TY - JOUR
T1 - Kemeny's constant for several families of graphs and real-world networks
AU - Kooij, Robert E.
AU - Dubbeldam, Johan L.A.
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Effective graph resistance
KW - Kemeny's constant
KW - Pseudo inverse Laplacian
KW - Random walks
KW - Spectral graph theory
UR - http://www.scopus.com/inward/record.url?scp=85086397220&partnerID=8YFLogxK
U2 - 10.1016/j.dam.2020.05.033
DO - 10.1016/j.dam.2020.05.033
M3 - Article
AN - SCOPUS:85086397220
SN - 0166-218X
VL - 285
SP - 96
EP - 107
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
ER -