Many methods have been proposed to detect communities, not only in plain, but also in attributed, di- rected, or even dynamic complex networks. From the modeling point of view, to be of some utility, the com- munity structure must be characterized relatively to the properties of the studied system. However, most of the existing works focus on the detection of communities, and only very few try to tackle this interpretation problem. Moreover, the existing approaches are limited either by the type of data they handle or by the nature of the results they output. In this work, we see the interpretation of commu- nities as a problem independent from the detection process, consisting in identifying the most characteristic features of communities. We give a formal definition of this problem and propose a method to solve it. To this aim, we first define a sequence-based representation of networks, com- bining temporal information, community structure, topo- logical measures, and nodal attributes. We then describe how to identify the most emerging sequential patterns of this dataset and use them to characterize the communities. We study the performance of our method on artificially generated dynamic attributed networks. We also em- pirically validate our framework on real-world systems: a DBLP network of scientific collaborations, and a LastFM network of social and musical interactions.
More recently, several authors used topological measures to characterize community structures in plain networks. In @cite_15 , Lancichinetti . visually examined the distribution of some community-based topological measures, both at local and intermediary levels. Their goal was to understand the general shape of communities belonging to networks modeling various types of real-world systems. In @cite_32 , Leskovec . proposed to study the community structure as a whole, by considering it at various scales, thanks to a global measure called conductance. These two studies are valuable, however, from the interpretation perspective, they are limited by the fact they consider the network as a whole. Communities are studied and characterized collectively, in order to identify trends in the whole network, or even a collection of networks.
@cite_15 @cite_32
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Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks. Uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents. Many algorithms have been proposed so far, but none of them has been subjected to strict tests to evaluate their performance. Most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure, which is very uncommon in real systems. Here we test several methods against a recently introduced class of benchmark graphs, with heterogeneous distributions of degree and community size. The methods are also tested against the benchmark by Girvan and Newman [Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002)] and on random graphs. As a result of our analysis, three recent algorithms introduced by Rosvall and Bergstrom [Proc. Natl. Acad. Sci. U.S.A. 104, 7327 (2007); Proc. Natl. Acad. Sci. U.S.A. 105, 1118 (2008)], [J. Stat. Mech.: Theory Exp. (2008), P10008], and Ronhovde and Nussinov [Phys. Rev. E 80, 016109 (2009)] have an excellent performance, with the additional advantage of low computational complexity, which enables one to analyze large systems. The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses. A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the "best" possible community - according to the conductance measure - over a wide range of size scales, and we study over 70 large sparse real-world networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large real-world networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually "blend in" with the rest of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are well-embeddable in a low-dimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community structure similar to our observations.