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Constrained graph generation: Preserving diameter and clustering coefficient simultaneously

Dávid Ferenczi, Alexander Grigoriev

TL;DR

This paper designs a layered ACO heuristic to perform a guided global search, effectively locating valid graphs with prescribed diameter and clustering coefficient and uses these ACO-designed graphs as structurally distinct seed states for an MCMC rewiring algorithm.

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.

Constrained graph generation: Preserving diameter and clustering coefficient simultaneously

TL;DR

This paper designs a layered ACO heuristic to perform a guided global search, effectively locating valid graphs with prescribed diameter and clustering coefficient and uses these ACO-designed graphs as structurally distinct seed states for an MCMC rewiring algorithm.

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.
Paper Structure (21 sections, 2 theorems, 14 equations, 4 figures, 2 algorithms)

This paper contains 21 sections, 2 theorems, 14 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Simple edge swaps
  4. Algorithmic description
  5. Metropolis-Hastings algorithm
  6. ACO for Graph Construction
  7. Layered construction
  8. Pheromones and edge probabilities
  9. Pheromone score update
  10. Hybrid Framework: Initializing MCMC with ACO
  11. Experimental results
  12. Success ratio and structural diversity
  13. Difference between MCMC and Hybrid MCMC
  14. Discussion
  15. MCMC Ergodicity
  16. ...and 6 more sections

Key Result

lemma 1

Let $G$ be a graph with degree sequence $(\delta_i)_{i=1}^n$, and $A$ its adjacency matrix. Then

Figures (4)

  • Figure 1: Impact of an edge swap on diameter. Moving a single red edge from a central position (Left) to a boundary position (Right) fundamentally alters the diameter.
  • Figure 2: Illustration of the connectivity barrier in the configuration space. To transition the clique from the left side (Step 0, $\mathop{\mathrm{gcc}}\nolimits \approx 0.8$) to the right side (Step 2, $\mathop{\mathrm{gcc}}\nolimits \approx 0.8$), the graph must pass through an intermediate state (Step 1) with significantly lower clustering ($\mathop{\mathrm{gcc}}\nolimits \approx 0.64$). If the constraint is strictly set to $\mathop{\mathrm{gcc}}\nolimits \in [0.7, 0.9]$, the move to Step 1 is rejected, leaving the algorithm trapped in the initial topology.
  • Figure 3: Success ratio (left) and spectral distance (right) for edge density 0.2.
  • Figure 5: Estimated density functions of the spectral distance (drift from seed). Left: ($n=40$, $m=78$, $\mathop{\mathrm{diam}}\nolimits=12$, $\mathop{\mathrm{gcc}}\nolimits=0.35$) Both methods find structurally similar solutions. Right: ($n=40$, $m=195$, $\mathop{\mathrm{diam}}\nolimits=4$, $\mathop{\mathrm{gcc}}\nolimits=0.4$) A different set of constraints allowing for a richer solution space. The standard MCMC (blue) remains clustered around the seed structure, while the hybrid method (orange) reveals the true structural diversity of the feasible set.

Theorems & Definitions (2)

  • lemma 1: Hofstad_2016
  • lemma 2: Hofstad_2016