Large graphs arise in a number of contexts and understanding their structure and extracting information from them is an important research area. Early algorithms on mining communities have focused on the global structure, and often run in time functional to the size of the entire graph. Nowadays, as we often explore networks with billions of vertices and find communities of size hundreds, it is crucial to shift our attention from macroscopic structure to microscopic structure when dealing with large networks. A growing body of work has been adopting local expansion methods in order to identify the community from a few exemplary seed members. In this paper, we propose a novel approach for finding overlapping communities called LEMON (Local Expansion via Minimum One Norm). Different from PageRank-like diffusion methods, LEMON finds the community by seeking a sparse vector in the span of the local spectra such that the seeds are in its support. We show that LEMON can achieve the highest detection accuracy among state-of-the-art proposals. The running time depends on the size of the community rather than that of the entire graph. The algorithm is easy to implement, and is highly parallelizable. Moreover, given that networks are not all similar in nature, a comprehensive analysis on how the local expansion approach is suited for uncovering communities in different networks is still lacking. We thoroughly evaluate our approach using both synthetic and real-world datasets across different domains, and analyze the empirical variations when applying our method to inherently different networks in practice. In addition, the heuristics on how the quality and quantity of the seed set would affect the performance are provided.
Various community detection algorithms have been developed in the past decade. And most of the algorithms fall into the category of global approach. One stream of global algorithms attempt to find communities by optimizing an objective function. For example, OSLOM @cite_3 is based on the optimization of a fitness function, which expresses the statistical significance of clusters with respect to random fluctuations (i.e., the random graph generated by the configuration model @cite_25 during community expansion). However, the communities identified by mathematical construction may structurally diverge from real communities as pointed out in @cite_4 . Another main stream of research adopts the label propagation approach @cite_24 , which defines rules that simulate the spread of labels of vertices in the network. The DEMON algorithm @cite_0 , for example, democratically lets each vertex vote for the communities it sees surrounding it in its limited view of the global system using a label propagation algorithm, and then merges the local communities into a global collection. Other approaches such as Link Community (LC) @cite_7 partitions the graph by first building a hierarchical link dendrogram according to the link similarity and then cutting the dendrogram at some threshold to yield link communities.
@cite_4 @cite_7 @cite_3 @cite_24 @cite_0 @cite_25
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Four major factors govern the intricacies of community extraction in networks: (1) the literature offers a multitude of disparate community detection algorithms whose output exhibits high structural variability across the collection, (2) communities identified by algorithms may differ structurally from real communities that arise in practice, (3) there is no consensus characterizing how to discriminate communities from noncommunities, and (4) the application domain includes a wide variety of networks of fundamentally different natures. In this article, we present a class separability framework to tackle these challenges through a comprehensive analysis of community properties. Our approach enables the assessment of the structural dissimilarity among the output of multiple community detection algorithms and between the output of algorithms and communities that arise in practice. In addition, our method provides us with a way to organize the vast collection of community detection algorithms by grouping those that behave similarly. Finally, we identify the most discriminative graph-theoretical properties of community signature and the small subset of properties that account for most of the biases of the different community detection algorithms. We illustrate our approach with an experimental analysis, which reveals nuances of the structure of real and extracted communities. In our experiments, we furnish our framework with the output of 10 different community detection procedures, representative of categories of popular algorithms available in the literature, applied to a diverse collection of large-scale real network datasets whose domains span biology, online shopping, and social systems. We also analyze communities identified by annotations that accompany the data, which reflect exemplar communities in various domain. We characterize these communities using a broad spectrum of community properties to produce the different structural classes. As our experiments show that community structure is not a universal concept, our framework enables an informed choice of the most suitable community detection method for identifying communities of a specific type in a given network and allows for a comparison of existing community detection algorithms while guiding the design of new ones. We consider the problem of community detection in the Stochastic Block Model with a finite number @math of communities of sizes linearly growing with the network size @math . This model consists in a random graph such that each pair of vertices is connected independently with probability @math within communities and @math across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed @math with high probability as @math grows large, whenever @math , @math and where @math and @math denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by and , this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than @math misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios. Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods. Community detection in graphs is the problem of finding groups of vertices which are more densely connected than they are to the rest of the graph. This problem has a long history, but it is currently motivated by social and biological networks. While there are many ways to formalize it, one of the most popular is as an inference problem, where there is a planted “ground truth” community structure around which the graph is generated probabilistically. Our task is then to recover the ground truth knowing only the graph. Recently it was discovered, first h euristically i n p hysics a nd then rigorously in probability and computer science, that this problem has a phase transition at which it suddenly becomes impossible. Namely, if the graph is too sparse, or the probabilistic process that generates it is too noisy, then no algorithm can find a partition t hat i s correlated with the planted one—or even tell if there are communities, i.e., distinguish the graph from a purely random one with high probability. Above this information-theoretic threshold, there is a second threshold beyond which polynomial-time algorithms are known to succeed; in between, there is a regime in which community detection is possible, but conjectured to be exponentially hard. For computer scientists, this field offers a wealth of new ideas and open questions, with connections to probability and combinatorics, message-passing algorithms, and random matrix theory. Perhaps more importantly, it provides a window into the cultures of statistical physics and statistical inference, and how those cultures think about distributions of instances, landscapes of solutions, and hardness.