A General Definition of Network Communities and the Corresponding Detection Algorithm

Network structures, consisting of nodes and edges, have applications in almost all subjects. A set of nodes is called a community if the nodes have strong interrelations. Industries (including cell phone carriers and online social media companies) need community structures to allocate network resources and provide proper and accurate services. However, all the current detection algorithms are motivated by the practical problems, whose applicabilities in other fields are open to question. Thence, for a new community problem, researchers need to derive algorithms ad hoc, which is arduous and even unnecessary. In this paper, we represent a general procedure to find community structures in practice. We mainly focus on two typical types of networks: transmission networks and similarity networks. We reduce them to a unified graph model, based on which we propose a general method to define and detect communities. Readers can specialize our general algorithm to accommodate their problems. In the end, we also give a demonstration to show how the algorithm works

Defining and identifying communities in networks

The investigation of community structures in networks is an important issue in many domains and disciplines. This problem is relevant for social tasks (objective analysis of relationships on the web), biological inquiries (functional studies in metabolic, cellular or protein networks) or technological problems (optimization of large infrastructures). Several types of algorithm exist for revealing the community structure in networks, but a general and quantitative definition of community is still lacking, leading to an intrinsic difficulty in the interpretation of the results of the algorithms without any additional non-topological information. In this paper we face this problem by introducing two quantitative definitions of community and by showing how they are implemented in practice in the existing algorithms. In this way the algorithms for the identification of the community structure become fully self-contained. Furthermore, we propose a new local algorithm to detect communities which outperforms the existing algorithms with respect to the computational cost, keeping the same level of reliability. The new algorithm is tested on artificial and real-world graphs. In particular we show the application of the new algorithm to a network of scientific collaborations, which, for its size, can not be attacked with the usual methods. This new class of local algorithms could open the way to applications to large-scale technological and biological applications.Comment: Revtex, final form, 14 pages, 6 figure

Community detection for correlation matrices

A challenging problem in the study of complex systems is that of resolving, without prior information, the emergent, mesoscopic organization determined by groups of units whose dynamical activity is more strongly correlated internally than with the rest of the system. The existing techniques to filter correlations are not explicitly oriented towards identifying such modules and can suffer from an unavoidable information loss. A promising alternative is that of employing community detection techniques developed in network theory. Unfortunately, this approach has focused predominantly on replacing network data with correlation matrices, a procedure that tends to be intrinsically biased due to its inconsistency with the null hypotheses underlying the existing algorithms. Here we introduce, via a consistent redefinition of null models based on random matrix theory, the appropriate correlation-based counterparts of the most popular community detection techniques. Our methods can filter out both unit-specific noise and system-wide dependencies, and the resulting communities are internally correlated and mutually anti-correlated. We also implement multiresolution and multifrequency approaches revealing hierarchically nested sub-communities with `hard' cores and `soft' peripheries. We apply our techniques to several financial time series and identify mesoscopic groups of stocks which are irreducible to a standard, sectorial taxonomy, detect `soft stocks' that alternate between communities, and discuss implications for portfolio optimization and risk management.Comment: Final version, accepted for publication on PR

Community Detection via Maximization of Modularity and Its Variants

In this paper, we first discuss the definition of modularity (Q) used as a metric for community quality and then we review the modularity maximization approaches which were used for community detection in the last decade. Then, we discuss two opposite yet coexisting problems of modularity optimization: in some cases, it tends to favor small communities over large ones while in others, large communities over small ones (so called the resolution limit problem). Next, we overview several community quality metrics proposed to solve the resolution limit problem and discuss Modularity Density (Qds) which simultaneously avoids the two problems of modularity. Finally, we introduce two novel fine-tuned community detection algorithms that iteratively attempt to improve the community quality measurements by splitting and merging the given network community structure. The first of them, referred to as Fine-tuned Q, is based on modularity (Q) while the second one is based on Modularity Density (Qds) and denoted as Fine-tuned Qds. Then, we compare the greedy algorithm of modularity maximization (denoted as Greedy Q), Fine-tuned Q, and Fine-tuned Qds on four real networks, and also on the classical clique network and the LFR benchmark networks, each of which is instantiated by a wide range of parameters. The results indicate that Fine-tuned Qds is the most effective among the three algorithms discussed. Moreover, we show that Fine-tuned Qds can be applied to the communities detected by other algorithms to significantly improve their results

A shadowing problem in the detection of overlapping communities: lifting the resolution limit through a cascading procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.Comment: 14 pages, 12 figures + supporting information (5 pages, 6 tables, 3 figures