Characterizing the community structure of complex networks

Community structure is one of the key properties of complex networks and plays a crucial role in their topology and function. While an impressive amount of work has been done on the issue of community detection, very little attention has been so far devoted to the investigation of communities in real networks. We present a systematic empirical analysis of the statistical properties of communities in large information, communication, technological, biological, and social networks. We find that the mesoscopic organization of networks of the same category is remarkably similar. This is reflected in several characteristics of community structure, which can be used as ``fingerprints'' of specific network categories. While community size distributions are always broad, certain categories of networks consist mainly of tree-like communities, while others have denser modules. Average path lengths within communities initially grow logarithmically with community size, but the growth saturates or slows down for communities larger than a characteristic size. This behaviour is related to the presence of hubs within communities, whose roles differ across categories. Also the community embeddedness of nodes, measured in terms of the fraction of links within their communities, has a characteristic distribution for each category. Our findings are verified by the use of two fundamentally different community detection methods.Comment: 15 pages, 20 figures, 4 table

A novel framework for community modeling and characterization in directed temporal networks

Abstract We deal with the problem of modeling and characterizing the community structure of complex systems. First, we propose a mathematical model for directed temporal networks based on the paradigm of activity driven networks. Many features of real-world systems are encapsulated in our model, such as hierarchical and overlapping community structures, heterogeneous attitude of nodes in behaving as sources or drains for connections, and the existence of a backbone of links that model dyadic relationships between nodes. Second, we develop a method for parameter identification of temporal networks based on the analysis of the integrated network of connections. Starting from any existing community detection algorithm, our method enriches the obtained solution by providing an in-depth characterization of the very nature of the role of nodes and communities in generating the temporal link structure. The proposed modeling and characterization framework is validated on three synthetic benchmarks and two real-world case studies

Scaling theory of fractal complex networks

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach - grounded in both scaling theory of phase transitions and renormalization group theory - leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information - the World Wide Web, biological - the human brain, and social - scientific collaboration networks) and in several fractal network models.Comment: 18 pages, 7 figures; the paper is theoretical in nature; theoretical predictions have been succesfully verified in real networks (WWW, DBLP, human brain) and in several fractal network models (SHM-model, (u,v)-flowers , nested BA networks

Evolution of Network Architecture in a Granular Material Under Compression

As a granular material is compressed, the particles and forces within the system arrange to form complex and heterogeneous collective structures. Force chains are a prime example of such structures, and are thought to constrain bulk properties such as mechanical stability and acoustic transmission. However, capturing and characterizing the evolving nature of the intrinsic inhomogeneity and mesoscale architecture of granular systems can be challenging. A growing body of work has shown that graph theoretic approaches may provide a useful foundation for tackling these problems. Here, we extend the current approaches by utilizing multilayer networks as a framework for directly quantifying the progression of mesoscale architecture in a compressed granular system. We examine a quasi-two-dimensional aggregate of photoelastic disks, subject to biaxial compressions through a series of small, quasistatic steps. Treating particles as network nodes and interparticle forces as network edges, we construct a multilayer network for the system by linking together the series of static force networks that exist at each strain step. We then extract the inherent mesoscale structure from the system by using a generalization of community detection methods to multilayer networks, and we define quantitative measures to characterize the changes in this structure throughout the compression process. We separately consider the network of normal and tangential forces, and find that they display a different progression throughout compression. To test the sensitivity of the network model to particle properties, we examine whether the method can distinguish a subsystem of low-friction particles within a bath of higher-friction particles. We find that this can be achieved by considering the network of tangential forces, and that the community structure is better able to separate the subsystem than a purely local measure of interparticle forces alone. The results discussed throughout this study suggest that these network science techniques may provide a direct way to compare and classify data from systems under different external conditions or with different physical makeup

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

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

Interdisciplinary and physics challenges of Network Theory

Network theory has unveiled the underlying structure of complex systems such as the Internet or the biological networks in the cell. It has identified universal properties of complex networks, and the interplay between their structure and dynamics. After almost twenty years of the field, new challenges lie ahead. These challenges concern the multilayer structure of most of the networks, the formulation of a network geometry and topology, and the development of a quantum theory of networks. Making progress on these aspects of network theory can open new venues to address interdisciplinary and physics challenges including progress on brain dynamics, new insights into quantum technologies, and quantum gravity.Comment: (7 pages, 4 figures

A complex network approach reveals pivotal sub-structure of genes linked to Schizophrenia

Research on brain disorders with a strong genetic component and complex heritability, like schizophrenia and autism, has promoted the development of brain transcriptomics. This research field deals with the deep understanding of how gene-gene interactions impact on risk for heritable brain disorders. With this perspective, we developed a novel data-driven strategy for characterizing genetic modules, i.e., clusters, also called community, of strongly interacting genes. The aim is to uncover a pivotal module of genes by gaining biological insight upon them. Our approach combined network topological properties, to highlight the presence of a pivotal community, matchted with information theory, to assess the informativeness of partitions. Shannon entropy of the complex networks based on average betweenness of the nodes is adopted for this purpose. We analyzed the publicly available BrainCloud dataset, containing post-mortem gene expression data and we focused on the Dopamine Receptor D2, encoded by the DRD2 gene. To parse the DRD2 community into sub-structure, we applied and compared four different community detection algorithms. A pivotal DRD2 module emerged for all procedures applied and it represented a considerable reduction, compared with the beginning network size. Dice index 80% for the detected community confirmed the stability of the results, in a wide range of tested parameters. The detected community was also the most informative, as it represented an optimization of the Shannon entropy. Lastly, we verified that the DRD2 was strongly connected to its neighborhood, stronger than any other randomly selected community and more than the Weighted Gene Coexpression Network Analysis (WGCNA) module, commonly considered the standard approach for these studies