A factorization lemma for the agreement dynamics

In this note, we examine the extent by which the trajectories of the agreement protocol (also known as the Laplacian dynamics) over large-scale networks can be decomposed, or factored, in terms of the agreement trajectories over smaller networks. In this venue, we identify the cartesian product of graphs as a viable means for synthesizing large networks, or decomposing them to smaller-sized ones. Specifically, due to an intricate connection between the Laplacians of a connected graph and those of its "factors," we are able to prove the following two results: (1) the Laplacian dynamics over the cartesian product of a finite set of graphs is the Kronecker product of the Laplacian trajectories over the individual (atomic) graphs, and (2) the Laplacian dynamics over any connected graph admits a factorization in terms of the Laplacian dynamics over its "prime" decomposition.