Learning Interpretable Collective Variables for Spreading Processes on Networks

TL;DR

This Letter presents a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology and delivers evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Abstract

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this work, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Figures (13)

• Figure 1: Illustration of our method. Left: The random process is described by its distribution, sampled through $S$ samples per initial network state (anchor). Middle: These distributions are used to learn a low-dimensional parametrization of the transition manifold. Right: A regression step allows for interpretability of the learned CV.
• Figure 2: For the stochastic block model network (left), the transition manifold is a 3-dimensional cuboid (right). The vertices of the cuboid correspond to extreme states $\bm{x}$ where for each cluster either all (filled circle) or no nodes (empty circle) have state $1$.
• Figure 3: Optimal $\Lambda$ from \ref{['eq:zeta']} for Example 1. (a): Data $\varphi_{1}(\bm{x}^k)$ versus optimal fit $\bar{\varphi}_1$. (b)-(d): Optimal $\Lambda$ entries for the respective coordinates plotted as color values on the network.
• Figure 4: Left: optimal $\Lambda_{i,:}$ plotted as color values on the ring-shaped network. Right: $\Lambda_{i,:}$ (blue crosses) and a sine fit (orange line). The collective variables $\varphi_i$ represent the real Fourier coefficients of the distribution of 1's on the ring, since $\varphi_i(\bm{x}) \approx \Lambda_{i,:} \, \bm{x}$ with the $\Lambda_{i,:}$ being sines and cosines of increasing frequencies.
• Figure 5: (a) For the Albert--Barabási network, the optimal $\Lambda$ as in \ref{['eq:zeta']} assigns a large weight to nodes with high degree. (b) After pre-weighting with node degree, cf. \ref{['eq:zeta_weight']}, the optimal $\Lambda$ is constant. Hence, the collective variable describes the degree-weighted count of 1's.