[Paper Review] Average Shortest Path Length of Graphs of Diameter 3

Nobutaka Shimizu, Ryuhei Mori|arXiv (Cornell University)|Jun 16, 2016

Interconnection Networks and Systems7 references3 citations

TL;DR

This paper proposes an efficient evaluation function, $3\triangle + 2\Box$, for stochastic local search (SLS) algorithms to minimize average shortest path length (ASPL) in regular graphs of diameter 3. By leveraging triangle and square counts, the method enables $O(1)$-time evaluation and enables the construction of low-ASPL graphs—achieving the best-known results in the Graph Golf competition for 10,000-node graphs with degrees 60 and 64 using simulated annealing and iterative first improvement.

ABSTRACT

A network topology with low average shortest path length (ASPL) provides efficient data transmission while the number of nodes and the number of links incident to each node are often limited due to physical constraints. In this paper, we consider the construction of low ASPL graphs under these constraints by using stochastic local search (SLS) algorithms. Since the ASPL cannot be calculated efficiently, the ASPL is not suitable for the evaluation function of SLS algorithms. We first derive an equality and bounds for the ASPL of graphs of diameter 3. Then, we propose use the simpliest upper bound represented by the number of triangles and squares in the graph as an evaluation function for graphs of diameter 3. We show that the proposed evaluation function can be evaluated in O(1) time as the number of nodes and the maximum degree tend to infinity by using some data tables. By using the simulated annealing with the proposed evaluation function, we construct low ASPL regular graphs of diameter 3 with 10 000 nodes.

Motivation & Objective

• To address the challenge of efficiently minimizing average shortest path length (ASPL) in graphs of diameter 3 under fixed order and maximum degree constraints.
• To overcome the computational infeasibility of directly evaluating ASPL in stochastic local search (SLS) algorithms.
• To develop a fast, proxy evaluation function based on graph substructures that correlates strongly with ASPL.
• To construct low-ASPL regular graphs of diameter 3 using SLS algorithms with the proposed evaluation function.
• To achieve state-of-the-art results in the Graph Golf competition for large-scale graphs (10,000 nodes).

Proposed method

• Derives exact equality and tight bounds for ASPL in diameter-3 graphs using counts of triangles ($\triangle$), squares ($\Box$), and higher-order substructures like $\triangle^{(2)}$ and $\Box^{(2)}$.
• Proposes $3\triangle + 2\Box$ as a computationally efficient evaluation function for SLS, justified by its strong correlation with ASPL and $O(1)$ evaluation time using precomputed data tables.
• Implements the evaluation function using data structures that store per-edge counts of triangles and squares, enabling constant-time updates during local search.
• Applies iterative first improvement (IFI) and simulated annealing (SA) with the proposed evaluation function to explore the solution space efficiently.
• Optimizes IFI by re-sorting edges based on $3\triangle_e + 2\Box_e$ at intervals to prioritize promising edge pairs for switching.
• Employs an annealing schedule $T(k) = 11 / \ln(k+1)$ to balance exploration and exploitation in simulated annealing.

Experimental results

Research questions

• RQ1 Can a fast, substructure-based proxy function accurately approximate ASPL in diameter-3 graphs?
• RQ2 Is $3\triangle + 2\Box$ sufficient as an evaluation function for SLS to minimize ASPL in regular graphs of diameter 3?
• RQ3 Can $O(1)$-time evaluation of the proxy function enable efficient search for low-ASPL graphs?
• RQ4 What level of ASPL improvement can be achieved using SLS with the proposed function compared to random regular graphs?
• RQ5 Can the method outperform existing solutions in large-scale graph construction, such as in the Graph Golf competition?

Key findings

• The proposed evaluation function $3\triangle + 2\Box$ enables $O(1)$-time evaluation and $O(d^2)$-time update per local move, making it highly efficient for large-scale SLS.
• For $n = 10,000$ and $d = 60$, the simulated annealing with the proposed function achieved an ASPL of 2.6502, with an ASPL gap of $6.2 \times 10^{-3}$, the best result in the Graph Golf competition.
• For $n = 10,000$ and $d = 64$, the method achieved an ASPL of 2.6099, with an ASPL gap of $10.0 \times 10^{-3}$, also the best in Graph Golf.
• The modified IFI algorithm reduced the ASPL gap for $n=10,000, d=60$ from $21.31 \times 10^{-3}$ in random graphs to $6.886 \times 10^{-3}$ in local optima, achieving a 68% improvement.
• The method significantly outperformed random regular graphs, with the ASPL gap reduced by over 50% for $n=10,000, d=64$ after optimization.
• The bounds derived in the paper, especially those involving $\triangle^{(2)}$ and $\Box^{(2)}$, were found to be more accurate approximations of ASPL than the $3\triangle + 2\Box$ function, suggesting potential for even better evaluation functions.