mathematics//graph theory

How a network organizes collective behavior. A graph records which nodes are connected and how strongly; everything else (what quantity lives on the nodes, what law moves it, what the eigenvalues mean) is a separate decision layered on top.

How a network organizes collective behavior. A graph records which nodes are connected and how strongly; everything else (what quantity lives on the nodes, what law moves it, what the eigenvalues mean) is a separate decision layered on top.

adjacency and degree describes the connections. The graph Laplacian sums differences across them, and its normalized versions turn absolute sums into local averages.

nodal capacity adds storage, which is what turns a balance into a rate. diffusion and consensus describes the relaxation that follows, and the spectral gap says which patterns persist.

spectral coordinates separates two uses of the same eigenvectors: decomposing a state into modes, and giving each node new coordinates.

Two worked cases hold the pieces together: three-tank network (balance, capacity, modes and a random walk on one topology) and two-community graph (a slow mode that separates two groups).

The same topology can carry a physical network of levels, a random walk of probabilities, or a similarity graph over data (spectral clustering). State, units, capacities and clock must be declared each time; an edge can be a conductance, a dependence or a similarity, and its meaning does not travel automatically between models.