Notation
Let $A$ be the adjacency matrix of a given graph/network. (For simplicity we can assume an undirected graph, but this should work for weighted, directed graphs as well.) Define Katz Centrality as $$ C_{\text{Katz}}(i) = (I - \alpha A')^{-1} \vec{\mathbf{1}}= \sum_{k=0}^\infty \sum_{j=1}^n \alpha^k (A^k)_{ji}, $$ where $n$ is the number of nodes in the graph (so that $A$ is an $n \times n$ matrix) and $\alpha$ is an attenuation factor chosen so that $0 < \alpha <|\kappa|^{-1}$ where $\kappa$ is the principal eigenvalue of $A$.
Since $(A^k)_{ji}$ represents a measure of the number of walks of length $k$ from node $i$ to $j$, the definition of Katz centrality measures the number of walks from one node to another, where each walk of length $k$ is penalized by $\alpha^k$.
Now, consider a generalization of Katz centrality in which we restrict our count of walks to those of length $t$ or less: $$ C_{\text{TKatz}}(i;t) := (I - \alpha A')^{-1}(I - (\alpha A')^{t+1}) \mathbf{\vec 1} = \sum_{k=0}^t \sum_{j=1}^n \alpha^k (A^k)_{ji}. $$ Obviously, $C_{\text{Katz}}(i) = \lim_{t\rightarrow \infty} C_{\text{TKatz}}(i;t)$.
Question
I have a project in which the definition that I've called $C_{\text{TKatz}}(i;t)$ arises. The motivation is to create a variant of Katz centrality in which each "step" in a walk is penalized by $\alpha$, but walks longer than $t$ steps are impossible. Then, for a given graph represented by $A$, I'm interested in analyzing how this centrality $C_{\text{TKatz}}(i;t)$ changes with different values of $t=0,1,2...$. However, I am currently not aware of a definition like $C_{\text{TKatz}}(i;t)$ in the graph theory literature. Has anyone seen any terminology for a concept like this? And, if so, can you point me to some references that discuss it?