Contact process for the spread of knowledge

TL;DR

This work studies a spatial variant of the contact process where each site holds a knowledge level $\xi_t(x)\in[0,1]$, updated through rate-$\lambda$ interactions transferring a fraction $\mu$ of knowledge and deaths resetting sites to zero. The total knowledge $\Xi_t=\sum_x \xi_t(x)$ dictates extinction or survival, leading to a phase structure: extinction for $\lambda\le\lambda_c$ or $\mu=0$, and, for $\lambda>\lambda_c$, a unique phase transition in $\mu$; likewise, for fixed $\mu>0$, a unique transition in $\lambda$. The authors prove survival via block constructions that compare the process to supercritical oriented percolation, and extinction via martingale methods showing $\Xi_t$ decays exponentially when $\mu$ is small. They further show that for every $\lambda>\lambda_c$ there exists $\mu_c(\lambda)\in(0,1)$, with survival above and extinction below, and for every $\mu>0$ a corresponding $\lambda_c(\mu)$ with a similar phase transition in $\lambda$. The results yield a detailed phase diagram and highlight how knowledge persistence hinges on the balance between learning and forgetting, with tools ranging from graphical representations and coupling to block constructions and martingale techniques.

Abstract

This paper is concerned with a natural variant of the contact process modeling the spread of knowledge on the integer lattice. Each site is characterized by its knowledge, measured by a real number ranging from 0 = ignorant to 1 = omniscient. Neighbors interact at rate $λ$, which results in both neighbors attempting to teach each other a fraction $μ$ of their knowledge, and individuals die at rate one, which results in a new individual with no knowledge. Starting with a single omniscient site, our objective is to study whether the total amount of knowledge on the lattice converges to zero (extinction) or remains bounded away from zero (survival). The process dies out when $λ\leq λ_c$ and/or $μ= 0$, where $λ_c$ denotes the critical value of the contact process. In contrast, we prove that, for all $λ> λ_c$, there is a unique phase transition in the direction of $μ$, and for all $μ> 0$, there is a unique phase transition in the direction of $λ$. Our proof of survival relies on block constructions showing more generally convergence of the knowledge to infinity, while our proof of extinction relies on martingale techniques showing more generally an exponential decay of the knowledge.

Figures (3)

• Figure 1: Snapshots at time 1000 of the two-dimensional process with various values of the interaction rate $\lambda$ and the fraction $\mu$. When $\mu = 1$, the process reduces to the contact process with 0 = ignorant and 1 = omniscient, but the spatial distribution of knowledge becomes more uniform as $\lambda$ increases and $\mu$ decreases.
• Figure 2: Phase structure of the process $\xi_t$.
• Figure 3: Picture of a path (in black), of its $i$th overlap, and of a double interaction (at time $t_i$). The dashed line at site $x_{i - 1}$ moves upward starting at time $s_i$ until the first death mark, and $\tau_i = s_{i + 1}$ because the death mark appears after time $s_{i + 1}$ moving upward. The dashed line at site $x_i$ moves downward starting at time $s_i$ until the first death mark, and $\sigma_i$ is equal to the time of the death mark because it appears before time $s_{i - 1}$ moving downward.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8