Shihua Zhang, Junfei Zhao, Xiang-Sun Zhang
In this report, we introduce the concept of co-community structure in time-varying networks. We propose a novel optimization algorithm to rapidly detect co-community structure in these networks. Both theoretical and numerical results show that the proposed method not only can resolve detailed co-communities, but also can effectively identify the dynamical phenomena in these networks.
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Identifying community structure of a complex network provides insight to the interdependence between the network topology and emergent collective behaviors of networks, while detecting such invariant communities in a time-varying network is more challenging. In this paper, we define the temporal stable community and newly propose the concept of dynamic modularity to evaluate the stable community structures in time-varying networks, which is robust against small changes as verified by several empirical time-varying network datasets. Besides, using the volatility features of temporal stable comm…
Time-varying community structures widely exist in various real-world networks. However, the spreading dynamics on this kind of network has not been fully studied. To this end, we systematically study the effects of time-varying community structures on social contagions. We first propose a non-Markovian social contagion model on time-varying community networks based on the activity driven network model, in which an individual adopts a behavior if and only if the accumulated behavioral information it has ever received reaches a threshold. Then, we develop a mean-field theory to describe the prop…
One of the most widely studied problem in mining and analysis of complex networks is the detection of community structures. The problem has been extensively studied by researchers due to its high utility and numerous applications in various domains. Many algorithmic solutions have been proposed for the community detection problem but the quest to find the best algorithm is still on. More often than not, researchers focus on developing fast and accurate algorithms that can be generically applied to networks from a variety of domains without taking into consideration the structural and topologic…
Much of the community detection literature studies structural communities, communities defined solely by the connectivity patterns of the network. Often, networks contain additional metadata which can inform community detection such as the grade and gender of students in a high school social network. In this work, we introduce a tuning parameter to the content map equation that allows users of the Infomap community detection algorithm to control the metadata's relative importance for identifying network structure. On synthetic networks, we show that our algorithm can overcome the structural de…
Community detection of network flows conventionally assumes one-step dynamics on the links. For sparse networks and interest in large-scale structures, longer timescales may be more appropriate. Oppositely, for large networks and interest in small-scale structures, shorter timescales may be better. However, current methods for analyzing networks at different timescales require expensive and often infeasible network reconstructions. To overcome this problem, we introduce a method that takes advantage of the inner-workings of the map equation and evades the reconstruction step. This makes it pos…
Social interactions vary in time and appear to be driven by intrinsic mechanisms, which in turn shape the emerging structure of the social network. Large-scale empirical observations of social interaction structure have become possible only recently, and modelling their dynamics is an actual challenge. Here we propose a temporal network model which builds on the framework of activity-driven time-varying networks with memory. The model also integrates key mechanisms that drive the formation of social ties - social reinforcement, focal closure and cyclic closure, which have been shown to give ri…
Community detection on directed networks with missing edges
本文扩展 Flow Stability 框架，利用节点出度的已知不确定性水平，在含缺失链接的加权有向网络中提升社区检测的鲁棒性，并通过合成网络与 Telegram 真实网络验证其可靠性。
Clustering, assortativity, and communities are key features of complex networks. We probe dependencies between these attributes and find that ensembles with strong clustering display both high assortativity by degree and prominent community structure, while ensembles with high assortativity are much less biased towards clustering or community structure. Further, clustered networks can amplify small homophilic bias for trait assortativity. This marked asymmetry suggests that transitivity, rather than homophily, drives the standard nonsocial/social network dichotomy.
Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the Wigner semicircle distribution and phase transitions in the spectra of community structured networks. In this paper we generalize both calculations, giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution. In general the spectrum has two parts, a continuous spectral band, which can dep…
Community structure can naturally emerge in paths to synchronization, and scratching it from the paths is a tough issue that accounts for the diverse dynamics of synchronization. In this paper, with assumption that the synchronization on complex networks is made up of local and collective processes, we proposed a scheme to lock the local synchronization (phase locking) at a stable state meanwhile suppress the collective synchronization based on Kuramoto model. Through this scheme, the network dynamics only contains the local synchronization, which suggests that the nodes in the same community …
Community detecting is one of the main approaches to understanding networks \cite{For2010}. However it has been a longstanding challenge to give a definition for community structures of networks. Here we found that community structures are definable in networks, and are universal in real world. We proposed the notions of entropy- and conductance-community structure ratios. It was shown that the definitions of the modularity proposed in \cite{NG2004}, and our entropy- and conductance-community structures are equivalent in defining community structures of networks, that randomness in the ER mode…
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 maximu…
We found that neither randomness in the ER model nor the preferential attachment in the PA model is the mechanism of community structures of networks, that community structures are universal in real networks, that community structures are definable in networks, that communities are interpretable in networks, and that homophyly is the mechanism of community structures and a structural theory of networks. We proposed the notions of entropy- and conductance-community structures. It was shown that the two definitions of the entropy- and conductance-community structures and the notion of modularity…
In this work we study diffusion in networks with community structure. We first replicate and extend work on networks with non-overlapping community structure. We then study diffusion on network models that have overlapping community structure. We study contagions in the standard SIR model, and complex contagions thought to be better models of some social diffusion processes. Finally, we investigate diffusion on empirical networks with known overlapping community structure, by analysing the structure of such networks, and by simulating contagion on them. We find that simple and complex contagio…
The international trade network is a complex system where multiple trade blocs with varying sizes coexist and overlap with each other. However, the resulting structures of community detection in trade networks are often inconsistent and fails to capture the complex landscape of international trade. To address these problems, we propose a multiresolution framework that aggregates all the configuration information from a range of resolutions. This allows us to consider trade communities of different sizes and illuminate the underlying hierarchical structure of trade networks and its constituting…
Bipartite networks composed of dichotomous node sets are ubiquitous in nature and society. Partly for simplicity's sake, many studies have focused on their projection onto their unipartite versions where one only needs to care about a single type of node. When it comes to mesoscale structures such as communities, however, properly incorporating a priori structural restrictions such as bipartivity is ever more important. In this paper, as a case study, we take the community structure of bipartite networks in various scales to examine the amount of information of bipartivity encoded in the commu…
Spectral analysis has been successfully applied at the detection of community structure of networks, respectively being based on the adjacency matrix, the standard Laplacian matrix, the normalized Laplacian matrix, the modularity matrix, the correlation matrix and several other variants of these matrices. However, the comparison between these spectral methods is less reported. More importantly, it is still unclear which matrix is more appropriate for the detection of community structure. This paper answers the question through evaluating the effectiveness of these five matrices against the ben…
In recent years, community structure has emerged as a key component of complex network analysis. As more data has been collected, researchers have begun investigating changing community structure across multiple networks. Several methods exist to analyze changing communities, but most of these are limited to evolution of a single network over time. In addition, most of the existing methods are more concerned with change at the community level than at the level of the individual node. In this paper, we introduce scaled inclusivity, which is a method to quantify the change in community structure…
How to determine the community structure of complex networks is an open question. It is critical to establish the best strategies for community detection in networks of unknown structure. Here, using standard synthetic benchmarks, we show that none of the algorithms hitherto developed for community structure characterization perform optimally. Significantly, evaluating the results according to their modularity, the most popular measure of the quality of a partition, systematically provides mistaken solutions. However, a novel quality function, called Surprise, can be used to elucidate which is…
The past decade has seen tremendous growth in the field of Complex Social Networks. Several network generation models have been extensively studied to develop an understanding of how real world networks evolve over time. Two important applications of these models are to study the evolution dynamics and processes that shape a network, and to generate benchmark networks with known community structures. Research has been conducted in both these directions, relatively independent of the other. This creates a disjunct between real world networks and the networks generated as benchmarks to study com…
We introduce a new conception of community structure, which we refer to as hidden community structure. Hidden community structure refers to a specific type of overlapping community structure, in which the detection of weak, but meaningful, communities is hindered by the presence of stronger communities. We present Hidden Community Detection HICODE, an algorithm template that identifies both the strong, dominant community structure as well as the weaker, hidden community structure in networks. HICODE begins by first applying an existing community detection algorithm to a network, and then remov…
While there has been a plethora of approaches for detecting disjoint communities from real-world complex networks, some methods for detecting overlapping community structures have also been recently proposed. In this work, we argue that, instead of developing separate approaches for detecting overlapping communities, a promising alternative is to infer the overlapping communities from multiple disjoint community structures. We propose an ensemble-based approach, called EnCoD, that leverages the solutions produced by various disjoint community detection algorithms to discover the overlapping co…
When an epidemic spreads in a population, individuals may adaptively change the structure of their social contact network to reduce risk of infection. Here we study the spread of an epidemic on an adaptive network with community structure. We model the effect of two communities with different average degrees. The disease model is susceptible-infected-susceptible (SIS), and adaptation is rewiring of links between susceptibles and infectives. The bifurcation structure is obtained, and a mean field model is developed that accurately predicts the steady state behavior of the system. We show that a…
Community structures detection in signed network is very important for understanding not only the topology structures of signed networks, but also the functions of them, such as information diffusion, epidemic spreading, etc. In this paper, we develop a joint nonnegative matrix factorization model to detect community structures. In addition, we propose modified partition density to evaluate the quality of community structures. We use it to determine the appropriate number of communities. The effectiveness of our approach is demonstrated based on both synthetic and real-world networks.
Community detection and edge prediction are both forms of link mining: they are concerned with discovering the relations between vertices in networks. Some of the vertex similarity measures used in edge prediction are closely related to the concept of community structure. We use this insight to propose a novel method for improving existing community detection algorithms by using a simple vertex similarity measure. We show that this new strategy can be more effective in detecting communities than the basic community detection algorithms.
Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both \textit{enforce} as well as \textit{inhibit} diffusion processes. Our…
Core-periphery structure and community structure are two typical meso-scale structures in complex networks. Though the community detection has been extensively investigated from different perspectives, the definition and the detection of core-periphery structure have not received much attention. Furthermore, the detection problems of the core-periphery and community structure were separately investigated. In this paper, we develop a unified framework to simultaneously detect core-periphery structure and community structure in complex networks. Moreover, there are several extra advantages of ou…
One of the most prominent properties in real-world networks is the presence of a community structure, i.e. dense and loosely interconnected groups of nodes called communities. In an attempt to better understand this concept, we study the relationship between the strength of the community structure and the network transitivity (or clustering coefficient). Although intuitively appealing, this analysis was not performed before. We adopt an approach based on random models to empirically study how one property varies depending on the other. It turns out the transitivity increases with the community…
Studying temporal dynamics of topics in social media is very useful to understand online user behaviors. Most of the existing work on this subject usually monitors the global trends, ignoring variation among communities. Since users from different communities tend to have varying tastes and interests, capturing community-level temporal change can improve the understanding and management of social content. Additionally, it can further facilitate the applications such as community discovery, temporal prediction and online marketing. However, this kind of extraction becomes challenging due to the…
The relationship of friends in social networks can be strong or weak. Some research works have shown that a close relationship between friends conducts good community structure. Based on this result, we propose an effective method in detecting community structure in social networks based on the closeness of relations among neighbors. This method calculates the gravity between each neighbor node to core nodes, then makes judgement if the node should be classified in the community or not, and finally form the process of community detection. The experimental results show that the proposed method …
We present a new layout algorithm for complex networks that combines a multi-scale approach for community detection with a standard force-directed design. Since community detection is computationally cheap, we can exploit the multi-scale approach to generate network configurations with close-to-minimal energy very fast. As a further asset, we can use the knowledge of the community structure to facilitate the interpretation of large networks, for example the network defined by protein-protein interactions.
The increasing availability of temporal network data is calling for more research on extracting and characterizing mesoscopic structures in temporal networks and on relating such structure to specific functions or properties of the system. An outstanding challenge is the extension of the results achieved for static networks to time-varying networks, where the topological structure of the system and the temporal activity patterns of its components are intertwined. Here we investigate the use of a latent factor decomposition technique, non-negative tensor factorization, to extract the community-…
We study the community structure of the multi-network of commodity-specific trade relations among world countries over the 1992-2003 period. We compare structures across commodities and time by means of the normalized mutual information index (NMI). We also compare them with exogenous community structures induced by geographical distances and regional trade agreements. We find that commodity-specific community structures are very heterogeneous and much more fragmented than that characterizing the aggregate ITN. This shows that the aggregate properties of the ITN may result (and be very differe…
Although community structure is ubiquitous in complex networks, few works exploit this topological property to control epidemics. In this work, devoted to networks with non-overlapping community structure (i.e, a node belongs to a single community), we propose and investigate three deterministic immunization strategies. In order to characterize the influence of a node, various pieces of information are used such as the number of communities that the node can reach in one hop, the nature of the links (intra community links, inter community links), the size of the communities, and the interconne…
Network communities represent mesoscopic structure for understanding the organization of real-world networks, where nodes often belong to multiple communities and form overlapping community structure in the network. Due to non-triviality in finding the exact boundary of such overlapping communities, this problem has become challenging, and therefore huge effort has been devoted to detect overlapping communities from the network. In this paper, we present PVOC (Permanence based Vertex-replication algorithm for Overlapping Community detection), a two-stage framework to detect overlapping communi…
The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure. We propose a suitable method for perturbing networks and a measure of the resulting change in comm…
Most real-world networks exhibit community structure, a phenomenon characterized by existence of node clusters whose intra-edge connectivity is stronger than edge connectivities between nodes belonging to different clusters. In addition to facilitating a better understanding of network behavior, community detection finds many practical applications in diverse settings. Communities in online social networks are indicative of shared functional roles, or affiliation to a common socio-economic status, the knowledge of which is vital for targeted advertisement. In buyer-seller networks, community d…
Recent years have seen a growing interest in the modeling and simulation of social networks to understand several social phenomena. Two important classes of networks, small world and scale free networks have gained a lot of research interest. Another important characteristic of social networks is the presence of community structures. Many social processes such as information diffusion and disease epidemics depend on the presence of community structures making it an important property for network generation models to be incorporated. In this paper, we present a tunable and growing network gener…
Community detection emerges as an important task in the discovery of network mesoscopic structures. However, the concept of a "good" community is very context-dependent and it is relatively complicated to deduce community characteristics using available community detection techniques. In reality, the existence of a gap between structural goodness quality metrics and expected topological patterns creates a confusion in evaluating community structures. In this paper, we introduce an empirical multivariate analysis of different structural goodness properties in order to characterize several detec…
Link streams model interactions over time in a wide range of fields. Under this model, the challenge is to mine efficiently both temporal and topological structures. Community detection and change point detection are one of the most powerful tools to analyze such evolving interactions. In this paper, we build on both to detect stable community structures by identifying change points within meaningful communities. Unlike existing dynamic community detection algorithms, the proposed method is able to discover stable communities efficiently at multiple temporal scales. We test the effectiveness o…
Though much work has been done on ensemble clustering in data mining, the application of ensemble methods to community detection in networks is in its infancy. In this paper, we propose two ensemble methods: ENDISCO and MEDOC. ENDISCO performs disjoint community detection. In contrast, MEDOC performs disjoint, overlapping, and fuzzy community detection and represents the first ever ensemble method for fuzzy and overlapping community detection. We run extensive experiments with both algorithms against both synthetic and several real-world datasets for which community structures are known. We sh…
Various models have been recently proposed to reflect and predict different properties of complex networks. However, the community structure, which is one of the most important properties, is not well studied and modeled. In this paper, we suggest a principle called "preferential placement", which allows to model a realistic clustering structure. We provide an extensive empirical analysis of the obtained structure as well as some theoretical results.
In this paper we compare the social network structure of people talking about Crohn's disease, Cystic Fibrosis, and Type 1 diabetes on Facebook and Twitter. We find that the Crohn's community's contributors are most emotional on Facebook and Twitter and most negative on Twitter, while the T1D community's communication network structure is most cohesive.
Community structure is of paramount importance for the understanding of complex networks. Consequently, there is a tremendous effort in order to develop efficient community detection algorithms. Unfortunately, the issue of a fair assessment of these algorithms is a thriving open question. If the ground-truth community structure is available, various clustering-based metrics are used in order to compare it versus the one discovered by these algorithms. However, these metrics defined at the node level are fairly insensitive to the variation of the overall community structure. To overcome these l…
Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly distort the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algori…
This study relates the local property of node dominance to local and global properties of a network. Iterative removal of dominated nodes yields a distributed algorithm for computing a core-periphery decomposition of a social network, where nodes in the network core are seen to be essential in terms of network flow and global structure. Additionally, the connected components in the periphery give information about the community structure of the network, aiding in community detection. A number of explicit results are derived, relating the core and periphery to network flow, community structure …
Most real-world social networks are inherently dynamic, composed of communities that are constantly changing in membership. To track these evolving communities, we need dynamic community detection techniques. This article evaluates the performance of a set of game theoretic approaches for identifying communities in dynamic networks. Our method, D-GT (Dynamic Game Theoretic community detection), models each network node as a rational agent who periodically plays a community membership game with its neighbors. During game play, nodes seek to maximize their local utility by joining or leaving the…
Community structure plays a significant role in the analysis of social networks and similar graphs, yet this structure is little understood and not well captured by most models. We formally define a community to be a subgraph that is internally highly connected and has no deeper substructure. We use tools of combinatorics to show that any such community must contain a dense Erdös-Rényi (ER) subgraph. Based on mathematical arguments, we hypothesize that any graph with a heavy-tailed degree distribution and community structure must contain a scale free collection of dense ER subgraphs. These the…
The analysis of stability of community structure is an important problem for scientists from many fields. Here, we propose a new framework to reveal hidden properties of community structure by quantitatively analyzing the dynamics of Potts model. Specifically we model the Potts procedure of community structure detection by a Markov process, which has a clear mathematical explanation. Critical topological information regarding to multivariate spin configuration could also be inferred from the spectral significance of the Markov process. We test our framework on some example networks and find it…
The goal of community detection algorithms is to identify densely-connected units within large networks. An implicit assumption is that all the constituent nodes belong equally to their associated community. However, some nodes are more important in the community than others. To date, efforts have been primarily driven to identify communities as a whole, rather than understanding to what extent an individual node belongs to its community. Therefore, most metrics for evaluating communities, for example modularity, are global. These metrics produce a score for each community, not for each indivi…
In this paper a simple but efficient real-time detecting algorithm is proposed for tracking community structure of dynamic networks. Community structure is intuitively characterized as divisions of network nodes into subgroups, within which nodes are densely connected while between which they are sparsely connected. To evaluate the quality of community structure of a network, a metric called modularity is proposed and many algorithms are developed on optimizing it. However, most of the modularity based algorithms deal with static networks and cannot be performed frequently, due to their high c…
Community structure is one of the most important properties of networks. Most community algorithms are not suitable for large networks because of their time consuming. In fact there are lots of networks with millons even billons of nodes. In such case, most algorithms running in time O(n2logn) or even larger are not practical. What we need are linear or approximately linear time algorithm. Rising in response to such needs, we propose a quick methods to evaluate community structure in networks and then put forward a local community algorithm with nearly linear time based on random walks. Using …
Constrained clustering has been well-studied in the unsupervised learning society. However, how to encode constraints into community structure detection, within complex networks, remains a challenging problem. In this paper, we propose a semi-supervised learning framework for community structure detection. This framework implicitly encodes the must-link and cannot-link constraints by modifying the adjacency matrix of network, which can also be regarded as de-noising the consensus matrix of community structures. Our proposed method gives consideration to both the topology and the functions (bac…
The discovery of community structure in networks is a problem of considerable interest in recent years. In online social networks, often times, users are simultaneously involved in multiple social media sites, some of which share common social relationships. It is of great interest to uncover a shared community structure across these networks. However, in reality, users typically identify themselves with different usernames across social media sites. This creates a great difficulty in detecting the community structure. In this paper, we explore several approaches for community detection across…
Finding community structure using the ordered random graph model
本文针对经典矩阵排序算法难以揭示网络社区结构的问题，提出基于有序随机图模型最大似然估计的排序算法，实验表明其能更清晰地识别社区结构。
Modularity is widely used to effectively measure the strength of the disjoint community structure found by community detection algorithms. Although several overlapping extensions of modularity were proposed to measure the quality of overlapping community structure, there is lack of systematic comparison of different extensions. To fill this gap, we overview overlapping extensions of modularity to select the best. In addition, we extend the Modularity Density metric to enable its usage for overlapping communities. The experimental results on four real networks using overlapping extensions of mo…
Community structure discovery in complex networks is a quite challenging problem spanning many applications in various disciplines such as biology, social network and physics. Emerging from various approaches numerous algorithms have been proposed to tackle this problem. Nevertheless little attention has been devoted to compare their efficiency on realistic simulated data. To better understand their relative performances, we evaluate systematically eleven algorithms covering the main approaches. The Normalized Mutual Information (NMI) measure is used to assess the quality of the discovered com…
Community detection, which focuses on clustering vertex interactions, plays a significant role in network analysis. However, it also faces numerous challenges like missing data and adversarial attack. How to further improve the performance and robustness of community detection for real-world networks has raised great concerns. In this paper, we explore robust community detection by enhancing network structure, with two generic algorithms presented: one is named robust community detection via genetic algorithm (RobustECD-GA), in which the modularity and the number of clusters are combined in a …
Many networks display community structure which identifies groups of nodes within which connections are denser than between them. Detecting and characterizing such community structure, which is known as community detection, is one of the fundamental issues in the study of network systems. It has received a considerable attention in the last years. Numerous techniques have been developed for both efficient and effective community detection. Among them, the most efficient algorithm is the label propagation algorithm whose computational complexity is O(|E|). Although it is linear in the number of…
A grand challenge in network science is apparently the missing of a structural theory of networks. The authors have showed that the existence of community structures is a universal phenomenon in real networks, and that neither randomness nor preferential attachment is a mechanism of community structures of network \footnote{A. Li, J. Li, and Y. Pan, Community structures are definable in networks, and universal in the real world, To appear.}. This poses a fundamental question: What are the mechanisms of community structures of real networks? Here we found that homophyly is the mechanism of comm…
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