Accessibility and Delay in Random Temporal Networks

In a wide range of complex networks, the links between the nodes are temporal and may sporadically appear and disappear. This temporality is fundamental to analyze the formation of paths within such networks. Moreover, the presence of the links between the nodes is a random process induced by nature in many real-world networks. In this paper, we study random temporal networks at a microscopic level and formulate the \emph{probability of accessibility} from a node \emph{i} to a node \emph{j} after a certain number of discrete time units $T$. While solving the original problem is computationally intractable, we provide an upper and two lower bounds on this probability for a very general case with arbitrary time-varying probabilities of links’ existence. Moreover, for a special case where the links have identical probabilities across the network at each time slot, we obtain the exact probability of accessibility between any two nodes. Finally, we discuss scenarios where the information regarding the presence and absence of links is initially available in the form of time duration (of presence or absence intervals) continuous probability distributions rather than discrete probabilities over time slots. We provide a method for transforming such distributions to discrete probabilities which enables us to apply the given bounds in this paper to a broader range of problem settings.

💡 Research Summary

The paper investigates the fundamental problem of determining the probability that a source node i can reach a target node j within a finite observation window of T discrete time steps in a random temporal network. In such networks, each possible directed edge (u, v) at time t appears independently with a possibly time‑varying probability p_uv(t). Because edges can appear and disappear, a temporal path must respect causality: a traveler can only move forward when the required edge becomes available, and may have to wait at intermediate nodes. The event “i is accessible to j by time T” (denoted i T → j) is the complement of the event that no open temporal path exists. Direct computation of this probability is infeasible: the number of possible paths grows exponentially with T, and different paths are positively correlated when they share edges in the same time slot.

The authors first treat a special case where all edges share the same existence probability p(t) at each time step (the network is an Erdős‑Rényi graph that changes over time). By tracking the set W(t) of nodes that have been visited for the first time at step t, they model |W(t)| as a binomial random variable. A recursive formula (Eq. 2) gives the distribution of |W(t)|, and the exact accessibility probability follows from a simple weighting (Eq. 3). This exact solution exploits the homogeneity of edge probabilities and the independence across time slots, providing a baseline for validation.

For the general case with arbitrary, possibly heterogeneous edge probabilities, the paper derives a computationally tractable upper bound using the Harris‑FKG correlation inequality. They introduce a recursively defined quantity α_ij(t) (Eq. 4) that aggregates the influence of all intermediate nodes. The proof proceeds by induction: assuming P(i t → j) ≤ α_ij(t), they show that the probability of reaching j in the next step is bounded by α_ij(t+1). The key step is to treat the family of events “there exists an open path from i to k by time t” as a monotone decreasing family, which allows the application of the FKG inequality to bound the joint probability of multiple such events. Consequently, α_ij(T) serves as a rigorous, yet efficiently computable, upper bound for any network size and any time‑varying probability profile.

Two lower bounds are also presented. The first lower bound searches for a clique subgraph \hat V in which every edge maintains a probability at least p_min throughout the whole observation window. Because a complete subgraph of size | \hat V | guarantees that any node can reach any other in a single time step if all its edges are present, the probability that the clique stays fully connected for T steps can be computed as 1 − (1 − p_min)^{| \hat V |·T}. This yields a conservative but analytically simple lower bound that is especially useful when the network exhibits dense, high‑probability regions. The second lower bound considers a specific set of candidate paths (for example, the shortest‑hop paths) and multiplies their individual existence probabilities, ignoring inter‑path correlations. While this approach can underestimate the true probability, it is computationally cheap and can be tightened by selecting a richer path set.

A practical obstacle in many real‑world applications is that the data often provide continuous‑time distributions for contact durations (e.g., the distribution of how long two individuals stay within communication range). The authors propose a transformation method: given a probability density f_uv(τ) for the length of a contact interval, the discrete probability that an edge is present during time slot t of length Δt is obtained by integrating f_uv over the interval