Finding a good clustering of vertices in a network, where vertices in the same cluster are more tightly connected than those in different clusters, is a useful, important, and well-studied task. Many clustering algorithms scale well, however they are not designed to operate upon internet-scale networks with billions of nodes or more. We study one of the fastest and most memory efficient algorithms possible - clustering based on the connected components in a random edge-induced subgraph. When defining the cost of a clustering to be its distance from such a random clustering, we show that this surprisingly simple algorithm gives a solution that is within an expected factor of two or three of optimal with either of two natural distance functions. In fact, this approximation guarantee works for any problem where there is a probability distribution on clusterings. We then examine the behavior of this algorithm in the context of social network trust inference.
Frequently we cluster networks in order to find inherent communities in the data. perform an extensive study on the best communities of different sizes in many large social networks @cite_36 . They use conductance (or the normalized cut metric @cite_17 ), defined as the ratio of edges between the community and the outside world to edges within the community, as a measure of community strength. For all of the networks they examine, regardless of size, maximum community conductance drops off considerably for community sizes greater than one hundred. This results suggests that there may be no clusterings of large social networks which help us understand the networks structure. However even if clustering such networks does not reveal anything important about them, it may still be useful in getting better application-specific results or efficiency.
@cite_36 @cite_17
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In local community detection by seed expansion a single cluster concentrated around few given query nodes in a graph is discovered in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. Algorithms that directly optimize conductance usually add or remove one node at a time to find a local optimum. This amounts to fix a specific neighborhood structure over clusters. A natural way to avoid the problem of choosing a specific neighborhood structure is to use a continuous relaxation of conductance. This paper studies such a continuous relaxation of conductance. We show that in this setting continuous optimization leads to hard clusters. We investigate the relation of conductance with weighted kernel k-means for a single cluster, which leads to the introduction of a weighted kernel 1-mean objective function, called -conductance, where is a parameter which influences the size of the community. Conductance is obtained by setting to 0. Two algorithms for local optimization of -conductance based on the expectation maximization and the projected gradient descend method are developed, called EMc and PGDc, respectively. We show that for =0 EMc corresponds to gradient descend with an infinite step size at each iteration. We design a procedure to automatically select a value for . Performance guarantee for these algorithms is proven for a class of dense communities centered around the seeds and well separated from the rest of the network. On this class we also prove that our algorithms stay localized. A comparative experimental analysis on networks with ground-truth communities is performed using state-of-the-art algorithms based on the graph diffusion method. Our experiments indicate that EMc and PGDc stay localized and produce communities most similar to the ground. Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks. A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the "best" possible community - according to the conductance measure - over a wide range of size scales, and we study over 70 large sparse real-world networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large real-world networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually "blend in" with the rest of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are well-embeddable in a low-dimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community structure similar to our observations. Communities of vertices within a giant network such as the World-Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barth ' e lemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than @math edges, where @math is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barth ' e lemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with fewer than @math total edge weight, where @math is the total edge weight in the network and @math is the maximum weight of an inter-community edge. If @math is small, then small communities can be resolved. Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low @math , we modify the CNM'' community detection algorithm to maximize weighted modularity, and show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed.