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 in large networks. A growing body of work has been adopting local expansion methods in order to identify the community members 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). The algorithm 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. We further provide theoretical analysis on the local spectral properties, bounding the measure of tightness of extracted community in terms of the eigenvalues of graph Laplacian. 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 seed set quality and quantity would affect the performance are provided.
As noted in the preceding section, among the divergent approaches, random walks tend to reveal communities that bear the closest resemblance to the ground truth communities in nature @cite_4 . In the following, we briefly review some methods that have adopted the random walk technique in finding communities. Speaking of methods that focus on the global structure, Pons @cite_12 proposed a hierarchical agglomerative algorithm, , that quantified the similarity between vertices using random walks and then partitioned the network into non-overlapping communities. Meil @math @cite_19 presented a clustering approach by viewing the pairwise similarities as edge flows in a random walk and studied the eigenvectors and values of the resulting transition matrix. A later successful algorithm, , proposed by by Rosvall & Bergstrom @cite_5 enables uncovering hierarchical structures in networks by compressing a description of a random walker as a proxy for real flow on networks. Variants of this technique such as biased random walk @cite_9 has also been employed in community finding.
@cite_4 @cite_9 @cite_19 @cite_5 @cite_12
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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 present a new view of clustering and segmentation by pairwise similarities. We interpret the similarities as edge ows in a Markov random walk and study the eigenvalues and eigenvectors of the walk's transition matrix. This view shows that spectral methods for clustering and segmentation have a probabilistic foundation. We prove that the Normalized Cut method arises naturally from our framework and we provide a complete characterization of the cases when the Normalized Cut algorithm is exact. Then we discuss other spectral segmentation and clustering methods showing that several of them are essentially the same as NCut. To comprehend the hierarchical organization of large integrated systems, we introduce the hierarchical map equation, which reveals multilevel structures in networks. In this information-theoretic a ... In a representative embodiment of the invention described herein, a well logging system for investigating subsurface formations is controlled by a general purpose computer programmed for real-time operation. The system is cooperatively arranged to provide for all aspects of a well logging operation, such as data acquisition and processing, tool control, information or data storage, and data presentation as a well logging tool is moved through a wellbore. The computer controlling the system is programmed to provide for data acquisition and tool control commands in direct response to asynchronous real-time external events. Such real-time external events may occur, for example, as a result of movement of the logging tool over a selected depth interval, or in response to requests or commands directed to the system by the well logging engineer by means of keyboard input.