Graph Limits via Quotients

TL;DR

The paper introduces grapheurs, a new limit object for growing weighted directed graphs defined via convergence of random quotients, and proves a precise duality with graphons. It develops a complete theory: quotient densities and homomorphism numbers characterize quotient convergence, grapheurs provide compact limit objects with a Wasserstein-type metric, and a random-edge sampling framework yields an edge-based Szemerédi-like regularity lemma and a practical method for testing hub-related properties. It also shows that equipartition-consistent random graph models correspond to mixtures of grapheurs, connecting finite-graph quotients to underlying random measures on $[0,1]^2$. The framework emphasizes hub structure and global connectivity, offering new tools for modeling large networks, and provides a rigorous bridge to graphon theory while highlighting distinctive capabilities for hub detection and global graph properties.

Abstract

We present a new notion of limits of weighted directed graphs of growing size based on convergence of their random quotients. These limits are specified in terms of random exchangeable measures on the unit square. We call our limits grapheurs and show that these are dual to graphons in a precise sense. Grapheurs are well-suited to modeling hubs and connections between them in large graphs; previous notions of graph limits based on subgraph densities fail to adequately model such global structures as subgraphs are inherently local. Using our framework, we present an edge-based sampling approach for testing properties pertaining to hubs in large graphs. This method relies on an edge-based analog of the Szemerédi regularity lemma, whereby we show that sampling a small number of edges from a large graph approximately preserves its quotients. Finally, we observe that the random quotients of a graph are related to each other by equipartitions, and we conclude with a characterization of such random graph models.

Figures (3)

• Figure 1: Traffic network for the Gold Coast, Australia, and an associated random quotient. The city is divided into 1068 zones, and the weight of an edge is the fraction of trips from a dataset with given start and end zones traffic_networks. Here we plot the first 300 of these zones and their interconnections, displaying the width of an edge proportionally to its weight. We color the vertices according to their image in the quotient, and color and edge connecting two vertices with the same image by the color of that image. The nonuniformity in the edge weights of the quotient indicates the presence of hubs that we can see in the plot of the entire network.
• Figure 2: Network of chemical connections among the 575 neurons of adult male C. elegans (a species of roundworm), and its random quotient. Edges are directed from pre-synaptic to post-synaptic cells, and their weight is the total number of EM serial sections of connectivity cook2019wholeworms_brain. The nonuniform distribution of self-loops and the unequal in-degrees and out-degrees among the four vertices in the quotient again indicate the presence of hubs.
• Figure 3: Hasse diagram for the refinement poset consisting of multigraphs with two edges. Here there is an upward path from $K$ to $H$ precisely when $K\leq_R H$.

Theorems & Definitions (72)

• Definition 1.1
• Theorem 1.2
• Definition 1.3: Random exchangeable measures
• Definition 1.4
• Theorem 1.5
• Theorem 1.6: Informal; see Theorem \ref{['thm:graphon_duality']}
• Theorem 1.7
• Definition 1.8: Equipartition-consistent models
• Theorem 1.9
• Example 2.1: Convergence of complete graphs