Space-Time Connectivity Theorem

• Space-time connectivity theorems are rigorous frameworks that formalize the interplay between spatial and temporal connections using topological, percolation, and categorical methods.
• They provide systematic approaches to generate connectivity structures via transfinite induction and categorical constructions, enabling precise modeling of complex systems.
• Applications span mobile networks, continuum mechanics, and relativistic causal reconstructions, offering robust tools for analyzing dynamic and evolving systems.

Space-time connectivity theorems formalize the relationships between spatial and temporal properties of connectivity in diverse mathematical and applied contexts, including topology, geometric measure theory, percolation models, and network theory. Across these domains, such theorems characterize how connectedness, deformation, and propagation through space and time can be established, witnessed, or reconstructed. The following sections synthesize rigorous results and model frameworks from the mathematical, computational, and physical literature.

1. Abstract Connectivity Structures and Generation

Connectivity structures generalize the concept of connectedness beyond classical topology, allowing for systematic definitions of "connected" subsets without requiring an underlying topology. A connectivity space is defined as a pair $(X, \kappa)$, with $X$ a set and $\kappa$ a family of subsets of $X$ designated as connected. In the "integral" case, every singleton belongs to $\kappa$.

A canonical generation process utilizes a "completion" operator $\Phi$ acting on families of subsets: for a given $\mathcal{A} \subset \mathcal{P}(X)$,

$$\Phi(\mathcal{A}) = \left\{ \bigcup \mathcal{F} : \mathcal{F} \subset \mathcal{A},\, \mathcal{F}\ \text{finite},\, \bigcap \mathcal{F} \neq \emptyset \right\}.$$

Via transfinite induction ($\Phi^{\alpha}$), the minimal connectivity structure containing $\mathcal{A}$ is generated. This framework captures the union and fusion of overlapping regions, relevant for modeling the propagation of connectivity in physical or abstract "space-time" systems (Dugowson, 2010).

2. Categorical Constructions: Limits, Colimits, and Monoidal Products

Connectivity spaces form a category where limits and colimits mirror the gluing and decomposition of local regions:

• Limits of diagrams yield connectivity on the set-theoretic limit, with

$$\kappa(L) = \{ K \subset |L| : \forall i,\, \ell_i(K) \in \kappa(D_i) \}.$$

• Colimits use pushforward structures, taking the minimal connectivity structure making all canonical injections connectivity morphisms:

$$\kappa(C) = [ \{ c_i(K) : i \in I,\, K \in \kappa(D_i) \} ].$$

Critically, Dugowson introduces a tensor product $\boxtimes$ on integral connectivity spaces:

$$\kappa(X \boxtimes Y) = \left[ \{ K_1 \times K_2 : K_1 \in \kappa(X),\, K_2 \in \kappa(Y) \} \right],$$

with associated internal homs establishing a closed symmetric monoidal structure. These categorical features provide analogs for encoding the composite connectivity of multiple interacting space-time regions (Dugowson, 2010).

3. Dynamic and Space-Time Models: Percolation and Currents

Dynamic models formalize space-time connectivity in settings where both spatial and temporal variability are intrinsic. Notable examples include:

• Space-Time Percolation in Mobile Networks: In continuum percolation with mobile nodes, each participant's location evolves over time, and inter-node connections are established if they are within a specified communication radius. The connectivity time between two nodes is measured by

$$\tau_T^{(N)} = \int_0^T 1\left\{ X_s^1 \longleftrightarrow_s^N X_s^2 \right\}\, ds,$$

with limiting behavior captured by

$$\tau_T^{(>)}(X^1, X^2) \le \liminf_{N \to \infty} \tau_T^{(N)} \le \limsup_{N \to \infty} \tau_T^{(N)} \le \tau_T^{(\geq)}(X^1, X^2),$$

where the functions $\tau_T^{(\diamond)}$ depend on spatial percolation thresholds and local percolation probabilities. This result elucidates how temporal windows of connectivity are determined jointly by global (deterministic) spatial constraints and local (probabilistic) cluster attachment probabilities. Robustness is demonstrated through large deviation estimates indicating that extended durations of poor connectivity are exponentially rare (Döring et al., 2013).

• Space-Time Connectivity for Normal Currents: In geometric measure theory, the space-time connectivity theorem for normal currents constructs, for sequences of boundaryless normal cycles $(T_j)$ weakly converging to $T$, a family of space-time trajectories $S_j$ in $[0,1]\times K$ such that

$$\partial S_j = \delta_1 \times T - \delta_0 \times T_j,\qquad \mathrm{Var}(S_j) \to 0$$

and

$$\limsup_{j\to\infty} \|S_j\|_{L^\infty} \le C\cdot \limsup_{j\to\infty} \mathbb{M}(T_j).$$

This result provides not only a convergence criterion (in flat/homogeneous distance) but also a "witness"—a space-time current interpolating between configurations in time, with vanishing variation. The evolutionary, time-parameterized nature distinguishes this from classical static fillings and is valuable in the analysis of defect motion, dislocation dynamics, and rate-independent evolution (Bonicatto et al., 9 Oct 2025).

4. Effective and Algorithmic Notions of Connectivity

For computably compact continua in $\mathbb{R}^n$, effective local connectivity (ELC) is characterized by the existence of an LC-function $f:\mathbb{N}\to \mathbb{N}$ such that small enough neighborhoods are connected inside slightly larger balls. Uniformly local arcwise connectivity (EULAC and SEULAC) provides, respectively, functions (and witnesses) ensuring close points can be connected by arcs of small diameter, with computable parametrizations in the strong form. Under computable compactness, ELC, EULAC, and SEULAC are equivalent—crucially, this provides algorithms to construct explicit connecting arcs given sufficient effective local data. These principles apply broadly in computational topology and inform algorithmic approaches to navigating and simulating space-time models where connectivity must be guaranteed or constructed (Daniel et al., 2011).

5. Topological and Causal Reconstruction in Space-Time Geometry

In Lorentzian geometry, the causal structure of a space-time can be reconstructed from the topology and differential geometry of its space of light rays $\mathcal{N}$ and the family of skies $\Sigma$. The sky map $S : M\rightarrow \Sigma$ is a diffeomorphism under strong causality and sky-distinguishing hypotheses, and each sky $S(p)$ forms a Legendrian submanifold in the canonical contact manifold of null geodesics. Celestial curves in $\mathcal{N}$ with tangent vectors in sky directions correspond precisely to causal, null curves in $M$ up to the sign of associated Legendrian isotopies. Theorems demonstrate that a celestial causal map between light ray spaces of two spacetimes (with null non-conjugacy) must arise from a conformal immersion. Therefore, the totality of skies and the induced contact topology suffices to recover both the differential and causal structure of $M$, illustrating a profound linkage between topological configuration and physical connectivity (Bautista et al., 2013).

6. Numerical and Invariant Measures of Connectivity

Quantitative invariants offer another lens on connectivity:

• For finite integral connectivity spaces, the connectivity index $\omega(X)$ is defined as the maximum height in the generic directed acyclic graph of irreducible connected subsets, measuring the "depth" or layering of connectivity. This invariant, which serves as the connectivity order for links when the splittability structure is transferred, quantifies the complexity of entanglement beyond simple connectedness (Dugowson, 2010).
• Such metrics may plausibly be adapted to cosmological or network models to quantify hierarchical or multi-scale connectivity.

The Brunn–Debrunner–Kanenobu theorem underlies a striking representability result: every finite integral connectivity structure is realized by the splittability structure of a tame link in $\mathbb{R}^3$ or $S^3$, connecting abstract connectivity—possibly including those of space-time discretizations—to concrete embedded topological objects.

7. Practical Applications and Implications

The above frameworks find application in:

• Mobile ad-hoc network analysis via percolation-based theorems guaranteeing time-averaged connectivity robustness (Döring et al., 2013);
• Existence and construction of energy-minimizing evolutions in continuum mechanics, plasticity, and dislocation frameworks by exploiting explicit space-time deformations with controlled variation (Bonicatto et al., 9 Oct 2025);
• Algorithmic navigation, simulation, and path planning in continuous or discretized space-time models through effective local connectivity and computable arcwise connectedness (Daniel et al., 2011);
• Rigorous reconstruction and causal identification in mathematical relativity via the light-ray and sky bundle machinery, connecting contact geometry and conformal class recovery (Bautista et al., 2013);
• Abstract modeling and quantification of space-time linkage complexity using numerical invariants and categorical constructions (Dugowson, 2010).

These developments exemplify the use of topological, categorical, geometric, and algorithmic tools to understand, quantify, and exploit connectivity in static and evolving space-time systems.