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For a random walk on a network, the mean first-passage time from a node $i$to another node $j$ chosen stochastically according to the equilibriumdistribution of Markov chain representing the random walk is called Kemenyconstant, which is closely related to the navigability on the network. Thus,the configuration of a network that provides optimal or suboptimal navigationefficiency is a question of interest. It has been proved that complete graphshave the exact minimum Kemeny constant over all graphs. In this paper, by usinganother method we first prove that complete graphs are the optimal networkswith a minimum Kemeny constant, which grows linearly with the network size.Then, we study the Kemeny constant of a class of sparse networks that exhibitremarkable scale-free and fractal features as observed in many real-lifenetworks, which cannot be described by complete graphs. To this end, wedetermine the closed-form solutions to all eigenvalues and their degeneraciesof the networks. Employing these eigenvalues, we derive the exact solution tothe Kemeny constant, which also behaves linearly with the network size for someparticular cases of networks. We further use the eigenvalue spectra todetermine the number of spanning trees in the networks under consideration,which is in complete agreement with previously reported results. Our workdemonstrates that scale-free and fractal properties are favorable for efficientnavigation, which could be considered when designing networks with highnavigation efficiency.

Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks