Quality functions in community detection

Community structure represents the local organization of complex networks and the single most important feature to extract functional relationships between nodes. In the last years, the problem of community detection has been reformulated in terms of the optimization of a function, the Newman-Girvan modularity, that is supposed to express the quality of the partitions of a network into communities. Starting from a recent critical survey on modularity optimization, pointing out the existence of a resolution limit that poses severe limits to its applicability, we discuss the general issue of the use of quality functions in community detection. Our main conclusion is that quality functions are useful to compare partitions with the same number of modules, whereas the comparison of partitions with different numbers of modules is not straightforward and may lead to ambiguities.Comment: 10 pages, 4 figures, invited paper to appear in the Proceedings of SPIE International Conference "Fluctuations and Noise 2007", Florence, Italy, 20-24 May, 200

Community detection algorithms: a comparative analysis

Uncovering the community structure exhibited by real networks is a crucial step towards 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 and on random graphs. As a result of our analysis, three recent algorithms introduced by Rosvall and Bergstrom, Blondel et al. and Ronhovde and Nussinov, respectively, have an excellent performance, with the additional advantage of low computational complexity, which enables one to analyze large systems.Comment: 12 pages, 8 figures. The software to compute the values of our general normalized mutual information is available at http://santo.fortunato.googlepages.com/inthepress

Scale-free network growth by ranking

Network growth is currently explained through mechanisms that rely on node prestige measures, such as degree or fitness. In many real networks those who create and connect nodes do not know the prestige values of existing nodes, but only their ranking by prestige. We propose a criterion of network growth that explicitly relies on the ranking of the nodes according to any prestige measure, be it topological or not. The resulting network has a scale-free degree distribution when the probability to link a target node is any power law function of its rank, even when one has only partial information of node ranks. Our criterion may explain the frequency and robustness of scale-free degree distributions in real networks, as illustrated by the special case of the Web graph.Comment: 4 pages, 2 figures. We extended the model to account for ranking by arbitrarily distributed fitness. Final version to appear on Physical Review Letter

Coevolution of Glauber-like Ising dynamics and topology

We study the coevolution of a generalized Glauber dynamics for Ising spins, with tunable threshold, and of the graph topology where the dynamics takes place. This simple coevolution dynamics generates a rich phase diagram in the space of the two parameters of the model, the threshold and the rewiring probability. The diagram displays phase transitions of different types: spin ordering, percolation, connectedness. At variance with traditional coevolution models, in which all spins of each connected component of the graph have equal value in the stationary state, we find that, for suitable choices of the parameters, the system may converge to a state in which spins of opposite sign coexist in the same component, organized in compact clusters of like-signed spins. Mean field calculations enable one to estimate some features of the phase diagram.Comment: 5 pages, 3 figures. Final version published in Physical Review