Work

Further Exploration of an Upper Bound for Kemeny’s Constant

Year: 2025

Type: article

Abstract: Even though Kemeny's constant was first discovered in Markov chains and expressed by Kemeny in terms of mean first passage times on a graph, it can also be expressed using the pseudo-inverse of the Laplacian matrix representing the graph, which facilitates the calculation of a sharp upper bound of Kemeny's constant. We show that for certain classes of graphs, a previously found bound is tight, which generalises previous results for bipartite and (generalised) windmill graphs. Moreover, we show numerically that for real-world networks, this bound can be used to find good numerical approximations for Kemeny's constant. For certain graphs consisting of up to 100 K nodes, we find a speedup of a factor 30, depending on the accuracy of the approximation that can be achieved. For networks consisting of over 500 K nodes, the approximation can be used to estimate values for the Kemeny constant, where exact calculation is no longer feasible within reasonable computation time.

Source: Entropy

DOI: 10.3390/e27040384

Authors: Robert E. Kooij, Johan L. A. Dubbeldam

Institutions: Netherlands Organisation for Applied Scientific Research, Delft University of Technology

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funders: Marie Curie, European Commission, Nederlandse Organisatie voor Wetenschappelijk Onderzoek