dynammat

Dynammat is a term used in theoretical and applied mathematics to denote a matrix-based representation of a system that changes over time. It is designed to capture time-dependent relationships in an algebraic form, generalizing the adjacency or interaction matrix of a graph to a dynamic context.

Formally, a dynammat can be described as either a sequence {A(t)} of n×n matrices indexed by time T, or as a three-dimensional array D ∈ R^{n×n×|T|} with entries D_{ij}(t) representing the strength of the relation from i to j at time t. In some formulations, D(t) = A + B(t) where A is a baseline matrix and B(t) encodes temporal changes.

For a social network with n individuals observed over T days, D(t) stores adjacency weights; for a transportation network, D(t) encodes travel times or capacities between hubs at time t. Dynammat can be sparse, symmetric, or directed, and may include negative entries to represent inhibitory effects in certain models.

Analysis: Static methods extend to the dynamic case through instantaneous analysis of each D(t) or through joint treatments such as dynamic eigen decomposition, temporal centrality measures, or tensor factorization of the full D. Laplacian-based dynamics on dynammat yield diffusion models and stability criteria for evolving networks.

Applications include modeling social, communication, and transportation networks, systems biology, finance, and control theory. Related concepts include temporal networks, dynamic graphs, and tensor decompositions used to extract patterns across time.