Admissibility, Mean, and Spectral Contraction of the Fixed-Epoch Averaging Matrix

Abstract

This note isolates the fixed-epoch averaging result used as Lemma~2.1 of Priority-State Dynamics in Open Multi-Agent Systems: Dynamic Membership and History-Dependent Consensus Endpoints. For a connected simple undirected graph with Laplacian $L$, maximum degree $d_{\max}$, and gain $0<c<d_{\max}^{-1}$, the matrix $P=I-cL$ is nonnegative, symmetric, doubly stochastic, and has positive diagonal. It preserves the nonnegative and strictly positive row-stochastic state spaces and their population mean. On the disagreement subspace it contracts geometrically in Frobenius norm with factor $\rho=\max_{j\ge 2}|1-c\lambda_j(L)|<1$. We give a self-contained proof, identify the exact spectral-contraction range $0<c<2/\lambda_n(L)$ when stochasticity is not required, show why the strict governing gain bound cannot be replaced uniformly by $c\le d_{\max}^{-1}$, exhibit the attainable case $\rho=0$, and separate positive-state invariance from positive-diagonal and contraction requirements. These statements concern one fixed connected averaging epoch only; they do not establish results for delayed, asynchronous, open-membership, or distributed membership-transition dynamics.