255 research outputs found
Cover and Hitting Times of Hyperbolic Random Graphs
We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the
regime when the degree distribution obeys a power law with exponent in the range (2, 3). In
particular, we focus on the expected times for a random walk to hit a given vertex or visit, i.e. cover,
all vertices. We show that up to multiplicative constants: the cover time is n(log n)
2
, the maximum
hitting time is n log n, and the average hitting time is n. The first two results hold in expectation
and a.a.s. and the last in expectation (with respect to the HRG).
We prove these results by determining the effective resistance either between an average vertex
and the well-connected “center” of HRGs or between an appropriately chosen collection of extremal
vertices. We bound the effective resistance by the energy dissipated by carefully designed network
flows associated to a tiling of the hyperbolic plane on which we overlay a forest-like structure
Dirichlet Densifier Bounds : Densifying Beyond the Spectral Gap Constraint
In this paper, we characterize the universal bounds of our recently reported Dirichlet Densifier. In particular we aim to study the impact of densification on the bounding of intra-class node similarities. To this end we derive a new bound for commute time estimation. This bound does not rely on the spectral gap, but on graph densification (or graph rewiring). Firstly, we explain how our densifier works and we motivate the bound by showing that implicitly constraining the spectral gap through graph densification cannot fully explain the cluster structure in real-world datasets. Then, we pose our hypothesis about densification: a graph densifier can only deal with a moderate degradation of the spectral gap if the inter-cluster commute distances are significantly shrunk. This points to a more detailed bound which explicitly accounts for the shrinking effect of densification. Finally, we formally develop this bound, thus revealing the deeper implications of graph densification in commute time estimation
Sensitivity of mixing times
In this note, we demonstrate an instance of bounded-degree graphs of size
, for which the total variation mixing time for the random walk is decreased
by a factor of if we multiply the edge-conductances by
bounded factors in a certain way.Comment: 7pages, 2 figure
Dirichlet Densifiers for Improved Commute Times Estimation
In this paper, we develop a novel Dirichlet densifier that can be used to increase the edge density in undirected graphs. Dirichlet densifiers are implicit minimizers of the spectral gap for the Laplacian spectrum of a graph. One consequence of this property is that they can be used improve the estimation of meaningful commute distances for mid-size graphs by means of topological modifications of the original graphs. This results in a better performance in clustering and ranking. To do this, we identify the strongest edges and from them construct the so called line graph, where the nodes are the potential q −step reachable edges in the original graph. These strongest edges are assumed to be stable. By simulating random walks on the line graph, we identify potential new edges in the original graph. This approach is fully unsupervised and it is both more scalable and robust than recent explicit spectral methods, such as the Semi-Definite Programming (SDP) densifier and the sufficient condition for decreasing the spectral gap. Experiments show that our method is only outperformed by some choices of the parameters of a related method, the anchor graph, which relies on pre-computing clusters representatives, and that the proposed method is effective on a variety of real-world datasets.M. Curado, F. Escolano and M.A. Lozano are funded by the projects TIN2015-69077-P and BES2013-064482 of the Spanish Government
Graph similarity through entropic manifold alignment
In this paper we decouple the problem of measuring graph similarity into two sequential steps. The first step is the linearization of the quadratic assignment problem (QAP) in a low-dimensional space, given by the embedding trick. The second step is the evaluation of an information-theoretic distributional measure, which relies on deformable manifold alignment. The proposed measure is a normalized conditional entropy, which induces a positive definite kernel when symmetrized. We use bypass entropy estimation methods to compute an approximation of the normalized conditional entropy. Our approach, which is purely topological (i.e., it does not rely on node or edge attributes although it can potentially accommodate them as additional sources of information) is competitive with state-of-the-art graph matching algorithms as sources of correspondence-based graph similarity, but its complexity is linear instead of cubic (although the complexity of the similarity measure is quadratic). We also determine that the best embedding strategy for graph similarity is provided by commute time embedding, and we conjecture that this is related to its inversibility property, since the inverse of the embeddings obtained using our method can be used as a generative sampler of graph structure.The work of the first and third authors was supported by the projects TIN2012-32839 and TIN2015-69077-P of the Spanish Government. The work of the second author was supported by a Royal Society Wolfson Research Merit Award
Cover and Hitting Times of Hyperbolic Random Graphs
We study random walks on the giant component of Hyperbolic Random Graphs
(HRGs), in the regime when the degree distribution obeys a power law with
exponent in the range . In particular, we focus on the expected times
for a random walk to hit a given vertex or visit, i.e. cover, all vertices. We
show that up to multiplicative constants: the cover time is , the
maximum hitting time is , and the average hitting time is . The
first two results hold in expectation and a.a.s. and the last in expectation
(with respect to the HRG). We prove these results by determining the effective
resistance either between an average vertex and the well-connected "center" of
HRGs or between an appropriately chosen collection of extremal vertices. We
bound the effective resistance by the energy dissipated by carefully designed
network flows associated to a tiling of the hyperbolic plane on which we
overlay a forest-like structure.Comment: 34 pages, 2 figures. To appear at the conference RANDOM 202
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