Eigentime Identity Research Articles

Eigenvalue spectrum of transition matrix of dual Sierpinski gaskets and its applications

The eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it. In this paper, we study the eigenvalues of the transition matrix of the dual Sierpinski gaskets embedded in d-dimensional Euclidean spaces. We obtain all the eigenvalues, as well as their corresponding degeneracies, by making use of the spectral decimation technique. We then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them.

Mean first-passage time for random walks on undirected networks

In this paper, by using two different techniques we derive an explicit formula for the mean first-passage time (MFPT) between any pair of nodes on a general undirected network, which is expressed in terms of eigenvalues and eigenvectors of an associated matrix similar to the transition matrix. We then apply the formula to derive a lower bound for the MFPT to arrive at a given node with the starting point chosen from the stationary distribution over the set of nodes. We show that for a correlated scale-free network of size $N$ with a degree distribution $P(d)\sim d^{-\gamma}$, the scaling of the lower bound is $N^{1-1/\gamma}$. Also, we provide a simple derivation for an eigentime identity. Our work leads to a comprehensive understanding of recent results about random walks on complex networks, especially on scale-free networks.

Eigentime identity for asymmetric finite Markov chains

Two kinds of eigentime identity for asymmetric finite Markov chains are proved both in the ergodic case and the transient case.

Eigentime identity for transient Markov chains

An eigentime identity is proved for transient symmetrizable Markov chains. For general Markov chains, if the trace of Green matrix is finite, then the expectation of first leap time is uniformly bounded, both of which are proved to be equivalent for single birth processes. For birth–death processes, the explicit formulas are presented. As an application, we give the bounds of exponential convergence rates of (sub-) Markov semigroup P t from l ∞ to l ∞ .

The eigentime identity for continuous-time ergodic Markov chains

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generatorL. When the essential spectrum is empty, let {0 = λ0< λ1≤ λ2≤ ···} be the whole spectrum ofLin L2. Then ∑n≥1λn-1< ∞ implies the existence of the spectral gapαofLin L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1λn-1< ∞ if and only ifα> 0.

The eigentime identity for continuous-time ergodic Markov chains

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generator L. When the essential spectrum is empty, let {0 = λ0 < λ1 ≤ λ2 ≤ ···} be the whole spectrum of L in L2. Then ∑n≥1 λn-1 < ∞ implies the existence of the spectral gap α of L in L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1 λn-1 < ∞ if and only if α > 0.