Lower bounds for transition probabilities on graphs

The paper presents two results. The first one provides separate conditions for the upper and lower estimates of the distribution of the time of exit from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.     

Spectra of graphs and fractal dimensions II

This paper continues the study of exponentsd(x), d (x), d _R (x) andd (x) for graphG; and the nearest neighbor random walk {X _n } _nN onG, if the starting pointX ₀=x is fixed. These exponents are responsible for the geometric, resistance, diffusion and spectral properties of the graph. The main concern of this paper is the relation of these exponents to the spectral density of the transition matrix. A series of new exponentse, e ,e _R ,e are introduced by allowingx to vary along the vertices. The results suggest that the geometric and resistance properties of the graph are responsible for the diffusion speed on the graph.     

Random walk in an inhomogeneous medium with local impurities

In spite of Sinai's result that the decay of the velocity autocorrelation function for a random walk on ^d (d=2) can drastically change if local impurities are present, it is shown that local impurities can not abolish weak convergence to the Brownian motion if d2.     

Resolvent metric and the heat kernel estimate for random walks

In this paper we introduce the resolvent metric, the generalization of the resistance metric used for strongly recurrent walks. By using the properties of the resolvent metric we show heat kernel estimates for recurrent and transient random walks.     

Characterization and statistical test using truncated expectations for a class of skew distributions

The expectation of left truncated Waring and Pareto distributions is a linear function of the point of truncation. Based on this property, a characterization theorem and statistical tests can be constructed.     

The Einstein Relation for Random Walks on Graphs

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs.     

Load balanced diffusive capture process on homophilic scale-free networks

Diffusive capture processes are known to be an effective method for information search on complex networks. The biased N$N$ lions–lamb model provides quick search time by attracting random walkers to high degree nodes, where most capture events take place. The price of the efficiency is extreme traffic concentration on top hubs. We propose traffic load balancing provided by type specific biased random walks. For that we introduce a multi-type scale-free graph generation model, which embeds homophily structure into the network by utilizing type dependent random walks. We show analytically and with simulations that by augmenting the biased random walk method with a simple type homophily rule, we can alleviate the traffic concentration on high degree nodes by spreading the load proportionally between hubs with different types of our generated multi-type scale-free topologies.     

Spectra of graphs and fractal dimensions. I

Summary In this paper we consider the nearest neighbour Random Walk on infinite graphs. We discuss the connection between the two smallest eigenvalues of the Laplacian of the graph and the diffusion speed of the RW.