Synchronization and modularity in complex networks

We investigate the connection between the dynamics of synchronization and the modularity on complex networks. Simulating the Kuramoto's model in complex networks we determine patterns of meta-stability and calculate the modularity of the partition these patterns provide. The results indicate that the more stable the patterns are, the larger tends to be the modularity of the partition defined by them. This correlation works pretty well in homogeneous networks (all nodes have similar connectivity) but fails when networks contain hubs, mainly because the modularity is never improved where isolated nodes appear, whereas in the synchronization process the characteristic of hubs is to have a large stability when forming its own community.     

Graph spectra and the detectability of community structure in networks

We study networks that display community structure--groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods, we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.     

蛋白质相互作用网络特征的理论再现

无标度性、小世界性、功能模块结构及度负关联性是大量生物网络共同的特征. 为了理解生物网络无标度性、小世界性和度负关联性的形成机制, 研究者已经提出了各种各样基于复制和变异的网络增长模型. 在本文中,我们从生物学的角度通过引入偏爱小复制原则及变异和非均匀的异源二聚作用构建了一个简单的蛋白质相互作用网络演化模型.数值模拟结果表明,该演化模型几乎可以再现现在实测结果所公认的蛋白质相互作用网络的性质：无标度性、小世界性、度负关联性和功能模块结构. 我们的演化模型对理解蛋白质相互作用网络演化过程中的可能机制提供了一定的帮助. 关键词： 蛋白质相互作用网络 偏爱小 非均匀的异源二聚作用 功能模块结构     

Modularity density of network community divisions

The problem of dividing a network into communities is extremely complex and grows very rapidly with the number of nodes and edges that are involved. In order to develop good algorithms to identify optimal community divisions it is extremely beneficial to identify properties that are similar for most networks. We introduce the concept of modularity density, the distribution of modularity values as a function of the number of communities, and find strong indications that the general features of this modularity density are quite similar for different networks. The region of high modularity generally has very low probability density and occurs where the number of communities is small. The properties and shape of the modularity density may give valuable information and aid in the search for efficient algorithms to find community divisions with high modularities.     

Community structure and modularity in networks of correlated brain activity

Functional connectivity patterns derived from neuroimaging data may be represented as graphs or networks, with individual image voxels or anatomically-defined structures representing the nodes, and a measure of correlation between the responses in each pair of nodes determining the edges. This explicit network representation allows network-analysis approaches to be applied to the characterization of functional connections within the brain. Much recent research in complex networks has focused on methods to identify community structure, i.e. cohesive clusters of strongly interconnected nodes. One class of such algorithms determines a partition of a network into 'sub-networks' based on the optimization of a modularity parameter, thus also providing a measure of the degree of segregation versus integration in the full network. Here, we demonstrate that a community structure algorithm based on the maximization of modularity, applied to a functional connectivity network calculated from the responses to acute fluoxetine challenge in the rat, can identify communities whose distributions correspond to anatomically meaningful structures and include compelling functional subdivisions in the brain. We also discuss the biological interpretation of the modularity parameter in terms of segregation and integration of brain function.     

Maximum modular graphs

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.     

Deterministic modularity optimization

We study community structure of networks. We have developed a scheme for maximizing the modularity Q [Newman and Girvan, Phys. Rev. E 69, 026113 (2004)] based on mean field methods. Further, we have defined a simple family of random networks with community structure; we understand the behavior of these networks analytically. Using these networks, we show how the mean field methods display better performance than previously known deterministic methods for optimization of Q.     

Community structure in directed networks

We consider the problem of finding communities or modules in directed networks. In the past, the most common approach to this problem has been to ignore edge direction and apply methods developed for community discovery in undirected networks, but this approach discards potentially useful information contained in the edge directions. Here we show how the widely used community finding technique of modularity maximization can be generalized in a principled fashion to incorporate information contained in edge directions. We describe an explicit algorithm based on spectral optimization of the modularity and show that it gives demonstrably better results than previous methods on a variety of test networks, both real and computer generated.     

Modular networks with hierarchical organization: The dynamical implications of complex structure

Several networks occurring in real life have modular structures that are arranged in a hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.      

A smart local moving algorithm for large-scale modularity-based community detection

We introduce a new algorithm for modularity-based community detection in large networks. The algorithm, which we refer to as a smart local moving algorithm, takes advantage of a well-known local moving heuristic that is also used by other algorithms. Compared with these other algorithms, our proposed algorithm uses the local moving heuristic in a more sophisticated way. Based on an analysis of a diverse set of networks, we show that our smart local moving algorithm identifies community structures with higher modularity values than other algorithms for large-scale modularity optimization, among which the popular “Louvain algorithm”. The computational efficiency of our algorithm makes it possible to perform community detection in networks with tens of millions of nodes and hundreds of millions of edges. Our smart local moving algorithm also performs well in small and medium-sized networks. In short computing times, it identifies community structures with modularity values equally high as, or almost as high as, the highest values reported in the literature, and sometimes even higher than the highest values found in the literature.     

Detecting hierarchical and overlapping network communities using locally optimal modularity changes

Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the preferred edge that edge for which the modularity change is maximal. When an edge is preferred by both vertices that it links, it appears to be the optimal choice from the local viewpoint. We use the locally optimal edges to define the algorithm: simultaneously merge all pairs of communities that are connected by locally optimal edges that would increase the modularity, redetermining the locally optimal edges after each step and continuing so long as the modularity can be further increased. We apply the algorithm to model and empirical networks, demonstrating that it can efficiently produce high-quality community solutions. We relate the performance and implementation details to the structure of the resulting community hierarchies. We additionally consider a complementary local clustering algorithm, describing how to identify overlapping communities based on the local optimality condition.     

复杂网络中社团结构发现的多分辨率密度模块度

现实中的许多复杂网络呈现出明显的模块性或社团性.模块度是衡量社团结构划分优劣的效益函数, 它也通常被用作社团结构探测的目标函数,但最为广泛使用的Newman-Girvan模块度却存在着分辨率限制问题,多分辨率模块度也不能克服误合并社团和误分裂社团同时存在的缺陷. 本文在网络密度的基础上提出了多分辨率的密度模块度函数, 通过实验和分析证实了该函数能够使社团结构的误划分率显著降低, 而且能够体现出网络社团结构是一个有机整体,不是各个社团的简单相加.     

Modularity-like objective function in annotated networks

We ascertain the modularity-like objective function whose optimization is equivalent to the maximum likelihood in annotated networks. We demonstrate that the modularity-like objective function is a linear combination of modularity and conditional entropy. In contrast with statistical inference methods, in our method, the influence of the metadata is adjustable; when its influence is strong enough, the metadata can be recovered. Conversely, when it is weak, the detection may correspond to another partition. Between the two, there is a transition. This paper provides a concept for expanding the scope of modularity methods.     

Directed LPA: Propagating labels in directed networks

Ignoring edge directionality and considering the graph as undirected is a common approach to detect communities in directed networks. However, it's not a meaningful way due to the loss of information captured by the edge property. Even if Leicht and Newman extended the original modularity to a directed version to address this issue, the problem of distinguishing the directionality of the edges still exists in maximizing modularity algorithms. To this direction, we extend one of the most famous scalable algorithms, namely label propagation algorithm (LPA), to a directed case, which can recognize the flow direction among nodes. To explore what properties the directed modularity should have, we also use another directed modularity, called LinkRank, and provide an empirical study. The experimental results on both real and synthetic networks demonstrate that the proposed directed algorithms can not only make use of the edge directionality but also keep the same time complexity as LPA.     

小世界网络与无标度网络的社区结构研究

模块性(modularity)是度量网络社区结构(community structure)的主要参数.探讨了Watts和Strogatz的小世界网络(简称W-S模型)以及Barabàsi 等的B-A无标度网络(简称B-A模型)两类典型复杂网络模块性特点.结果显示，网络模块性受到网络连接稀疏的影响，W-S模型具有显著的社区结构，而B-A模型的社区结构特征不明显.因此，应用中应该分别讨论网络的小世界现象和无标度特性.社区结构不同于小世界现象和无标度特性，并可以利用模块性区别网络类型，因此网络复杂性指标应该包括 关键词： 模块性 社区结构 小世界网络 无标度网络     

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.     

Comparison of methods for the detection of node group membership in bipartite networks

Most real-world networks considered in the literature have a modular structure. Analysis of these real-world networks often are performed under the assumption that there is only one type of node. However, social and biochemical systems are often bipartite networks, meaning that there are two exclusive sets of nodes, and that edges run exclusively between nodes belonging to different sets. Here we address the issue of module detection in bipartite networks by comparing the performance of two classes of group identification methods – modularity maximization and clique percolation – on an ensemble of modular random bipartite networks. We find that the modularity maximization methods are able to reliably detect the modular bipartite structure, and that, under some conditions, the simulated annealing method outperforms the spectral decomposition method. We also find that the clique percolation methods are not capable of reliably detecting the modular bipartite structure of the bipartite model networks considered.