Meta-Causal Graphs

• Meta-causal graphs are formal structures that generalize cellular automata to arbitrary time-varying graphs, encoding both state evolution and topology changes.
• They enforce strict local interaction rules through axioms of conjugacy, freshness, and uniform continuity, ensuring finite information propagation.
• The framework supports robust operations like composition, invertibility, and local rule implementation, making it applicable to complex systems and dynamic networks.

A meta-causal graph is a formal structure that generalizes the theory of cellular automata to arbitrary, time-varying graphs, encoding both the evolution of vertex states and the topology itself under strong constraints on information propagation and invariance. This framework, designed to formalize and analyze systems where causality and local interactions dictate the dynamics on evolving networks, rests on a set of axioms and theorems that ensure finite-speed information travel, generalizes translation invariance via isomorphism, and demonstrates robust closure properties analogous to those of cellular automata. The resulting mathematical object serves as a unifying language and analytic tool across domains from theoretical computer science and network theory to complex systems and physics.

1. Core Definitions and Axiomatic Foundations

A meta-causal graph is constructed by considering graphs $G$ defined on a countable set of uniquely named vertices $V$, each with a maximum number $\pi$ of ports, vertex states drawn from a set $\Sigma$, and edge states from a set $\Delta$. Each graph is a tuple $(V(G), \sigma, \delta)$, with $\sigma$ a partial function assigning vertex states, and $\delta$ defining edge connectivity via ports.

The central notion is that of a dynamics $F$—a function mapping the set of all such graphs $\mathcal{G}_{(\Sigma, \Delta, \pi)}$ to itself, adhering to:

• Conjugacy (Isomorphism-invariance): For every vertex renaming isomorphism $R$, there exists a conjugate $R'$ such that

$F(R(G)) = R'(F(G)), \, \forall G,$

generalizing translation invariance on grids to arbitrary graphs.

• Freshness: Disjoint graphs remain disjoint under the dynamics:

$$\bigcap V(G^{(i)}) = \emptyset \implies \bigcap V(F(G^{(i)})) = \emptyset,$$

preventing spurious vertex identifications under parallel update.

A causal dynamic additionally incorporates:

• Uniform Continuity: For every output vertex $v'$, the state and connectivity depend only on a disk $G^r(v)$ of radius $r$ in the input graph $G$, centered at any of its antecedents $v \in a(v')$:

$G^r(v) = H^r(v) \implies F(G)_{v'} = F(H)_{v'}.$

• Boundedness: Each input vertex $v$ can be an antecedent of at most $b$ output vertices in any image graph.

Antecedence is formalized via an antecedent codynamics mapping: $$v \in a(v') \Leftrightarrow \forall G, v' \in F(G) \implies v \in G$$.

2. Structure Theorems and Robustness Properties

Several key theorems underpin the theoretical foundation:

• Structure Theorem: A dynamic $F$ is causal (in the sense above) if and only if it is localizable: there exists a local rule $f$ (of radius $r$) such that

$F(G) = \bigcup_{v \in G} f(G^r_v),$

and the local images are glued globally under necessary consistency constraints. This gives an explicit parallel “local rule” implementation, akin to cellular automata.

• Composition and Universality: The composition of two causal dynamics with radii $r_1$ and $r_2$ yields a new causal dynamic with an explicitly computable radius, $r'' = 2 r_1 r_2 + r_1 + r_2$. Furthermore, any causal dynamic of arbitrary radius can be simulated by a radius-1 dynamic on an enlarged state space, making radius-1 universal.
• Reversibility and Invertibility: For finite state spaces, if a causal dynamic $F$ is invertible ($F^\dagger \circ F = \text{Id}$), then $F^\dagger$ is also causal. Proofs use local consistency and compactness arguments, adapting classical results from cellular automata theory.

3. Generalized Symmetries: Isomorphism-Invariance

Classical cellular automata enforce translation invariance by requiring local rules to act identically at every grid point. On arbitrary graphs, this symmetry is generalized to invariance under graph isomorphisms:

$F(R(G)) = R'(F(G)),\quad R \text{ isomorphism},$

where the “conjugate” $R'$ ensures that the dynamics is independent of vertex naming.

This invariance propagates to the antecedent function, enforcing

$R \circ a = a \circ R',$

so that the structure of causal dependencies remains coherent under any relabeling of vertices. This symmetry is foundational for defining local, name-agnostic updates on arbitrary, possibly irregular, graphs.

4. Examples and Domains of Application

Meta-causal graphs subsume and strictly generalize cellular automata, enabling modeling of a much broader class of dynamical systems:

• Cellular Automata on Grids: When the underlying graph is fixed and regular, meta-causal graphs reproduce classical CA, with the familiar local rule acting on a static neighborhood.
• Time-Varying and Inflating Graphs: Dynamics such as “the inflating grid,” where each vertex spawns new vertices and edges at each step, illustrate the framework’s natural accommodation of dynamic topologies. Applications include modeling of network growth, distributed computation, or spacetime discretization (e.g., Regge calculus in theoretical physics).
• Boolean and Dynamical Networks: Encompassing Boolean networks and generative network automata, meta-causal graphs formalize systems where vertices update according to evolving local environments, capturing both state changes and rewiring.

Table 1: Examples of Dynamical Systems Addressed

System Type	Underlying Topology	State Dynamics	Topology Dynamics
Cellular automata	Fixed regular grid	Local update, discrete	None (static)
Boolean network	Arbitrary, possibly dynamic	Local Boolean functions	Allowed
Inflating grid	Expanding grid	Local expansion	Graph grows over time
Regge calculus discretization	Dynamical triangulation	Local geometric rules	Time-varying mesh

5. Comparison with Classical Cellular Automata

While inspired by cellular automata, meta-causal graphs show both deep continuities and significant generalizations:

Similarities:

• Both enforce strictly local information propagation (bounded by radius $r$).
• Both permit the description of global evolution as synchronous application of a local rule.
• On fixed regular graphs, meta-causal graph dynamics reduce to CA, supporting theorems like Curtis–Hedlund–Lyndon in spirit.

Differences:

• Meta-causal graphs allow the topology to evolve (edges and vertices can be created or deleted) under local rules, while classical CA operate on a fixed topology.
• Translation invariance in CA is generalized to isomorphism invariance, enabling local rules that are “agnostic” to vertex naming or graph structure.
• State update and topological rewrite are both natively supported, facilitating modeling of more complex systems such as dynamically reconfiguring networks or discrete physical spaces.

This flexibility is particularly important when modeling scenarios such as mobile ad hoc networks, biochemical reaction/diffusion systems, or spatially distributed physics models where the geometry is not fixed.

6. Conceptual Terms and Applications

The framework naturally subsumes various established concepts:

• Dynamical Networks / Boolean Networks: Time-varying local interactions with either fixed or dynamically changing connectivity.
• Graph Automata / Generative Network Automata: Parallel application of a local update rule—including both state and connection changes.
• Time-Varying / Amalgamated Graphs: Descriptions of evolving topologies, including processes such as node/edge formation or dissolution.
• Applications: Spanning complex systems science (e.g., social or biological network dynamics), distributed computation, and models in discrete spacetime physics.

7. Broader Significance and Theoretical Consequences

The meta-causal graph framework provides an axiomatic, name-agnostic approach for studying how causal dynamics—imposing strict locality and bounded influence—play out in systems with evolving connectivity. Theoretical results ensuring closure under composition and inversion, as well as the universality of radius-1 dynamics, align this theory closely with cellular automata but vastly generalize its scope.

The localizability theorem establishes that, even for the most general causal graph dynamics, local rule constructions suffice—a powerful result with practical implications for simulation and model design. The generalized symmetry principles clarify how locality and “homogeneity” can be appropriately defined and preserved in irregular or dynamic architectures.

In summary, meta-causal graphs yield a unified mathematical apparatus for analyzing and simulating systems where both the state and structure of networks evolve in accordance with strong locality and symmetry constraints. This makes them an essential tool for interrogating phenomena ranging from complex adaptive networks through distributed algorithms to discrete models of fundamental physics (Arrighi et al., 2012).