Detecting periodic time scales in temporal networks

TL;DR

Temporal networks exhibit multi-scale dynamics, including periodic variations in activity and group structure. The authors introduce a framework that maps temporal slices into lossless static representations—Supra-Adjacency ($G_{SA}$) and Event-Graph ($G_{EG}$)—computes a tensor portrait-based dissimilarity $D_*(m)$ via KL divergence, and analyzes its Fourier spectrum to identify dominant time scales. Validation on synthetic Activity-Driven Temporal Networks shows that SA excels at activity-driven periods while EG captures structural changes, and applications to real datasets (school, conference, flights, resistance game) confirm the method's ability to recover meaningful circadian and intra-day scales where baseline activity analysis fails. The approach provides a principled, extensible tool for uncovering time scales in complex time-evolving systems and suggests avenues for extending to other static representations and higher-order network forms.

Abstract

Temporal networks are commonly used to represent dynamical complex systems like social networks, simultaneous firing of neurons, human mobility or public transportation. Their dynamics may evolve on multiple time scales characterising for instance periodic activity patterns or structural changes. The detection of these time scales can be challenging from the direct observation of simple dynamical network properties like the activity of nodes or the density of links. Here we propose two new methods, which rely on already established static representations of temporal networks, namely supra-adjacency matrices and temporal event graphs. We define dissimilarity metrics extracted from these representations and compute their Fourier Transform to effectively identify dominant periodic time scales characterising the original temporal network. We demonstrate our methods using synthetic and real-world data sets describing various kinds of temporal networks. We find that while in all cases the two methods outperform the reference measures, the supra-adjacency based method identifies more easily periodic changes in network density, while the temporal event graph based method is better suited to detect periodic changes in the group structure of the network. Our methodology may provide insights into different phenomena occurring at multiple time-scales in systems represented by temporal networks.

Figures (11)

• Figure 1: Methodology pipeline to measure the time scales of a temporal network $G_T$. From top to bottom: the initial temporal network is divided into sub-temporal networks through a sliding window. The $m^{th}$ sub-network is denoted $G_T^m$. A static representation of each sub-network ($G^m_*$) is generated through the method *. Each $G^m_*$ is described by a 3-dimensional tensor $B_{*}^m(j,k,\tau)$ that encodes information about the paths and distances in the sub-network (see Appendix \ref{['appendix:tensor']}). We compare consecutive tensors with a dissimilarity measure, obtaining the dissimilarity function $D_{*}$. Finally, we compute the Fourier transform of $D_{*}$ and measure the frequencies of the main harmonics.
• Figure 2: Schematic representation of three settings simulated with the Activity-Driven temporal network model with periodic changes of parameters ($N=100$, $\epsilon=0.001$, $\eta = 4$). (a) The Change of activity case presents networks with activity periods of $T_a=200$; (b) the Change of grouping case presents recurrent structural changes with period $T_g=150$; while (c) the Change of activity and grouping setting is defined as a mix of both dynamics. Panels (a-c) display the number of events as a function of time for a realization of each experiment; Gray areas in panels b and c indicate the intervals in which interactions can only occur within groups. Panels (d-f) depict the Fourier transforms of these networks obtained respectively through the SA-method and EG-method, as well as the Fourier transform of the activity timeline. The first and second harmonics of each Fourier transform are shown respectively with a star and a diamond symbol. In each case, the SA-method and EG-method are able to retrieve the correct period of the networks, while the Fourier transform of the activity signal fails in measuring temporal structural changes. In the Change of activity and grouping case, the SA-method identifies the frequency of activity changes as the main harmonic, while the EG-method detects the structural changes frequency as the dominant one.
• Figure 3: Periods corresponding to the two first harmonics measured through the SA-method (panel a) and the EG-method (panel b), for periodic synthetic temporal networks generated through the Change of activity and grouping setting ($N=100$, $\epsilon=0.001$, $\eta = 3$, $\mid T \mid=9200$) with respective periods $T_a$ (x-axis) and $T_g$ (y-axis). For each pair of values ($T_a$, $T_g$), we generate $100$ realizations of the temporal network and apply the SA- and EG-method to extract the two main harmonics. We show in blue around a small black disk (resp. grey disk) the fraction of realizations in which the main frequency (resp. the second main) corresponds to $T_a$, in pink the fraction of cases in which it yields $T_g$, and in yellow the cases in which it corresponds to neither (we consider a tolerance of $10\%$ for both periods). In most cases, both periods are correctly inferred, with the main frequency corresponding to $T_a$ in the SA-method and to $T_g$ in the EG-method.
• Figure 4: Fourier transforms of dissimilarity and activity functions of four real-world data sets (a) a US middle-school, (b) the US flight network, (c) a conference, and (d) the resistance game networks. Dissimilarity functions were calculated by the SA-method (in orange) and the EG-method (in blue), while results computed for the baseline model using activity signals are shown in purple. The highest harmonics are highlighted with a star symbol for each FT, and the corresponding values of the period is indicated below each panel. The parameters of the sliding windows $(t_w, \Delta t_w)$ are (2 minutes, 5 minutes) for the US middle school, (1/3 minute, 1 minute) for the Resistance game, (2 minutes, 10 minutes) for the US flight and (2 minutes, 5 minutes) for the Conference.
• Figure 5: Fourier transform for the data sets US school, Resistance game, US flight and Conference networks shuffled using the two shuffling methods $P_p(\Gamma)$ (panels a-d) and $P_t$ (panels e-h), obtained with the SA-method (orange curve) and the EG-method (blue curve). The period of each original data set is indicated with a black vertical line. For data shuffled using the $P_p(\Gamma)$ method, the original period is never recovered. In the case of the $P_t$ shuffling instead, the SA- and EG methods still measure original periods if the network presents large activity changes (US flight and Conference data sets). In the case of the US middle school network, only the SA-method is able to assess the original time scale as this method performs better to detect activity changes. Finally, none of the method can measure the original period of the Resistance game network shuffled with the $P_t$ method as it does not present any periodic variations.