Continuous-time process for human contact dynamics

Building on the work of Almasan et al. [IEEE Trans. Netw. Sci. Eng. 12, 1649 (2025)10.1109/TNSE.2025.3537162], we propose a continuous-time Markov model for human contact dynamics denoted the continuous random walkers induced temporal graph model (CRWIG). In CRWIG, M walkers move randomly and independently of each other on a Markov graph with N nodes in continuous time. If walkers are in the same state (node of the Markov graph) at time t, a link is created between them in their temporal contact graph G(t), where each walker corresponds to one of the M nodes. We define the exact Markov governing equation that describes the movement of the ensemble of M walkers. We investigate the consequences of the time discretization of CRWIG. We prove that CRWIG is characterized by exponential decay of the initial condition and exponentially tailed intermeeting times of the walkers. We investigate two special cases of CRWIG and derive analytical results supported by simulations. We extend the model to allow for nonexponential sojourn times for the single walkers. The non-Markovian model extension of CRWIG is able to reproduce empirical properties of human mobility observed on data: arbitrary flight length distribution, arbitrary pause-time distribution, and intermeeting time distributions that are power-law with an exponential tail.