Community detection has arisen as one of the most relevant topics in the field of graph data mining due to its importance in many fields such as biology, social networks or network traffic analysis. The metrics proposed to shape communities are generic and follow two approaches: maximizing the internal density of such communities or reducing the connectivity of the internal vertices with those outside the community. However, these metrics take the edges as a set and do not consider the internal layout of the edges in the community. We define a set of properties oriented to social networks that ensure that communities are cohesive, structured and well defined. Then, we propose the Weighted Community Clustering (WCC), which is a community metric based on triangles. We proof that analyzing communities by triangles gives communities that fulfill the listed set of properties, in contrast to previous metrics. Finally, we experimentally show that WCC correctly captures the concept of community in social networks using real and syntethic datasets, and compare statistically some of the most relevant community detection algorithms in the state of the art.
There are basically two types of metrics to evaluate the quality of a community. First, those that focus on the internal density of the community. The most widely used metric that falls into this category is the modularity , which was proposed by @cite_0 . Modularity measures the internal connectivity of the community (omitting the external connectivity) compared to an Erd "os-R 'enyi graph model. It has become very popular in the literature, and a lot of algorithms are based on maximizing it. Algorithms apply several optimization procedures: agglomerative greedy @cite_15 , simulated annealing strategy @cite_21 or multistep approaches @cite_3 .
@cite_0 @cite_15 @cite_21 @cite_3
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Modularity is widely used to effectively measure the strength of the disjoint community structure found by community detection algorithms. Although several overlapping extensions of modularity were proposed to measure the quality of overlapping community structure, there is lack of systematic comparison of different extensions. To fill this gap, we overview overlapping extensions of modularity to select the best. In addition, we extend the Modularity Density metric to enable its usage for overlapping communities. The experimental results on four real networks using overlapping extensions of modularity, overlapping modularity density, and six other community quality metrics show that the best results are obtained when the product of the belonging coefficients of two nodes is used as the belonging function. Moreover, our experiments indicate that overlapping modularity density is a better measure of the quality of overlapping community structure than other metrics considered. Modularity maximization is the most popular technique for the detection of community structure in graphs. The resolution limit of the method is supposedly solvable with the introduction of modified versions of the measure, with tunable resolution parameters. We show that multiresolution modularity suffers from two opposite coexisting problems: the tendency to merge small subgraphs, which dominates when the resolution is low; the tendency to split large subgraphs, which dominates when the resolution is high. In benchmark networks with heterogeneous distributions of cluster sizes, the simultaneous elimination of both biases is not possible and multiresolution modularity is not capable to recover the planted community structure, not even when it is pronounced and easily detectable by other methods, for any value of the resolution parameter. This holds for other multiresolution techniques and it is likely to be a general problem of methods based on global optimization. It has been shown that the communities of complex networks often overlap with each other. However, there is no effective method to quantify the overlapping community structure. In this paper, we propose a metric to address this problem. Instead of assuming that one node can only belong to one community, our metric assumes that a maximal clique only belongs to one community. In this way, the overlaps between communities are allowed. To identify the overlapping community structure, we construct a maximal clique network from the original network, and prove that the optimization of our metric on the original network is equivalent to the optimization of Newman's modularity on the maximal clique network. Thus the overlapping community structure can be identified through partitioning the maximal clique network using any modularity optimization method. The effectiveness of our metric is demonstrated by extensive tests on both artificial networks and real world networks with a known community structure. The application to the word association network also reproduces excellent results. We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.