Dynamical Cavity Method for Hypergraphs and its Application to Quenches in the k-XOR-SAT Problem

TL;DR

The paper extends the dynamical cavity method and its backtracking variant to hypergraphs and applies the framework to quenches in the sparse k-XOR-SAT problem, revealing how deterministic, locally greedy dynamics traverse the high-dimensional constraint landscape. By formulating a factor-graph-based BP approach and a dual representation, the authors derive RS and reduced-complexity equations to predict transient evolution and attractor energies, comparing them against mean-field predictions and numerical simulations. They show that DCM outperforms naive mean-field in capturing early-time dynamics and that BDCM can quantify the energy reached by typical trajectories within attractor basins, with distinct behavior for even versus odd degrees and for different k. The results provide a robust methodological tool for analyzing dynamical processes on hypergraphs and identify regimes where quench dynamics approach low-energy configurations or stall in complex attractors, offering insights relevant to both constraint satisfaction and disordered systems. The work thus offers a principled, scalable way to study dynamical processes on multi-variable interactions, with potential applications to a variety of CSPs and spin-glass-related models.

Abstract

The dynamical cavity method and its backtracking version provide a powerful approach to studying the properties of dynamical processes on large random graphs. This paper extends these methods to hypergraphs, enabling the analysis of interactions involving more than two variables. We apply them to analyse the $k$-XOR-satisfiability ($k$-XOR-SAT) problem, an important model in theoretical computer science which is closely related to the diluted $p$-spin model from statistical physics. In particular, we examine whether the quench dynamics -- a deterministic, locally greedy process -- can find solutions with only a few violated constraints on $d$-regular $k$-uniform hypergraphs. Our results demonstrate that the methods accurately characterize the attractors of the dynamics. It enables us to compute the energy reached by typical trajectories of the dynamical process in different parameter regimes. We show that these predictions are accurate, including cases where a classical mean-field approach fails.

Figures (12)

• Figure 1: Factor graph representation of the probability distribution \ref{['eq:prob']} for 3-uniform 3-regular hypergraph. The variable nodes can take value in $S^{k(p+c)}$. The factor nodes denoted by $i,j,k$ have a function $\mathcal{A}_i$ and enforce the evolution constraints defined by the local functions $f_i^{\mathrm{inner}}$ and $f_i^{\mathrm{outer}}$ and the chosen attractor constraints factorized on the nodes $\Xi_i$. The factor nodes denoted by $a,b,c$ have function $\mathcal{B}_a$ who implement the constraints factorized on the hyperedges $\Gamma_a$. The messages $\chi^\rightarrow$ and $\psi^\rightarrow$ are propagated in every direction between the variable and factor nodes of type $i,j,k$.
• Figure 2: Factor graph representation of the probability distribution \ref{['eq:prob-2']} for a 3-uniform 3-regular hypergraph. The variable nodes represent tuples $(i,a)$ of a node and a neighboring hyperedge, taking value in $S^{p+c}\times S^{p+c}$. The factor nodes denoted by $i,j,k$ have function $\mathcal{A}_i$ and the factor nodes denoted by $a,b,c$ have function $\mathcal{B}_a$. The messages $\chi^\rightarrow$ and $\psi^\rightarrow$ are propagated between the factor nodes.
• Figure 3: Evolution of the energy from numerical simulations compared to the dynamical cavity method and the mean-field model. 10 numerical trajectories of the energy of the system simulated on random hypergraphs of size $n=10^4$ with degree (a) $d=4$, and (b) $d=3$ are shown in colors, for each value of the order $k$ considered. The simulations are initialized at random with zero magnetization. The energies obtained from the DCM with $p=7$ and the mean-field energy computed by iterating equations \ref{['eq:mean-field-begin']}-\ref{['eq:mean-field-end']} are shown.
• Figure 4: Convergence to fixed points of trajectories on $4$-regular $k$-uniform hypergraphs. (a) Numerical results - Fraction of rattlers, i.e. nodes that do not stay constant during a cycle, $\rho$, after convergence to an attractor, averaged over $7000$ simulations initialized at random with zero magnetization, as a function of the graph size $n$, (b) BDCM results - $1 - s_{\rm norm}$ with $s_{\rm norm}=\frac{s}{\log{2}}$ the normalized entropy of attractors of length $c=1$ as a function of the incoming path length $p$, obtained from \ref{['eq:free-entropy-2']}.
• Figure 5: Energy of the attractor for typical trajectories obtained from the BDCM and numerical simulations. Comparison of the BDCM result for the energy $e_{\mathrm{attr}}$ of a typical attractor of length $c=1$, with the energy obtained from numerical simulations initialized at random with zero magnetization performed for $n$ going from $10^2$ to $10^4$ and linearly extrapolated at $\frac{1}{n}\rightarrow 0$. For the BDCM, both the result obtained with the largest $p$ considered and the extrapolated value at $\frac{1}{p}\rightarrow 0$ are presented. The ground state energies and the energies at which the phase transition between replica symmetry and dynamical 1-step replica symmetry breaking occurs (d1RSB) for the $k$-XOR-SAT model with spin glass interactions are shown as a reference. (a) The results are shown for $4$-regular hypergraphs with varying degree $k$. The procedure to compute the d1RSB energies is detailed in Appendix \ref{['app:RSB']} and the ground state energies can be found in spinglass-antiferromagnetick=3PITTEL_SORKIN_2016. (b) The degree is fixed at $k=3$ and the results are plotted for growing degrees $d$. The energy is multiplied by $\sqrt{\frac{d}{2k}}$ compared to its definition in (\ref{['eq:energy_def']}) to make the limit $d\rightarrow\infty$ converge to the fully connected $p$-spin model Wang_2021-fully-connectedGuiselin_2022-fully-connectedMontanari_2003-energies-k=3. The ground state and 1dRSB energy for the fully connected model are shown as dashed lines. Those energies, both for growing $d$ and for the fully connected case, were computed in Montanari_2003-energies-k=3 and are here multiplied by $\sqrt{\frac{k}{2d}}$ to match the rescaling of the BDCM and numerical energies. The complete results of the BDCM and the numerical simulations, that were used to obtain the extrapolated values, are shown in Fig. \ref{['app::fig:convergence-energy']} in Appendix \ref{['app:additional-even']}, and all BDCM results are detailed in Table. \ref{['app::tab:BDCM-even-degree-evenk-stay']}. The uncertainty on the linear fits used to obtain the $\frac{1}{n}\rightarrow 0$ extrapolation for the numerical results are too small to be visually observed. The BDCM energy with the largest $p$ considered represents an upper bound for the exact value at $p\rightarrow \infty$.