Constrained graph generation: Preserving diameter and clustering coefficient simultaneously

TL;DR

This paper introduces a hybrid ACO-MCMC framework for generating graphs that satisfy specific diameter and clustering coefficient constraints, overcoming ergodicity issues and producing diverse graph structures.

Contribution

The paper presents a novel hybrid approach combining Ant Colony Optimization and MCMC to efficiently generate structurally diverse graphs with prescribed properties.

Findings

The hybrid ACO-MCMC method outperforms standard MCMC in exploring feasible graph spaces.

The approach effectively locates valid graphs across various density and constraint regimes.

Generated graphs exhibit greater structural diversity compared to traditional methods.

Abstract

Generating graphs subject to strict structural constraints is a fundamental computational challenge in network science. Simultaneously preserving interacting properties-such as the diameter and the clustering coefficient- is particularly demanding. Simple constructive algorithms often fail to locate vanishingly small sets of feasible graphs, while traditional Markov-chain Monte Carlo (MCMC) samplers suffer from severe ergodicity breaking. In this paper, we propose a two-step hybrid framework combining Ant Colony Optimization (ACO) and MCMC sampling. First, we design a layered ACO heuristic to perform a guided global search, effectively locating valid graphs with prescribed diameter and clustering coefficient. Second, we use these ACO-designed graphs as structurally distinct seed states for an MCMC rewiring algorithm. We evaluate this framework across a wide range of graph edge densities and varying diameter-clustering-coefficient constraint regimes. Using the spectral distance of the normalized Laplacian to quantify structural diversity of the resulting graphs, our experiments reveal a sharp contrast between the methods. Standard MCMC samplers remain rigidly trapped in an isolated subset of feasible graphs around their initial seeds. Conversely, our hybrid ACO-MCMC approach successfully bridges disconnected configuration landscapes, generating a vastly richer and structurally diverse set of valid graphs.