Extreme Reachability of Continuous Time Random Walks on Networks

Abstract

Extreme events play a central role in networked stochastic processes, governing the fastest information spreading, quickest search, and earliest arrival in many real-world systems. In this paper, we develop a unified short-time framework for extreme reachability of continuous-time random walks on networks. We show that both the many-walker limit and the frequent-resetting limit are controlled by the short-time asymptotics of the first-passage time distribution, which is determined by the network's shortest-path structure and transition rates. By introducing an instantaneous arrival intensity based on shortest-path contributions, we derive explicit scaling laws for extreme first-passage times and demonstrate that these seemingly distinct regimes are governed by the same underlying network quantity. Our framework reveals a universal mechanism by which network topology shapes extreme events in continuous-time dynamics, independent of long-time diffusion properties. We further propose an efficient method to compute the short-time arrival intensity on large networks and validate our predictions on both synthetic and real-world graphs. The results provide a principled basis for quantifying extreme reachability, fastest search, and robustness under restarting in networked systems.