Cellular automata · finite graphs

Life on graphs

Put a living-or-dead state on every vertex, update all vertices at once, and rotate the graph in three dimensions while the population evolves. The familiar square-grid rule becomes a different dynamical system when the network has irregular degrees or community structure.

Choose the experiment

Changing the topology, size, or cluster settings builds a fresh network. Click or tap any dead vertex to make it alive.

Graph Topology

Edges

Edge turnover Off

After each Life update, this share of the current edges is replaced by randomly chosen non-edges. The edge count stays fixed. In cluster mode, sustained turnover gradually mixes the communities.

Dynamics Life-like rule

A dead vertex is born with 3 living neighbors; a living vertex survives with 2 or 3.

Scale

Side length 9

Degree 8

Initial state and time

Initial density 32%

Speed 5 gen/s

Current state

Rotate the graph. Seed a vertex.

Generation 0 · preparing graph

Loading the 3D graph…

Alive Dead

Paused

Generation 0

Living vertices 0 / 0

Edges 0

Average degree 0.0

The rule uses absolute neighbor counts. On the Moore torus every vertex has eight neighbors, so Conway’s B3/S23 rule is exactly the usual finite periodic Game of Life. On an irregular graph, hubs and low-degree vertices obey the same numerical thresholds despite having different numbers of neighbors; that is why changing the topology—or separating vertices into weakly joined clusters—can change the behavior so sharply. With edge turnover enabled, the displayed graph determines the next state before rewiring prepares the following generation.