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An efficient method for mining the maximal α-quasi-clique-community of a given node in complex networks

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Abstract

Detecting communities in large complex networks is important to understand their structure and to extract features useful for visualization or prediction of various phenomena like the diffusion of information or the dynamic of the network. A community is defined by a set of strongly interconnected nodes. An α-quasi-clique is a group of nodes where each member is connected to more than a proportion α of the other nodes. By construction, an α-quasi-clique has a density greater than α. The size of an α-quasi-clique is limited by the degree of its nodes. In complex networks whose degree distribution follows a power law, usually α-quasi-cliques are small sets of nodes for high values of α. In this paper, we present an efficient method for finding the maximal α-quasi-clique of a given node in the network. Therefore, the resulting communities of our method have two main characteristics: they are α-quasi-cliques (very dense for high α) and they are local to the given node. Detecting the local community of specific nodes is very important for applications dealing with huge networks, when iterating through all nodes would be impractical or when the network is not entirely known. The proposed method, called RANK-NUM-NEIGHS (RNN), is evaluated experimentally on real and computer-generated networks in terms of quality (community size), execution time and stability. We also provide an upper bound on the optimal solution.

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Notes

  1. The density of links \(\delta \) of a graph G with |E| edges et |V| nodes is given by \(\frac{2|E|}{|V|(|V|-1)}\).

  2. A complete clique is a set of node such as every two distinct nodes are connected to each other.

  3. This Definition of an \(\alpha \)-quasi-clique is not unique. Most authors define an \(\alpha \)-quasi-clique as a set of nodes that have a density greater than \(\alpha \), see for instance (Abello et al. 2002). The Definition considered in this paper constitutes a relative relaxation of a complete clique as it depends on the size of the quasi-clique.

  4. Notice that we used the word maximal instead of maximum. In graph theory a maximal clique is a clique which is not a proper subset of another clique whereas a maximum clique is a clique of the maximum cardinality in the graph. Since we aim to find \(\alpha \)-quasi-cliques containing a given node of interest we are looking for maximal \(\alpha \)-quasi-cliques instead of for the maximum \(\alpha \)-quasi-clique.

  5. If \(\alpha <0.5\), Theorem 1 does not hold anymore, then the input of the Algorithm will not be limited to the second neighborhood, but the whole graph specially for alpha small.

  6. The average degree is 20, the maximum degree 50, the exponent of the degree distribution is − 2 and that of the community size distribution is − 1. We chose three values of mixing parameter \(\lambda \), 0.10, 0.20 and 0.30. The results presented in this paper are those for \(\lambda =0.10\) to evaluate size, density and stability. The results have nearly the same behavior for the other values of mixing parameter.

  7. The results obtained for other network sizes have nearly the same behavior.

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Acknowledgements

This work is supported by REQUEST project.

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Authors and Affiliations

  1. LISITE - ISEP, 10 rue de Vanves, 92130, Issy-les-Moulineaux, France

    Patricia Conde-Cespedes

  2. L2TI, Université Paris 13, 99 Avenue Jean Baptiste Clément, 93430, Villetaneuse, France

    Patricia Conde-Cespedes, Blaise Ngonmang & Emmanuel Viennet

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  1. Patricia Conde-Cespedes
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  2. Blaise Ngonmang
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Correspondence to Patricia Conde-Cespedes.

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Conde-Cespedes, P., Ngonmang, B. & Viennet, E. An efficient method for mining the maximal α-quasi-clique-community of a given node in complex networks. Soc. Netw. Anal. Min. 8, 20 (2018). https://doi.org/10.1007/s13278-018-0497-y

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