[Paper Review] Using Markov chains to determine expected propagation time for probabilistic zero forcing

Yu Chan, Emelie Curl+5 more|arXiv (Cornell University)|Jun 25, 2019

Game Theory and Applications12 references5 citations

TL;DR

This paper uses Markov chains to derive an exact formula for the expected propagation time (ept) in probabilistic zero forcing on graphs. It establishes that ept(G) = O(n) for any connected graph G of order n, and provides closed-form expressions and bounds for ept on complete graphs, complete bipartite graphs, and other families, resolving open questions about edge addition effects and logarithmic growth in random graphs.

ABSTRACT

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact formula for expected propagation time. We apply Markov chains to determine bounds on expected propagation time for various families of graphs.

Motivation & Objective

• To develop an exact method for computing expected propagation time (ept) in probabilistic zero forcing using Markov chains.
• To characterize the Markov transition matrix for specific graph families, including complete graphs and complete bipartite graphs.
• To establish tight upper bounds on ept(G) for connected graphs of order n, showing ept(G) = O(n).
• To resolve open questions about whether adding an edge can increase expected propagation time, and to analyze logarithmic growth in random graph models.

Proposed method

• Derives an exact formula for ept(G,B) using the Markov transition matrix M: ept(G,B) = Σr=1∞ r(M^r - M^{r-1})_{1s}, where s is the absorbing state (all vertices blue).
• Constructs Markov matrices for complete graphs K_n, complete bipartite graphs K_{m,n}, n-sun graphs, and n-comb graphs to compute ept values.
• Applies Chebyshev’s inequality and Bernoulli’s inequality to bound the expected number of new blue vertices per round and analyze convergence speed.
• Uses reverse strong induction on |S|, the number of initially blue vertices, to prove ept(G,S) ≤ (e/(e-1))(n - |S|) for any initial set S.
• Leverages the fact that the probability of at least one force in a round is at least (e-1)/e when at least one blue vertex has a white neighbor, leading to a constant expected time for the first force.
• Analyzes the growth rate of blue vertices in rounds, showing that from √n to n/2 blue vertices requires Ω(log n) rounds with high probability, establishing Ω(log n) lower bound for ept(K_{c,n}).

Experimental results

Research questions

• RQ1 Can an exact formula be derived for expected propagation time in probabilistic zero forcing using Markov chains?
• RQ2 Does adding an edge to a graph always decrease or preserve expected propagation time, or can it increase it?
• RQ3 What are the asymptotic bounds on expected propagation time for complete graphs, complete bipartite graphs, and other structured families?
• RQ4 Is the expected propagation time of a connected graph O(n), and can this be proven via probabilistic and matrix-based methods?

Key findings

• The expected propagation time for any connected graph G of order n is O(n), with ept(G) ≤ (e/(e-1))(n - 1) when starting from a single vertex.
• There exist arbitrarily large graphs for which adding an edge increases the expected propagation time, answering an open question from prior work.
• For the complete graph K_n, the expected propagation time grows logarithmically with n, and a closed-form Markov matrix is constructed to compute it exactly.
• For complete bipartite graphs K_{c,n}, the expected propagation time is Ω(log n), showing a logarithmic lower bound that matches known upper bounds up to constants.
• The expected number of rounds to color all vertices in a subgraph R is O(1) when starting with one blue vertex in R, due to a constant expected time for the first force and bounded growth thereafter.
• The method using reverse strong induction on |S| yields a clean upper bound: ept(G,S) ≤ (e/(e-1))(n - |S|), which implies ept(G) = O(n) for any connected graph G.