Scalar embedding of temporal network trajectories

TL;DR

The paper presents a principled framework to convert temporal networks into low-dimensional, particularly scalar, time series by embedding network snapshots in a way that preserves inter-snapshot distances. It develops PCA- and MDS-based strategies to produce a scalar embedding from a distance-based feature representation and validates these embeddings against both synthetic dynamics (white noise, pulsating, periodic, memory, and chaos) and empirical networks (emails, sociopatterns). The results show that the scalar embeddings retain nontrivial dynamical signatures such as autocorrelation structures, memory timescales, and change points, enabling standard time-series analysis on temporal networks. This approach provides a versatile, interpretable, and broadly applicable pathway to analyze complex temporal networks with linear dimensionality reduction, while highlighting limitations and avenues for extension to nonlinear and kernel-based methods.

Abstract

A temporal network -- a collection of snapshots recording the evolution of a network whose links appear and disappear dynamically -- can be interpreted as a trajectory in graph space. In order to characterize the complex dynamics of such trajectory via the tools of time series analysis and signal processing, it is sensible to preprocess the trajectory by embedding it in a low-dimensional Euclidean space. Here we argue that, rather than the topological structure of each network snapshot, the main property of the trajectory that needs to be preserved in the embedding is the relative graph distance between snapshots. This idea naturally leads to dimensionality reduction approaches that explicitly consider relative distances, such as Multidimensional Scaling (MDS) or identifying the distance matrix as a feature matrix in which to perform Principal Component Analysis (PCA). This paper provides a comprehensible methodology that illustrates this approach. Its application to a suite of generative network trajectory models and empirical data certify that nontrivial dynamical properties of the network trajectories are preserved already in their scalar embeddings, what enables the possibility of performing time series analysis in temporal networks.

Figures (16)

• Figure 1: One-dimensional validation set. Distance matrices from one-dimensional trajectories of different complexity, used to provide a preliminary validation of the methods.
• Figure 2: Autocorrelation analysis of white network trajectories and its embeddings. (A) Distance matrix of a network trajectory of $T=500$ snapshots, where each snapshot is an Erdős-Rényi graph of $N=20$ nodes sampled from $\mathscr{G}(20,0.3)$. (B) Network Autocorrelation Function $\text{nACF}(\tau)$ estimated from the network trajectory, showing the characteristic Dirac-delta shape of uncorrelated signals. (C) Scalar autocorrelation function $\text{ACF}(\tau)$ (alongside a $95\%$ confidence interval of a null model where the one-dimensional signal was shuffled $10^3$ times) associated to the PCA-embedding, correctly capturing the uncorrelated nature of the network trajectory. All four embeddings display a similar $\text{ACF}(\tau)$ as the one displayed in this panel.
• Figure 3: Noisy pulsating network trajectories. We generate a noisy pulsating temporal network trajectory of $N=500$ snapshots, where each link consists in an asynchronous sinusoidal dynamics of period $P=100$ polluted with extrinsic Gaussian noise of standard deviation $\sigma$. The left panels display the distance matrix between network snapshots, for two cases: $\sigma=1$ and $\sigma=4$. The middle panels display the time series of an individual link, showing that whereas for the case $\sigma=1$ the individual links still show some (noisy) periodicity, for $\sigma=4$ any sign of periodicity has been washed off by the noise. The right panels display the scalar embedding of the full network trajectory using the classical-MDS strategy (embeddings from the PCA-based strategies are similar or marginally poorer). The three viable strategies (both PCA-based embeddings and the one based on classic-MDS) are capable to capture the periodic nature of the network trajectory even when at the individual link dynamics the trajectory is essentially indistinguishable from noise.
• Figure 4: Scalar embedding of DARN(p) network trajectories. Scalar embedding of a correlated temporal network trajectory of $T=10^3$ network snapshots generated by a DARN(3) model with parameters $(q,y)=(0.6,0.1)$ and $N=20$ nodes per snapshot. (A) Distance matrix between network snapshots, based on Eq. \ref{['eq:d_snapshots']}. Panels (B)--(D) report the scalar embedding $(z_t)_{t=1}^T, z_t\in \mathbb{R}$ of the network trajectory based on the three viable procedures: (B) classical-MDS embedding, (C) PCA-embedding, and (D) PCA-projection.
• Figure 5: Autocorrelation analysis of DARN(p) network trajectories and its embeddings. (A) Semi-log plot of the Network Autocorrelation Function $\text{nACF}(\tau)$ estimated from the network trajectory $(G_t)_{t=1}^{1000}$, where each snapshot is a network of 20 nodes) generated by the DARN(3) model reported in Fig. \ref{['fig:DARN3signal']}. The network trajectory has memory order $p=3$, and thus we have $\text{nACF}(1)=\text{nACF}(2)=\text{nACF}(3)$, followed by an exponentially decaying tail for $\tau>p$, as indicated by the dashed lines. Panels (B)--(D) report a semi-log plot of the (scalar) autocorrelation function $\text{ACF}(\tau)$ of the scalar time series $(z_t)_{t=1}^{1000}$ (alongside a gray area reporting the $95\%$ confidence interval of a null model which consists in computing such scalar autocorrelation function in an ensemble of $10^3$ times shuffled signals). Each panel reports results for the different scalar embeddings of Fig. \ref{['fig:DARN3signal']}: (B) obtained via classic-MDS embedding,(C) obtained via PCA-embedding, (D) obtained via PCA-projection. All scalar embeddings seem qualitatively capture the two components of the autocorrelation function, with a specially good performance by the PCA-embedding and PCA-projection strategies.