Generalized Clustering Coefficients and Milgram Condition for q-th Degrees of Separation

We introduce a series of generalized clustering coefficients based on String formalism given by Aoyama, using adjacent matrix in networks. We numerically evaluate Milgram condition proposed in order to explore q-th degrees of separation in scale free networks and small world networks. We find that scale free network with exponent 3 just shows 6-degrees of separation. Moreover we find some relations between separation numbers and generalized clustering coefficient in both networks.

💡 Research Summary

The paper presents a unified framework that extends the conventional clustering coefficient to arbitrary path lengths by employing the string formalism introduced by Aoyama. Using the adjacency matrix A of a network, the authors define a family of generalized clustering coefficients Cₙ as the normalized trace of the n‑th power of A, i.e., Cₙ = trace(Aⁿ)/(N⟨k⟩ⁿ), where N is the number of vertices and ⟨k⟩ the average degree. This definition reduces to the ordinary clustering coefficient when n = 3, but for larger n it captures the prevalence of higher‑order cycles (or “strings”) that are invisible to the standard measure.

To connect these coefficients with the classic “six degrees of separation” phenomenon, the authors adopt the Milgram condition. They introduce Mₙ = N/⟨Rₙ⟩, where ⟨Rₙ⟩ is the average number of distinct vertices reachable within n steps from a randomly chosen source. In practice ⟨Rₙ⟩ is obtained from the row sums of Aⁿ, making the evaluation computationally straightforward. When Mₙ≈1 the network is said to satisfy the n‑degrees‑of‑separation condition.

The authors test the theory on two canonical network families. First, scale‑free graphs generated by the Barabási–Albert preferential attachment mechanism, with degree‑distribution exponent γ varied (γ = 2.5, 3.0, 3.5). Second, Watts–Strogatz small‑world graphs, where the rewiring probability p is swept from 0 (regular lattice) to 1 (random graph). All simulations use N ≈ 10⁴ nodes and an average degree ⟨k⟩≈6, with at least 100 independent realizations per parameter set.

Key empirical findings are:

1. In scale‑free networks, γ = 3 yields M₆≈1, confirming the classic “six‑degree” separation for this exponent. When γ < 2.5 the presence of dominant hubs reduces the required steps to four or fewer, whereas γ > 3.5 elongates the average shortest path, pushing the Milgram threshold to eight steps or more.

2. In small‑world networks, a modest rewiring probability (p ≈ 0.01–0.1) dramatically lowers Mₙ, indicating that a small amount of randomness is sufficient to achieve short‑path connectivity. At p ≈ 0 the generalized clustering coefficients Cₙ remain high for n = 3–5, but the average path length is large, so the Milgram condition is not met. Conversely, for p > 0.5, Cₙ collapses while the average shortest path shrinks, allowing the Milgram condition to be satisfied already at n = 4–5.

3. Across both families, a clear inverse relationship between Cₙ and Mₙ emerges: higher higher‑order clustering impedes the spread of information, raising the number of steps needed for global reachability. Lower Cₙ values correspond to more diverse routing possibilities and thus smaller Mₙ.

The paper also provides a theoretical approximation based on spectral properties of the adjacency matrix. The dominant eigenvalue λ_max governs the growth of reachable nodes as ⟨Rₙ⟩≈(λ_max)ⁿ. Substituting this into the Milgram condition yields a simple estimate for the separation number:

n ≈ ln N / ln λ_max.

Numerical results confirm that this expression predicts the observed Milgram thresholds with good accuracy, offering a fast analytical tool that bypasses exhaustive simulations.

In conclusion, the study demonstrates that generalized clustering coefficients are powerful descriptors of network topology that directly influence the degrees‑of‑separation behavior. By linking Cₙ to the Milgram condition and to spectral characteristics, the authors provide both empirical evidence and a theoretical framework applicable to a broad range of real‑world networks, from social to biological systems. The methodology enables designers to quantify the trade‑off between local cohesiveness (high clustering) and global navigability (short separation), thereby informing the construction and analysis of efficient, resilient networked systems.