[Paper Review] Warmth and mobility of random graphs

Sukhada Fadnavis, Matthew Kahle+1 more|arXiv (Cornell University)|Sep 4, 2010

Topological and Geometric Data Analysis18 references3 citations

TL;DR

This paper studies the warmth of Erdős–Rényi random graphs $ G(n,p) $, introducing a nearly sharp threshold for the existence of 'cold maps' from infinite $ d $-branching trees to $ G(n,p) $, which determines the warmth parameter. It shows that for $ p = O(n^{-eta}) $, warmth is concentrated on at most two values, and verifies a conjecture of Lovász on mobility and chromatic number for 'almost all' graphs, while suggesting deep connections between statistical physics and topological graph invariants.

ABSTRACT

A graph homomorphism from the rooted $d$-branching tree $ϕ: T^d o H$ is said to be cold if the values of $ϕ$ for vertices arbitrarily far away from the root can restrict the value of $ϕ$ at the root. Warmth is a graph parameter that measures the non-existence of cold maps. We study warmth of random graphs $G(n,p)$, and for every $d \ge 1$, we exhibit a nearly-sharp threshold for the existence of cold maps. As a corollary, for $p=O(n^{-α})$ warmth of $G(n,p)$ is concentrated on at most two values. As another corollary, a conjecture of Lovász relating mobility to chromatic number holds for "almost all" graphs. Finally, our results suggest new conjectures relating graph parameters from statistical physics with graph parameters from equivariant topology.

Motivation & Objective

• To analyze the warmth parameter of random graphs $ G(n,p) $, particularly its concentration and phase transitions.
• To investigate the existence of 'cold maps' from infinite $ d $-branching trees to $ G(n,p) $, which determine the non-existence of constraints on root coloring.
• To verify a conjecture by Lovász relating mobility and chromatic number for 'almost all' graphs.
• To explore connections between graph parameters from statistical physics (warmth, mobility) and topological invariants (neighborhood complex connectivity).

Proposed method

• Define warmth $ w(H) $ as the largest $ d $ such that no cold map $ heta: T^{d-2} o H $ exists, where a cold map restricts root values via distant vertices.
• Use the equivalence from Brightwell and Winkler: $ H $ is not $ (d+2) $-warm iff there exists a $ d $-stable family of subsets in $ H $.
• Analyze the edge probability $ p = n^{-eta} $ to identify a nearly sharp threshold for the existence of cold maps, using probabilistic methods and graph properties.
• Establish that $ G(n,p) $ a.a.s. has property $ ho^s_3 $ (local connectivity) for $ p $ above a threshold, implying high connectivity of the neighborhood complex $ ext{conn}[ u(G)] $.
• Apply topological tools to show $ w(G) o ext{conn}[ u(G)] + 3 $ under certain $ p $-regimes, linking warmth to topological invariants.
• Use the neighborhood complex $ u(G) $ and its connectivity to derive bounds on warmth, leveraging results from equivariant topology and combinatorial topology.

Experimental results

Research questions

• RQ1 For which values of $ p = n^{-eta} $ does the random graph $ G(n,p) $ a.a.s. admit cold maps from $ T^d $, and what is the threshold behavior?
• RQ2 How concentrated is the warmth parameter $ w(G(n,p)) $ for $ p = O(n^{-eta}) $, and why is it limited to at most two values?
• RQ3 Does the conjecture of Lovász that mobility is a lower bound on chromatic number hold for 'almost all' graphs, and can it be verified in the random graph setting?
• RQ4 Is there a deep structural connection between graph parameters from statistical physics (warmth, mobility) and topological invariants (connectivity of neighborhood complex) in random graphs?
• RQ5 Can the inequality $ w(H) o ext{conn}[ u(H)] + 3 $ be proven or extended as a general conjecture for all finite graphs?

Key findings

• For $ p = O(n^{-eta}) $, the warmth $ w(G(n,p)) $ is concentrated on at most two values, despite warmth not being a monotone graph property.
• A nearly sharp threshold exists for the existence of cold maps $ T^d o G(n,p) $, analogous to a solid-liquid phase transition, with the threshold depending on $ d $ and $ eta $.
• The conjecture of Lovász that mobility is a lower bound on chromatic number holds asymptotically almost surely for $ G(n,p) $, confirming it for 'almost all' graphs.
• For $ rac{1}{s+1} < eta < rac{1+s}{1+s+s^2} $, a.a.s. $ w(G(n,p)) o ext{conn}[ u(G(n,p))] + 3 $, establishing a strong link between topological and statistical physics parameters.
• The inequality $ w(H) o ext{conn}[ u(H)] + 3 $ is verified for complete graphs, Kneser graphs $ KG_{n,k} $ with $ k o 3 $, and graphs with $ ho(H) o 3 $, supporting a general conjecture.
• The paper conjectures a sharp threshold for the warmth of $ G(n,p) $, even though warmth is not monotone, suggesting a universal phase transition behavior in random graphs.