Community Structure Characterization

This entry discusses the problem of describing some communities identified in a complex network of interest, in a way allowing to interpret them. We suppose the community structure has already been detected through one of the many methods proposed in the literature. The question is then to know how to extract valuable information from this first result, in order to allow human interpretation. This requires subsequent processing, which we describe in the rest of this entry

Deciphering Network Community Structure by Surprise

The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that maximization of a simple global parameter, which we call Surprise (S), leads to a very efficient characterization of the community structure of complex synthetic networks. Particularly, S qualitatively outperforms the most commonly used criterion to define communities, Newman and Girvan's modularity (Q). Applying S maximization to real networks often provides natural, well-supported partitions, but also sometimes counterintuitive solutions that expose the limitations of our previous knowledge. These results indicate that it is possible to define an effective global criterion for community structure and open new routes for the understanding of complex networks.Comment: 7 pages, 5 figure

Characterizing the dynamical importance of network nodes and links

The largest eigenvalue of the adjacency matrix of the networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.Comment: 4 pages, 4 figure

Diverse reductive dehalogenases are associated with Clostridiales-enriched microcosms dechlorinating 1,2-dichloroethane

The achievement of successful biostimulation of active microbiomes for the cleanup of a polluted site is strictly dependent on the knowledge of the key microorganisms equipped with the relevant catabolic genes responsible for the degradation process. In this work, we present the characterization of the bacterial community developed in anaerobic microcosms after biostimulation with the electron donor lactate of groundwater polluted with 1,2-dichloroethane (1,2-DCA). Through a multilevel analysis, we have assessed (i) the structural analysis of the bacterial community; (ii) the identification of putative dehalorespiring bacteria; (iii) the characterization of functional genes encoding for putative 1,2-DCA reductive dehalogenases (RDs). Following the biostimulation treatment, the structure of the bacterial community underwent a notable change of the main phylotypes, with the enrichment of representatives of the order Clostridiales. Through PCR targeting conserved regions within known RD genes, four novel variants of RDs previously associated with the reductive dechlorination of 1,2-DCA were identified in the metagenome of the Clostridiales-dominated bacterial community

Hierarchical mutual information for the comparison of hierarchical community structures in complex networks

The quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks. However, it is not clear if such methods provide consistent, robust and meaningful results when considering hierarchies as a whole. Part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures. In this work we give a contribution by introducing the {\it hierarchical mutual information}, which is a generalization of the traditional mutual information, and allows to compare hierarchical partitions and hierarchical community structures. The {\it normalized} version of the hierarchical mutual information should behave analogously to the traditional normalized mutual information. Here, the correct behavior of the hierarchical mutual information is corroborated on an extensive battery of numerical experiments. The experiments are performed on artificial hierarchies, and on the hierarchical community structure of artificial and empirical networks. Furthermore, the experiments illustrate some of the practical applications of the hierarchical mutual information. Namely, the comparison of different community detection methods, and the study of the consistency, robustness and temporal evolution of the hierarchical modular structure of networks.Comment: 14 pages and 12 figure

Accuracy and Precision of Methods for Community Identification in Weighted Networks

Based on brief review of approaches for community identification and measurement for sensitivity characterization, the accuracy and precision of several approaches for detecting communities in weighted networks are investigated. In weighted networks, the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity $Q^w$. The results reveal that link weight has important effects on communities especially in dense networks. Potts model and Weighted Extremal Optimization (WEO) algorithm work well on weighted networks. Then Potts model and WEO algorithms are used to detect communities in Rhesus monkey network. The results gives nice understanding for real community structure.Comment: 14 pages, 15 figure

Closed benchmarks for network community structure characterization

Characterizing the community structure of complex networks is a key challenge in many scientific fields. Very diverse algorithms and methods have been proposed to this end, many working reasonably well in specific situations. However, no consensus has emerged on which of these methods is the best to use in practice. In part, this is due to the fact that testing their performance requires the generation of a comprehensive, standard set of synthetic benchmarks, a goal not yet fully achieved. Here, we present a type of benchmark that we call "closed", in which an initial network of known community structure is progressively converted into a second network whose communities are also known. This approach differs from all previously published ones, in which networks evolve toward randomness. The use of this type of benchmark allows us to monitor the transformation of the community structure of a network. Moreover, we can predict the optimal behavior of the variation of information, a measure of the quality of the partitions obtained, at any moment of the process. This enables us in many cases to determine the best partition among those suggested by different algorithms. Also, since any network can be used as a starting point, extensive studies and comparisons can be performed using a heterogeneous set of structures, including random ones. These properties make our benchmarks a general standard for comparing community detection algorithms.Comment: 18 pages, 5 figures. Available at http://pre.aps.org/abstract/PRE/v85/i2/e02610