Abstract
We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponential matrices, under norm-preserving and sparsity constraints. Variants of the distance metric are introduced to consider such optimized maps under sparsity constraints as well as fixed time-scaling between the two Laplacians. The objective function of this optimization is multimodal and has discontinuous slope, and is hence difficult for univariate optimizers to solve. We demonstrate a novel procedure for efficiently calculating these optima for two of our distance measure variants. We present numerical experiments demonstrating that (a) upper bounds of our distance metrics can be used to distinguish between lineages of related graphs; (b) our procedure is faster at finding the required optima, by as much as a factor of 103; and (c) the upper bounds satisfy the triangle inequality exactly under some assumptions and approximately under others. We also derive an upper bound for the distance between two graph products, in terms of the distance between the two pairs of factors. Additionally, we present several possible applications, including the construction of infinite "graph limits" by means of Cauchy sequences of graphs related to one another by our distance measure.
Highlights
Structure comparison, as well as structure summarization, is a ubiquitous problem, appearing across multiple scientific disciplines
In this work we present a graph distance metric, based on the Laplacian exponential kernel of a graph
This measure generalizes the work of Hammond et al [1] on graph diffusion distance for graphs of equal size; crucially, our distance measure allows for graphs of inequal size
Summary
As well as structure summarization, is a ubiquitous problem, appearing across multiple scientific disciplines. In this work we present a graph distance metric, based on the Laplacian exponential kernel of a graph. A comparison of the two graph Laplacians This problem is a nested optimization problem with the innermost layer consisting of multivariate optimization subject to matrix constraints (e.g. orthogonality). To compute this dissimilarity score efficiently, we develop and demonstrate the lower computational cost of an algorithm which calculates upper bounds on the distance. This algorithm produces a prolongation/restriction operator, P, which produces an optimally coarsened version of the Laplacian matrix of a graph.
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