Continuum graph dynamics via population dynamics: well-posedness, duality and equilibria

TL;DR

The paper develops grapheme theory to model dynamic graphs as continuum objects embedded in ultrametric measure spaces, extending graphon limits by incorporating full space-time histories via genealogy-valued population dynamics. Graphemes are defined as equivalence classes of triples $(\mathcal{I}^*,h,\mu)$ with a binary edge function and sampling measure, endowed with strong equivalences to enable a martingale-problem formulation. The main contributions prove well-posed martingale problems, strong Markov and Feller properties, and diffusion-type path behavior for grapheme dynamics, establishing duality with Kingman-like coalescents and identifying non-trivial equilibrium laws (Poisson-Dirichlet and Moran-gamma related) for the long-time behavior. The framework supports finite-graph approximations, marked and varying-size extensions, and non-completely connected equilibria, offering a rigorous route to space-time graph limits and connecting with population genetics methods. Overall, graphemes provide a principled, analyzable bridge between dynamic networks and genealogical-population models, enabling precise control of equilibrium statistics and long-term behavior in complex evolving graphs.

Abstract

This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph densities, graphemes are capable of modeling the full space-time evolution of graphs. A grapheme is an equivalence class of triples: (Polish space, symmetric {0,1}-valued connection function, sampling probability measure). We focus on embeddings in ultrametric spaces, encoding the graph history and linking directly to population dynamics models. Graphemes utilize stronger equivalences (homeomorphism, isometry) than graphons. We construct grapheme-valued Markov processes as limits of finite graph evolutions, driven by Fleming-Viot, Dawson-Watanabe, and McKean-Vlasov analogues. We establish characterization via well-posed martingale problems, yielding strong Markov processes with the Feller property and continuous paths (diffusions). Duality relations involving coalescent processes are derived. We identify non-trivial equilibria, linked to classical distributions from population genetics. This framework extends [arXiv:1908.06241] by incorporating history, enabling rigorous analysis via martingale problems, and characterizing non-trivial long-term behavior.

Figures (4)

• Figure 1: Evolution of the graph. Vertices are represented as black dots on the segment $[0,1]$, and edges are represented by arcs above the segment.
• Figure 5: Representation of a graph as a grapheme: $\mathcal{I}$ space, $X$ vertices, $H$ edges.
• Figure 6: Left: A finite connected graph $G$ with 5 vertices and 8 edges. Right: Emprirical graphon $h^G$ representation of $G$, as a heat map with two levels: black = 1, white = 0.
• Figure 7: Convergence of (black and white) empirical graphons to a (grayscale) limiting graphon as $n \to \infty$. From left to right $n = 5, 8, 12, \infty$.

Theorems & Definitions (45)

• Example 1.1: A Dynamic Social Network
• Definition 1.2: Grapheme
• Remark 2.1: Connection to AHR21
• Definition 2.2: Grapheme processes associated with $\mathbb{U}$-valued diffusions $\mathcal{U}$
• Remark 2.3: $\mathbb{G}_n$- or $G_\infty^{[]}$-grapheme dynamics
• Remark 2.4: General initial states
• Definition 2.5: Order of operators
• Theorem 2.6: Grapheme diffusions
• Corollary 2.7: State properties
• Theorem 2.8: Grapheme approximations