Online Community Detection for Large Complex Networks

Complex networks describe a wide range of systems in nature and society. To understand complex networks, it is crucial to investigate their community structure. In this paper, we develop an online community detection algorithm with linear time complexity for large complex networks. Our algorithm processes a network edge by edge in the order that the network is fed to the algorithm. If a new edge is added, it just updates the existing community structure in constant time, and does not need to re-compute the whole network. Therefore, it can efficiently process large networks in real time. Our algorithm optimizes expected modularity instead of modularity at each step to avoid poor performance. The experiments are carried out using 11 public data sets, and are measured by two criteria, modularity and NMI (Normalized Mutual Information). The results show that our algorithm''s running time is less than the commonly used Louvain algorithm while it gives competitive performance.     

Discovering Link Communities in Complex Networks by an Integer Programming Model and a Genetic Algorithm

Identification of communities in complex networks is an important topic and issue in many fields such as sociology, biology, and computer science. Communities are often defined as groups of related nodes or links that correspond to functional subunits in the corresponding complex systems. While most conventional approaches have focused on discovering communities of nodes, some recent studies start partitioning links to find overlapping communities straightforwardly. In this paper, we propose a new quantity function for link community identification in complex networks. Based on this quantity function we formulate the link community partition problem into an integer programming model which allows us to partition a complex network into overlapping communities. We further propose a genetic algorithm for link community detection which can partition a network into overlapping communities without knowing the number of communities. We test our model and algorithm on both artificial networks and real-world networks. The results demonstrate that the model and algorithm are efficient in detecting overlapping community structure in complex networks.     

Module detection in complex networks using integer optimisation

Background  

The detection of modules or community structure is widely used to reveal the underlying properties of complex networks in biology, as well as physical and social sciences. Since the adoption of modularity as a measure of network topological properties, several methodologies for the discovery of community structure based on modularity maximisation have been developed. However, satisfactory partitions of large graphs with modest computational resources are particularly challenging due to the NP-hard nature of the related optimisation problem. Furthermore, it has been suggested that optimising the modularity metric can reach a resolution limit whereby the algorithm fails to detect smaller communities than a specific size in large networks.     

A Stochastic Model for Detecting Overlapping and Hierarchical Community Structure

Community detection is a fundamental problem in the analysis of complex networks. Recently, many researchers have concentrated on the detection of overlapping communities, where a vertex may belong to more than one community. However, most current methods require the number (or the size) of the communities as a priori information, which is usually unavailable in real-world networks. Thus, a practical algorithm should not only find the overlapping community structure, but also automatically determine the number of communities. Furthermore, it is preferable if this method is able to reveal the hierarchical structure of networks as well. In this work, we firstly propose a generative model that employs a nonnegative matrix factorization (NMF) formulization with a l_2,1 norm regularization term, balanced by a resolution parameter. The NMF has the nature that provides overlapping community structure by assigning soft membership variables to each vertex; the l_2,1 regularization term is a technique of group sparsity which can automatically determine the number of communities by penalizing too many nonempty communities; and hence the resolution parameter enables us to explore the hierarchical structure of networks. Thereafter, we derive the multiplicative update rule to learn the model parameters, and offer the proof of its correctness. Finally, we test our approach on a variety of synthetic and real-world networks, and compare it with some state-of-the-art algorithms. The results validate the superior performance of our new method.     

Followers Are Not Enough: A Multifaceted Approach to Community Detection in Online Social Networks

In online social media networks, individuals often have hundreds or even thousands of connections, which link these users not only to friends, associates, and colleagues, but also to news outlets, celebrities, and organizations. In these complex social networks, a ‘community’ as studied in the social network literature, can have very different meaning depending on the property of the network under study. Taking into account the multifaceted nature of these networks, we claim that community detection in online social networks should also be multifaceted in order to capture all of the different and valuable viewpoints of ‘community.’ In this paper we focus on three types of communities beyond follower-based structural communities: activity-based, topic-based, and interaction-based. We analyze a Twitter dataset using three different weightings of the structural network meant to highlight these three community types, and then infer the communities associated with these weightings. We show that interesting insights can be obtained about the complex community structure present in social networks by studying when and how these four community types give rise to similar as well as completely distinct community structure.     

Active Semi-Supervised Community Detection Based on Must-Link and Cannot-Link Constraints

Community structure detection is of great importance because it can help in discovering the relationship between the function and the topology structure of a network. Many community detection algorithms have been proposed, but how to incorporate the prior knowledge in the detection process remains a challenging problem. In this paper, we propose a semi-supervised community detection algorithm, which makes full utilization of the must-link and cannot-link constraints to guide the process of community detection and thereby extracts high-quality community structures from networks. To acquire the high-quality must-link and cannot-link constraints, we also propose a semi-supervised component generation algorithm based on active learning, which actively selects nodes with maximum utility for the proposed semi-supervised community detection algorithm step by step, and then generates the must-link and cannot-link constraints by accessing a noiseless oracle. Extensive experiments were carried out, and the experimental results show that the introduction of active learning into the problem of community detection makes a success. Our proposed method can extract high-quality community structures from networks, and significantly outperforms other comparison methods.     

DyCoNet: A Gephi Plugin for Community Detection in Dynamic Complex Networks

Community structure detection has proven to be important in revealing the underlying organisation of complex networks. While most current analyses focus on static networks, the detection of communities in dynamic data is both challenging and timely. An analysis and visualisation procedure for dynamic networks is presented here, which identifies communities and sub-communities that persist across multiple network snapshots. An existing method for community detection in dynamic networks is adapted, extended, and implemented. We demonstrate the applicability of this method to detect communities in networks where individuals tend not to change their community affiliation very frequently. When stability of communities cannot be assumed, we show that the sub-community model may be a better alternative. This is illustrated through test cases of social and biological networks. A plugin for Gephi, an open-source software program used for graph visualisation and manipulation, named “DyCoNet”, was created to execute the algorithm and is freely available from https://github.com/juliemkauffman/DyCoNet.     

Multi-Objective Community Detection Based on Memetic Algorithm

Community detection has drawn a lot of attention as it can provide invaluable help in understanding the function and visualizing the structure of networks. Since single objective optimization methods have intrinsic drawbacks to identifying multiple significant community structures, some methods formulate the community detection as multi-objective problems and adopt population-based evolutionary algorithms to obtain multiple community structures. Evolutionary algorithms have strong global search ability, but have difficulty in locating local optima efficiently. In this study, in order to identify multiple significant community structures more effectively, a multi-objective memetic algorithm for community detection is proposed by combining multi-objective evolutionary algorithm with a local search procedure. The local search procedure is designed by addressing three issues. Firstly, nondominated solutions generated by evolutionary operations and solutions in dominant population are set as initial individuals for local search procedure. Then, a new direction vector named as pseudonormal vector is proposed to integrate two objective functions together to form a fitness function. Finally, a network specific local search strategy based on label propagation rule is expanded to search the local optimal solutions efficiently. The extensive experiments on both artificial and real-world networks evaluate the proposed method from three aspects. Firstly, experiments on influence of local search procedure demonstrate that the local search procedure can speed up the convergence to better partitions and make the algorithm more stable. Secondly, comparisons with a set of classic community detection methods illustrate the proposed method can find single partitions effectively. Finally, the method is applied to identify hierarchical structures of networks which are beneficial for analyzing networks in multi-resolution levels.     

Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit

In recent years, there has been a surge of interest in community detection algorithms for complex networks. A variety of computational heuristics, some with a long history, have been proposed for the identification of communities or, alternatively, of good graph partitions. In most cases, the algorithms maximize a particular objective function, thereby finding the 'right' split into communities. Although a thorough comparison of algorithms is still lacking, there has been an effort to design benchmarks, i.e., random graph models with known community structure against which algorithms can be evaluated. However, popular community detection methods and benchmarks normally assume an implicit notion of community based on clique-like subgraphs, a form of community structure that is not always characteristic of real networks. Specifically, networks that emerge from geometric constraints can have natural non clique-like substructures with large effective diameters, which can be interpreted as long-range communities. In this work, we show that long-range communities escape detection by popular methods, which are blinded by a restricted 'field-of-view' limit, an intrinsic upper scale on the communities they can detect. The field-of-view limit means that long-range communities tend to be overpartitioned. We show how by adopting a dynamical perspective towards community detection [1], [2], in which the evolution of a Markov process on the graph is used as a zooming lens over the structure of the network at all scales, one can detect both clique- or non clique-like communities without imposing an upper scale to the detection. Consequently, the performance of algorithms on inherently low-diameter, clique-like benchmarks may not always be indicative of equally good results in real networks with local, sparser connectivity. We illustrate our ideas with constructive examples and through the analysis of real-world networks from imaging, protein structures and the power grid, where a multiscale structure of non clique-like communities is revealed.     

Netgram: Visualizing Communities in Evolving Networks

Real-world complex networks are dynamic in nature and change over time. The change is usually observed in the interactions within the network over time. Complex networks exhibit community like structures. A key feature of the dynamics of complex networks is the evolution of communities over time. Several methods have been proposed to detect and track the evolution of these groups over time. However, there is no generic tool which visualizes all the aspects of group evolution in dynamic networks including birth, death, splitting, merging, expansion, shrinkage and continuation of groups. In this paper, we propose Netgram: a tool for visualizing evolution of communities in time-evolving graphs. Netgram maintains evolution of communities over 2 consecutive time-stamps in tables which are used to create a query database using the sql outer-join operation. It uses a line-based visualization technique which adheres to certain design principles and aesthetic guidelines. Netgram uses a greedy solution to order the initial community information provided by the evolutionary clustering technique such that we have fewer line cross-overs in the visualization. This makes it easier to track the progress of individual communities in time evolving graphs. Netgram is a generic toolkit which can be used with any evolutionary community detection algorithm as illustrated in our experiments. We use Netgram for visualization of topic evolution in the NIPS conference over a period of 11 years and observe the emergence and merging of several disciplines in the field of information processing systems.     

A Shadowing Problem in the Detection of Overlapping Communities: Lifting the Resolution Limit through a Cascading Procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.     

Characterizing the Community Structure of Complex Networks

Background

Community structure is one of the key properties of complex networks and plays a crucial role in their topology and function. While an impressive amount of work has been done on the issue of community detection, very little attention has been so far devoted to the investigation of communities in real networks.

Methodology/Principal Findings

We present a systematic empirical analysis of the statistical properties of communities in large information, communication, technological, biological, and social networks. We find that the mesoscopic organization of networks of the same category is remarkably similar. This is reflected in several characteristics of community structure, which can be used as “fingerprints” of specific network categories. While community size distributions are always broad, certain categories of networks consist mainly of tree-like communities, while others have denser modules. Average path lengths within communities initially grow logarithmically with community size, but the growth saturates or slows down for communities larger than a characteristic size. This behaviour is related to the presence of hubs within communities, whose roles differ across categories. Also the community embeddedness of nodes, measured in terms of the fraction of links within their communities, has a characteristic distribution for each category.

Conclusions/Significance

Our findings, verified by the use of two fundamentally different community detection methods, allow for a classification of real networks and pave the way to a realistic modelling of networks'' evolution.     

Spatial correlations in attribute communities

Community detection is an important tool for exploring and classifying the properties of large complex networks and should be of great help for spatial networks. Indeed, in addition to their location, nodes in spatial networks can have attributes such as the language for individuals, or any other socio-economical feature that we would like to identify in communities. We discuss in this paper a crucial aspect which was not considered in previous studies which is the possible existence of correlations between space and attributes. Introducing a simple toy model in which both space and node attributes are considered, we discuss the effect of space-attribute correlations on the results of various community detection methods proposed for spatial networks in this paper and in previous studies. When space is irrelevant, our model is equivalent to the stochastic block model which has been shown to display a detectability-non detectability transition. In the regime where space dominates the link formation process, most methods can fail to recover the communities, an effect which is particularly marked when space-attributes correlations are strong. In this latter case, community detection methods which remove the spatial component of the network can miss a large part of the community structure and can lead to incorrect results.     

Time Development in the Early History of Social Networks: Link Stabilization,Group Dynamics,and Segregation

Studies of the time development of empirical networks usually investigate late stages where lasting connections have already stabilized. Empirical data on early network history are rare but needed for a better understanding of how social network topology develops in real life. Studying students who are beginning their studies at a university with no or few prior connections to each other offers a unique opportunity to investigate the formation and early development of link patterns and community structure in social networks. During a nine week introductory physics course, first year physics students were asked to identify those with whom they communicated about problem solving in physics during the preceding week. We use these students'' self reports to produce time dependent student interaction networks. We investigate these networks to elucidate possible effects of different student attributes in early network formation. Changes in the weekly number of links show that while roughly half of all links change from week to week, students also reestablish a growing number of links as they progress through their first weeks of study. Using the Infomap community detection algorithm, we show that the networks exhibit community structure, and we use non-network student attributes, such as gender and end-of-course grade to characterize communities during their formation. Specifically, we develop a segregation measure and show that students structure themselves according to gender and pre-organized sections (in which students engage in problem solving and laboratory work), but not according to end-of-coure grade. Alluvial diagrams of consecutive weeks'' communities show that while student movement between groups are erratic in the beginnning of their studies, they stabilize somewhat towards the end of the course. Taken together, the analyses imply that student interaction networks stabilize quickly and that students establish collaborations based on who is immediately available to them and on observable personal characteristics.     

Community Structure and Multi-Modal Oscillations in Complex Networks

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.     

Dynamics and Control of Diseases in Networks with Community Structure

The dynamics of infectious diseases spread via direct person-to-person transmission (such as influenza, smallpox, HIV/AIDS, etc.) depends on the underlying host contact network. Human contact networks exhibit strong community structure. Understanding how such community structure affects epidemics may provide insights for preventing the spread of disease between communities by changing the structure of the contact network through pharmaceutical or non-pharmaceutical interventions. We use empirical and simulated networks to investigate the spread of disease in networks with community structure. We find that community structure has a major impact on disease dynamics, and we show that in networks with strong community structure, immunization interventions targeted at individuals bridging communities are more effective than those simply targeting highly connected individuals. Because the structure of relevant contact networks is generally not known, and vaccine supply is often limited, there is great need for efficient vaccination algorithms that do not require full knowledge of the network. We developed an algorithm that acts only on locally available network information and is able to quickly identify targets for successful immunization intervention. The algorithm generally outperforms existing algorithms when vaccine supply is limited, particularly in networks with strong community structure. Understanding the spread of infectious diseases and designing optimal control strategies is a major goal of public health. Social networks show marked patterns of community structure, and our results, based on empirical and simulated data, demonstrate that community structure strongly affects disease dynamics. These results have implications for the design of control strategies.     

A novel method of spectral clustering in attributed networks by constructing parameter-free affinity matrix

The most basic and significant issue in complex network analysis is community detection, which is a branch of machine learning. Most current community detection approaches, only consider a network's topology structures, which lose the potential to use node attribute information. In attributed networks, both topological structure and node attributed are important features for community detection. In recent years, the spectral clustering algorithm has received much interest as one of the best performing algorithms in the subcategory of dimensionality reduction. This algorithm applies the eigenvalues of the affinity matrix to map data to low-dimensional space. In the present paper, a new version of the spectral cluster, named Attributed Spectral Clustering (ASC), is applied for attributed graphs that the identified communities have structural cohesiveness and attribute homogeneity. Since the performance of spectral clustering heavily depends on the goodness of the affinity matrix, the ASC algorithm will use the Topological and Attribute Random Walk Affinity Matrix (TARWAM) as a new affinity matrix to calculate the similarity between nodes. TARWAM utilizes the biased random walk to integrate network topology and attribute information. It can improve the similarity degree among the pairs of nodes in the same density region of the attributed network, without the need for parameter tuning. The proposed approach has been compared to other primary and new attributed graph clustering algorithms based on synthetic and real datasets. The experimental results show that the proposed approach is more effective and accurate compared to other state-of-the-art attributed graph clustering techniques.

     

Deciphering network community structure by surprise

The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that maximization of a simple global parameter, which we call Surprise (S), leads to a very efficient characterization of the community structure of complex synthetic networks. Particularly, S qualitatively outperforms the most commonly used criterion to define communities, Newman and Girvan's modularity (Q). Applying S maximization to real networks often provides natural, well-supported partitions, but also sometimes counterintuitive solutions that expose the limitations of our previous knowledge. These results indicate that it is possible to define an effective global criterion for community structure and open new routes for the understanding of complex networks.     

Towards online multiresolution community detection in large-scale networks

The investigation of community structure in networks has aroused great interest in multiple disciplines. One of the challenges is to find local communities from a starting vertex in a network without global information about the entire network. Many existing methods tend to be accurate depending on a priori assumptions of network properties and predefined parameters. In this paper, we introduce a new quality function of local community and present a fast local expansion algorithm for uncovering communities in large-scale networks. The proposed algorithm can detect multiresolution community from a source vertex or communities covering the whole network. Experimental results show that the proposed algorithm is efficient and well-behaved in both real-world and synthetic networks.     

Lewis Carroll's Doublets Net of English Words: Network Heterogeneity in a Complex System

Lewis Carroll''s English word game Doublets is represented as a system of networks with each node being an English word and each connectivity edge confirming that its two ending words are equal in letter length, but different by exactly one letter. We show that this system, which we call the Doublets net, constitutes a complex body of linguistic knowledge concerning English word structure that has computable multiscale features. Distributed morphological, phonological and orthographic constraints and the language''s local redundancy are seen at the node level. Phonological communities are seen at the network level. And a balancing act between the language''s global efficiency and redundancy is seen at the system level. We develop a new measure of intrinsic node-to-node distance and a computational algorithm, called community geometry, which reveal the implicit multiscale structure within binary networks. Because the Doublets net is a modular complex cognitive system, the community geometry and computable multi-scale structural information may provide a foundation for understanding computational learning in many systems whose network structure has yet to be fully analyzed.