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.
However, the lazy random walk endured a much slower mixing speed and it usually took more than 500 hundred steps to converge to a local structure compared with several steps of rapid mixing in a regular random walk. Featuring on the seeding strategies, Whang @cite_17 established several sophisticated methods for choosing the seed set, and then used similar PageRank scheme as that in @cite_18 to expand the seeds until a community with optimal conductance is found. Nonetheless, the performance gained by adopting these intricate seeding methods was not significantly better than that by using random seeds. This implies that a better scheme of expanding the seeds is also needed aside from a good seeding strategy. A recent work by Kloumann & Kleinberg @cite_11 provided a systematic understanding of variants of PageRank-based seed set expansion. They showed many insightful findings regarding the heuristics on seed set. However, the drawback of lacking a proper stop criterion has limited its functionality in practice. Even though a recently proposed heat kernel algorithm @cite_8 advances PageRank by introducing a sophisticated diffusion method, the detection accuracy achieved by heat kernel approach is still much lower than that of , which we will show in Section .
@cite_8 @cite_18 @cite_11 @cite_17
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In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any @math and for any exponent @math giving the rate of decay of the percolation process, we show that the return probability decays like @math up to logarithmic corrections, where @math denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster. We propose a randomized data mining method that finds clusters of spatially overlapping images. The core of the method relies on the min-Hash algorithm for fast detection of pairs of images with spatial overlap, the so-called cluster seeds. The seeds are then used as visual queries to obtain clusters which are formed as transitive closures of sets of partially overlapping images that include the seed. We show that the probability of finding a seed for an image cluster rapidly increases with the size of the cluster. The properties and performance of the algorithm are demonstrated on data sets with 104, 105, and 5 × 106 images. The speed of the method depends on the size of the database and the number of clusters. The first stage of seed generation is close to linear for databases sizes up to approximately 234 ? 1010 images. On a single 2.4 GHz PC, the clustering process took only 24 minutes for a standard database of more than 100,000 images, i.e., only 0.014 seconds per image. The heat kernel is a type of graph diffusion that, like the much-used personalized PageRank diffusion, is useful in identifying a community nearby a starting seed node. We present the first deterministic, local algorithm to compute this diffusion and use that algorithm to study the communities that it produces. Our algorithm is formally a relaxation method for solving a linear system to estimate the matrix exponential in a degree-weighted norm. We prove that this algorithm stays localized in a large graph and has a worst-case constant runtime that depends only on the parameters of the diffusion, not the size of the graph. On large graphs, our experiments indicate that the communities produced by this method have better conductance than those produced by PageRank, although they take slightly longer to compute. On a real-world community identification task, the heat kernel communities perform better than those from the PageRank diffusion. We consider an analogue of the Kac random walk on the special orthogonal group SO(N)SO(N), in which at each step a random rotation is performed in a randomly chosen 2-plane of RNRN. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large NN limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the L2L2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L2L2 norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.