In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a piecewise constant estimation of the density of the edges across the clusters, and address the problem of automatically and reliably inferring the number of clusters, which is the granularity of the coclustering. We use a model selection technique with data-dependent prior and obtain an exact evaluation criterion for the posterior probability of edge density estimation models. We demonstrate, both theoretically and empirically, that our data-dependent modeling technique is consistent, resilient to noise, valid non asymptotically and asymptotically behaves as an universal approximator of the true edge density in directed multigraphs. We evaluate our method using artificial graphs and present its practical interest on real world graphs. The method is both robust and scalable. It is able to extract insightful patterns in the unsupervised learning setting and to provide state of the art accuracy when used as a preparation step for supervised learning.
@cite_40: The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.
@cite_14: Abstract Blockmodel approaches to network analysis as developed by Harrison White are shown to fall in a broader class of established data analysis methods based on matrix permutations (e.g., clique detection, seriation, permutation algorithms for sparse matrices). Blockmodels are seen as an important generalization of these earlier methods since they permit the data to characterize their own structure, instead of seeking to manifest some preconceived structure which is imposed by the investigator (e.g., cliques, hierarchies, or structural balance). General algorithms for the inductive construction of blockmodels thus occupy a central position in the development of the area. We discuss theoretical and practical aspects of the blockmodel search procedure which has been most widely used (CONCOR algorithm). It is proposed that the distinctive and advantageous feature of CONCOR is that it solves what is initially presented as a combinatorial problem (permutations of matrices to reveal zeroblocks) by representing the problem as a continuous one (analysis of correlation matrices). When this representation strategy receives further development, it is predicted that the fairly crude empirical approach of CONCOR will be supplanted by more powerful procedures within this same class.
More expressive graph models aim at searching a partition of both the source and target vertices into clusters, with different types of interaction between clusters. The cross-product of the two partitions of vertices form a partition of the edges into or . This modeling approach is called blockmodeling and has been thoroughly studied for decades. In early approaches @cite_40 @cite_14 , non-stochastic blocks are considered, with a focus on predefined types of block patterns. The blockmodel is searched either indirectly using a (dis)similarity measure between pairs of vertices and then applying a standard clustering algorithm, or directly by optimizing an ad hoc function measuring the fit of real blocks to the corresponding predefined types of blocks. The limit of these approaches is that they do not cope with the stochastic nature of many real world datasets.