Diffusion profile embedding as a basis for graph vertex similarity

Abstract

AbstractWe describe here a notion ofdiffusion similarity, a method for defining similarity between vertices in a given graph using the properties of random walks on the graph to model the relationships between vertices. Using the approach of graph vertex embedding, we characterize a vertexviby considering two types of diffusion patterns: the ways in which random walks emanate from the vertexvito the remaining graph and how they converge to the vertexvifrom the graph. We define the similarity of two verticesviandvjas the average of the cosine similarity of the vectors characterizingviandvj. We obtain these vectors by modifying the solution to a differential equation describing a type of continuous time random walk.This method can be applied to any dataset that can be assigned a graph structure that is weighted or unweighted, directed or undirected. It can be used to represent similarity of vertices within community structures of a network while at the same time representing similarity of vertices within layered substructures (e.g., bipartite subgraphs) of the network. To validate the performance of our method, we apply it to synthetic data as well as the neural connectome of theC. elegansworm and a connectome of neurons in the mouse retina. A tool developed to characterize the accuracy of the similarity values in detecting community structures,the uncertainty index, is introduced in this paper as a measure of the quality of similarity methods.