Nodal-statistics-based equivalence relation for graph collections

Lucrezia Carboni, Michel Dojat, and Sophie Achard
Phys. Rev. E 107, 014302 – Published 18 January 2023

Abstract

Node role explainability in complex networks is very difficult yet is crucial in different application domains such as social science, neurosciences, or computer science. Many efforts have been made on the quantification of hubs revealing particular nodes in a network using a given structural property. Yet, in several applications, when multiple instances of networks are available and several structural properties appear to be relevant, the identification of node roles remains largely unexplored. Inspired by the node automorphically equivalence relation, we define an equivalence relation on graph nodes associated with any collection of nodal statistics (i.e., any functions on the node set). This allows us to define new graph global measures: the power coefficient and the orthogonality score to evaluate the parsimony and heterogeneity of a given nodal statistics collection. In addition, we introduce a new method based on structural patterns to compare graphs that have the same vertices set. This method assigns a value to a node to determine its role distinctiveness in a graph family. Extensive numerical results of our method are conducted on both generative graph models and real data concerning human brain functional connectivity. The differences in nodal statistics are shown to be dependent on the underlying graph structure. Comparisons between generative models and real networks combining two different nodal statistics reveal the complexity of human brain functional connectivity with differences at both global and nodal levels. Using a group of 200 healthy controls connectivity networks, our method computes high correspondence scores among the whole population to detect homotopy and finally quantify differences between comatose patients and healthy controls.

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  • Received 24 May 2022
  • Revised 3 October 2022
  • Accepted 7 November 2022

DOI:https://doi.org/10.1103/PhysRevE.107.014302

©2023 American Physical Society

Physics Subject Headings (PhySH)

NetworksPhysics of Living Systems

Authors & Affiliations

Lucrezia Carboni1,2, Michel Dojat2, and Sophie Achard1

  • 1Université Grenoble Alpes, CNRS, Inria, Grenoble INP, LJK, 38000 Grenoble, France
  • 2Université Grenoble Alpes, Inserm, U1216, Grenoble Institut Neurosciences, GIN, 38000 Grenoble, France

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Issue

Vol. 107, Iss. 1 — January 2023

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