Statistical Mechanics of Community Detection

Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity $Q$ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure

Comparison of methods to identify modules in noisy or incomplete brain networks

open6siCommunity structure, or "modularity," is a fundamentally important aspect in the organization of structural and functional brain networks, but their identification with community detection methods is confounded by noisy or missing connections. Although several methods have been used to account for missing data, the performance of these methods has not been compared quantitatively so far. In this study, we compared four different approaches to account for missing connections when identifying modules in binary and weighted networks using both Louvain and Infomap community detection algorithms. The four methods are "zeros," "row-column mean," "common neighbors," and "consensus clustering." Using Lancichinetti-Fortunato-Radicchi benchmark-simulated binary and weighted networks, we find that "zeros," "row-column mean," and "common neighbors" approaches perform well with both Louvain and Infomap, whereas "consensus clustering" performs well with Louvain but not Infomap. A similar pattern of results was observed with empirical networks from stereotactical electroencephalography data, except that "consensus clustering" outperforms other approaches on weighted networks with Louvain. Based on these results, we recommend any of the four methods when using Louvain on binary networks, whereas "consensus clustering" is superior with Louvain clustering of weighted networks. When using Infomap, "zeros" or "common neighbors" should be used for both binary and weighted networks. These findings provide a basis to accounting for noisy or missing connections when identifying modules in brain networks.openWilliams N.; Arnulfo G.; Wang S.H.; Nobili L.; Palva S.; Palva J.M.Williams, N.; Arnulfo, G.; Wang, S. H.; Nobili, L.; Palva, S.; Palva, J. M

The map equation

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.Comment: 9 pages and 3 figures, corrected typos. For associated Flash application, see http://www.tp.umu.se/~rosvall/livemod/mapequation

Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations

Human brain anatomy and function display a combination of modular and hierarchical organization, suggesting the importance of both cohesive structures and variable resolutions in the facilitation of healthy cognitive processes. However, tools to simultaneously probe these features of brain architecture require further development. We propose and apply a set of methods to extract cohesive structures in network representations of brain connectivity using multi-resolution techniques. We employ a combination of soft thresholding, windowed thresholding, and resolution in community detection, that enable us to identify and isolate structures associated with different weights. One such mesoscale structure is bipartivity, which quantifies the extent to which the brain is divided into two partitions with high connectivity between partitions and low connectivity within partitions. A second, complementary mesoscale structure is modularity, which quantifies the extent to which the brain is divided into multiple communities with strong connectivity within each community and weak connectivity between communities. Our methods lead to multi-resolution curves of these network diagnostics over a range of spatial, geometric, and structural scales. For statistical comparison, we contrast our results with those obtained for several benchmark null models. Our work demonstrates that multi-resolution diagnostic curves capture complex organizational profiles in weighted graphs. We apply these methods to the identification of resolution-specific characteristics of healthy weighted graph architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom

Community mining with graph filters for correlation matrices

International audienceCommunities are an important type of structure in networks. Graph filters, such as wavelet filterbanks, have been used to detect such communities as groups of nodes more densely connected together than with the outsiders. When dealing with times series, it is possible to build a relational network based on the correlation matrix. However, in such a network, weights assigned to each edge have different properties than those of usual adjacency matrices. As a result, classical community detection methods based on modularity optimization are not consistent and the modularity needs to be redefined to take into account the structure of the correlation from random matrix theory. Here, we address how to detect communities from correlation matrices, by filtering global modes and random parts using properties that are specific to the distribution of correlation eigenval-ues. Based on a Louvain approach, an algorithm to detect multiscale communities is also developed, which yields a weighted hierarchy of communities. The implementation of the method using graph filters is also discussed