Detecting Communities in Complex Networks Using Formal Concept Analysis

Abstract

The complex nature of many real-world networks is motivating researchers to investigate or extend network analysis methods such as centrality computation, link prediction, and community detection. One of these complex structures is the multilayer network in which each layer contains a network. Multilayer networks frequently possess complex local structures of multimodal data and interlinked relations. Thus, efficient detection of local communities in such networks often remains a key challenge. In this paper, we propose a community detection strategy, called CoDeBi, which leverages Formal Concept Analysis (FCA) to find possibly overlapping and nested communities in multilayer networks. At the preprocessing stage, we exploit operations such as apposition, subposition and composition on formal contexts—associated with individual layers—to generate a global formal context representing the whole multilayer network. At the first step of CoDeBi, we extract the formal concepts that capture groups in the global formal context while in the second step, we filter the extracted formal concepts to keep only the ones that have a high harmonic mean of stability and separation indices. Such groups represent core communities. In the third step, we detect final communities by refining the core groups using Silhouette Analysis. Our validation study shows that CoDeBi can accurately identify communities in bipartite graphs, and hence can be exploited for community detection in multilayer networks. Another contribution of this paper is the application of the attractive features of Triadic Concept Analysis and the adaptation of our approach to the analysis of tridimensional networks represented by a tridimensional adjacency matrix.

Appendix

Omega index (Collins and Dent 1988) counts the number of node pairs without community assignment as well as those which are in exactly one community, two communities, and so on Chakraborty et al. (2017).

Let $R=\{R_1, R_2, \ldots , R_J\}$ be the set of the J ground-truth communities in the graph of size N, and $C=\{C_1, C_2, \ldots , C_K\}$ the set of detected communities. The Omega index is then defined as follows:

$$\begin{aligned} Omega \left( C,R\right) = \frac{Omega_u \left( C,R\right) -Omega_e \left( C,R\right) }{1-Omega_e \left( C,R\right) } \end{aligned}$$

(7)

where the unadjusted omega index Omega$_u$ is defined as

$$\begin{aligned} Omega_u \left( C,R\right) = \frac{1}{M} \sum _{j=1} \max (|C|,|R|)|t_j(c_i)\cap t_j(r_j)| \end{aligned}$$

(8)

where $M = N(N - 1)/2$ represents the number of node pairs, and $t_j (R)$ is the set of pairs that appear exactly j times in the ground-truth set R. Finally, the expected omega index Omega$_e$ is given by

$$\begin{aligned} Omega_e \left( C, R\right) = \frac{1}{M^{2}} \sum _{j=1} \max (|C|,|R|)|t_j(c_i)|\cdot | t_j(r_j)| \end{aligned}$$

(9)

The computation of the overlapping Normalized Mutual Information is as follows. For each node i in the detected community structure C, its community membership can be declared as a binary vector of length |C|, where $(x_i)_k$ is set to 1 if node i is a member of the k-th cluster $C_k$, and 0 otherwise. The k-th entry of this vector can be viewed as a random variable $X_k$ whose probability distribution is given by:

$P(X_k = k) = N_k/N$, $P(X_k = 0)= 1 - P(X_k =1)$, where $N_k= |C|$, and N is the number of nodes in the graph. The same holds for the random variable $Y_l$ associated with the $l-$th cluster in the ground truth community structure R.

To define the entropy measures H(X) and $H(X_k, Y_l)$, both the empirical marginal probability distribution $P(X_k)$ and the joint probability distribution $P(X_k, Y_l)$ are needed. The conditional entropy of a cluster $X_k$ given $Y_l$ is defined as $H(X_k|Y_l) = H(X_k, Y_l) - H(Y_l)$. The entropy of $X_k$ with respect to the entire vector Y is based on the best matching between $X_k$ and any component of Y:

$$\begin{aligned} H (X_k|Y) = min_{l=1,..,|R|} H (X_k|Y_l) \end{aligned}$$

(10)

The normalized conditional entropy of a community X with respect to Y is

$$\begin{aligned} H(X|Y) = \frac{1}{R} \sum _{k} \frac{H(X_k|y)}{H(X_k)} \end{aligned}$$

(11)

Similarly, we define H(Y|X).

Then, the final Overlapping Normalized Mutual Information formula for two community structures C and R is given by :

$$\begin{aligned} ONMI(X|Y ) = 1 - [H(X|Y ) + H(Y |X)]/2 \end{aligned}$$

(12)