Disentangling the Timescales of a Complex System: A Bayesian Approach to Temporal Network Analysis

TL;DR

The work tackles the challenge of disentangling multiple overlapping timescales in temporal networks. It introduces a Bayesian framework that couples a Hypergeometric Temporal Configuration Model (HTCM) with Minimum Description Length (MDL) change-point inference to identify both the time-resolution that best captures data-generating processes and their associated timescales. By constructing a timescale spectrum based on local minima of the description length and ranking them with topographic prominence, the method reveals fundamental timescales, their harmonics, and resonances, validated on synthetic data and real-world cases (ENRON and DEVS). The approach advances temporal-network analysis by moving beyond discrete change-point detection to a principled, nonparametric, generative description of the full spectrum of dynamic scales, with practical implications for crisis detection, resilience assessment, and the monitoring of complex adaptive systems.

Abstract

Changes in the timescales at which complex systems evolve are essential to predicting critical transitions and catastrophic failures. Disentangling the timescales of the dynamics governing complex systems remains a key challenge. With this study, we introduce an integrated Bayesian framework based on temporal network models to address this challenge. We focus on two methodologies: change point detection for identifying shifts in system dynamics, and a spectrum analysis for inferring the distribution of timescales. Applied to synthetic and empirical datasets, these methologies robustly identify critical transitions and comprehensively map the dominant and subsidiaries timescales in complex systems. This dual approach offers a powerful tool for analyzing temporal networks, significantly enhancing our understanding of dynamic behaviors in complex systems.

Figures (6)

• Figure 1: Can we infer and disentangle timescales? Interaction data originates from the complex interplay of multiple processes, highlighted in (green) and (yellow) in the figure. On the right side of the figure, we visualise how events originating from two distinct processes merge into the time-unfolded temporal network visualised in the centre. As the data-generating processes evolve at different timescales, we show the interactions expected from each process at any point in time as static networks (top and bottom rows). For the observer, edges coming from process A (green) or process B (yellow) are indistinguishable. However, the information about change points (dashed red lines in the plot), i.e., points where data generating processes change, remains hidden in the data. In this article, we show how we can disentangle the different timescales at which complex systems evolve by means of Bayesian network models.
• Figure 2: Investigating the resolution limits of change-point detection. (Left) Inferred time-windows from synthetic data with ground truth $\Delta_{GT} = (22,22,22,33,33,33,33,22,...)$. Different colors are assigned to the two different timescales $\psi_1=22$ and $\psi_2=33$ used to generate the data. The inferred partition is shown as a dashed line. When the number of interactions $M$ is large enough (here $M = 100\,000$, $N=20$, $T=462$), we find the correct partition. (Right) This contour plot illustrates the relative loss in bits between the ground truth partition and the optimal partition found by our change-point detection method. We vary the number of interactions ($M$) and the heterogeneity of the data, represented by the coefficient of variance of degrees. Green regions denote high loss, transitioning to yellow for lower loss. Adequate data volume ($M$ large) ensures precise model performance, accurately identifying the true decomposition. However, as the data volume diminishes and degree variance increases, the model's accuracy in detecting the correct partitions decreases, leading to more pronounced errors. This analysis is conducted with $N = 20$ and $T=105$.
• Figure 3: Optimal time-window partitions for ENRON (Left) and of the fastlane project in the DEVS dataset (Right). In ENRON, there is an evident acceleration in the communication dynamics from approximately 2001 onwards, marking the beginning of the accounting scandal. For DEVS, we observe a regular pattern in the time-window partition. The red line indicates when a core-developer suddenly stopped committing to the GitHub project. This event is identified as a change-point in the optimal partition that sets a transition in the project dynamics right after the shock.
• Figure 4: Timescale Spectrogram. The process described in \ref{['fig:problem']} leaves us with a temporal network $\mathcal{G}_{T}$ recording the interactions generated by processes A (green) and B (yellow). We partition $\mathcal{G}_{T}$ into adjacent time-windows of size $\Delta$. By plotting the description length of a temporal network model as a function of the time-window size $\Delta$, we obtain the (top) plot. The minimum description length (normalised to 0 in the plot) gives us the optimal time-window partitioning for the data. This value ($\Delta=11$) matches with the theoretical optimum, the greatest common divisor between the timescales of process A ($\psi=22$) and B ($\psi=33$). Further, we can observe different local minima in the description length. By plotting their topographical prominence (bottom), we obtain a spectrogram for the different timescales present in the temporal spectrum of the data. Similarly to a classical spectrogram, this plot allows us to identify the most important frequency in the temporal data that we study. The second most prominent minima match the two timescales of the data-generating processes. The remaining local minima correspond to harmonics and sub-harmonics of these fundamental frequencies, shown in (green) and (yellow), or combinations of them, shown in (gray).
• Figure 5: Spectrograms for the ENRON dataset computed on the communication network obtained on the data before 2001 (left) and after 2001 (right). In the top part of the plot, we visualise the normalised DL as a function of the fixed window size $\Delta$ that defines the timescale analysed. In the bottom part, we visualise relative prominence of the timescale contained in the spectrum of the system. The dominant timescale and its harmonics are highlighted in yellow.

Theorems & Definitions (4)

• Definition 1: Change-point
• Theorem 1
• Theorem 2
• Definition 2: Spectrum of a temporal network's dynamics