complexingow

Complexingow is a fictional mathematical construct used in theoretical computer science and network theory. It describes a two-layer model that pairs a graph with complex-valued node states. Given a graph G = (V,E) and a complex value z_v assigned to each vertex v in V, a complexing operation updates the configuration by computing a linear combination of neighboring values with complex coefficients. The resulting state z' defines a complex-valued dynamics on the network that can be iterated to study convergence, oscillation, or other dynamic regimes.

Origin and usage: The term is not part of standard mathematical literature but is used in didactic and exploratory contexts to illustrate how linear operators on graphs with complex weights behave. The notion echoes standard graph diffusion and spectral methods while emphasizing the role of complex arithmetic.

Formally, a complexingow configuration consists of a graph G=(V,E) and a labeling of vertices by complex numbers z ∈ C^V. A complexing operator is given by a matrix A ∈ C^{V×V} with entries a_{vu} that encode influence of vertex u on v for adjacent pairs; z' = A z. Often A is supported on edges, with a_{vu}=0 when (v,u)∉E.

Key properties include linearity, closure under iteration, and dependence on the spectrum of A. If A is Hermitian or unitary, the dynamics have particular stability or oscillatory behavior. When coefficients are real, the model reduces to ordinary diffusion on graphs; with nonreal coefficients it exhibits richer complex trajectories.

Example: on a triangle with unit neighbor weights, starting from z=(1,1,1), z'=(2,2,2). With z=(1,i,-1), z' has complex components showing nontrivial phase changes, illustrating how A acts on different configurations. Such simple cases illustrate the concept and its potential for exploring complex-valued dynamics on networks.