Tensor network dynamical message passing for epidemic models

TL;DR

The paper addresses the trade-off between exact stochastic epidemic simulations and scalable analytical approximations by introducing Tensor Network Dynamical Message Passing (TNDMP), built on Susceptible-Induced Factorization which decouples neighborhoods of susceptible nodes. This factorization enables exact local tensor-network updates and a tunable N-based approximation, unifying PA and DMP as low-order limits while improving accuracy on synthetic and real networks; notably, N-based partitions achieve near-exact results with runtimes comparable to classical heuristics. The method demonstrates superior performance in predicting epidemic thresholds and steady states, including burn-out phenomena, and provides a scalable framework that can extend to temporal dynamics and reverse-time inference. Overall, TNDMP bridges the efficiency of message passing with the expressiveness of tensor networks, offering a flexible, extensible tool for rigorous epidemic analysis on complex networks.

Abstract

While epidemiological modeling is pivotal for informing public health strategies, a fundamental trade-off limits its predictive fidelity: exact stochastic simulations are often computationally intractable for large-scale systems, whereas efficient analytical approximations typically fail to account for essential short-range correlations and network loops. Here, we resolve this trade-off by introducing Tensor Network Dynamical Message Passing (TNDMP), a framework grounded in a rigorous property we term \textit{Susceptible-Induced Factorization}. This theoretical insight reveals that a susceptible node acts as a dynamical decoupler, factorizing the global evolution operator into localized components. Leveraging this, TNDMP provides a dual-mode algorithmic suite: an exact algorithm that computes local observables with minimal redundancy on tractable topologies and a scalable and tunable approximation for complex real-world networks. We demonstrate that widely adopted heuristics, such as Dynamical Message Passing (DMP) and Pair Approximation (PA), are mathematically recoverable as low-order limits of our framework. Numerical validation in synthetic and real-world networks confirms that TNDMP significantly outperforms existing methods to predict epidemic thresholds and steady states, offering a rigorous bridge between the efficiency of message passing and the accuracy of tensor network formalisms.

Figures (5)

• Figure 1: Schematic illustration of tensor network dynamical message passing. (a) The Susceptible-Infected-Recovered (SIR) model, featuring three states and two transition processes. The three states are encoded as three vectors. (b) The tensor $X_G^t$ with $n$ indices representing the probability of $3^n$ states in an $n$-node system. The state tensor evolves by contracting a tensor operator, which can be decomposed to local evolution of each node. (c) Susceptible-Induced Factorization: contracting projector $S$ factorizes the global operator into sub operators with projector $S$ itself and consequently contract onto initial state at last. With an independent initial state, the joint probability given a node susceptible can be split into independent parts. (d) Updating the tensor of a local area by contracting local operator with messages passed in, which only establishes on specific area based on the factorization property. (e) Illustration of exact and approximate partition, where the later is a repartition on each exact region. Obtained regions are circled by gray dashed lines, where the naive calculation is equivalent to assigning entire network as a region.
• Figure 2: Performance evaluation on synthetic networks with feasible regions. (a) A network consisting of $n = 43$ nodes with an ideal exact region partition. The structure contains one intermediate node and seven loops with sizes ranging from 3 to 9. The intermediate node is patient zero, representing the only infectious individual at $t=0$. (b) Temporal evolution of the cumulative infection fraction $f = \frac{1}{n} \sum^{n-1}_{i=0} P(C_i)$, where $P(C_i) = P(I_i) + P(R_i)$. We compare the ground-truth marginals derived via Monte Carlo (MC) simulations with Pair Approximation (PA), Dynamical Message Passing (DMP), and our rigorous method ($N9$). (c) The error in cumulative infection marginals, $E = \frac{1}{n}\sum_{i=0}^{n-1} |P(C_{i}) - P^{MC}(C_i)|$, obtained by our approximation for all non-trivial values of $N \in [3, 9]$, as well as PA and DMP. (d) Final errors contributed by single loops, arranged according to their lengths. (e) A network of $n=313$ nodes generated from a random tree augmented with cliques of sizes 3 to 6; the respective counts for these cliques are [35, 10, 2, 1]. The experiment originates from a single patient zero (colored red). (f) The fraction of cumulative infection $f$ on network in panel (e) over time. (g) Error in cumulative infection marginals through time for existing algorithms compared to our approximation with various $N$. (h) Probability density distribution of cumulative infection marginals at the steady state, comparing rigorous TNDMP ($N6$) with other algorithms.
• Figure 3: Comparative analysis of epidemic dynamics and steady-state distributions in real-world networks. Experimental results for four real-world networks are presented in columns, respectively. (a) Structural sketches of the networks annotated with their average degrees $\langle k \rangle$. (b) Temporal evolution of the cumulative infection fraction $f$. (c) Temporal evolution of the $L_1$ error $E$ in cumulative infection marginals. (d) Temporal evolution of the the derivative $df/dt$. (e) Probability density function of the cumulative infection marginals at the steady state. (f) Scatter plot of the steady-state marginal probabilities $C_i$ for 50 randomly selected nodes. The $y$-axis represents the MC ground truth, while the $x$-axis represents results obtained via TNDMP ($N=9$) and PA. The green dashed line indicates the ideal $y=x$ agreement with the ground truth.
• Figure 4: The pictorial illustration of Susceptible-Induced Factorization on a 1-dimensional chain. Sketch of the tensor contraction of a susceptible projector $\bra{S}$ and a state tensor $\mathcal{P}(t)$. (a) The contraction breaks factorized $\mathcal{T}$ into $\mathcal{T}_{\mathcal{C}}^{i}$. (b) Pictorial function of contracting $\bra{S}$ onto factorized $\mathcal{T}$.
• Figure S1: Exact and approximate partition on an example network and the dolphin social network. (a-c) An application of exact partition and approximate partition with $L$ and $N$. Dashed lines circle non-trivial regions with different colors and other gray edges are trivial regions, while the common nodes are in gradient coloring. (a): Exact partition, (b): Approximation with $L=3$, (c): Approximation with $N=4$. (d) The size of largest region of a real-world social network of dolphins, obtained by approximation with $L$ and $N$. The network has $n=62$ nodes and the largest exact region has a size of 53.