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.     

Modeling network growth with assortative mixing

We propose a model of an underlying mechanism responsible for the formation of assortative mixing in networks between “similar” nodes or vertices based on generic vertex properties. Existing models focus on a particular type of assortative mixing, such as mixing by vertex degree, or present methods of generating a network with certain properties, rather than modeling a mechanism driving assortative mixing during network growth. The motivation is to model assortative mixing by non-topological vertex properties, and the influence of these non-topological properties on network topology. The model is studied in detail for discrete and hierarchical vertex properties, and we use simulations to study the topology of resulting networks. We show that assortative mixing by generic properties directly drives the formation of community structure beyond a threshold assortativity of r ∼0.5, which in turn influences other topological properties. This direct relationship is demonstrated by introducing a new measure to characterise the correlation between assortative mixing and community structure in a network. Additionally, we introduce a novel type of assortative mixing in systems with hierarchical vertex properties, from which a hierarchical community structure is found to result. Electronic supplementary material Supplementary Online Material     

A class of improved algorithms for detecting communities in complex networks

Detecting communities in complex networks is of considerable importance for understanding both the structure and function of the networks. Here, we propose a class of improved algorithms for community detection, by combining the betweenness algorithm of Girvan and Newman with the edge weight defined by the edge-clustering coefficient. The improved algorithms are tested on some artificial and real-world networks, and the results show that they can detect communities of networks more effectively in both unweighted and weighted cases. In addition, the technique for improving the betweenness algorithm in this paper, thanks to its compatibility, can directly be applied to various detection algorithms.     

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.     

Finding community structures in complex networks using mixed integer optimisation

The detection of community structure has been used to reveal the relationships between individual objects and their groupings in networks. This paper presents a mathematical programming approach to identify the optimal community structures in complex networks based on the maximisation of a network modularity metric for partitioning a network into modules. The overall problem is formulated as a mixed integer quadratic programming (MIQP) model, which can then be solved to global optimality using standard optimisation software. The solution procedure is further enhanced by developing special symmetry-breaking constraints to eliminate equivalent solutions. It is shown that additional features such as minimum/maximum module size and balancing among modules can easily be incorporated in the model. The applicability of the proposed optimisation-based approach is demonstrated by four examples. Comparative results with other approaches from the literature show that the proposed methodology has superior performance while global optimum is guaranteed.     

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.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

Detection of node group membership in networks with group overlap

Most networks found in social and biochemical systems have modular structures. An important question prompted by the modularity of these networks is whether nodes can be said to belong to a single group. If they cannot, we would need to consider the role of “overlapping communities.” Despite some efforts in this direction, the problem of detecting overlapping groups remains unsolved because there is neither a formal definition of overlapping community, nor an ensemble of networks with which to test the performance of group detection algorithms when nodes can belong to more than one group. Here, we introduce an ensemble of networks with overlapping groups. We then apply three group identification methods – modularity maximization, k-clique percolation, and modularity-landscape surveying – to these networks. We find that the modularity-landscape surveying method is the only one able to detect heterogeneities in node memberships, and that those heterogeneities are only detectable when the overlap is small. Surprisingly, we find that the k-clique percolation method is unable to detect node membership for the overlapping case.     

Epidemic threshold and phase transition in scale-free networks with asymmetric infection

We study the SIS epidemic dynamics on scale-freeweighted networks with asymmetric infection, by both analysis andnumerical simulations, with focus on the epidemic threshold aswell as critical behaviors. It is demonstrated that the asymmetryof infection plays an important role: we could redistribute theasymmetry to balance the degree heterogeneity of the network andthen to restore the epidemic threshold to a fnite value. On theother hand, we show that the absence of the epidemic threshold isnot so bad as commented previously since the prevalence grows veryslowly in this case and one could only protect a few vertices toprevent the diseases propagation.     

Opinion Dynamics on Complex Networks with Communities

The Ising or Potts models of ferromagnetism have been widely used to describe locally interacting social or economic systems. We consider a related model, introduced by Sznajd to describe the evolution of consensus in the scale-free networks with the tunable strength （noted by Q） of community structure. In the Sznajd model, the opinion or state of any spins can only be changed by the influence of neighbouring pairs of similar connection spins. Such pairs can polarize their neighbours. Using asynchronous updating, it is found that the smaller the community strength Q, the larger the slope of the exponential relaxation time distribution. Then the effect of the initial upspin concentration p as a function of the final all up probability E is investigated by taking different initialization strategies, the random node-chosen initialization strategy has no difference under different community strengths, while the strategies of community node-chosen initialization and hub node-chosen initialization are different in fina/probability under different Q, and the latter one is more effective in reaching final state.     

Decline of minorities in stubborn societies

Opinion compromise models can give insight into how groups of individuals may either come to form consensus or clusters of opinion groups, corresponding to parties. We consider models where randomly selected individuals interact pairwise. If the opinions of the interacting agents are not within a certain confidence threshold, the agents retain their own point of view. Otherwise, they constructively dialogue and smooth their opinions. Persuasible agents are inclined to compromise with interacting individuals. Stubborn individuals slightly modify their opinion during the interaction. Collective states for persuasible societies include extremist minorities, which instead decline in stubborn societies. We derive a mean field approximation for the compromise model in stubborn populations. Bifurcation and clustering analysis of this model compares favorably with Monte Carlo analysis found in the literature.     

Limitation of multi-resolution methods in community detection

Community detection is of considerable interest for analyzing the structure and function of complex networks. Recently, a type of multi-resolution methods in community detection was introduced, which can adjust the resolution of modularity by modifying the modularity function with tunable resolution parameters, such as those proposed by Arenas, Fernández and Gómez and by Reichardt and Bornholdt. In this paper, we show that these methods still have the intrinsic limitation–large communities may have been split before small communities become visible–because it is at the cost of the community stability that the enhancement of the modularity resolution is obtained. The theoretical results indicated that the limitation depends on the degree of interconnectedness of small communities and the difference between the sizes of small communities and of large communities, while independent of the size of the whole network. These findings have been confirmed in several example networks, where communities even are full-completed sub-graphs.     

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.     

Universal power laws in the threshold network model: A theoretical analysis based on extreme value theory

We theoretically and numerically investigated the threshold network model with a generic weight function where there were a large number of nodes and a high threshold. Our analysis was based on extreme value theory, which gave us a theoretical understanding of the distribution of independent and identically distributed random variables within a sufficiently high range. Specifically, the distribution could be generally expressed by a generalized Pareto distribution, which enabled us to formulate the generic weight distribution function. By using the theorem, we obtained the exact expressions of degree distribution and clustering coefficient which behaved as universal power laws within certain ranges of degrees. We also compared the theoretical predictions with numerical results and found that they were extremely consistent.     

Developmental time windows for spatial growth generate multiple-cluster small-world networks

Many networks extent in space, may it be metric (e.g. geographic) or non-metric (ordinal). Spatial network growth, which depends on the distance between nodes, can generate a wide range of topologies from small-world to linear scale-free networks. However, networks often lacked multiple clusters or communities. Multiple clusters can be generated, however, if there are time windows during development. Time windows ensure that regions of the network develop connections at different points in time. This novel approach could generate small-world but not scale-free networks. The resulting topology depended critically on the overlap of time windows as well as on the position of pioneer nodes.     

Connectivity degrees in the threshold preferential attachment model

In this paper we present a study of the connectivity degrees of the threshold preferential attachment model, a generalization of the Barabási-Albert model to heterogeneous complex networks. The threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states. We numerically analyze the connectivity degrees of the model, studying the influence of the main parameters on the distribution of connectivity degrees and its statistics, the average degree and highest degree of the network. We show that such statistics exhibit markedly different behaviors in the dependence on the model parameters, particularly as regards the interaction threshold. Nevertheless, we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.     

The backbone of a city

Recent studies have revealed the importance of centrality measures to analyze various spatial factors affecting human life in cities. Here we show how it is possible to extract the backbone of a city by deriving spanning trees based on edge betweenness and edge information. By using as sample cases the cities of Bologna and San Francisco, we show how the obtained trees are radically different from those based on edge lengths, and allow an extended comprehension of the “skeleton” of most important routes that so much affects pedestrian/vehicular flows, retail commerce vitality, land-use separation, urban crime and collective dynamical behaviours.     

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.