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 analyzing 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 probability of accessibility from a node $i$ to a node $j$ 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 the 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.

Figure 1

In the upper row, the adjacency graph at each time slot is given. The lower row shows the evolution of the accessibility graph over time.

Figure 2

Node $j$ has not been visited by time $t$. In time slot $t+1$, node $j$ falls into the set of nodes labeled as visited.

Figure 3

Lower bound I: Searching for a clique based on a given threshold $p_{\min }=0.5$. The triplet on each edge represents the probabilities of links being open in time slots $t=1,t=2$, and $t=3$.

Figure 4

Comparing two paths $R_a=I,K,J$ and $R_b=I,K,L,J$ based on their quality (10), $f(R_a\left(10\right))=0.26$ (on the left) and $f(R_b\left(10\right))=0.98$ (on the right).

Figure 5

Different possibilities (events) based on starting from on or off position for a given edge.

Figure 6

Probability of accessibility between two (randomly selected) nodes in a network with $N=20$ nodes. The probability of each edge is uniformly selected from the range [0.05, 1] and remains constant during the observation window of 16 time slots.

Figure 7

Probability of accessibility between two randomly selected nodes for the partially connected network. In this network of $N=20$ nodes, a fraction $\alpha =0.5$ of the edges are assumed to be open with a fixed probability $p=0.1$ and the rest of edges will not be open at any time.

Figure 8

Comparison between the upper bound and Monte Carlo simulations of the probability of accessibility for 40 randomly selected pairs in four time slots, $t=2,t=4,t=6$, and $t=8$.

Figure 9

Accessibility delay distribution (for the experiment setting of Fig. 6). Such delay is defined as the elapsed time for a node $j$ to become accessible from $i$ for the first time (for uniform probabilities of edges).

Figure 10

Accessibility delay distribution (for the partially connected network setting experiment in Fig. 7).

Figure 11

Comparing probability of an edge existing between two specific nodes (randomly selected) obtained from simulations and on-off analytical model for exponentially distributed on-off periods.

Figure 12

Comparing the expected delay derived from the upper bound and estimated edge probabilities over the training phase, and the average delay from the 10 experiments. Due to the relatively small size of the dataset and possibility of unexpected events (e.g., absence of a vehicle from the network) a few points of discrepancy are not surprising.

Figure 13

Pearson's correlation coefficient between the predicted probabilities of accessibility and the empirically estimated values.