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Diffusion profile embedding as a basis for graph vertex similarity

Published online by Cambridge University Press:  07 October 2021

Scott Payne ,
Edgar Fuller Open the ORCID record for Edgar Fuller [Opens in a new window] ,
George Spirou  and
Cun-Quan Zhang
Scott Payne
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV, USA (e-mails: spayne7@mix.wvu.edu, cun-quan.zhang@mail.wvu.edu),
Edgar Fuller*
Affiliation:
Department of Mathematics and Statistics, Florida International University, Miami, FL, USA
George Spirou
Affiliation:
Department of Medical Engineering, University of South Florida, Tampa, FL, USA (e-mail: gspirou@usf.edu)
Cun-Quan Zhang
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV, USA (e-mails: spayne7@mix.wvu.edu, cun-quan.zhang@mail.wvu.edu),
*
*Corresponding author. Email: ejfuller@gmail.com
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Abstract

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We describe here a notion of diffusion 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 vertex vi by considering two types of diffusion patterns: the ways in which random walks emanate from the vertex vi to the remaining graph and how they converge to the vertex vi from the graph. We define the similarity of two vertices vi and vj as the average of the cosine similarity of the vectors characterizing vi and vj. 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 the C. elegans worm 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.

Keywords

similarity functionvertex similaritydiffusion profilerandom walksdiffusion kernel
Type
Research Article
Information
Network Science , Volume 9 , Issue 3 , September 2021 , pp. 328 - 353
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Action Editor: Ulrik Brandes

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