Directed Percolation in Temporal Networks

Directed Percolation in Random Temporal Network Models with Heterogeneities

“…Furthermore, there is typically no discrete structure in the temporal dimension, which is usually modeled as a continuous axis. Nevertheless, if the various concepts such as order parameter, control parameter and cluster sizes are defined carefully, temporal networks and limited waitingtime connectivity can be mapped to directed percolation [22].…”

“…Section III provides an overview of the theoretical results from Ref. [22], that we extend in Section III A by explicitly deriving some critical exponents and scaling relations. Section IV A will lay down the algorithmic techniques that make large-scale simulations of spreading processes on temporal networks possible.…”

“…Similar to static networks, the connectivity of temporal networks passes through a phase-transition. However, close to this critical threshold, temporal networks exhibit different critical behavior as compared to static structures [20][21][22]. For limited-waiting-time connectivity, where the control parameter is δt, this phase-transition can be theoretically understood under some simplifying assumptions about the homogeneous dynamics of connectivity [22].…”

“…However, close to this critical threshold, temporal networks exhibit different critical behavior as compared to static structures [20][21][22]. For limited-waiting-time connectivity, where the control parameter is δt, this phase-transition can be theoretically understood under some simplifying assumptions about the homogeneous dynamics of connectivity [22]. Since there is an embedded direction (or flow) of time, the microscopic dynamics can be fundamentally irreversible with a broken detailed balance and non-equilibrium steady-state.…”

“…The basic approximations taken for these derivations, however, become less valid when the underlying structure deviates from a random graph or if the interaction dynamics become inhomogeneous. In this paper, our goal is to build on the theory laid down in [22] to investigate further the relation between temporal networks and directed percolation. In its epidemic interpretation, directed percolation can be one of the most basic nonequilibrium second-order phase transitions from fluctuating states into so-called absorbing states, which exhibit universal features, determined by symmetry properties and conservation laws.…”