First passage percolation (SI epidemics) on finite graphs

This is a surprisingly little-studied topic, somewhat analogous to the topic of Markov chain mixing times. I became interested in the topic via a very special motivation (as one component in a setting with extra structure: game-theoretic analysis of a "gossip process") which you can find in this 2007 talk and this 2007 draft paper. Here let me stick to the basic "first passage percolation" aspect, viewed as an (SI) epidemic process, as follows.

There are n states i,j,k ....
A symmetric irreducible matrix \theta_{ij} \ge 0 of "infection rates".
Start at time t = 0 with some state i_0 infected.
Any non-infected state j is subject to infection from any infected state i at stochastic rate \theta_{ij}.

In other words, we study the continuous-time Markov chain on subsets, with S(0) = {i_0} and
P(S(t +dt) = S \cup \{j\}|S(t) = S) = \sum_{i \in S} \theta_{ij} dt.
For simplicity let us suppose the matrix is transitive (invariant under some transitive group acting on the state space) so that the initial state does not matter. We are interested in the random process
F_n(t) = n^{-1}(n - |S(t)|)
giving the proportion of population not infected.

Analogy with the MC mixing time window

We could alternatively use the matrix \theta_{ij} to define a continuous-time Markov chain, in which context a standard object of study is the (deterministic) function d(t) giving the variation distance between the time-t distribution (starting from i_0) and the uniform stationary distribution. Write T(a) = \min \{t: d(t) = a}. For many families of chains indexed by n, there are known values t_n and w_n = o(t_n) such that
T_n(a) = (1 + o(1)) t_n for each 0 < a < 1
W_n(a):= T_n(a) - T_n(1-a) is order s_n for each 0 < a < 1/2.
Here t_n is called the mixing time and w_n is the window width.

The General Project is to study the analogs of t_n and w_n for the FPP process. This is slightly more complicated in that F_n(t) is random, so (using the same notation as for the MC case) we now have random T_n(a) and W_n(a). So there are two distinct notions of window: the size of W_n(a) in a typical realization; or the typical spread of T_n(a) between realizations.

By analogy with the theory of MC mixing times, one would like a theory of FPP:

Describe general methods to upper and lower bound T_n(a)
Under what general conditions is W_n(a) of smaller order of magnitude than T_n(a)?
Is there any precise relation between this FP process and MC mixing, or is it just an analogy?

Particular families

Obviously a natural way to start is by considering particular families.

The complete graph case (\theta_{ij} = constant) is well understood -- the model has been often rediscovered - see e.g. sec 2.3 of the 2007 draft paper.
On the discrete N x N torus, take \theta_{ij} = 1 for neighbors {i,j}. This is the usual "first passage percolation on the infinite lattice" model, well understood.
On the discrete N x N torus, take \theta_{ij} = 1 for neighbors {i,j} and = N^{- \alpha} for non-neighbors. This has intresting behavior, described heuristically in the 2007 draft paper and then rigorously by Shirshendu Chatterjee and Rick Durrett: Asymptotic Behavior of Aldous' Gossip Process.
A possible next step is to study the hypercube {0,1}^k, with \theta_{ij} = 1 for neighbors {i,j}. It is natural to conjecture that in this model (and similar models where the RW is rapidly mixing) the FPP behavior is qualitatvely the same as for the complete graph.