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graph dynamical system
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
The work on GDSs considers finite graphs and finite state spaces. As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or probabilistic cellular automata over or interacting particle systems when some randomness is included), as well as GDSs with infinite state space (e.g. as in coupled map lattices); see, for example, Wu. In the following, everything is implicitly assumed to be finite unless stated otherwise.
==Formal definition==
A graph dynamical system is constructed from the following components:
* A finite ''graph'' ''Y'' with vertex set v() = . Depending on the context the graph can be directed or undirected.
* A state ''xv'' for each vertex ''v'' of ''Y'' taken from a finite set ''K''. The ''system state'' is the ''n''-tuple ''x'' = (''x''1, ''x''2, ... , ''xn''), and ''x''() is the tuple consisting of the states associated to the vertices in the 1-neighborhood of ''v'' in ''Y'' (in some fixed order).
* A ''vertex function'' ''fv'' for each vertex ''v''. The vertex function maps the state of vertex ''v'' at time ''t'' to the vertex state at time ''t'' + 1 based on the states associated to the 1-neighborhood of ''v'' in ''Y''.
* An ''update scheme'' specifying the mechanism by which the mapping of individual vertex states is carried out so as to induce a discrete dynamical system with map ''F'': ''Kn → Kn''.
The ''phase space'' associated to a dynamical system with map ''F'': ''Kn → Kn'' is the finite directed graph with vertex set ''Kn'' and directed edges (''x'', ''F''(''x'')). The structure of the phase space is governed by the properties of the graph ''Y'', the vertex functions (''fi'')''i'', and the update scheme. The research in this area seeks to infer phase space properties based on the structure of the system constituents. The analysis has a local-to-global character.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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