Kemeny's constant for several families of graphs and real-world networks

Kemeny's constant for several families of graphs and real-world networks

Kooij, Robert (TU Delft Network Architectures and Services)
Dubbeldam, J.L.A. (TU Delft Mathematical Physics)

2020

The linear relation between Kemeny's constant, a graph metric directly linked with random walks, and the effective graph resistance in a regular graph has been an incentive to calculate Kemeny's constant for various networks. In this paper we consider complete bipartite graphs, (generalized) windmill graphs and tree networks with large diameter and give exact expressions of Kemeny's constant. For non-regular graphs we propose two approximations for Kemeny's constant by adding to the effective graph resistance term a linear term related to the degree heterogeneity in the graph. These approximations are exact for complete bipartite graphs, but show some discrepancies for generalized windmill and tree graphs. However, we show that a recently obtained upper-bound for Kemeny's constant in Wang et al. (2017) based on the pseudo inverse Laplacian gives the exact value of Kemeny's constant for generalized windmill graphs. Finally, we have evaluated Kemeny's constant, its two approximations and its upper bound, for 243 real-world networks. This evaluation reveals that the upper bound is tight, with average relative error of only 0.73%. In most cases the upper bound clearly outperforms the other two approximations.

Effective graph resistance
Kemeny's constant
Pseudo inverse Laplacian
Random walks
Spectral graph theory

http://resolver.tudelft.nl/uuid:8a072f6d-f105-4193-aa2e-276ee560f3e9

https://doi.org/10.1016/j.dam.2020.05.033

0166-218X

Discrete Applied Mathematics, 285, 96-107

Institutional Repository

journal article

© 2020 Robert Kooij, J.L.A. Dubbeldam