Effective dimensional reduction of complex systems based on tensor networks

TL;DR

The paper addresses the challenge of representing the steady-state distribution of Markovian spreading on networks with an exponentially large state space. It introduces a tunable Matrix Product State (MPS) framework, using entanglement entropy as a compressibility diagnostic to control the effective dimensionality via the bond dimension $\\chi$. The authors benchmark the approach on small random graphs against exact solutions and on a 55-node real-world inspired network against Markov Chain Monte Carlo, showing that the MPS can outperform second-order mean-field methods for moderate to large $\\chi$ and reliably capture rare-event statistics. This tensor-network approach provides a principled, scalable method to study non-equilibrium steady states and potential edge-of-chaos behavior in complex networks.

Abstract

The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components $n$. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this work, we introduce a tunable approximation scheme for Markovian spreading models on networks based on Matrix Product States (MPS). By controlling the bond dimensions of the MPS, we can investigate the effective dimensionality needed to accurately represent the exact $2^n$ dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the 'edge of chaos'. We compare the accuracy of the MPS with exact methods on different types of small random networks and with Markov Chain Monte Carlo methods for a simplified version of the railway network of the Netherlands with 55 nodes. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.

Figures (12)

• Figure 1: Overview of the MPS approximation method. a) The allowed transitions of the stochastic $\epsilon$-SIS process: connected nodes can transmit an infection (purple) to their susceptible neighbors (yellow) with rate $\beta$, while they recover with rate $\gamma$. Susceptible nodes are spontaneously infected with rate $\epsilon$. b) The probability vector of a network state (top) is mapped to a Matrix Product State (MPS, bottom). The MPS consists of an array of tensors contracted over bond indices. The unconnected (physical) indices represent the two states the node may be in. c) Each bond in the MPS is optimized by performing a singular value decomposition and then truncating the singular values below a threshold $\delta$, or keeping a maximal of $\chi_{\rm max}$ singular values. The plot on the right shows the singular values of this particular bond, illustrating that we keep the largest singular values (in green) in the truncated diagonal matrix $s^*$. In this way a compressed, lower-dimensional, approximation of the state vector can be obtained.
• Figure 2: A randomly selected Barabási-Albert network with $16$ nodes, displayed on the left with nodes labeled by the original ordering and colored by the Fiedler vector ordering. We compute the MPS representation of the steady-state once without reordering the nodes and once with an ordering based on the first Fiedler vector. As illustrated in the middle, the MPO bond dimensions are decreased by reordering. On the right we see that the Fielder vector ordering leads to a concentration of the mutual information on the diagonal, meaning that more highly correlated nodes are placed closer to each other in the MPS.
• Figure 3: The density (blue, right axis), variance in the density (red, left axis) and the entanglement entropy (black, left axis) for a randomly generated Erdös-Rényi network with $16$ nodes. The network used is shown on the left and colored and labeled according to its first Fiedler vector. The variance in density peaks at a lower $\lambda$ value compared to the entanglement entropy, indicating that the state is less compressible on the endemic side of the phase transition. These results were obtained with $\delta = 10^{-10}$.
• Figure 4: Comparing the accuracy of estimators for local densities $\langle\hat{n}^i\rangle$ using the MPS representation as a function of maximal bond dimension $\chi$ with the mean-field approximations for four random graph generators. We pick four values of $\lambda$, such that the graphs are in the inactive regime ($\lambda = 0.1$), close to maximal variance ($\lambda = 0.55$), close to maximal entanglement entropy ($\lambda = 0.85$) and in the endemic regime ($\lambda = 2$). The error is defined as the Euclidean distance between the approximation of the single node expectation values and those computed from exact diagonalization.
• Figure 5: The entanglement entropy as defined in \ref{['See']}, averaged over 100 realizations of random graphs generated by four graph generators (with parameters given in Fig \ref{['fig:meanfield_comparison']}). We observe that the entanglement entropy converges for bond dimensions of the order of $n$, which is also where the accuracy starts to improve over the second order mean-field approximation.