Fluid dynamics meet network science: two cases of temporal network eigendecomposition

TL;DR

Two distinct eigendecompositions of temporal networks are proposed, one of which proposes a numerical approximation of the Koopman operator, a linear operator acting on a suitable observable of the graph space which provides the best linear approximation of the latent graph dynamics.

Abstract

Temporal networks, defined as sequences of time-aggregated adjacency matrices, sample latent graph dynamics and trace trajectories in graph space. By interpreting each adjacency matrix as a different time snapshot of a scalar field, fluid-mechanics theories can be applied to construct two distinct eigendecompositions of temporal networks. The first builds on the proper orthogonal decomposition (POD) of flowfields and decomposes the evolution of a network in terms of a basis of orthogonal network eigenmodes which are ordered in terms of their relative importance, hence enabling compression of temporal networks as well as their reconstruction from low-dimensional embeddings. The second proposes a numerical approximation of the Koopman operator, a linear operator acting on a suitable observable of the graph space which provides the best linear approximation of the latent graph dynamics. Its eigendecomposition provides a data-driven spectral description of the temporal network dynamics, in terms of dynamic modes which grow, decay or oscillate over time. Both eigendecompositions are illustrated and validated in a suite of synthetic generative models of temporal networks with varying complexity.

Figures (6)

• Figure 1: Compression of noisy, multiscale periodic TN dynamics. (A,B,D,E) The first two network eigenmodes obtained from the eigendecomposition of the covariance matrix $\mathscr{S}\mathscr{S}^\top$ of a temporal network of $m=100$ network snapshots of $n=50$ nodes with asynchronous, noisy sinusoidal link dynamics of the form $A_{ij}(t)= \cos(2\pi t/T_1 + \eta_{ij}) + \cos(2\pi t/T_2 + \eta_{ij}) + \xi$, with two concomitant periods $T_1$ and $T_2$ (see the text for details). The first two eigenmodes (out of a total of $n^2=2500$) already concentrate about $80\%$ of the variance. (C,F) Projection of the temporal network trajectory onto the space spanned by the first two network eigenmodes, resulting in a periodic orbit with period LCM($T_1,T_2$).
• Figure 2: Illustration of 2-dimensional POD projections of the dynamics of different TNs. In every case TNs consist of $m=5000$ time snapshots, each snapshot having $n=20$ nodes. Projections are made into the two leading POD network eigenmodes (out of the total of 400 eigenmodes). Dynamics include: (A) white temporal network, (B) A Random Walk TN, (C) a DARN(1) process. The energy stored in the first two eigenmodes amounts to (A) $1.5\%$, (B) $10\%$, (C) $2.6\%$. (D) Mean square displacement of the projected trajectory, for the three projections. Results match the behavior of MSD for iid ($\text{MSD}(\tau)\sim \tau^0$), random walks ($\text{MSD}(\tau)\sim \tau$) and autorregressive processes ($\text{MSD}(\tau)\sim \tau$ at short times, and $\text{MSD}(\tau)\sim \tau^0$ at long times), hence suggesting that the projections preserve the dynamical fingerprint of the original TN dynamics, even if the eigenmodes accumulate very modest variance.
• Figure 3: Characterisation of TN dynamics from their scalar POD-projection. (A) ACF($\tau$) of the projected network trajectory into the leading eigenmode ($6\%$ variance) for a periodic TN with period $T=50$ of $m=500$ time snapshots, where each snapshot has a fixed number of nodes $n=50$ and links (i.e. its periodicity is not related to a periodically fluctuating link density). (B) Same as (A), but applied to a DARN(4) model for a TN of $m=10^4$ time snapshots and $n=10$ nodes per snapshot (the leading eigenmode only captures $0.8\%$ of the variance). The panel shows the autocorrelation function $\text{ACF}(\tau)$ in semi-log, correctly retrieving the stylised shape of an autorregressive process (flat autocorrelation up to memory order, followed by an exponential decay), even if the leading eigenmode only captures $0.8\%$ of the variance. (C) Semi-log plot of the ensemble-averaged distance of nearby network initial conditions in the projection of the chaotic TN network (first POD eigenspace, $67\%$ variance) over time, computed via the Wolf method (see methods), applied to a chaotic temporal network of $n=50$ nodes and $m=1000$ time snapshots generated by the dictionary trick from the fully chaotic logistic map (see methods). The distance increases exponentially with a slope in good agreement with the (network) Lyapunov exponent $\lambda=\ln 2$. (D) Empirical temporal network of proximity in a workspace, over roughly 4 days (discarding transients, 163 snapshots where each snapshot time-aggregates links over a window of $1987.08$). In order to remove any trace of link density fluctuations (which could flag easy-to-spot periodicity), the network has been dynamically polluted with substantial amounts of noise by adding many links at random to each snapshot, so as to keep each snapshot with the same number of links. Despite such contamination, the autocorrelation function of the network trajectory in the leading POD mode projection (only $0.9\%$ variance) still captures a clear periodic backbone with period $T=44$, corresponding to 24.3 hours, i.e. daily periodicity.
• Figure 4: Conway's Game of Life as a temporal network. We set $n=30$ nodes (akin to a $30\times 30$ lattice) and run Life for $m=1000$ time steps (network snapshots). (A) Evolution of Life in the space spanned by the first two network eigenmodes (the actual eigenmodes are depicted at the sides, and account for $15\%$ and $4\%$ variance, respectively). This trajectory is similar to a random walk, see Fig. \ref{['fig:eigens']}B. The mean-square displacement is plotted in Panel (C), showing a linear scaling regime up to a cut-off time. (B) Projecting into the leading eigenmode detects the onset of the fixed point in the dynamics at $t\approx 350$, and this coincides with the cut-off of the linear MSD scaling.
• Figure 5: Permutation TN dynamics ${\bf A}(t+1)=\mathbb{P} {\bf A}(t)\mathbb{P}^\top$ for a network of $n=20$ nodes over a total of 500 time snapshots. $\mathbb{P}$ is a randomly chosen permutation matrix which induces a cyclic dynamics of period 17. (A) $20\times 20$ permutation matrix $\mathbb{P}$. (B) Projection of the TN dynamics onto the plane spanned by the first two network eigenmodes. The trajectory is periodic, mimicking the periodic cycle induced $\mathbb{P}$, recovering the periodicity. Observe that such periodicity can already be retrieved from the periodic behavior of the projection in the first POD network eigenmode (which accumulates about 9% of the variance). (C) Spectrum of $\tilde{\mathscr{K}}$ and $\mathbb{P}$. The leading $n$ eigenvalues of $\tilde{\mathscr{K}}$ coincide with those of $\mathbb{P}$, being roots of 1 that yield no growth or decay of dynamic modes, as expected for a permutation operator.