Tolerating the community detection resolution limit with edge weighting

New Algorithms for Counting Temporal Graph Pattern

Temporal networks can describe multiple types of complex systems with temporal information in the real world. As an effective method for analyzing such network, temporal graph pattern (TGP) counting has received extensive attention and has been applied in diverse domains. In this paper, we study the problem of counting the TGP in the temporal network. Then, an exact algorithm is proposed based on the time first search (TFS) algorithm. This algorithm can reduce the intermediate results generated in the graph isomorphism and has high computational efficiency. To further improve the algorithm performance, we design an estimation algorithm by applying the edge sampling strategy to the exact algorithm. Finally, we evaluate the performances of the two algorithms by counting both the symmetric and asymmetric TGP. Extensive experiments on real datasets demonstrated that the exact algorithm is faster than the existing algorithm and the estimation algorithm can greatly reduce the running time while guaranteeing the accuracy.

Multi-scale detection of hierarchical community architecture in structural and functional brain networks

Community detection algorithms have been widely used to study the organization of complex networks like the brain. These techniques provide a partition of brain regions (or nodes) into clusters (or communities), where nodes within a community are densely interconnected with one another. In their simplest application, community detection algorithms are agnostic to the presence of community hierarchies: clusters embedded within clusters of other clusters. To address this limitation, we exercise a multi-scale extension of a common community detection technique, and we apply the tool to synthetic graphs and to graphs derived from human neuroimaging data, including structural and functional imaging data. Our multi-scale community detection algorithm links a graph to copies of itself across neighboring topological scales, thereby becoming sensitive to conserved community organization across levels of the hierarchy. We demonstrate that this method is sensitive to topological inhomogeneities of the graph's hierarchy by providing a local measure of community stability and inter-scale reliability across topological scales. We compare the brain's structural and functional network architectures, and we demonstrate that structural graphs display a more prominent hierarchical community organization than functional graphs. Finally, we build an explicitly multimodal multiplex graph that combines both structural and functional connectivity in a single model, and we identify the topological scales where resting state functional connectivity and underlying structural connectivity show similar versus unique hierarchical community architecture. Together, our results demonstrate the advantages of the multi-scale community detection algorithm in studying hierarchical community structure in brain graphs, and they illustrate its utility in modeling multimodal neuroimaging data.

Global vs local modularity for network community detection

Community structures are ubiquitous in various complex networks, implying that the networks commonly be composed of groups of nodes with more internal links and less external links. As an important topic in network theory, community detection is of importance for understanding the structure and function of the networks. Optimizing statistical measures for community structures is one of most popular strategies for community detection in complex networks. In the paper, by using a type of self-loop rescaling strategy, we introduced a set of global modularity functions and a set of local modularity functions for community detection in networks, which are optimized by a kind of the self-consistent method. We carefully compared and analyzed the behaviors of the modularity-based methods in community detection, and confirmed the superiority of the local modularity for detecting community structures on large-size and heterogeneous networks. The local modularity can more quickly eliminate the first-type limit of modularity, and can eliminate or alleviate the second-type limit of modularity in networks, because of the use of the local information in networks. Moreover, we tested the methods in real networks. Finally, we expect the research can provide useful insight into the problem of community detection in complex networks.

Micro-blog user community discovery using generalized SimRank edge weighting method

Community discovery is one of the most popular issues in analyzing and understanding a network. Previous research suggests that the discovery can be enhanced by assigning weights to the edges of the network. This paper proposes a novel edge weighting method, which balances both local and global weighting based on the idea of shared neighbor ranging between users and the interpersonal significance of the social network community. We assume that users belonging to the same community have similar relationship network structures. By controlling the measure of "neighborhood", this method can adequately adapt to real-world networks. Therefore, the famous similarity calculation method-SimRank-can be regarded as a special case of our method. According to the practical significance of social networks, we propose a new evaluation method that uses the communication rate to measure its divided demerit to better express users' interaction relations than the ordinary modularity Q. Furthermore, the fast Newman algorithm is extended to weighted networks. In addition, we use four real networks in the largest Chinese micro-blog website Sina. The results of experiments demonstrate that the proposed method easily meets the balancing requirements and is more robust to different kinds of networks. The experimental results also indicate that the proposed algorithm outperforms several conventional weighting methods.

On the Detection of Overlapped Network Communities via Weight Redistributions

A community is an important attribute of networking, since people who join networks tend to join communities. Community detection is used to identify and understand the structure and organization of real-world networks, thus, it has become a problem of considerable interest. The study of communities is highly related to network partitioning, which is defined as the division of a network into a set of groups of approximately equal sizes with minimum number of edges. Since this is an NP-hard problem, unconventional computation methods have been widely applied. This work addresses the problem of detecting overlapped communities (communities with common nodes) in weighted networks with irregular topologies. These communities are particularly interesting, firstly because they are more realistic, i.e., researchers may belong to more than one research community, and secondly, because they reveal hierarchies of communities: i.e., a medical community is subdivided into groups of certain specialties. Our strategy is based on weight redistribution: each node is examined against all communities and weights are redistributed between the edges. At the end of this process, these weights are compared to the total connectivity of each community, to determine if overlapping exists.

Clustering of 770,000 genomes reveals post-colonial population structure of North America

Despite strides in characterizing human history from genetic polymorphism data, progress in identifying genetic signatures of recent demography has been limited. Here we identify very recent fine-scale population structure in North America from a network of over 500 million genetic (identity-by-descent, IBD) connections among 770,000 genotyped individuals of US origin. We detect densely connected clusters within the network and annotate these clusters using a database of over 20 million genealogical records. Recent population patterns captured by IBD clustering include immigrants such as Scandinavians and French Canadians; groups with continental admixture such as Puerto Ricans; settlers such as the Amish and Appalachians who experienced geographic or cultural isolation; and broad historical trends, including reduced north-south gene flow. Our results yield a detailed historical portrait of North America after European settlement and support substantial genetic heterogeneity in the United States beyond that uncovered by previous studies.

Network clustering and community detection using modulus of families of loops

We study the structure of loops in networks using the notion of modulus of loop families. We introduce an alternate measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link usage optimally. We propose weighting networks using these expected link usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks.

Estimating the resolution limit of the map equation in community detection

A community detection algorithm is considered to have a resolution limit if the scale of the smallest modules that can be resolved depends on the size of the analyzed subnetwork. The resolution limit is known to prevent some community detection algorithms from accurately identifying the modular structure of a network. In fact, any global objective function for measuring the quality of a two-level assignment of nodes into modules must have some sort of resolution limit or an external resolution parameter. However, it is yet unknown how the resolution limit affects the so-called map equation, which is known to be an efficient objective function for community detection. We derive an analytical estimate and conclude that the resolution limit of the map equation is set by the total number of links between modules instead of the total number of links in the full network as for modularity. This mechanism makes the resolution limit much less restrictive for the map equation than for modularity; in practice, it is orders of magnitudes smaller. Furthermore, we argue that the effect of the resolution limit often results from shoehorning multilevel modular structures into two-level descriptions. As we show, the hierarchical map equation effectively eliminates the resolution limit for networks with nested multilevel modular structures.

Algorithm for parametric community detection in networks

Modularity maximization is extensively used to detect communities in complex networks. It has been shown, however, that this method suffers from a resolution limit: Small communities may be undetectable in the presence of larger ones even if they are very dense. To alleviate this defect, various modifications of the modularity function have been proposed as well as multiresolution methods. In this paper we systematically study a simple model (proposed by Pons and Latapy [Theor. Comput. Sci. 412, 892 (2011)] and similar to the parametric model of Reichardt and Bornholdt [Phys. Rev. E 74, 016110 (2006)]) with a single parameter α that balances the fraction of within community edges and the expected fraction of edges according to the configuration model. An exact algorithm is proposed to find optimal solutions for all values of α as well as the corresponding successive intervals of α values for which they are optimal. This algorithm relies upon a routine for exact modularity maximization and is limited to moderate size instances. An agglomerative hierarchical heuristic is therefore proposed to address parametric modularity detection in large networks. At each iteration the smallest value of α for which it is worthwhile to merge two communities of the current partition is found. Then merging is performed and the data are updated accordingly. An implementation is proposed with the same time and space complexity as the well-known Clauset-Newman-Moore (CNM) heuristic [Phys. Rev. E 70, 066111 (2004)]. Experimental results on artificial and real world problems show that (i) communities are detected by both exact and heuristic methods for all values of the parameter α; (ii) the dendrogram summarizing the results of the heuristic method provides a useful tool for substantive analysis, as illustrated particularly on a Les Misérables data set; (iii) the difference between the parametric modularity values given by the exact method and those given by the heuristic is moderate; (iv) the heuristic version of the proposed parametric method, viewed as a modularity maximization tool, gives better results than the CNM heuristic for large instances.

Identifying communities by influence dynamics in social networks

Communities are not static; they evolve, split and merge, appear and disappear, i.e., they are the product of dynamical processes that govern the evolution of a network. A good algorithm for community detection should not only quantify the topology of the network but incorporate the dynamical processes that take place on the network. We present an algorithm for community detection that combines network structure with processes that support the creation and/or evolution of communities. The algorithm does not embrace the universal approach but instead tries to focus on social networks and model dynamic social interactions that occur on those networks. It identifies leaders and communities that form around those leaders. It naturally supports overlapping communities by associating each node with a membership vector that describes the node's involvement in each community. This way, in addition to the overlapping communities, we can identify nodes that are good followers of their leader and also nodes with no clear community involvement that serve as proxies between several communities and that are equally as important. We run the algorithm for several real social networks which we believe represent a good fraction of the wide body of social networks and discuss the results, including other possible applications.

Spike-train communities: finding groups of similar spike trains

Identifying similar spike-train patterns is a key element in understanding neural coding and computation. For single neurons, similar spike patterns evoked by stimuli are evidence of common coding. Across multiple neurons, similar spike trains indicate potential cell assemblies. As recording technology advances, so does the urgent need for grouping methods to make sense of large-scale datasets of spike trains. Existing methods require specifying the number of groups in advance, limiting their use in exploratory analyses. I derive a new method from network theory that solves this key difficulty: it self-determines the maximum number of groups in any set of spike trains, and groups them to maximize intragroup similarity. This method brings us revealing new insights into the encoding of aversive stimuli by dopaminergic neurons, and the organization of spontaneous neural activity in cortex. I show that the characteristic pause response of a rat's dopaminergic neuron depends on the state of the superior colliculus: when it is inactive, aversive stimuli invoke a single pattern of dopaminergic neuron spiking; when active, multiple patterns occur, yet the spike timing in each is reliable. In spontaneous multineuron activity from the cortex of anesthetized cat, I show the existence of neural ensembles that evolve in membership and characteristic timescale of organization during global slow oscillations. I validate these findings by showing that the method both is remarkably reliable at detecting known groups and can detect large-scale organization of dynamics in a model of the striatum.

Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics

Background

Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult.

Methodology/Principal Findings

Here we introduce the novel concept of ModuLand, an integrative method family determining overlapping network modules as hills of an influence function-based, centrality-type community landscape, and including several widely used modularization methods as special cases. As various adaptations of the method family, we developed several algorithms, which provide an efficient analysis of weighted and directed networks, and (1) determine pervasively overlapping modules with high resolution; (2) uncover a detailed hierarchical network structure allowing an efficient, zoom-in analysis of large networks; (3) allow the determination of key network nodes and (4) help to predict network dynamics.

Conclusions/Significance

The concept opens a wide range of possibilities to develop new approaches and applications including network routing, classification, comparison and prediction.