Sections 14.3 and 14.4 treat some models whose behavior relates “by duality” to random-walk-type processes. It is possible (see Notes) to fit all our examples into an abstract duality framework, but for the sake of concreteness I haven’t done so. Note that for simplicity we work in the setting of regular graphs, though the structural results go over to general graphs and indeed to weighted graphs.
Fix a -regular -vertex graph . In the voter model we envisage a person at each vertex. Initially each person has a different opinion (person has opinion , say). As time passes, opinions change according to the following rule. For each person and each time interval , with chance the person chooses uniformly at random a neighbor (, say) and changes (if necessary) their opinion to the current opinion of person . Note that the total number of existing opinions can only decrease with time, and at some random time there will be only one “consensus” opinion.
In the coalescing random walk process, at time there is one particle at each vertex. These particles perform independent continuous-time random walks on the graph, but when particles meet they coalesce into clusters and the cluster thereafter sticks together and moves as a single random walk. So at time there are clusters, composed of one or more particles, at distinct vertices, and during each cluster has chance to move to a random neighbor and (if that neighbor is occupied by another cluster) to coalesce with that other cluster. Note that the total number of clusters can only decrease with time, and at some random time the particles will have all coalesced into a single cluster.
Remarkably, the two random variables and associated with the two models turn out to have the same distribution, depending only on the graph . The explanation is that the two processes can be obtained by looking at the same picture in two different ways. Here’s the picture. For each edge and each direction on , create a Poisson process of rate . In the figure, is the -cycle, “time” is horizontal and an event of the Poisson process for edge at time is indicated by a vertical arrow at time .
In the voter model, we interpret time as increasing left-to-right from to , and we interpret an arrow at time as meaning that person adopts ’s opinion a time . In the coalescing random walk model, we interpret time as increasing right-to-left from to , and we interpret an arrow at time as meaning that the cluster (if any) at state at time jumps to state , and coalesces with the cluster at (if any).
So for fixed , we can regard both processes as constructed from the same Poisson process of “arrows”. For any vertices the event (for the voter model)
The opinions of persons and at time are both the opinion initially held by
is exactly the same as the event (for the coalescing random walk process)
The particles starting at and at have coalesced before time and their cluster is at vertex at time .
The horizontal lines in the figure indicate part of the trajectories. In terms of the coalescing random walks, the particles starting at and coalesce, and the cluster is at at time . In terms of the voter model, the opinion initially held by person is held by persons and at time . The reader may (provided this is not a library book) draw in the remaining trajectories, and will find that exactly of the initial opinions survive, i.e. that the random walks coalesce into clusters.
In particular, the event (for the voter model)
By time everyone’s opinion is the opinion initially held by person
is exactly the same as the event (for the coalescing random walk process)
All particles have coalesced by time , and the cluster is at at time .
So , and these two times (which we shall now call just ) do indeed have the same distribution.
We now discuss bounds on . It is interesting that the two models give us quite different ways to prove bounds. Bounding here is somewhat analogous to the problem of bounding mean cover time, discussed in Chapter 6.
Recall from Chapter 4 yyy the definition of the Cheeger time constant . In the present setting of a -regular graph, the definition implies that for any subset of vertices
(14.11) |
(a) If is -edge-connected then .
(b) .
Proof. The proof uses two ideas. The first is a straightforward comparison lemma.
Let be a continuous-time chain on states . Let be such that never jumps by more than , and such that there exist strictly positive constants such that, for each and each state with ,
Then
where refers to mean hitting time for the chain on states with transition rates
The second idea is that our voter model can be used to define a less-informative “two-party” model. Fix an initial set of vertices, and group the opinions of the individuals in into one political party (“Blues”) and group the remaining opinions into a second party (“Reds”). Let be the number of Blues at time and let be the first time at which everyone belongs to the same party. Then
(14.12) |
Cases (a) and (b) now use Lemma 14.10 with different comparison chains. For (a), while both parties coexist, the number of edges being counted in (14.12) is at least . To see this, fix two vertices of different parties, and consider (c.f. Chapter 6 yyy) a collection of edge-disjoint paths from to . Each path must contain at least one edge linking Blue to Red. Thus the quantity (14.12) is at least . If that quantity were then would be continuous time random walk on and the quantity would be the mean time, starting at , for simple random walk to hit or , which by Chapter 5 yyy we know equals . So using Lemma 14.10
(14.13) |
For (b), use (14.11) to see that the quantity (14.12) must be at least . Consider for comparison the chain on with transition rates . For this chain
where is the mean occupation time for simple symmetric random walk and the second term is the speed-up factor for the comparison chain under consideration. Using the formula for from Chapter 5 yyy,
So using Lemma 14.10
(14.14) |
Finally, imagine choosing at random by letting each individual initially be Blue or Red with probability each, independently for different vertices. Then by considering some two individuals with different opinions at time ,
Integrating over gives . But , so the Proposition follows from (14.13) and (14.14).
The following result bounds the mean coalescing time in terms of mean hitting times of a single random walk.
Proof. We can construct the coalescing random walk process in two steps. Order the vertices arbitrarily as . First let the particles perform independent random walks for ever, with the particles starting at first meeting at time , say. Then when two particles meet, let them cluster and follow the future path of the lower-labeled particle. Similarly, when two clusters meet, let them cluster and follow the future path of the lowest-labeled particle in the combined cluster. Using this construction, we see
(14.15) |
Now let . Using subexponentiality as in Chapter 2 section yyy,
(14.16) |
and so
where the final equality is the calculus fact
The result now follows from Proposition 14.5.
The complete graph. On the complete graph, the number of clusters at time in the coalescing random walk model is itself the continuous-time chain with transition rates
Since is the time taken for to reach state ,
Recall from Chapter 7 yyy that in a vertex-transitive graph with small, the first hitting time to a typical vertex has approximately exponential distribution with mean . Similarly, the meeting time for typical has approximately exponential distribution with mean . It seems intuitively clear that, for fixed small , when the number of clusters first reaches these clusters should be approximately uniformly distributed, so that the mean further time until one of the pairs coalesce should be about . Repeating the analysis of the complete graph suggests
Prove that for a sequence of vertex-transitive graphs with , we have .
In the general setting, there is good reason to believe that the log term in Proposition 14.11 can be removed.
Prove there exists an absolute constant such that on any graph
The assertion of Open Problem 14.12 in the case of the torus for was proved by Cox [104]. A detailed outline is given in [132] Chapter 10b, so we will not repeat it here, but see the remark in section 14.3.5 below.
xxx discuss ?
For a simple variation of the voter model, fix a parameter and suppose that each individual independently decides at rate (i.e. with chance in each time interval ) to adopt a new opinion, not previously held by anyone. For this process we may take as state space the set of partitions of the vertex-set of the underlying graph , where two individuals have the same opinion iff they are in the same component of . The duality relationship holds with the following modification. In the dual process of coalescing random walks, each cluster “dies” at rate . Thus in the dual process run forever, each “death” of a cluster involves particles started at some set of vertices, and this partition of vertices into components is (by duality) distributed as the stationary distribution of the voter model with new opinions. This is the unique stationary distribution, even though (e.g. on the -cycle) the Markov chain may not be irreducible because of the existence of transient states.
The time to approach stationarity in this model is controlled by the time for the dual process to die out completely. Clearly , where is the coalescing time discussed in previous sections, and we do not have anything new to say beyond what is implied by previous results. Instead, we study properties of the stationary distribution . A natural parameter is the chance, say, that two random individuals have the same opinion, i.e.
(14.17) |
where is the number of edges with endpoints in different components, under the stationary distribution.
Proof. Run the stationary process, and let and be the partition and the number of edges linking distinct components, at time , and let . Then
(14.18) |
The first term arises from the “voter” dynamics. If an opinion change involves an edge linking components of sizes and , then the change in has expectation
and for each of the edges linking distinct components, opinion changes occur at rate . The second term arises from new opinions. A new opinion occurs in a component of size at rate , and the resulting change in is
Stationarity implies that the expectation of (14.18) equals zero, and so
and the lemma follows.
.
Proof. Clearly is at most the total number of edges, , so the upper bound follows from the lemma. For the lower bound, (14.11) implies
and hence
and the bound follows from the lemma after brief manipulation.
We now consider bounds on obtainable by working with the dual process. Consider the meeting time of two independent random walks started with the stationary distribution. Then by duality (xxx explain)
where denotes a random variable with exponential distribution independent of the random walks. Now is the hitting time of the stationary “product chain” (i.e. two independent continuous-time random walks) on the diagonal , so by Chapter 3 yyy has completely monotone distribution, and we shall use properties of complete monotonicity to get
Proof. We can write , where has exponential distribution and is independent of . Then
For the upper bound, apply Chapter 3 yyy to the product chain to obtain
(recall that is the same for the product chain as for the underlying random walk). So
and the upper bound follows after rearrangement.
xxx discuss coalescent, GEM and population genetics.
xxx genetics already implicit in xxx
Fix . take independent random variables with distribution
and define
so that .
Consider a sequence of vertex-transitive graphs for which . Consider the stationary distribution of the voter model with new opinions, presented in size-biased order. If then
xxx proof
Remark. The same argument goes halfway to proving Open Problem 14.12, by showing
Consider a sequence of vertex-transitive graphs for which . Let be the coalescing time for walks started at independent uniform positions. Then, for fixed , .
xxx argument similar (?) to part of the proof in Cox [104] for the torus.
xxx result