Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

Depth-based Hypergraph Complexity Traces from Directed Line Graphs

In this paper, we aim to characterize the structure of hypergraphs in terms of structural complexity measure. Measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders. We thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis. To locate the dominant substructure within a line graph, we identify a centroid vertex by computing the minimum variance of its shortest path lengths. A family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex. We compute the depth-based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs. The resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs. We perform (hyper)graph classification in the principal component space of the complexity trace vectors. Experiments on (hyper)graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces.This work is supported by National Natural Science Foundation of China (Grant no. 61503422). This work is supported by the Open Projects Program of National Laboratory of Pattern Recognition. Francisco Escolano is supported by the project TIN2012-32839 of the Spanish Government. Edwin R. Hancock is supported by a Royal Society Wolfson Research Merit Award

Isospectral graphs with identical nodal counts

According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

Novikov-Shubin invariants and asymptotic dimensions for open manifolds

The Novikov-Shubin numbers are defined for open manifolds with bounded geometry, the Gamma-trace of Atiyah being replaced by a semicontinuous semifinite trace on the C*-algebra of almost local operators. It is proved that they are invariant under quasi-isometries and, making use of the theory of singular traces for C*-algebras developed in math/9802015, they are interpreted as asymptotic dimensions since, in analogy with what happens in Connes' noncommutative geometry, they indicate which power of the Laplacian gives rise to a singular trace. Therefore, as in geometric measure theory, these numbers furnish the order of infinitesimal giving rise to a non trivial measure. The dimensional interpretation is strenghtened in the case of the 0-th Novikov-Shubin invariant, which is shown to coincide, under suitable geometric conditions, with the asymptotic counterpart of the box dimension of a metric space. Since this asymptotic dimension coincides with the polynomial growth of a discrete group, the previous equality generalises a result by Varopoulos for covering manifolds. This paper subsumes dg-ga/9612015. In particular, in the previous version only the 0th Novikov-Shubin number was considered, while here Novikov-Shubin numbers for all p are defined and studied.Comment: 43 pages, LaTex2