Graph dynamical system¶
A discrete dynamical system built from a graph, local vertex states and neighborhood functions, and an update scheme that induces a global state-transition map.
Core Idea¶
A graph dynamical system constructs global discrete dynamics from local interactions. Vertices carry states; each vertex has a function of its neighborhood; and an update scheme says when or in what order those functions act. Their combination induces a map F on the product of all vertex state sets.
Two graphs matter and must not be confused. The dependency graph specifies who can influence whom, while the phase-space graph has complete configurations as vertices and edges x→F(x). Research seeks local-to-global connections: how connectivity, local functions, and scheduling determine fixed points, attractors, cycles, and reachability. Synchronous and sequential schemes are different model classes, not merely different software implementations.
Scope of Application¶
- Generalized cellular automata. Synchronous local updates operate on arbitrary finite graphs.
- Sequential dynamical systems. An ordered composition of local maps defines evolution.
- Network processes. Finite-state spreading, regulation, or interaction models use graph-local dependence.
- Phase-space analysis. Global fixed points, cycles, basins, and reachability are derived from constituents.
Clarity¶
Specify graph directionality, local state sets, neighborhood convention, every vertex function, and the update scheme. State whether randomness or infinite spaces extend the standard finite setting. Never use the dependency graph and the phase-space graph interchangeably. Inclusion test: A model is a graph dynamical system when graph-local state functions plus a specified update scheme determine a global discrete transition map. Exclusion test: A static graph with labels or a dynamical process whose dependencies are not encoded by graph neighborhoods is excluded. Nearest boundary: A cellular automaton on a regular lattice is a special case, while a generic agent-based model is only a near neighbor unless its local functions and schedule have this explicit graph construction. Exit condition: The identity exits if the graph, local state/function assignment, or update scheme no longer determines the global transition. Common misclassifications: It is not any time-varying data displayed on a graph. It is not the phase-space graph alone. It is not restricted to identical local functions or regular lattices. It is not fully specified until the update scheme is given. Nearest named distinctions: Dynamic graph: Changes its own vertices or edges; a GDS can use a fixed dependency graph. Phase-space graph: Represents global state transitions rather than local dependency. Cellular automaton: A typically regular and homogeneous special case of synchronous GDS. Agent-based model: May use agents and networks without the explicit local-function/update-scheme formalism.
Manages Complexity¶
The framework compresses an exponential global map into a graph plus local rules and schedule. This makes construction and comparison tractable, but the induced phase space can still be enormous. Structural results aim to infer global behavior without enumerating every configuration.
Abstract Reasoning¶
- Choose the vertices and encode allowed dependencies as a graph.
- Assign a state set to each vertex and form the product state space.
- Define each vertex function on its ordered neighborhood state.
- Choose synchronous, sequential, stochastic, or another explicit update scheme.
- Compose local updates to obtain the global transition map F.
- Construct or characterize the phase space and its invariant behavior.
- Vary graph, functions, or schedule separately to attribute changes in dynamics.
Knowledge Transfer¶
The framework transfers across networked processes when interactions can be represented by local functions and a discrete update protocol. It stops at merely graph-indexed observations or continuous dynamics unless those are deliberately reformulated. The portable cargo is local dependency plus scheduled composition into a global map.
Relationships to Other Abstractions¶
Current abstraction Graph dynamical system Domain-specific
Parents (1) — more general patterns this builds on
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Graph dynamical system is a kind of Formal Model Domain-specific
It is explicitly defined by graph-coupled state and update rules.
Hierarchy path (1) — routes to 1 parentless root
- Graph dynamical system → Formal Model → Representation → Abstraction
Neighborhood in Abstraction Space¶
Graph dynamical system sits in a crowded region of the domain-specific corpus (39th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Structures & Algorithms (24 abstractions)
Nearest neighbors
- Prim’s Algorithm — 0.88
- Graphical Game Theory — 0.88
- Loop (Graph Theory) — 0.88
- Prism graph — 0.87
- Cycle Graph (Algebra) — 0.87
Computed from structural-signature embeddings · 2026-10-08