[Paper Review] WARM percolation on a regular tree in the strong reinforcement regime

Christian Hirsch, Mark Holmes+1 more|arXiv (Cornell University)|Sep 16, 2020

Stochastic processes and statistical mechanics7 references4 citations

TL;DR

This paper constructs a WARM (Weighted Average Reinforcement Model) process on a regular rooted tree with uniformly bounded firing rates, demonstrating that for any α > 1, infinite connected components of edges reinforced linearly often (i.e., surviving edges) almost surely emerge. The key result establishes a phase transition in the strong reinforcement regime: while finite-degree graphs typically suppress infinite components, this work constructs explicit counterexamples on regular trees where infinite survival components persist due to strategic reinforcement dynamics governed by α > 1.

ABSTRACT

We consider a class of reinforcement processes, called WARMs, on tree graphs. These processes involve a parameter $α$ which governs the strength of the reinforcement, and a collection of Poisson processes indexed by the vertices of the graph. It has recently been proved that for any fixed bounded degree graph with Poisson firing rates that are uniformly bounded above, in the very strong reinforcement regime ($α\gg 1$ sufficiently large depending on the maximal degree), the set of edges that "survive" (i.e. that are reinforced infinitely often by the process) has only finite connected components. The present paper is devoted to the construction of an example in the opposite direction, that is, with the set of surviving edges having infinite connected components. Namely, we show that for each fixed $α>1$ one can find a regular rooted tree and firing rates that are uniformly bounded from above, for which there are infinite components almost surely. Joining such examples, we find a graph (with unbounded degrees) on which for any $α>1$ almost surely there are infinite connected components of surviving edges.

Motivation & Objective

• To resolve a referee’s query from prior work questioning whether infinite connected components of surviving edges can exist in WARM processes on bounded-degree graphs.
• To construct explicit examples of WARM processes on regular trees with uniformly bounded firing rates where infinite components of linearly reinforced edges almost surely occur.
• To demonstrate that for any α > 1, such infinite components can be constructed, even when the graph has bounded degree.
• To establish a critical threshold in reinforcement strength where the behavior shifts from finite to infinite component formation.

Proposed method

• The authors analyze a Pólya urn model with one dominant color (the 'cat') and many minor colors (the 'mice'), initialized with m balls of color 0 and 1 ball each for n colors, with reinforcement parameter α > 1.
• They prove that with high probability (greater than 4/5), only color 0 is selected more than 2m−1 times, while at least five other colors are selected exactly m−1 times.
• The proof uses a coupling with i.i.d. uniform random variables to define edge selection rules at each vertex firing event, ensuring the process is well-defined on the tree.
• A key technical tool is the analysis of the partition function of a sum of independent random variables, showing it grows as a power of the argument, establishing sub-exponential tail behavior.
• The authors derive lower and upper bounds on the tail probabilities of the sum of reinforcement contributions, using integral estimates and asymptotic expansions.
• They apply these bounds to show that the probability of a given edge being selected linearly often decays slowly enough to allow infinite connected components to form almost surely.

Experimental results

Research questions

• RQ1 Can infinite connected components of surviving edges exist in a WARM process on a bounded-degree graph with uniformly bounded firing rates, for α > 1?
• RQ2 What conditions on the reinforcement strength α and initial configuration allow for persistent, linearly reinforced edges to form infinite clusters?
• RQ3 Is it possible to construct a regular tree and firing rates such that the set of linearly reinforced edges has infinite components almost surely, even when α is not extremely large?
• RQ4 How does the behavior of a Pólya urn with one dominant color and many minor colors evolve under α > 1, and what does this imply for edge selection in WARM processes?
• RQ5 What is the minimal growth rate of the partition function that supports the existence of infinite surviving components in such models?

Key findings

• For any α > 1, there exists a regular rooted tree and uniformly bounded firing rates such that the set of edges reinforced linearly often has infinite connected components almost surely.
• With high probability (greater than 4/5), in a Pólya urn with initial state (m,1,…,1), only the dominant color (0) is selected more than 2m−1 times, while at least five other colors are selected exactly m−1 times.
• The survival of multiple minor colors at a fixed level (m−1) despite strong reinforcement of the dominant color enables the formation of persistent, linearly reinforced paths in the tree.
• The partition function of the reinforcement sum grows as a power of the argument, specifically as m^β for β < (α−1)/α, which ensures sub-exponential decay of tail probabilities.
• The probability that a given edge is selected linearly often decays slowly enough—slower than any exponential—that infinite connected components of such edges can form almost surely.
• The construction implies that for any α > 1, an unbounded-degree graph can be formed by joining such regular trees, on which infinite components of surviving edges exist almost surely.