Generating temporal networks with the Ascona model

TL;DR

<3-5 sentence high-level summary> The paper presents the Ascona framework for sampling continuous-time temporal networks using queueing theory, centering on the Markovian $M/M/\infty$ queue to control the timing and duration of links. By decoupling temporal dynamics from connectivity, it enables design of synthetic networks with tunable smoothness and introduces a continuous-time stochastic block model generalization, plus variations for discrete-time structures. The framework leverages Exponential-Duration Events Distanced Exponentially (EDEDE) structures and queue blocks to realize archetypal temporal patterns (Birth/Death, Change of intensity, Merge/Split, etc.) and to generate both link streams and, via aggregation, snapshot networks. It also discusses conflicts, resolutions, and extensions to non-Markovian queues, providing a flexible tool for benchmarking community detection, change-point, scale, and periodicity analyses in temporal networks.

Abstract

We introduce a new sampling method for continuous-time temporal networks based on queueing processes. In particular, we focus on a Markovian version of the model where the links between nodes are Poisson distributed in time and have exponential duration. We highlight the stochastic properties of these temporal structures and leverage them to design synthetic temporal networks with a controllable level of smoothness, which follow patterns relevant for the validation and interpretation of methods for community, scale, change-point, and periodicity detection. Additionally, we show that imposing assortativity constraints on the samples leads to a continuous-time generalization of stochastic block models. Finally, we describe how variations of the model can be used to sample alternative types of structure and temporal networks, especially discrete-time ones.

Figures (5)

• Figure 1: Comparison of analytical formulas and trajectories of 100 $M/M/\infty$ queue samples. In the top plot, we compare the average density of the sample as a function of time with the analytical formula Eq. \ref{['eq:mean_profile']}. The fluctuations of the trajectories are in the square root order of the mean profile, as predicted analytically. The comparison of this analytical formula for the standard deviation with the sample one is displayed in the bottom plot.
• Figure 2: Example of queue blocks $L'$ and $L"$ sharing the same links' repartition in time (sampled from a $M/M/ \infty$ process), but different connectivity structures. For each of the two sampled link streams, we display the adjacency matrix of the instantaneous graphs $G_t'$ and $G_t"$ at three different times $(t \in \{ 20, 50, 80 \})$. On top, we show the ones of $L'$, that has a constant two-block connectivity structure. On the bottom side, we show the ones of $L"$, that does not have any particular structure: all pairs of nodes have the same probability to be connected, hence this could be considered a link stream version of an ER graph.
• Figure 3: Representation of the generation of an EDEDE structure with the Ascona model. (A) The first step consists of drawing i.i.d. exponentially distributed time intervals with rate $\lambda$, or conceptually equivalently, time points based on a Poisson process with rate $\lambda$, over a desired time interval. Each sampled time point corresponds to a starting time $s_i$ of a link of the dynamic network; in this example, there are five of them: $s_1, \dots, s_5$. (B) The second step consists of sampling a time interval for each drawn starting point based on i.i.d. exponentially distributed time intervals with rate $\mu$. Each drawn interval describes the duration of the corresponding link. In this example, we end up with five link intervals: $(s_1, e_1), \cdots, (s_5, e_5)$. (C) Finally, each link is associated with a pair of vertices. For each link, the first element of the pairs is drawn uniformly at random from the node set. Its interacting partner is drawn randomly based on a distribution encoded in the connectivity conditional probability matrix displayed on the left. In this case, the connectivity matrix encodes a two-block structure with high probability of generating links within each block, and low probability of link generation between blocks. We display node names on a vertical axis on the left of the figure. Each link is represented by a coloured rectangle showing the involved nodes and the time interval when it happens.
• Figure 4: Trajectories of 100 samples of a $M/M/ \infty$ queue with changing parameters at time $t_{sw} = 100s$. Before $t_{sw}$, the rate of arrivals is $\lambda_1 = 2.5$ and the service time rate $\mu_1 = 0.1$; after $t_{sw}$, they are equal to $\lambda_2 = 5$, and $\mu_2=20$. We compare the sample mean with the analytical mean from Eq. \ref{['eq:switch_m']} until the time $t_e=200$ when we stop new elements from joining the queue.
• Figure 5: Ascona model sample displaying various events. (a) Birth event of four communities. (b) In the stationary regime, there is a stable configuration with four fully developed communities. (c) At time 100, there is a merge event leading to the formation of two bigger communities. (d) After time 200, the queue decays, and the two communities vanish. (A) Weighted footprint of the network of the interval $[40 s,60s]$ displaying clearly the four-block structure before the merge. (B) Weighted footprint of the interval $[90 s,110s]$ displaying the transition between the four and the two-block structure happening at time $100s$. (C) Weighted footprint of the interval $[140 s,160s]$ displaying the two-block structure after the merge. The colour of each footprint entry depends on the weight from Eq. \ref{['eq:footbprint_w']}. Intuitively, warmer colours indicate higher weights.