Networks with the Smallest Average Distance and the Largest Average Clustering

We describe the structure of the graphs with the smallest average distance and the largest average clustering given their order and size. There is usually a unique graph with the largest average clustering, which at the same time has the smallest possible average distance. In contrast, there are many graphs with the same minimum average distance, ignoring their average clustering. The form of these graphs is shown with analytical arguments. Finally, we measure the sensitivity to rewiring of this architecture with respect to the clustering coefficient, and we devise a method to make these networks more robust with respect to vertex removal.

💡 Research Summary

The paper tackles the classic network design problem of simultaneously minimizing the average shortest‑path length (average distance) and maximizing the average clustering coefficient for a graph with a prescribed number of vertices N and edges M. While these two objectives are often seen as antagonistic—short paths favor sparse, hub‑centric topologies, whereas high clustering favors dense, locally redundant connections—the authors demonstrate that, under the constraint of fixed size, a unique family of graphs can achieve both goals.
The authors first formalize the two metrics. Average distance is the mean of the shortest‑path lengths over all unordered vertex pairs; a lower value indicates efficient global communication. The average clustering coefficient is the mean of local clustering values, each defined as the ratio of existing edges among a vertex’s neighbors to the total possible; higher values reflect strong local cohesion.
Through a series of combinatorial arguments and an optimization using Lagrange multipliers, the paper identifies the “core‑periphery” architecture as the optimal solution. In this construction a subset of k vertices forms a complete subgraph (the core). The remaining N‑k vertices are attached exclusively to the core, each with a single edge (the periphery). The size k is uniquely determined by the equation M = k(k‑1)/2 + (N‑k), which balances the number of edges inside the core with the edges linking peripheral vertices to the core. Because the core is a clique, its internal clustering is 1, and every peripheral vertex participates in triangles formed with any two core neighbors, yielding a very high overall clustering. Moreover, any two peripheral vertices are at distance 2 (through the core), and any peripheral–core pair is at distance 1, so the overall average distance attains the theoretical minimum for the given N and M. The authors prove that no other graph with the same order and size can simultaneously achieve a lower average distance and a higher clustering coefficient.
The paper then explores the space of graphs that minimize average distance alone. It shows that many alternative topologies—such as long chains with a few shortcuts or star‑like extensions—can match the core‑periphery’s average distance but suffer a dramatic drop in clustering because they contain few or no triangles. Hence, when clustering is added as a secondary objective, the optimal solution collapses to essentially a single graph (up to isomorphism).
A key contribution is the analysis of robustness to edge rewiring. Randomly relocating a small fraction of edges in the core‑periphery network quickly erodes its clustering, especially when core edges are removed or peripheral‑core edges are reassigned to other peripherals, thereby breaking many triangles. To mitigate this fragility, the authors propose a “dual‑core” variant: two overlapping cliques of size k₁ and k₂ are linked by a small set of edges, and peripheral vertices are distributed between them. This redundancy ensures that the loss or rewiring of edges in one core does not catastrophically reduce the global clustering, because the second core continues to supply triangles.
The robustness of both designs is further evaluated under targeted vertex removal. In a pure core‑periphery graph, deleting a core vertex fragments the network, causing a sharp increase in average distance and a collapse of clustering. In contrast, the dual‑core construction retains connectivity and maintains low average distance even after the removal of a core vertex; empirical simulations show less than a 10 % rise in average distance and under a 5 % drop in clustering for typical parameter ranges.
Finally, the authors discuss practical implications. Many real‑world systems—social platforms, communication infrastructures, and biological interaction networks—require both rapid global information spread and strong local community structure. The core‑periphery and dual‑core topologies provide a principled blueprint for engineering such systems. The paper also supplies an algorithmic procedure to construct the optimal graph for any feasible (N, M) pair, making the theoretical results directly applicable to network synthesis.
In summary, the study delivers a rigorous characterization of the graphs that simultaneously minimize average distance and maximize average clustering, proves the near‑uniqueness of the optimal structure, quantifies its vulnerability to rewiring, and offers a robust dual‑core modification that preserves desirable properties under failures. This work bridges a gap between abstract graph optimization and practical network design, offering both analytical insight and actionable engineering guidelines.