Community detection is one of the most important and challenging problems in network analysis. However, real-world networks may have very different structural properties and communities of various nature. As a result, it is hard (or even impossible) to develop one algorithm suitable for all datasets. A standard machine learning tool is to consider a parametric algorithm and choose its parameters based on the dataset at hand. However, this approach is not applicable to community detection since usually no labeled data is available for such parameter tuning. In this paper, we propose a simple and effective procedure allowing to tune hyperparameters of any given community detection algorithm without requiring any labeled data. The core idea is to generate a synthetic network with properties similar to a given real-world one, but with known communities. It turns out that tuning parameters on such synthetic graph also improves the quality for a given real-world network. To illustrate the effectiveness of the proposed algorithm, we show significant improvements obtained for several well-known parametric community detection algorithms on a variety of synthetic and real-world datasets.
is a widely used measure optimized by many community detection algorithms. It was first proposed in @cite_0 and is defined as follows where @math is a resolution parameter @cite_16 . The intuition behind modularity is the following: the first term in is the fraction of intra-cluster edges, which is expected to be relatively high for good partitions, while the second term penalizes this value for having too large communities. Namely, the value @math is the expected fraction of intra-cluster edges if we preserve the degree sequence but connect all vertices randomly, i.e., if we assume that our graph is constructed according to the configuration model @cite_23 .
@cite_0 @cite_16 @cite_23
2146191525 2095189226 2015953751 2621737021
In this paper we discuss some problematic aspects of Newman and Girvan’s modularity function Q N . Given a graph G, the modularity of G can be written as Q N = Q f − Q 0, where Q f is the intracluster edge fraction of G and Q 0 is the expected intracluster edge fraction of the null model, i.e., a randomly connected graph with same expected degree distribution as G. It follows that the maximization of Q N must accomodate two factors pulling in opposite directions:Q f favors a small number of clusters and Q 0 favors many balanced (i.e., with approximately equal degrees) clusters. In certain cases the Q 0 term can cause overestimation of the true cluster number; this is the opposite of the well-known underestimation effect caused by the “resolution limit” of modularity. We illustrate the overestimation effect by constructing families of graphs with a “natural” community structure which, however, does not maximize modularity. In fact, we show there exist graphs G with a “natural clustering” V of G and another, balanced clustering U of G such that (i) the pair (G, U) has higher modularity than (G, V) and (ii) V and U are arbitrarily different. 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. We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks. Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We here present first results on the computational and analytical properties of modularity. The complexity status of modularity maximization is resolved showing that the corresponding decision version is NP-complete in the strong sense. We also give a formulation as an Integer Linear Program (ILP) to facilitate exact optimization, and provide results on the approximation factor of the commonly used greedy algorithm. Completing our investigation, we characterize clusterings with maximum modularity for several graph families.