I am working with the following matrix function: $\mathbf{b}=[I + \alpha \Gamma D]^{-1} (\beta \mathbf{1} + \alpha \Gamma \mathbf{1})$, which describes an equilibrium behavior of all nodes in some networked system.
Notation-wise, $I$ is an identity matrix of size $n \times n$; $\alpha>0, \beta$ are some scalars. Matrix $\Gamma$ is a network adjacency matrix (size $n\times n$), let's say it is symmetric with $0$ diagonal elements and off-diagonal elements $g_{ij}=1$ if there is some sort of connection between nodes $i$ and $j$ and $g_{ij}=0$ otherwise. $D$ is a diagonal matrix with elements $d_{ii}\in[0,1]$. $\mathbf{b}$ describes some actions which each of the nodes in this network undertakes.
To ensure invertibility of $I+\alpha \Gamma D$, I assumed $\alpha \cdot \max_i \sum_{j} g_{ij} < 1$ and hence this matrix is diagonally-dominant and invertible (I realize that this is a very strict condition for invertibility and the latter can be achieved under milder conditions on the minimal eigenvalue of $\Gamma D$, but I wanted to exclude $D$ from this condition).
In order to understand my system better, I was thinking of looking at the first and second order effects first. However, I have trouble understanding which terms to keep and in particular what is the residual term. In other words, I don't understand what do we assume to be small for this approximation (either $\alpha$ or interaction effects overall). I can see at least two ways to move on:
If we just use $[I+\alpha \Gamma D]^{-1} \sim I-\alpha \Gamma D + \alpha^2 (\Gamma D)^2 -\ldots$ then the first-order approximation for $\mathbf{b}$ is just $\beta \mathbf{1}+\alpha \Gamma \mathbf{1}$. What bothers me in this approach is that we excluded the term with the first power of $\alpha$: $-\alpha \beta \Gamma D \mathbf{1}$, which should be there if we assumed $\alpha$ to be small and expand around its powers. However, this approach seems to be better suiting situation when we assume overall interaction effects to be small (i.e. some combination of $\alpha$ and $\Gamma D$).
As I've said above, if we concentrate only on the powers of $\alpha$ then the first order approximation will be $\beta \mathbf{1} +\alpha \Gamma \mathbf{1} - \alpha \beta \Gamma D \mathbf{1}$.
Could you please give me some advice on which approach would be better to use?
Thanks in advance.