Modeling Temporal Networks Using Random Itineraries

We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce time-extended structures with adjustable features. The procedure is first described in a general framework. It is then illustrated in a case study inspired by a transportation system for which the resulting synthetic network is shown to accurately mimic the empirical phenomenology.

Figure 1

Schematic representation of the construction procedure: a weighted directed graph $\mathcal{G}$ can be regarded as a superposition of paths. Unfolding these paths in time results in itineraries (of variable characteristics) that generate a temporal network $\mathcal{N}$.

Figure 2

Prescribed (solid line) and realized (circles) distributions of random walk lengths in the temporal network $\mathcal{N}$. The discrepancy is due to the decay of the number of remaining edges in $\mathcal{G}$ as the construction proceeds.

Figure 3

Distributions of node in degrees, link weights, and node strengths for networks aggregated over intervals of various lengths. For integration lengths $<{10}^3$, the figures display distributions obtained by averaging over all corresponding intervals in $\mathcal{W}$. Power laws (thick black lines) with expected exponents are plotted for reference.

Figure 4

Distributions of activity (a) and inactivity (b) periods aggregated over intervals of various lengths $T_{\mathrm{agg}}=1$, 3, 10, 100, i.e., fractions of nodes that are active at least once in each of (respectively, inactive during) $n$ consecutive intervals of duration $T_{\mathrm{agg}}$ (the maximal value of $n$ is $T/T_{\mathrm{agg}}$).

Figure 5

Left: Autocorrelation function $C(t)$ vs $t$, averaged over 40 realizations of $\mathcal{N}$, for two distinct walk length averages and compared to a null model where all links have been temporally randomized, equivalent to imposing that all walks in the construction have length 1. Right: Number of $\tau$-tolerant temporal paths ($\tau =5$) occurring at least twice vs their length, both in $\mathcal{N}$ and in the null model.

Figure 6

$F_{\mathcal{C},r}(\tau )$ vs $\tau$ for 2 categories of nodes $\mathcal{C}=\mathcal{B}$ (busy nodes) and $\{d=2\}$, and two neighboring ball radii $r=1$ and $r=2$. Each panel displays the curves corresponding to 10 realizations of $\mathcal{N}$. Dashed lines refer to the average activity $\eta =|\mathcal{N}|/T/|\mathcal{S}|$ per day per node.