Diffusion of Information on Networked Lattices by Gossip

-- We study time-dependent dynamics on a network of order lattices, where structure-preserving lattice maps are used to fuse lattice-valued data over vertices and edges. The principal contribution is a novel asynchronous Laplacian, generalizing the usual graph Laplacian, adapted to a network of heterogeneous lattices. The resulting gossip algorithm is shown to converge asymptotically to stable "harmonic" distributions of lattice data. This general theorem is applicable to several general problems, including lattice-valued consensus, Kripke semantics, and threat detection, all using asynchronous local update rules. The use of the graph Laplacian to diffuse information over networks is well-established in classical and contemporary work ranging from opinion dynamics [1] to distributed multi-agent consensus [2] and control [3], synchronization [4], [5], flocking [6], and much more. In the past decade, Laplacians that are adapted to handle vector-valued data, such as graph connection Laplacians [7], [8] or matrix-weighted Laplacians [9], have been revolutionary in signal processing processing [10], [11] and machine learning [12], [13]. While the ultimate form of a generalized Laplacian is as yet not present in applications, there are hints of a broader theory finding its way from algebraic topology to data science. The Laplacian from calculus class and the graph Laplacian are two extreme examples of a Hodge Laplacian .