First-passage time distribution for random walks on complex networks using inverse Laplace transform and mean-field approximation

We obtain an exact formula for the first-passage time probability distribution for random walks on complex networks using inverse Laplace transform. We write the formula as the summation of finitely many terms with different frequencies corresponding to the poles of Laplace transformed function and separate the short-term and long-term behavior of the first-passage process. We give a formula of the decay rate $beta$, which is inversely proportional to the characteristic relaxation time $tau$ of the target node. This exact formula for the first-passage probability between two nodes at a given time can be approximately solved in the mean field approximation by estimation of the characteristic relaxation time $tau$. Our theoretical results compare well with numerical simulation on artificial as well as real networks.