Uncovering hidden community structures in evolving networks based on neighborhood similarity

Abstract

Detection of densely interconnected nodes also called modules or communities in static or dynamic networks has become a key approach to comprehend the topology, functions and organizations of the networks. Over the years, numerous methods have been proposed to detect the accurate community structure in the networks. State-of-the-art approaches only focus on finding non-overlapping and overlapping communities in a network. However, many networks are known to possess a hidden or embedded structure, where communities are recursively grouped into a hierarchical structure. Here, we reinvent such sub-communities within a community, which can be redefined based on nodes similarity. We term those derived communities as hidden or hierarchical communities. In this work, we present a method called Hidden Community based on Neighborhood Similarity Computation (HCNC) to uncover undetected groups of communities that embedded within a community. HCNC can detect hidden communities irrespective of density variation within the community. We define a new similarity measure based on the degree of a node and it’s adjacent nodes degree.

We evaluate the efficiency of HCNC by comparing it with several well-known community detectors through various real-world and synthetic networks. Results show that HCNC has better performance in comparison to the candidate community detectors concerning various statistical measures. The most intriguing findings of HCNC is that it became the first research work to report the presence of hidden communities in Les Miserables, Karate and Polbooks networks.

Keywords

Embedded Community Intrinsic structure Hidden community Neighborhood similarity Community Strength Social graphs

1 Introduction

Complex systems such as social media, biological interactions and internet can be represented as graphs (networks) $G (V, E)$ , where the individual entities (members or actors) are represented by the set of nodes $V$ and their affinity among themselves, depicted by the set of edges or links $E$ . Nodes in a sub-graphs or community [1] are more densely interconnected to each other but sparsely connected with the rest of the network and generally share common attributes or properties [2]. Community detection has cropped up as a fundamental task to unveil the underlying structure and potential functions of complex networks.

Over the last few decades, numerous community detection algorithms have been proposed for identifying disjoint sub-groups (communities) of related nodes within the networks [3]. In recent years, researchers have observed that some community members have significant external connections with the other community members and have proposed new methods for finding overlapping communities [4 –6]. Besides these two categories of communities, one form of overlapping community termed as a hidden (or embedded) community may also form due to the variation in the relationship within the community members. Therefore, one can further analyze the detected communities and can build a hierarchical dendrogram based on the neighborhood similarity present within a community. Best of our knowledge, very less attention has been given by the past researches to extract such communities.

During the identification of a community structure, the membership of a node towards the community is measured based on the connectivity or relationship possess by any node at that time. However, in the case of a real-world scenario, the connectivity retains by any node may fluctuate with time. An increase or decrease in connectivity within a community can transform a single community into multiple dense or sparse sub-communities or even leads to the formation of the death community. Contrarily, an increase or decrease in connectivity with other communities leads to the shifting of nodes from one community to others or may create overlapping nodes, which further may lead to the community merging. For a clear understanding of the above situations within a community, let us consider a community of baseball players created by the college students during their graduation. In the beginning, the community members are more close or interactive to each other leads to forming a single and compact community. After a few years, it seems that the interaction among the members of the community deliberately decreases over time, except a few of them, who continue to share similar interests among them. The reasons behind these behavioral changes within the members of the baseball community players depend on several aspects. The community members may either grow interests in other areas than baseball or lack of interest within the members themselves over time. Due to this two probable scenarios (not limited to), the group (including social groups such as Facebook, WhatsApp, etc) may split into sub-groups such as (i) members which are strongly involved in the community, (ii) members which are not fully involved in the community or (iii) members which existence are ideal within the group but still consuming few benefits and services created by the other members in the community.

It would be more understandable if we consider the formation as well as the transformation of a family community as shown in Fig.1. It is obvious that the male members in the family (brothers, cousin brothers & brothers-in-law) and the female members (wives, sisters & sisters-in-law) both communities are strongly involved in family interest. Communities of their sons and daughters show less involvement in family interests. Similarly, the community of kids and old persons have the least involvement, however, they still get the benefits of being a family member. If we observed the transition within the family communities for a particular period of time, it seems that a highly active community progressively transforms into the least active community. These underlying phenomena present within a community is more crucial and challenging task to uncover. For a similar type of example, we can consider any social media groups like WhatsApp, Facebook groups. In such social groups, a few members are strongly involved and share information in the group. Some of them share or post when they feel it necessary and some of them become inactive all the time, but still getting information from the group. Hence, instead of having a single community of family members or any social community, we can split the whole community into sub-communities based on the role, characteristics or similarity possesses by each community member.

Fig. 1

Presence of sub-communities within a community of Family members.

The advantage of revealing sub-communities within a community is to get a better understanding of the underlying properties and to provide better services. Since, the potential members of a community take the reins of almost all imperative activities. Thus, it is essential to impart with the potential members of a community to conceal precise services, rather than providing the services to all the members. Similarly, for the marketing of the commercial products, its’ necessary to target a group of potential users to expose in social networks, rather than targeting all users for optimizing the spread of product consumption. Since, the negative reviews of some people may impair the product value as well as long-term profit.

Uncovering the hidden communities within a community in complex networks come across various challenges. This work focuses on providing insight into the hidden structure and also address the following challenges.

Due to frequent manipulation (increase or decrease) of connectivity with in the community leads to the transformation of community structure from one to another.

Community objects may join or leave from one sub-group to another due to the alteration of their interests or connectedness.

To address the above issues, we propose a new community detection method HCNC. In our approach, we use an affinity score based on internal and external node similarity score exist among the members of a community. Our approach is initiated immediately after a community is identified. We believe that this is the first work that report to handle such situations. We compare the performance of our method with other community detectors in light of several artificial and real-world networks.

The rest of the paper is organized as follows. In Section 2, the detection of communities within a community is described in detail. The similarity measures for hidden community detection is presented in Section 3. Section 4, report the performance evaluation of the proposed model. Finally, we summarize our work in Section 5.

2 Detection of communities within a community

A sub-graphs with dense intra-community connectivity in comparison to sparse-community connectivity within the network is popularly defined as a community [7].

In community detection problem, each $v_{i \in V}$ is assigned to a community $C_{i}$ based on a mapping function $f : G \to {C_{1}, \dots, C_{k}}$ . Members or nodes are mapped to the community $C_{i}$ based on certain affinity or similarity scores. As per the community definition proposed by various researchers [8 –10], it is assumed that a community contains similar kind of members with high intra-connectivity among them. However, in reality, the relationship among the members is not constant over time that brings a significant variation in relationship or association within the community members. Members that retain analogous characteristics can further group into compact sub-community within the parent community.

In the recent past, a few proposals are put forward on identifying hierarchical communities. OSLOM [11] is an optimized fitness function based approach to detect overlapping and hierarchical community structures in diverse types of networks. EAGLE [12] is able to detect both hierarchical and overlapping communities. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. iLCD [13] is presented by [13] to detect overlapping community structures in dynamic networks. iLCD state that communities are independent of its context and it transforms from one state (origin) to other (current). LCDNN [14], is an NGC (Nodes with Greater Centrality) based local community detection approach. In this approach, a fuzzy relation is adopted to compute the affinity from nodes to their NGC nodes. TSB [15] is a local community detection method based on breadth first search (BFS), transfer similarity and local clustering coefficient. However, this approach ignores the presence of hidden communities within the communities. LGIEM [16] can uncover communities based on global and local influential nodes. First, initial communities are formed based on the most influential nodes. Then, the existing communities are expanded based on an expansion strategy. Finally, overlapping communities are merged to obtain the final communities. However, LGIEM does not report the concept of hidden community structure. Recently, research work like InDEN [17] and InOvIn [18] has proposed an integrated approach to uncover overlapping, non-overlapping and intrinsic communities in growing networks. However, they don’t consider those community structures which don’t have dense connectivity compare to other members yet they retain similar characteristics.

Most of the conventional methods use a mapping function based on certain features (e.g. properties or connectedness), which is remain the same for detecting both communities (from a network) and hierarchical communities (from a community). It fails to uncover hierarchical communities, where members do not retain such particular features. For example, if we need to group family members from a random crowd group (network). We will look into some particular features and identify some members as a family. However, within that family community, the features or connectedness upon which we consider them as a member of this group may not be the same. For example, connectedness between father-son and mother-son has a significant difference. Hence, a particular feature is incompetent to detect hidden or hierarchical community within a community.

Algorithms like InDEN and InOvIn can detect density variation only when there are significant changes (measured by a threshold) in nodes degree within a community. It can not detect communities which do not exhibit significant variations in degree distributions within a particular community. However, it has been observed that members, who do not acquire the essential features (such as connected density, exclusive characteristics, etc.) are also capable of forming communities within the parent community. In our approach, we are not considering only the dense connectivity as a feature. Rather, we are considering all similar kinds of connectivity (both dense and sparse) acquired by each node of the community. Nodes with similar connected density are further group together within a community, what we termed as hidden or embedded communities and it’s detection is our prime objective that has been discussed next.

3 Uncovering hidden community structures

For every community finding approach, the mapping function used in it assesses the efficiency of the approach. A community with robust similarity measure produces more rigorous community structures. In this work, we have adopted the Affinity score proposed in InDEN [17] for detecting communities in evolving networks. InDEN presents a new Affinity score based on intra-connectivity possess by the members of a community.

We examine the presence of hidden communities by seeking similar kinds of members within a community. In this regard, a few new measures are introduced to find the proximity of one node to another. A brief interpretation of these parameters is discussed below.

3.1 Calculating node similarity within a community

We create sub-communities within a community based on the Node Similarity Index (NSI) of a node with respect to other. Objects are assigned to the respective sub-communities with high NSI score. A high NSI score represent better proximity of a node towards a sub-community.

Definition 3.1. (Node Similarity Index) Node similarity index between any two nodes v_i and v_j is the ration of inter-nodes connectivity strength, Δ along with the common adjacent nodes, Θ to the degree of all the neighbors of both the nodes v_i and v_j.

$NSI (v_{i}, v_{j}) = \frac{Δ (v_{i}, v_{j}) + Θ}{\sum_{i = 1}^{n} φ (v_{i}) + \sum_{j = 1}^{m} φ (v_{j})}$ (1) where, Δ is the degree difference among the adjacent nodes of v_i with the adjacent nodes of v_j, where v_i, v_j ∈ C_k. On the other hand, Θ represent the number of common adjacent nodes present between v_i and v_j. φ (v_i) is the vector of degree of all the neighbours of v_i in $C_{k}$ i.e. $φ (v_{i}) = {δ_{v_{k}} | v_{k} \in {Neigb}_{C_{k}} (v_{i})}$ and φ (v_j) represents the vector of degree of all the neighbours of v_j in $C_{k}$ i.e. $φ (v_{j}) = {δ_{v_{l}} | v_{l} \in {Neigb}_{C_{k}} (v_{j})}$ .

Next, we discuss how we initiate our hierarchical or hidden community detection approach within a community with the help of neighborhood similarity computation.

3.2 Identification of hierarchical community based on neighborhood similarity

To uncovering the hidden communities within a community $C_{i}$ , we compute the adjacency matrix and a data structure called community degree vector ${CD}_{C_{i}}$ . The degree vector consist of each node degree δ_{v
_k} that belongs to a certain community $C_{i}$ i.e. ${CD}_{C_{i}} = {δ_{v_{1}}, δ_{v_{2}}, \dots, δ_{v_{m}}}, \forall δ_{v_{k}} \in C_{i}$ . For every insertion of a new node in $C_{i}$ , ${CD}_{C_{i}}$ and the adjacency matrix are updated accordingly. We initiate our propose model on a community after a certain period of time and keep on updating for every insertion that took place within the community. The steps involved in the detection of hierarchical communities are presented below.

Finding a leader: In this process, a leader is selected from the community $C_{i}$ . A leader can be defined as the highest degree node, which is not a member of any existing sub-communities. A leader is the initial member of a sub-community and gradually other members are included in the sub-communities over time.

Finding similar neighbors: After the selection of a leader, the neighbors of the leader are traversed one by one. Nodes which satisfy the NSI score along with threshold τ are considered as the member of the same community with the leader. The process of traversing the neighbors of a leader will continue until and unless it satisfies the value of NSI score and τ. After the termination of the step 2 process, the step 1 is initiated again within that community and so on.

3.3 Merging communities

Given two sub-communities c_p and c_q are merged together only when a significant number of members from both the communities have the NSI score which is measured with respect to both the communities is less than a threshold value λ. Mathematically, |NSI (v_k, v_p) - NSI (v_k, v_q) | < λ, where v_p ∈ c_p and v_q ∈ c_q.

3.4 HCNC: The Algorithm

Algorithm 1 represents the stepwise process of our proposed method HCNC. The proposed methods works in two phases, first community finding, for which we use the Affinity score reported in InDEN [17]. In second phase, hidden community finding, where we run HCNC within a certain interval of time. For each incoming edge, we update the community feature vector of ${CD}_{C_{j}}$ and adjacency matrix as per requirement.

We initiate the process by selecting the leader from a community and traverse the neighbors of neighbors (expand the neighbors) of the leader until NSI satisfy the threshold value (τ) (ref line no. 7-12 of Algorithm 1). Otherwise, we search for a new leader and create a new sub-community along with the new leader (ref line no. 20-24). We perform merging between two sub-communities based on the number of nodes (linking nodes) having sufficient NSI with both the participating sub-communities (ref line no. 13-17).

4 Performance evaluation

In this section, we examine the performance of the proposed HCNC on four (04) synthetic and eight (08) real-world networks. We compare our results with four well know community detection methods InDEN, iLCD, EAGLE and OSLOM. We consider various validity measures like Connectivity (Conn) [19], Davies-Bouldin (DB) [20], Dunn [21], Silhouette (Silht) [22], Extended Modularity (EQ) [12] and NMI [23] to appraise the performance of HCNC against the candidate community detectors. Moreover, we also consider a few overlapping community measures such as Overlapping Normalized Mutual Information (ONMI) [24], Average F-Score (AFS) [25] and Omega Index (OI) [26].

4.1 Testing data

To evaluate the computational efficiency of the proposed method, we consider four (04) synthetic and eight (08) real-world input networks. Due to unavailability of networks with hidden or embedded communities, we generate four synthetic networks (see Fig. 2) HiNet1 to HiNet4, with significant degree distribution among the nodes. Both network HiNet1 and HiNet4 have four hidden communities. HiNet2 have one hidden communities and network HiNet3 contains two hidden community. All synthetic HiNet networks are available on GitHub 1 for download.

We also use well known real-world networks, which are available at UCI 2 machine repository and SNAP 3 . Table 1 summarized both synthetic and real-world networks used in the experiment.

Fig. 2

Original Synthetic HiNet networks.

Table 1

Dataset used in the experiments

Network	Dataset	Available	#Nodes	#Edges	#Communities	#Hidden
Synthetic	HiNet1	GitHub ¹	86	222	1	5
	HiNet2		56	170	2	4
	HiNet3		101	189	2	4
	HiNet4		83	218	1	5
Real-World	Karate Network	UCI ²	34	78	2 [23]	–
	Dolphin Social Network		62	159	2 [23]	–
	Books about US Politics		105	441	3 [27]	–
	Les Miserables		76	254	7 [28]	–
	ca-GrQc	SNAP ³	5,242	14,496	–	–
	ca-HepTh		9,877	25,998	–	–
	Facebook		4,039	88,234	–	–
	ca-CondMat		23,133	93,497	–	–

– Information not available. Note: All these synthetic and real-world networks used in the experiment are undirected in nature.

Table 2

Qualitative Assessment of HCNC for Synthetic Networks

Dataset	Method	Dunn	Silhouette	Connectivity	Davies-Bouldin	EQ
HiNet1	HCNC	0.2305	0.3410	9.356	0.2013	0.3160
	iLCD	0.1434	0.2805	7.782	0.3108	0.2915
	EAGLE	0.1244	0.1923	10.103	0.4066	0.3078
	InDEN	0.2037	0.2374	9.130	0.3623	0.2875
	OSLOM	0.1204	0.2045	10.081	0.3727	0.2606
HiNet2	HCNC	0.3417	0.2313	7.424	0.3368	0.3618
	iLCD	0.3132	0.2816	9.843	0.3578	0.3381
	EAGLE	0.2889	0.2037	10.215	0.3804	0.3816
	InDEN	0.2945	0.2516	8.036	0.3468	0.4209
	OSLOM	0.3346	0.2196	9.565	0.3503	0.4077
HiNet3	HCNC	0.2406	0.2208	10.173	0.3145	0.2961
	iLCD	0.2507	0.2617	11.175	0.2878	0.2394
	EAGLE	0.2119	0.3543	13.162	0.3254	0.2804
	InDEN	0.3215	0.3229	11.803	0.3068	0.1685
	OSLOM	0.3027	0.2578	12.916	0.3224	0.2449
HiNet4	HCNC	0.3501	0.4251	6.856	0.2848	0.3665
	iLCD	0.3320	0.3185	7.845	0.2819	0.3492
	EAGLE	0.2785	0.3576	10.865	0.3314	0.2864
	InDEN	0.3154	0.4073	9.186	0.2464	0.3264
	OSLOM	0.2034	0.3836	8.169	0.3336	0.2717

4.2 Experimental results

A series of experiments were performed to quantify the efficiency of HCNC in comparison with other well-established community detectors. Table 2 represent the validity measures of various community detectors on synthetic networks. Results shows that, HCNC outperformed all other algorithms with respect to all parameters on HiNet1, HiNet2 and HiNet4. On network HiNet3, others approaches marginally outperform HCNC in Dunn, Silhouette and DB.

In the case of real-world network karate, HCNC perform better results than all other methods with respect to Dunn, Silhouette and Connectivity. However, iLCD and EAGLE outperform HCNC on karate network by about 2.08% and 3.21% respectively on DB index (see Fig. 3). On dolphin and Polbook networks, HCNC perform superior result than other community detectors with respect to all validity measures (see Fig. 3), except the EQ index. In case of dolphin network, algorithms iLCD and InDEN outperform HCNC with a performance of 1.8% and 2.6% based on EQ index. Whereas algorithms EAGLE, InDEN and OSLOM shows superior performance than HCNC on Polbook networks by about 2.4%, 1.6% and 2.7% on EQ index respectively. On Les Miserables network, HCNC produce satisfactory performance.

Fig. 3

Validity Measures for Karate, Dolphin, Polbook and Les Miserables networks.

To assess the performance of HCNC in detecting the overlapping communities concerning the different community detectors, we consider a few validity indexes such as ONMI, AFS, EQ and OI. Figure 4 demonstrate the performance of various community detectors on large datasets namely ca-CondMat, Ca-GrQc, ca-HepTh and Facebook. On networks Ca-GrQc, Ca-HepTh and Facebook, HCNC produce better results in comparison to other community detectors based on assessment measures ONMI, AFS and EQ. However, iLCD marginally outperform HCNC concerning the validity index Omega by 0.8% and 2.4% on Facebook and Ca-HepTh networks (see Fig. 4). On Ca-CondMat network, HCNC outperform EAGLE, iLCD, InDEN and OSLOM with respect to validity measure ONMI and AFS. Algorithms EAGLE and OSLOM produce almost similar results like HCNC with a performance difference of 0.4% and 1.2% on Ca-CondMat network on Omega and EQ index respectively.

Fig. 4

Validity Measures for ca-CondMat, CaGrQc, ca-HepTh and Facebook networks.

For quality measure, we consider the Normalized Mutual Information (NMI) [23]. In this experiment, we increase the percentage of the incoming edge by N/8, N/6, N/4 and N/2, where N represent the total number of edges. For each segment of incoming edges respective NMI values are computed. Figure 5 present the NMI score (Y-axis) as a function of percentage of incoming edge constraints (X-axis) for real-world networks Facebook and Ca-HepTh. On Facebook network, HCNC outperformed all other candidate methods for the edges segments N/8, N/7, N/5, N/3 and N and second best for the edges segments N/4 and N/2.

Fig. 5

Results generated by five algorithms on Facebook (left) and on Ca-HepTh network (right) with respect to NMI (Y-axis) and Percentage of incoming edge (X-axis).

4.2.1 Uncovering hidden communities

We initiate the process of uncovering hidden communities after a certain interval of time. Experimentally, we have successfully detected the presence of hidden communities in the real-world datasets namely Karate, Polbooks and Les Miserables. Network Karate contains one hidden communities, whereas Polbooks and Les Miserables contain two and four hidden communities respectively. To the best of our knowledge, no prior research reports the presence of hidden communities in these networks. Figures 6, 7 demonstrate the accuracy of HCNC in uncovering the hidden communities in both synthetic and real-world networks respectively. Table 3 reports the total number of communities, including both disjoint and hidden communities detected by each method. Results show that HCNC can detect meaningful communities in real and synthetic networks effectively.

Fig. 6

Sub-communities identified by HCNC on Artificial Network HiNet.

Fig. 7

HCNC outcomes on Karate network, American College Football, Dolphin Social Network, Books about US Politics, Les Miserables and Word Adjacencies.

Table 3

Total number of communities including both hidden and disjoint communities identified by various algorithms

#Datasets	#n	#m	# Communities	iLCD	EAGLE	InDEN	OSLOM	HCNC
ca-GrQc	5,242	14,496	–	55	44	52	48	41
ca-HepTh	9,877	25,998	–	61	47	54	58	49
Facebook	4,039	88,234	–	13	10	8	11	10
ca-CondMat	23,133	93,497	–	66	63	58	74	67
Zacharys karate club	34	78	2	2	2	2	2	3
American college football	115	613	12	12	12	11	12	12
Political books	105	441	3	3	3	2	3	5
Dolphin social network	62	160	2	2	2	2	2	3
HiNet1	86	222	1	1	1	4	2	5
HiNet2	56	170	2	2	2	3	2	4
HiNet3	101	189	2	2	2	3	2	4
HiNet4	83	218	1	1	2	4	2	5

– Information not available. Number of nodes are represent by n and m stands for edges.

5 Conclusion

We introduce an approach ttHCNC for uncovering both disjoint and hidden communities from evolving networks. We introduced a new measure called Node Similarity Index (ttNSI) to calculate the similarity between any two nodes. We test our proposed model against contemporary detectors on both synthetic and real-world networks. Results imply that ttHCNC produce superior results than all other community detectors concerning most of the assessment indices.

In our current work, we assume that networks are evolving in nature, where each incoming node is joining the network in a constant manner. However, in the real-world scenario, nodes may join the network in different time frames with varying velocity. In our future work, we will consider the possible effects that can crop up in detecting communities due to the varying volume and velocity of growing networks.

Footnotes

https://github.com/nathkeshab/HiNet

http://archive.ics.uci.edu/ml/datasets.html

https://snap.stanford.edu/data/#citnets

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