[Paper Review] An Estimation of the Shortest and Largest Average Path Length in Graphs of Given Density

László Gulyás, Gábor Horváth+2 more|arXiv (Cornell University)|Jan 13, 2011

Complex Network Analysis Techniques5 references4 citations

TL;DR

This paper provides analytical lower bounds for the shortest and longest possible average path length (ℓ) in undirected, connected graphs of given size N and density d. It derives a minimum ℓ = 2 − d for connected graphs and constructs a specialized 'tail-and-blob' graph family to estimate the maximum ℓ, yielding a closed-form expression that closely matches empirical and simulated data across various network types and densities.

ABSTRACT

Many real world networks (graphs) are observed to be 'small worlds', i.e., the average path length among nodes is small. On the other hand, it is somewhat unclear what other average path length values networks can produce. In particular, it is not known what the maximum and the minimum average path length values are. In this paper we provide a lower estimation for the shortest average path length (l) values in connected networks, and the largest possible average path length values in networks with given size and density. To the latter end, we construct a special family of graphs and calculate their average path lengths. We also demonstrate the correctness of our estimation by simulations.

Motivation & Objective

• To determine the theoretical minimum average path length (ℓ) achievable in connected graphs of given size N and density d.
• To identify the maximum possible average path length for graphs with fixed N and d, as existing models often assume short ℓ without quantifying upper limits.
• To construct a novel graph family—'tail-and-blob'—that maximizes path length under density constraints.
• To validate the analytical bounds through exhaustive enumeration of small graphs (N=10, d=0.2) and simulations on ER and BA networks.
• To provide a general framework applicable across the full density spectrum, unlike prior works limited to specific network types or narrow density ranges.

Proposed method

• Derives a lower bound for minimum average path length using extremal graph theory: ℓ_min = 2 − d, based on counting 1-length and at-least-2-length paths.
• Constructs a 'tail-and-blob' graph family with a linear tail of z nodes connected via a bridge to a dense central blob of N−z−1 nodes to maximize path lengths.
• Computes the total path length SUM(N,z) via nested summations over all node pairs, accounting for distances within the tail, from tail to bridge, from bridge to blob, and within the blob.
• Optimizes the path length by substituting z with the value that maximizes ℓ under the density constraint D = d·N(N−1)/2, leading to a closed-form expression in terms of N and d.
• Derives analytical expressions for the maximum ℓ as a function of N and d: l(N,d) = [expression involving square roots and polynomial terms], derived from SUM(N,D)/(N(N−1)/2).
• Validates the theoretical bounds by simulating Erdős–Rényi and Barabási–Albert networks across the density spectrum and comparing ℓ to the analytical estimates.

Experimental results

Research questions

• RQ1 What is the minimum possible average path length in a connected graph with given size N and density d?
• RQ2 What is the maximum possible average path length in a connected graph with fixed N and d, and what graph structure achieves it?
• RQ3 How do the analytical bounds for minimum and maximum ℓ compare to empirical and simulated networks across different densities?
• RQ4 Can a closed-form expression be derived for the maximum average path length under arbitrary N and d?
• RQ5 How accurate is the proposed 'tail-and-blob' graph family in approximating the theoretical upper bound of ℓ?

Key findings

• The shortest possible average path length in any connected graph with size N and density d is ℓ_min = 2 − d, achieved when as many pairs as possible are connected by direct edges.
• The constructed 'tail-and-blob' graph family achieves the largest known average path length for given N and d, with ℓ approaching values significantly above 2 for low densities.
• The analytical expression for maximum ℓ, derived as l(N,d) = [complex expression involving √(4dN² − (4d+8)N + 9)], closely matches simulated and enumerated values across N and d.
• For N=10 and d=0.2, exhaustive enumeration of all connected graphs confirms that the theoretical minimum ℓ = 2 − 0.2 = 1.8 is achieved, and the maximum ℓ is bounded by the derived formula.
• Empirical networks (e.g., food webs, social networks) generally cluster near the theoretical minimum ℓ, indicating that real-world networks tend to be highly efficient in path length.
• The maximum ℓ bound is valid only for connected graphs; disconnected graphs can have shorter average path lengths, but this is not considered in the main analysis.