Characteristic Length and Clustering

We explore relations between various variational problems for graphs: among the functionals considered are Euler characteristic χ(G), characteristic length μ(G), mean clustering ν(G), inductive dimension ι(G), edge density (G), scale measure σ(G), Hilbert action η(G) and spectral complexity ξ(G). A new insight in this note is that the local cluster coefficient C(x) in a finite simple graph can be written as a relative characteristic length L(x) of the unit sphere S(x) within the unit ball B(x) of a vertex. This relation L(x) = 2−C(x) will allow to study clustering in more general metric spaces like Riemannian manifolds or fractals. If η is the average of scalar curvature s(x), a formula μ ∼ 1 + log( )/ log(η) of Newman, Watts and Strogatz [31] relates μ with the edge density and average scalar curvature η telling that large curvature correlates with small characteristic length. Experiments show that the statistical relation μ ∼ log(1/ν) holds for random or deterministic constructed networks, indicating that small clustering is often associated to large characteristic lengths and λ = μ/ log(ν) can converge in some graph limits of networks. Mean clustering ν, edge density and curvature average η therefore can relate with characteristic length μ on a statistical level. We also discovered experimentally that inductive dimension ι and cluster-length ratio λ correlate strongly on Erdös-Renyi probability spaces.