Consider a population structured into $s$ categories, and a matrix $\mathbf{M}$ of size $s\times s$, that projects deterministically the population vector $\mathbf{n}$ of length $s$. All elements of $\mathbf{M}$ and of $\mathbf{n}$ are real and non-negative. We can project the population vector over time:$\mathbf{n}(t+1)=\mathbf{M}\mathbf{n}(t)$ and therefore $\mathbf{n}(t)=\mathbf{M}^t\mathbf{n}(0)$.
Matrix $\mathbf{M}$, the "population projection matrix", can be seen as expressing the "life-cycle" of the population, where $M(i,j)$ is the expected number of individuals in state $i$ at the next time-step that one individual in state $j$ at the current time-step will produce. I am not a graph specialist at all, but I have the impression that $\mathbf{M}$ may be the "incidence matrix" of a weighted directed graph; is that correct ?
Anyway, the question I have is: is there a specific field in spectral analysis or network analysis (or any other field) that studies the sensitivity of the asymptotic behavior of $\mathbf{M}^t$ to the entries of $\mathbf{M}$ ? I suspect this may come down to: what is the effect on the maximum eigenvalue of $\mathbf{M}$ of a change in $\mathbf{M}$?
In population ecology in general such "pertubation analysis" are done by considering that the ranking of magnitudes of eigenvalues are unchanged by the pertubation, and therefore the effect of the pertubation is measured by the effect on the maximum eigenvalue of $\mathbf{M}$ before the pertubation. Here, I am interested in a more general case where the second maximum eigenvalue can "take over" as the maximum eigenvalue as a result of the perturbation.
In particular, I would be interested in the specific cases where $\mathbf{M}$ is "almost" diagonal, that is $\mathbf{M}=\mathbf{D}+\mathbf{\epsilon}$ where $\mathbf{D}$ is a diagonal matrix and $\mathbf{\epsilon}$ a non-negative matrix, such that for instance $\mathbf{1}^T\mathbf{\epsilon}<<\mathbf{1}^T\mathbf{D}$. (by the way, does such a matrix have a name ?).