353 research outputs found
Direction Matters : On Influence-Preserving Graph Summarization and Max-Cut Principle for Directed Graphs
Summarizing large-scale directed graphs into small-scale representations is a useful but less-studied problem setting. Conventional clustering approaches, based on Min-Cut-style criteria, compress both the vertices and edges of the graph into the communities, which lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed-edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features, useful for efficient interventions of population behavior. In this letter, we present a model, based on minimizing reconstruction error with nonnegative constraints, which relates to a Max-Cut criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.Peer reviewe
Bipartite Mixed Membership Distribution-Free Model. A novel model for community detection in overlapping bipartite weighted networks
Modeling and estimating mixed memberships for un-directed un-weighted
networks in which nodes can belong to multiple communities has been well
studied in recent years. However, for a more general case, the bipartite
weighted networks in which nodes can belong to multiple communities, row nodes
can be different from column nodes, and all elements of adjacency matrices can
be any finite real values, to our knowledge, there is no model for such
bipartite weighted networks. To close this gap, this paper introduces a novel
model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. As a
special case, bipartite signed networks with mixed memberships can also be
generated from BiMMDF. Our model enjoys its advantage by allowing all elements
of an adjacency matrix to be generated from any distribution as long as the
expectation adjacency matrix has a block structure related to node memberships
under BiMMDF. The proposed model can be viewed as an extension of many previous
models, including the popular mixed membership stochastic blcokmodels. An
efficient algorithm with a theoretical guarantee of consistent estimation is
applied to fit BiMMDF. In particular, for a standard bipartite weighted network
with two row (and column) communities, to make the algorithm's error rates
small with high probability, separation conditions are obtained when adjacency
matrices are generated from different distributions under BiMMDF. The behavior
differences of different distributions on separation conditions are verified by
extensive synthetic bipartite weighted networks generated under BiMMDF.
Experiments on real-world directed weighted networks illustrate the advantage
of the algorithm in studying highly mixed nodes and asymmetry between row and
column communities.Comment: 33 pages, 12 figures, 4 table
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