# Accessibility and Delay in Random Temporal Networks

## TL;DR

This paper investigates the probability of node accessibility in random temporal networks, providing bounds and exact calculations for specific cases, and extends the analysis to networks with probabilistic link durations.

## Contribution

It introduces bounds and exact formulas for accessibility probabilities in random temporal networks, including methods to handle continuous-time link presence distributions.

## Key findings

- Derived upper and lower bounds for accessibility probability.
- Obtained exact accessibility probability for networks with identical link probabilities.
- Extended analysis to networks with probabilistic link durations.

## Abstract

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.