Adjusting from disjoint to overlapping community detection of complex networks

The investigation of community structures is one of the most important problems in the field of complex networks and has countless applications in different disciplines: biology, computer, social sciences, etc. Many community detection algorithms have been developed in various fields recently. The vast majority of these algorithms only find disjoint communities; however, in many real-world networks communities often overlap to some extent. In this paper, we propose an efficient method for adjusting these classical algorithms to match the requirement for discovering overlapping communities in complex networks, which is based on a local definition of community strength. The method can in principle be applied with any clustering algorithm. Tests on a set of computer generated and real-world networks give excellent results. In particular, we show that the method can also allow one to availably analyze the problem of unstable nodes in community detection, which is very helpful for understanding the structural properties of the networks correctly and comprehensively.     

Communicability betweenness in complex networks

Betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks. They allow us to study the extent to which a node takes part when information is passed around the network. Nodes with high betweenness may be regarded as key players that have a highly active role. At one extreme, betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes. At the other extreme, an alternative type of betweenness has been defined by considering all possible walks of any length. In this work, we propose a betweenness measure that lies between these two opposing viewpoints. We allow information to pass through all possible routes, but introduce a scaling so that longer walks carry less importance. This new definition shares a similar philosophy to that of communicability for pairs of nodes in a network, which was introduced by Estrada and Hatano [E. Estrada, N. Hatano, Phys. Rev. E 77 (2008) 036111]. Having defined this new communicability betweenness measure, we show that it can be characterized neatly in terms of the exponential of the adjacency matrix. We also show that this measure is closely related to a Fréchet derivative of the matrix exponential. This allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations. Using illustrative synthetic and real life networks, we show that the new betweenness measure behaves differently to existing versions, and in particular we show that it recovers meaningful biological information from a protein-protein interaction network.     

Methods to find community based on edge centrality

Divisive algorithms are of great importance for community detection in complex networks. One algorithm proposed by Girvan and Newman (GN) based on an edge centrality named betweenness, is a typical representative of this field. Here we studied three edge centralities based on network topology, walks and paths respectively to quantify the relevance of each edge in a network, and proposed a divisive algorithm based on the rationale of GN algorithm for finding communities that removes edges iteratively according to the edge centrality values in a certain order. In addition, we gave a comparison analysis of these measures with the edge betweenness and information centrality. We found the principal difference among these measures in the partition procedure is that the edge centrality based on walks first removes the edge connected with a leaf vertex, but the others first delete the edge as a bridge between communities. It indicates that the edge centrality based on walks is harder to uncover communities than other edge centralities. We also tested these measures for community detection. The results showed that the edge information centrality outperforms other measures, the edge centrality based on walks obtains the worst results, and the edge betweenness gains better performance than the edge centrality based on network topology. We also discussed our method’s efficiency and found that the edge centrality based on walks has a high time complexity and is not suitable for large networks.     

A very fast algorithm for detecting community structures in complex networks

In this paper, a new algorithm is proposed, which uses only local information to analyze community structures in complex networks. The algorithm is based on a table that describes a network and a virtual cache similar to the cache in the computer structure. When being tested on some typical computer-generated and real-world networks, this algorithm demonstrates excellent detection results and very fast processing performance, much faster than the existing comparable algorithms of the same kind.     

Betweenness centrality in finite components of complex networks

We use generating function formalism to obtain an exact formula of the betweenness centrality in finite components of random networks with arbitrary degree distributions. The formula is obtained as a function of the degree and the component size, and is confirmed by simulations for Poisson, exponential, and power-law degree distributions. We find that the betweenness centralities for the three distributions are asymptotically power laws with an exponent 1.5 and are invariant to the particular distribution parameters.     

Networks of interactions in the secondary and tertiary structure of ribosomal RNA

We construct four different structural networks for both the secondary and tertiary structures of the 16S and 23S ribosomal RNAs (rRNAs) in the high-resolution crystal structures of the Thermus thermophilus 30S and Haloarcula marismortui 50S ribosomal subunits, and investigate topological characteristics of the rRNA structures by determining relevant measures, such as the characteristic path length, the clustering coefficient, and the helix betweenness. This study reveals that the 23S rRNA network is more compact than the 16S rRNA networks, reflecting the more globular overall structure of the 23S rRNA relative to the 16S rRNA. In particular, the large number of tertiary interactions in the 23S rRNA tends to cluster, accounting for its small-world network properties. In addition, although the rRNA networks are not the scale-free network, their helix betweenness has a power-law distribution and is correlated with the phylogenetic conservation of helices. The higher the helix betweenness, the more conserved the helix. These results suggest a potential role of the rRNA network as a new quantitative approach in rRNA research.     

Modularity-maximizing graph communities via mathematical programming

In many networks, it is of great interest to identify communities, unusually densely knit groups of individuals. Such communities often shed light on the function of the networks or underlying properties of the individuals. Recently, Newman suggested modularity as a natural measure of the quality of a network partitioning into communities. Since then, various algorithms have been proposed for (approximately) maximizing the modularity of the partitioning determined. In this paper, we introduce the technique of rounding mathematical programs to the problem of modularity maximization, presenting two novel algorithms. More specifically, the algorithms round solutions to linear and vector programs. Importantly, the linear programing algorithm comes with an a posteriori approximation guarantee: by comparing the solution quality to the fractional solution of the linear program, a bound on the available “room for improvement” can be obtained. The vector programming algorithm provides a similar bound for the best partition into two communities. We evaluate both algorithms using experiments on several standard test cases for network partitioning algorithms, and find that they perform comparably or better than past algorithms, while being more efficient than exhaustive techniques.     

Detecting community structure from coherent oscillation of excitable systems

Many networks are proved to have community structures. On the basis of the fact that the dynamics on networks are intensively affected by the related topology, in this paper the dynamics of excitable systems on networks and a corresponding approach for detecting communities are discussed. Dynamical networks are formed by interacting neurons; each neuron is described using the FHN model. For noisy disturbance and appropriate coupling strength, neurons may oscillate coherently and their behavior is tightly related to the community structure. Synchronization between nodes is measured in terms of a correlation coefficient based on long time series. The correlation coefficient matrix can be used to project network topology onto a vector space. Then by the K-means cluster method, the communities can be detected. Experiments demonstrate that our algorithm is effective at discovering community structure in artificial networks and real networks, especially for directed networks. The results also provide us with a deep understanding of the relationship of function and structure for dynamical networks.     

Limited resolution in complex network community detection with Potts model approach

According to Fortunato and Barthélemy, modularity-based community detection algorithms have a resolution threshold such that small communities in a large network are invisible. Here we generalize their work and show that the q-state Potts community detection method introduced by Reichardt and Bornholdt also has a resolution threshold. The model contains a parameter by which this threshold can be tuned, but no a priori principle is known to select the proper value. Single global optimization criteria do not seem capable for detecting all communities if their size distribution is broad.     

Overlapping Community Detection Based on Attribute Augmented Graph

There is a wealth of information in real-world social networks. In addition to the topology information, the vertices or edges of a social network often have attributes, with many of the overlapping vertices belonging to several communities simultaneously. It is challenging to fully utilize the additional attribute information to detect overlapping communities. In this paper, we first propose an overlapping community detection algorithm based on an augmented attribute graph. An improved weight adjustment strategy for attributes is embedded in the algorithm to help detect overlapping communities more accurately. Second, we enhance the algorithm to automatically determine the number of communities by a node-density-based fuzzy k-medoids process. Extensive experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively detect overlapping communities with fewer parameters compared to the baseline methods.     

Detecting community structure in networks

There has been considerable recent interest in algorithms for finding communities in networks--groups of vertices within which connections are dense, but between which connections are sparser. Here we review the progress that has been made towards this end. We begin by describing some traditional methods of community detection, such as spectral bisection, the Kernighan-Lin algorithm and hierarchical clustering based on similarity measures. None of these methods, however, is ideal for the types of real-world network data with which current research is concerned, such as Internet and web data and biological and social networks. We describe a number of more recent algorithms that appear to work well with these data, including algorithms based on edge betweenness scores, on counts of short loops in networks and on voltage differences in resistor networks.Received: 10 November 2003, Published online: 14 May 2004PACS: 89.75.Hc Networks and genealogical trees - 87.23.Ge Dynamics of social systems - 89.20.Hh World Wide Web, Internet - 05.10.-a Computational methods in statistical physics and nonlinear dynamics     

Recursive weighted treelike networks

We propose a geometric growth model for weighted scale-free networks, which is controlled by two tunable parameters. We derive exactly the main characteristics of the networks, which are partially determined by the parameters. Analytical results indicate that the resulting networks have power-law distributions of degree, strength, weight and betweenness, a scale-free behavior for degree correlations, logarithmic small average path length and diameter with network size. The obtained properties are in agreement with empirical data observed in many real-life networks, which shows that the presented model may provide valuable insight into the real systems.     

Routing in scale-free networks based on expanding betweenness centrality

In this paper, an improved routing strategy is proposed for enhancing the traffic capacity of scale-free networks. Instead of using the information of degree and betweenness centrality, the new algorithm is derived on the basis of the expanding betweenness centrality of nodes, which gives an estimate of the traffic handled by the vertex for a certain route set. Since the nodes with large betweenness centrality are more susceptible to traffic congestion, the traffic can be improved by redistributing traffic loads from nodes with large betweenness centrality to nodes with small betweenness centrality in the process of computing the collective routing table. Comparing with results of previous routing strategies, it is shown that the present improved routing performs more effectively.     

Community detection in networks by using multiobjective evolutionary algorithm with decomposition

Community structure is an important property of complex networks. Most optimization-based community detection algorithms employ single optimization criteria. In this study, the community detection is solved as a multiobjective optimization problem by using the multiobjective evolutionary algorithm based on decomposition. The proposed algorithm maximizes the density of internal degrees, and minimizes the density of external degrees simultaneously. It can produce a set of solutions which can represent various divisions to the networks at different hierarchical levels. The number of communities is automatically determined by the non-dominated individuals resulting from our algorithm. Experiments on both synthetic and real-world network datasets verify that our algorithm is highly efficient at discovering quality community structure.     

Optimization of transport protocols in complex networks

In this paper, an optimal routing strategy is proposed to enhance the traffic capacity of complex networks. In order to avoid nodes overloading, the new algorithm is derived on the basis of generalized betweenness centrality which gives an estimate of traffic handled by the node for a route set. Since the nodes with large betweenness centrality are more susceptible to traffic congestion, the traffic can be improved, as our strategy, by redistributing traffic load from nodes with large betweenness centrality to nodes with small betweenness centrality in the proceeding of computing collective routing table. Particularly, depending on a parameter that controls the optimization scale, the new routing can not only enlarge traffic capacity of networks more, but also enhance traffic efficiency with smaller average path length. Comparing results of previous routing strategies, it is shown that the present improved routing performs more effectively.     

Detect overlapping and hierarchical community structure in networks

Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This paper proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results.     

Overlapping community detection using weighted consensus clustering

Many overlapping community detection algorithms have been proposed. Most of them are unstable and behave non-deterministically. In this paper, we use weighted consensus clustering for combining multiple base covers obtained by classic non-deterministic algorithms to improve the quality of the results. We first evaluate a reliability measure for each community in all base covers and assign a proportional weight to each one. Then we redefine the consensus matrix that takes into account not only the common membership of nodes, but also the reliability of the communities. Experimental results on both artificial and real-world networks show that our algorithm can find overlapping communities accurately.     

Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdo?s-Re?nyi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.     

Communities recognition in the Chesapeake Bay ecosystem by dynamical clustering algorithms based on different oscillators systems

We have recently introduced [Phys. Rev. E 75, 045102(R) (2007); AIP Conference Proceedings 965, 2007, p. 323] an efficient method for the detection and identification of modules in complex networks, based on the de-synchronization properties (dynamical clustering) of phase oscillators. In this paper we apply the dynamical clustering tecnique to the identification of communities of marine organisms living in the Chesapeake Bay food web. We show that our algorithm is able to perform a very reliable classification of the real communities existing in this ecosystem by using different kinds of dynamical oscillators. We compare also our results with those of other methods for the detection of community structures in complex networks.     

Optimal symmetric networks in terms of minimizing average shortest path length and their sub-optimal growth model

Homogeneous entangled networks characterized by small world, large girths, and no community structure have attracted much attention due to some of their favorable performances. However, the optimization algorithm proposed by Donetti et al. is very time-consuming and will lose its efficiency when the size of the target network becomes large. In this paper, an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length. It is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds, which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks. In order to overcome the time-consuming obstacle of the optimization algorithms proposed by us and Donetti et al., a growth model is proposed to get large scale sub-optimal symmetric networks. Numerical simulations show that the symmetric networks derived by our growth model will have small-world property, and besides, these networks will have many other similar favorable performances as entangled networks, e.g., robustness against errors and attacks, very good load balancing ability, and strong synchronizability.