Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities

Many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups. This feature is known as community structure and encodes precious information about the organization and the function of the nodes. Many algorithms have been proposed but it is not yet clear how they should be tested. Recently we have proposed a general class of undirected and unweighted benchmark graphs, with heterogenous distributions of node degree and community size. An increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links, which require suitable benchmark graphs for testing. In this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built-in community structure. We also consider the possibility that nodes belong to more communities, a feature occurring in real systems, like, e. g., social networks. As a practical application, we show how modularity optimization performs on our new benchmark.Comment: 9 pages, 13 figures. Final version published in Physical Review E. The code to create the benchmark graphs can be freely downloaded from http://santo.fortunato.googlepages.com/inthepress

Optimal map of the modular structure of complex networks

Modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and function of complex systems. Generally speaking, modules are islands of highly connected nodes separated by a relatively small number of links. Every module can have contributions of links from any node in the network. The challenge is to disentangle these contributions to understand how the modular structure is built. The main problem is that the analysis of a certain partition into modules involves, in principle, as many data as number of modules times number of nodes. To confront this challenge, here we first define the contribution matrix, the mathematical object containing all the information about the partition of interest, and after, we use a Truncated Singular Value Decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allow us to scrutinize the skeleton of the modular structure, revealing the structure of individual modules and their interrelations.Comment: 21 pages, 10 figure

Local multiresolution order in community detection

Community detection algorithms attempt to find the best clusters of nodes in an arbitrary complex network. Multi-scale ("multiresolution") community detection extends the problem to identify the best network scale(s) for these clusters. The latter task is generally accomplished by analyzing community stability simultaneously for all clusters in the network. In the current work, we extend this general approach to define local multiresolution methods, which enable the extraction of well-defined local communities even if the global community structure is vaguely defined in an average sense. Toward this end, we propose measures analogous to variation of information and normalized mutual information that are used to quantitatively identify the best resolution(s) at the community level based on correlations between clusters in independently-solved systems. We demonstrate our method on two constructed networks as well as a real network and draw inferences about local community strength. Our approach is independent of the applied community detection algorithm save for the inherent requirement that the method be able to identify communities across different network scales, with appropriate changes to account for how different resolutions are evaluated or defined in a particular community detection method. It should, in principle, easily adapt to alternative community comparison measures.Comment: 19 pages, 11 figure

Partitioning and modularity of graphs with arbitrary degree distribution

We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with . In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with ^1/2 [Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also generalize to the problem of q-partitioning. They can be used to find the expected modularity Q [Newman and Grivan, Phys. Rev. E, 69, 2004] of random graphs and allow for the assessment of statistical significance of the output of community detection algorithms.Comment: Revised version including new plots and improved discussion of some mathematical detail

Modular synchronization in complex networks with a gauge Kuramoto model

We modify the Kuramoto model for synchronization on complex networks by introducing a gauge term that depends on the edge betweenness centrality (BC). The gauge term introduces additional phase difference between two vertices from 0 to $\pi$ as the BC on the edge between them increases from the minimum to the maximum in the network. When the network has a modular structure, the model generates the phase synchronization within each module, however, not over the entire system. Based on this feature, we can distinguish modules in complex networks, with relatively little computational time of $\mathcal{O}(NL)$, where $N$ and $L$ are the number of vertices and edges in the system, respectively. We also examine the synchronization of the modified Kuramoto model and compare it with that of the original Kuramoto model in several complex networks.Comment: 10 pages, 7 figure

Outlier Edge Detection Using Random Graph Generation Models and Applications

Outliers are samples that are generated by different mechanisms from other normal data samples. Graphs, in particular social network graphs, may contain nodes and edges that are made by scammers, malicious programs or mistakenly by normal users. Detecting outlier nodes and edges is important for data mining and graph analytics. However, previous research in the field has merely focused on detecting outlier nodes. In this article, we study the properties of edges and propose outlier edge detection algorithms using two random graph generation models. We found that the edge-ego-network, which can be defined as the induced graph that contains two end nodes of an edge, their neighboring nodes and the edges that link these nodes, contains critical information to detect outlier edges. We evaluated the proposed algorithms by injecting outlier edges into some real-world graph data. Experiment results show that the proposed algorithms can effectively detect outlier edges. In particular, the algorithm based on the Preferential Attachment Random Graph Generation model consistently gives good performance regardless of the test graph data. Further more, the proposed algorithms are not limited in the area of outlier edge detection. We demonstrate three different applications that benefit from the proposed algorithms: 1) a preprocessing tool that improves the performance of graph clustering algorithms; 2) an outlier node detection algorithm; and 3) a novel noisy data clustering algorithm. These applications show the great potential of the proposed outlier edge detection techniques.Comment: 14 pages, 5 figures, journal pape

Comparing community structure identification

We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.Comment: 10 pages, 3 figures, 1 table. v2: condensed, updated version as appears in JSTA

Multiple Nuclear Polarization States in a Double Quantum Dot

We observe multiple stable states of nuclear polarization in a double quantum dot under conditions of electron spin resonance. The stable states can be understood within an elaborated theoretical rate equation model for the polarization in each of the dots, in the limit of strong driving. This model also captures unusual features of the data, such as fast switching and a `wrong' sign of polarization. The results reported enable applications of this polarization effect, including manipulation and control of nuclear fields.Comment: 5 pages, 4 figures, 7 pages supplementary materia

Community Structure in Congressional Cosponsorship Networks

We study the United States Congress by constructing networks between Members of Congress based on the legislation that they cosponsor. Using the concept of modularity, we identify the community structure of Congressmen, as connected via sponsorship/cosponsorship of the same legislation, to investigate the collaborative communities of legislators in both chambers of Congress. This analysis yields an explicit and conceptually clear measure of political polarization, demonstrating a sharp increase in partisan polarization which preceded and then culminated in the 104th Congress (1995-1996), when Republicans took control of both chambers. Although polarization has since waned in the U.S. Senate, it remains at historically high levels in the House of Representatives.Comment: 8 pages, 4 figures (some with multiple parts), to appear in Physica A; additional background info and explanations added from last versio