TOPICS

# Causal Graph

A causal graph (or causal network) is an acyclic digraph arising from an evolution of a substitution system (Wolfram 2002, pp. 486--524) or other update system (Wolfram). The nodes of a causal graph represent updating events and its edges represent their causal relationships (Wolfram). The graph itself therefore represent the history of the system. The illustration above shows a causal network corresponding to the rules (applied in a left-to-right scan) and initial condition (Wolfram 2002, p. 498, fig. a).

The figure above shows the procedure for diagrammatically creating a causal graph from a mobile automaton (Wolfram 2002, pp. 488-489).

In an evolution of a multiway system, each substitution event is a vertex in a causal graph. Two events which are related by causal dependence, meaning one occurs just before the other, have an edge between the corresponding vertices in the causal graph. More precisely, the edge is a directed edge leading from the past event to the future event.

Some causal graphs are independent of the choice of evolution, and these are called causally invariant.

Wolfram (2023) considers multicomputation using causal graphs. For example, the illustration above depicts the causal graph obtained by computing the Fibonacci number using the standard recursive definition. While all the events (or subevaluations) in any timelike chain must be done in sequence, spacelike-separated events (or subevaluations) don't immediately have a particular relative order. The whole graph can be therefore be thought of as defining a partial ordering for all events (Wolfram 2023).

Causal Invariance, Multiway System

Portions of this entry contributed by Todd Rowland

## References

Gorard, J. and Piskunov, M. "CausalConnectionGraph: Generate a Graph Giving the Relations Between Regions of Causally Equivalent Events in a Causal Graph." https://resources.wolframcloud.com/FunctionRepository/resources/CausalConnectionGraph/.Wolfram Physics Team. "SubstitutionSystemCausalGraph: Generate a Causal Graph From a Substitution System Using Standard Updating Order." https://resources.wolframcloud.com/FunctionRepository/resources/SubstitutionSystemCausalGraph/.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 486-524, 2002.Wolfram, S. "Appendix: Graph Types: Causal Graph ('Spacetime Causal Graph')." https://www.wolframphysics.org/technical-introduction/additional-material/appendix-graph-types/.Wolfram, S. "Expression Evaluation and Fundamental Physics: Causal Graphs and Spacetime." Sep. 29, 2023. https://writings.stephenwolfram.com/2023/09/expression-evaluation-and-fundamental-physics/#causal-graphs-and-spacetime.

## Cite this as:

Rowland, Todd and Weisstein, Eric W. "Causal Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CausalGraph.html