Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

Abstract

A simple lemma bounds s.d.(T )/ET for hitting times T in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result concerns a model of first passage percolation on a finite graph, where the traversal times of edges are independent Exponentials with arbitrary rates. Consider the percolation time X between two arbitrary vertices. We prove that s.d.(X)/EX is small if and only if Ξ/EX is small, where Ξ is the maximal edge-traversal time in the percolation path attaining X.MSC 2010 subject classifications: 60J27, 60K35, 05C82.

The incipient giant component in bond percolation on general finite weighted graphs

“…Bounds of this type are classical on Z d [16] and are at least folklore in more general settings: an explicit statement and martingale proof in our setting is given in [2]. But inspecting the proof of Proposition 7 in [2] shows that the same bound var T n (v n , s) ≤ ET n (v n , s)/w * n (21)…”

“…Rigorous mathematical treatment of bond percolation has focussed on infinite graphs, with "general theory" developed under the assumption of transitivity, that is spatial symmetry. As the 2006 survey [15] In our setting of a sequence of finite edge-weighted graphs, one can readily formalize the idea of giant components being unique as the property sup t C [2] n (t)/|V n | → p 0 as n → ∞…”

“…where C [2] n (t) is the size of the second-largest component at time t. Note that this can be restated in terms of the jumps of (C n (·)), as sup 0≤t<∞ |C n (t) − C n (t−)|/|V n | → p 0 as n → ∞.…”

“…The proof is based on a simple general variance bound described in section 2.1. A technically more complicated application of that bound to first passage percolation on general weighted graphs, plus other simple applications, can be found in [2]. "Big picture" discussions of various random processes over finite edge-weighted graphs can be found in [1] and [3].…”

A Framework for Imperfectly Observed Networks

“…This is a kind of "backwards" technique, in that such functionals may not be very natural in themselves, but one can then seek to relate them to more natural ones. This second technique and some simple examples (involving times to observe triangles or spanning trees) were introduced in [3] and are reviewed in section 4. Such results suggest a more detailed formulation of our estimation program, as follows.…”