Minority Takeover in Majority Dynamics: Searching for Rare Initializations via the History Passing Algorithm

TL;DR

This work investigates how little initial bias is needed to compel a synchronous majority dynamics to consensus on large random $d$-regular graphs. By applying the backtracking dynamical cavity method (BDCM) and its replica-symmetry-breaking extensions, it estimates minimal initial magnetizations that guarantee $+1$ consensus within a fixed horizon, revealing minority takeover for $d\ge 4$. It then introduces the History Passing Reinforcement (HPR) algorithm, which uses BD CM marginals to construct explicit minority initializations that lead to $+1$ consensus, performing better than simulated annealing but not reaching the lowest theoretically predicted biases due to replica-symmetry-breaking effects and potential clustering (OGP) of solutions. The results demonstrate an Mpemba-like effect where strategically chosen, seemingly unfavorable initial states reach consensus faster than typical random ones, and they provide insights into algorithmic hardness and the structure of the solution space in dynamical CSPs on sparse graphs. Overall, the work advances both the theoretical understanding of non-equilibrium dynamics on graphs and practical methods for steering complex systems toward desired global states.

Abstract

We investigate how much bias in the initial configuration is required to drive global agreement in synchronous, deterministic majority dynamics on large random $d$-regular graphs. Nodes take values $\pm 1$ and update their states at each discrete time step to align with the majority of their neighbors. Using the backtracking dynamical cavity method (BDCM), we estimate the minimal fraction of initial $+1$ nodes required to achieve a $+1$ consensus in $p$ time steps. Our analysis predicts that for $d\geq4$ an initial global minority of $+1$ nodes is sufficient to quickly steer the entire system toward consensus on $+1$. We then investigate whether such initial conditions can be determined explicitly for a given large random regular graph. To this end, we introduce a new algorithm, which we name history-passing reinforcement (HPR), designed to find such initial configurations with a minority of $+1$ nodes. We find, as a main result, that the HPR algorithm finds initial configurations where the minority takes over the majority for $d$-regular random graphs with $d\geq4$. The HPR algorithm outperforms standard simulated annealing-based methods, but does not reach the lowest densities predicted by the BDCM. Rather, the lowest density achievable by the algorithm is near the onset of a dynamical one-step replica symmetry breaking (d1RSB) phase, which we estimate using a one-step replica symmetry breaking (1RSB) formulation of the BDCM. While we focus on the majority dynamics and random $d$-regular graphs, the algorithm can be extended to other dynamical rules and classes of sparse graphs.

Figures (15)

• Figure 1: Always-stay majority dynamics on $4$-regular random graphs.(Left) Replica symmetric entropy on initializations going to a positive consensus in $p$ steps as a function of the initial magnetization obtained from for $d=4$ and $p=1,2,3$. We mark the replica symmetric lower bound $m^*_{\rm RS}(p)$, and the magnetization $m_{\rm sample}^*(p)$ as the one at which $+1$ consensus becomes the entropically dominating attractor BDCM. The gray line represents the total entropy of all configurations. (right) Zoom into the region where the entropy becomes negative for the $p=1$ case. The red line shows the complexity (eq. \ref{['complexity def']}) at Parisi parameter $r=1$. The dashed grey lines denote the initial magnetizations at which the and phases emerge, as well as the point where the entropy crosses $0$.
• Figure 2: Extrapolation of the RS minimal initial magnetization leading to $+1$ consensus under majority dynamics with always-stay tie breaking on $d$-. (Left) $m^*_{\rm RS}(d)$ for $p=1$ as a function of $1/\sqrt{d}$. The linear fit for points with $d\geq40$ ($1/\sqrt{d}\leq0.158$) is shown. (Right) $m^*_{\rm RS}(p)$ for $d=3$ as a function of $1/p$. All points are used in the fit.
• Figure 3: Thresholds of $m^*_{\rm RS}, m^*_{\rm sRSB}$ and $m^*_{\rm dRSB}$ (vertical lines from left to right) for $p=1$ and different values of $d$. Each black point represents the complexity $\Sigma$ obtained from the 1RSB equations for a single $\lambda$, and the error on the sample mean is indicated by error bars in $x$ and $y$ over 10 samples. The red curve is a quadratic fit of this data, which ends at $m^*_{\rm dRSB}$ from Table \ref{['tab:1RSB']}.
• Figure 4: Factor graph construction for the probability distribution. (Left.) Part of the original tree graph $G$. (Middle.) Naive construction of the factor graph, where variable nodes (circles) correspond directly to the nodes of $G$. The addition of factor nodes in this manner produces short cycles which is problematic for the procedure. (Right.) Factor graph in the edge-dual representation of the graph $G$ with pairs of trajectories $(\stackunder[1pt]{$x$}{}_i,\stackunder[1pt]{$x$}{}_j)$ as variable nodes. This construction preserves the tree structure.
• Figure 5: Derivation of the scheme in the setting. The tree structure of the factor graph is utilized to rewrite partition function $Z$ via the partial partition functions $V$ and $R$, for which we obtain the local iterative schemes \ref{['V via R']} and \ref{['R via V']}.