the rest of the paper is as follows:

We first present background about networks and community detection. Next, in the methodology section, we discuss the data sets and the communities using modularity are analyzed in the networks. Next in the results section, we discuss the implications of analyzing the network using modularity in the networks. Then we describe the discussion on the results and last, It is followed by conclusions section.

Background

Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. Community structure detection, is a data analysis technique used to analyze the structure of large-scale network data sets, such as social networks, internet and web data, or biochemical networks [1]. Community structure methods usually undertake that the network of interest divides naturally into subgroups and the experimenter’s job is to find those groups. The number and size of the groups are thus determined by the network itself and not by the experimenter.

2.1. Graph Theoretical Notation of the Network

A complex network is mapped to the graph G(V, E), where V represents the node set and E represents the edge set. The cardinality (order) of V and E is shown by n and m respectively. If v is the subset of V and e is the subset of E then it is said to be subnetwork of a network C(v, e).

2.2. Community detection

The broad applications of community detection in the area of complex network analysis, there are different definitions presented for the community among which two descriptions get popular by data scientists. (i) A set of nodes whose properties are more similar, called community [11]. (ii) A community is defined a set of nodes which has more edges in a group rather than the other nodes in the rest of graph [12]. The broad overview of the community detection methods can be found in [14-17].

2.2.1. Modularity

Modularity has been used as a quality metric often for resulting good partitions formed by these methods by comparing the number of links within and between communities. The range of value of modularity is from -1 to 1. The modularity value near 1 indicates good quality of partitions while value near -1 indicates bad quality of partitions. Researchers have been exploring to achieve reasonable approximate for modularity.

2.3. Community Detection Approaches

Uncovering of community structure helps to understand network functionality. However, community detection is computationally inflexible in large-scale networks.  No accepted solution of community detection exists yet, for which considerable number of techniques for the optimization of community detection has been introduced in the scientific literature.

We present an overview of the six state-of-the-art groups of community detection algorithms in this section. We provided techniques for the identification of static, dynamic, disjoint, and overlapping communities. The comprehensive overview of the community detection techniques can be found in [14, 15, 17].

2.3.1. Traditional Community Detection Techniques

We focus on traditional techniques of community detection.

2.3.1.1. Graph partitioning

It splits the graph into g clusters of predefined size, such that the number of links in a cluster is more denser than the number of edges between the clusters (15). Spectral Bisection method is well-known examples of graph partitioning techniques (27) and also Kernighan-Lin algorithm (26).

2.3.1.2. Hierarchical clustering

The graphs may hold hierarchical structure, where each community may be a collection of small groups at different levels (15; Friedman et al. 2001). In such cases, hierarchical clustering techniques may be used to classify the multilevel community structure of the graph. Hierarchical grouping techniques are based on the node similarity measure. As it does not need a predefined size and number of communities. Dendrograms can represent them better. Hierarchical clustering approaches can be divided into two classes.

  • Agglomerative algorithms

It is a bottom-up method as at the start it studies each node as a separate cluster and iteratively merges them based on high similarity and ends up with the unique community.

  • Divisive algorithms

It is a top-down method as at the start it considers the whole network as a single cluster and iteratively divides it by eliminating links joining nodes with low similarity and ends up with unique communities.

2.3.1.3. Partitional clustering

Partitional clustering [15] divides a dataset into a predefined number of k non-overlapping clusters. The aim is to partition the data points into k clusters in order to minimize /maximize the cost function based on dissimilarity measure between nodes. Some of the frequently used cost functions are minimum k median, k-clustering sum, k-clustering, and k-center. Examples of partional clustering methods include k-mean clustering and fuzzy k-mean clustering. In fuzzy k-mean, clustering is one node that may fit to several clusters.

2.3.1.4. Spectral clustering

Spectral clustering includes all methods which use eigenvectors of matrices to split the set of data points based on the pairwise similarity between them [15]. Examples contain a Laplacian spectral clustering method of Fiedler [28] and Donath

2.3.1.5. Divisive algorithms

It eliminates inter-cluster edges in a network based on low-similarity to separate communities from each other [31]. The key examples of this type include Girvan-Newman algorithm [30] where edges eliminated are iteratively based on edge-betweenness score and Radicchi et al. technique [32], where edges are eliminated iteratively based on the edge clustering coefficient.

2.3.2. Modularity Optimisation Based Community Detection Techniques

It presents network community detection techniques based on modularity optimization. Modularity is a measure of networks or graphs that was designed to measure the power of division of a network into modules or it is the quality to approximate the communities. The larger the modularity value gives the better partition.

2.3.2.1. Greedy techniques

  • Greedy method of Newman

Newman’s greedy search algorithm [33] was the first algorithm recommended for modularity optimization. It is an agglomerative method, where originally, each node belongs to a discrete module, then they are fused iteratively based on the modularity gain. On sparse networks, it has a time complexity of O(n)3 on sparse networks.

  • Fast Greedy algorithm (CNM)

It is the fast version of Newman’s algorithm [33], implemented by efficient data structures. On sparse networks.It has a computational complexity of O(nlog2n).

  • Blondal’s Louvain algorithm

Louvain [34] is a heuristic greedy algorithm for the discovery of communities in complex weighted networks. It is also based on the modularity optimization. It allocates different communities to each vertex; one per vertex. It iteratively combines the nodes based on the gain of modularity. If no gain, then node remains in its own community. The process is repeated until no more improvement is possible. It then rebuilds the network in the way that communities identified in the first phase are replaced by super nodes. Its time complexity is given like O(nlogn).

2.3.2.2. Simulated Annealing (Kirkpatrick et al. 1983)

It is a discrete stochastic method used for the global optimization of the given objective function. Guimer`a et al. [35] have used simulated annealing based modularity optimization method. Primarily, it divides the network into random clusters. The optimization is based on local and global changes. Local moves a node randomly from one cluster to another based on modularity gain. Global moves consist of dividing and merging of clusters.

2.3.2.3. Extremal Optimization

Boettcher et al. [36] have calculated extremal optimization as a general purpose heuristic search method for physical and combinatorial optimization problems. It is proposed to get accuracy comparable with genetic algorithm and simulated annealing. It emphases on the optimization of local variables. Duch et al. in [37] have used it for modularity optimization. It allocates fitness to each node; the fitness value is gained by taking the ratio of the local modularity of the node by its degree. It changes an individual solution within a single configuration and makes local variations. It starts by randomly dividing the network into two clusters of the same order. It iteratively moves vertices with the lowest fitness to the other partitions. After moving, the clusters change, that’s why it recalculates local fitness of many nodes. The procedure repeats until an optimal value of global modularity is got.

2.3.2.4. Spectral Optimization

Spectral optimization is the use of eigenvectors and eigenvalues of the modularity matrix for modularity optimization [15].This optimization is pretty fast. Examples of the spectral optimization of modularity contain [38, 12].

2.3.2.5. Evolutionary algorithms

Evolutionary algorithms are a branch of metaheuristic optimization algorithms based on artificial intelligence. They are known for their real local learning and global searching competences. These approaches are largely divided into two classes based on single and multi-objective optimization. Examples of the first kind include MAGA-Net [39], MLAMA-Net [40] etc. Examples of the second category include MOEA/D [39].

2.3.3. Overlapping community detection techniques

In real world networks, most of the vertices may concurrently belong to multiple communities. Traditional community detection methods fail in recognizing overlapping communities. Clique percolation is the most known technique used for the u uncovering of overlapping communities in the networks. It is based on the idea that cliques are more likely to be shaped from internal edges that are densely connected than from external edges which are sparsely connected. The communities are made up of k-cliques that refer to the complete subgraphs with k vertices. Two cliques are said to be adjacent if they share k − 1 nodes. The k-clique community is the giant component shaped of all the adjacent k-cliques that are connected as a k-clique series. Other examples of this category contain top graph clusters [41].

2.3.4. Dynamic Community Detection Algorithms

This focuses on the community detection methods in dynamic networks, such as Twitter, Facebook, LinkedIn, etc. These techniques study the community assignment of the changed or new nodes during temporal updates in the network.

2.3.4.1. Potts model

The Potts model is one of the renowned models used in statistical physics. It proves a system of spins which can be in q diverse states. The interaction among neighboring turns may be ferromagnetic or antiferromagnetic. Potts spin variables can be mapped to the nodes of the graph having community structure. From interactions between neighbouring spins, it is plausible that community structure may be recognized from like-valued spin clusters of the system, as there may be more interactions in the community and fewer interactions outside the community.

2.3.4.2. Random walk

The Random Walk is adopted to find communities. In a random walk, the walker initiates to walk inside a community from a node and at each time step it moves to the neighbouring node selected randomly and consistently. The walker gives a long time inside the dense communities because of high density and multiple paths. Examples of the most general techniques based on random walks are PageRank algorithm.

2.3.4.3. Diffusion Community

“A diffusion community in a complex network is a set of nodes that are gathered together by the propagation of the same property, action or information in the network.  Label and dynamic node coloring  are examples of this category.

2.3.4.4. Synchronization

Synchronization is an emerging method which has interest from different fields. It happens in interacting units and is convincing in nature, technology and society. In a synchronized state, the system units remain in same or alike states over time. Synchronization is also used in community discovery in networks. If oscillators are located at nodes with random initial phases and interact with nearest neighbors, then oscillators going to the same partition synchronize first, whereas longer time is obligatory for full synchronization. If evolution time of the process is followed, then states with synchronized communities of nodes can be much stable and prolonged, therefore can be known easily.

  1. Methodology

In this work, we have used undirected networks of international relationship (IR) where six networks of formal interstate alliance and one network of military dispute network are constructed. A method is used to find partitions or clusters or communities based on network structural properties. For this purpose, Louvain algorithm strategy of community detection is followed. Louvain algorithm takes each node as a community. By comparing each node with its neighbors, it resolves to fuse communities with a maximum possible gain in modularity. A question arises that is it possible to observe the changes in the concept of international relationship (IR) through the IR networks, their properties, and communities? It is an interesting domain that motivates researchers to work on it progressively.

3.1. Complex Network Tools: An Overview

An effective set of mathematical, statistical, and computational tools for social realities network analysis and modeling is available. Every tool has a different design and computational models; one size does not fit all. Some are specialized, some are common, some are single threaded, some are multi-threaded, some are having shared memory, some are having distributed memory. Network scientists use these tools for network construction, visualization, simulation, properties extraction, pattern recognition, community detection and complex queries. Some of the usually used network analysis and visualization tools include Pajek, MATLAB, R, Gephi, Visone, Wolfram Mathematica, NodeXL and CiteSpace etc.

3.2. Tool Used for Analysis and visualization

Gephi is an open-source software package written in Java on the NetBeans platform, used for network analysis and visualization  [4]. Gephi has been used in several research projects in academia, journalism and elsewhere, for example; in visualizing the global connectivity of New York Times content [5]and investigating Twitter network traffic during social unrest[6,7] along with more traditional network analysis topics [8]. Gephi is broadly used within the digital humanities [9], literature, political sciences, etc., a community where many of its developers are involved. Gephi inspired the LinkedIn InMaps [10] and was also used for the network visualizations for Truthy.

3.2. Data Description

In this work we use Correlates of War Formal Interstate Alliance Dataset (v4.1), as positive relationship that is provided by Gibler and Press [2] and is available publicly. This dataset defines alliance relationships between countries according to different types such as defense pact, neutrality or non-aggression treaty, and entente agreement. The negative relationships between countries are provided by the Dyadic Militarized Interstate Disputes Dataset Version 2.0,that is made by Maoz [3] and is available publicly.

3.3 The Louvain method for community detection in large networks

The Louvain method is a simple, efficient and easy to implement method for classifying communities in large networks. The method has been used many different types of networks and for sizes up to 100 million nodes and billions of links. The method reveals hierarchies of communities and allows to zoom within communities to discover sub-clusters, sub-sub-clusters, etc. For detecting communities in large networks, It is today one of the most broadly used technique [13].

Louvain [10] is a heuristic greedy algorithm for identifying communities in complex weighted graphs. It is based on the modularity optimisation too. It allocates different communities to each vertex; one per vertex. It iteratively combines the nodes based on the gain of modularity. If no gain, then node remains in its own community. The procedure is repeated until no more development is possible. It then reconstructs the network in the way that communities recognized in the first phase are substituted by super nodes. The time of this algorithm complexity is O(nlogn).

3.4. Network characteristics

In this work, undirected networks of IR are used. These seven networks are showing the dyad relationship between two countries. The networks are constructed in Gephi tool. The general characteristics of the networks are described in Table 1, such as nodes, No. of edges, average degree, average weighted degree, network diameter, density, average clustering coefficient, average path length, number of weakly connected components etc; of the Alliance By Directed, Alliance By Directed (Yearly), Alliance By Dyad, Alliance By Dyad (Yearly), Alliance By Member, Alliance By Member (Yearly), Dyadic MID networks respectively. Table 2 shows different centrality metrics such as Hubs, Authority, PageRank, Betweenness Centrality, Closeness Centrality, Eigenvector Centrality, Eccentricity, Harmonic closeness centrality of the seven networks.

Table 3 indicates the community detection of the Alliance By Directed, Alliance By Directed (Yearly), Alliance By Dyad, Alliance By Dyad (Yearly), Alliance By Member, Alliance By Member (Yearly), Dyadic MID networks respectively, using modularity. Louvain community detection technique is used keeping the resolution equal to 1, using the weighs and keeping it randomized. Different modularity classes are displayed showing the percentage by which the countries are in alliance or disputes relationships, making dense connected networks.

  1. Results and discussion

4.1. Alliance By Directed and Directed (Yearly) Network Analysis

The Alliance By Directed having 859 and Directed (Yearly) having 899 nodes respectively, seven communities are detected in both networks while the modularity value of AB directed is 0.279 which much better than the other one. It is because of the highly dense connected relationship of the countries that make alliance. Figure 1 and 2 and Table 3 shows the community detection graphs, modularity class distribution graph and no. of communities plus modularity value of the networks respectively.

4.2. Alliance By Dyad and Dyad (Yearly) Network Analysis

The Alliance By Dyad having 860 and Dyad (Yearly) having 899 nodes respectively, seven and six communities are detected in both networks respectively while the modularity value of AB dyad is 0.258 which much better than the other one. It is because of the highly dense connected relationship of the countries that make alliance. Figure 3, 4 and Table 3 shows the community detection graphs, modularity class distribution graph and no. of communities plus modularity value of the networks respectively.

4.3. Alliance By Member and Member (Yearly) Network Analysis

The Alliance By member having 872 and member (Yearly) having 912 nodes respectively, eleven and eight communities are detected in both networks respectively while the modularity value of  member (Yearly)  is 0.236  which much better than the other one. It is because of the highly dense connected relationship of the countries that make alliance. Figure 5, 6 and Table 3 shows the community detection graphs, modularity class distribution graph and no. of communities plus modularity value of the networks respectively.

4.4. Dyadic MID Network Analysis

The Dyadic MID network having 3301 nodes with 11 communities detecting having modularity value 0.143 that depicts the dispute relationship of the network. Consequently, it is clear and obvious that the IR concept is dynamic as there is a change in network topologies, structures, properties and as well as communities that are quite different from each other while making different kind of positive and negative relationships.

4.5. An Analysis of an Example of social Friendship Network of Elementary School (E1)

This analysis was conducted using the adjacency matrices obtained for the network that is depicted in Table 4. The properties that characterize this social network are also displayed. Louvain Algorithm and VOS clustering algorithm are used to find the community structure of the network and the relationship between the friendship concept and that structure is also investigated. To compare the community detection in same network (E1) with both different algorithms, it is resulted that VOS clustering algorithm results 0.507 that is much better based on modularity and eight communities are detected, while 6 communities are detected in Louvain method with modularity value of 0.433 that is less than the VOS clustering algorithm. Pajek software is being used for network analysis, community detection and visualization of this friendship network of elementary school (E1). Figure 8 depicts the friendship Network of E1, extraction of communities according to Louvain Method and its community detection.

 Table 1. Principal characteristics of International Relationship

Network Characteristics                                                  Formal Alliances (v4.1)

 

Military Disputes
Alliance By Directed Alliance By Directed (Yearly) Alliance By Dyad Alliance By Dyad (Yearly) Alliance By Member Alliance By Member (Yearly) Dyadic

MID

Nodes 859 899 860 899 872 912 3301
No. of Edges 5814 11676 5815 11674 6271 12132 40419
Average Degree 13.537

 

25.976 13.523 25.971 14.383 26.605 24.489
Average Weighted Degree 232.792 5422.578 123.779 2711.362 56.612 945.969 161.853
Network Diameter 4 4 4 4 4 4 4
Density 0.016 0.029 0.016 0.029 0.017 0.029 0.007
Average Clustering Coefficient 0.162 0.149

 

0.162 0.148 0.158 0.151 0.237
Average Path length 2.827 2.465 2.4668 2.822 2.4656 2.5168
Number of Weakly Connected Components 2 2 2 2 2 2 2
Components 0.979045

 

0.978865

 

0.977907

 

0.978865

 

0.973624

 

0.973684

 

0.983338

 

No. of triangles 33.48894

 

65.37486

 

33.45

 

65.03115

 

37.8945

 

69.33882

 

94.5468

 

Table 2. Different Centrality Metrics

Centrality Metrics AB Directed AB  Directed (Yearly) AB  Dyad AB  Dyad (Yearly) AB  Member AB  Member (Yearly) Dyadic

MID

Hubs 0.02106

 

0.022522

 

0.021036

 

0.022493

 

0.020875

 

0.02242

 

0.011341

 

Authority 0.02106

 

0.022522

 

0.021036

 

0.022493

 

0.020875

 

0.02242

 

0.011341

 

PageRank 0.001099

 

0.001155

 

0.00063

 

0.001147

 

0.001096

 

0.000303

 

Betweenness Centrality 751.596

 

631.2684

 

750.7419

 

631.3204

 

752.6411

 

633.3289

 

2420.817

 

Closeness Centrality 0.365394

 

0.417663

 

0.366288

 

0.418087

 

0.367481

 

0.418632

 

0.408386

 

Eigenvector Centrality 0.0067 0.072524

 

0.0068

0.060724

0.072661

 

0.062726

 

0.073134

 

0.040521

 

Eccentricity

 

3.464494

 

 

3.270601

 

3.462791

 

3.459399

 

3.458716

 

3.265351

 

3.269312

 

Harmonic closeness centrality 0.384718

 

0.439509

 

0.385567

 

0.439981

 

0.387019

 

0.440571

 

0.422883

 

Table 3. Number of Communities and Modularity Metrics

AB Directed AB Directed (Yearly) AB Dyad By Dyad (Yearly) AB  Member AB  Member (Yearly) Dyadic

MID

Number of Countries 859 899 860 899 872 912 3301
Modularity 0.279 0.252 0.258 0.251 0.187 0.236 0.143
Number of Communities 7 7 7 6 11 8 11

Table 4.  An Example of General Characteristics of an Elementary School Friendship network, its centralities and modularity.

Network Characteristics Elementary School (E1)  
 
Nodes 108
No. of Arcs 1006
Average Weighted Degree 18.62
No. Components 1
Density 0.087
Clustering Coefficient 0.294
Average Path length 2.477
Betweenness Centrality 0.035
Closeness Centrality 0.161
Degree Centrality 10.887
Modularity

(VOS Quality)

0.507
Number of Communities 8
Modularity

(Louvain Community Detection)

0.433
Number of Communities 6

Conclusion

From the study of IR networks presented we can conclude that, in general, the differences in topologies between the FIA networks and dyad MID networks are an effect attributable to the concept of IR, which changes over a lifetime. However, a bigger effect is visible when communities in the networks are studied as the basis of alliance types and disputes; it is concluded, on the basis of the networks analysis that the concept of IR is clearly different at both alliance and dispute level. This difference in the country size and its composition is interesting with respect to the behavior of the countries evaluated, when they are observed from the perspective of IR concept evaluation.

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