Assortative Mixing in Close-Packed Spatial Networks

A general relation for the dependence of nearest neighbor degree correlations on degree is derived. Dependence of local clustering on degree is shown to be the sole determining factor of assortative versus disassortative mixing in networks. The characteristics of networks derived from spatial atomic/molecular systems exemplified by self-organized residue networks and block copolymers, atomic clusters and well-compressed polymeric melts are studied. Distributions of statistical properties of the networks are presented. For these densely-packed systems, assortative mixing in the network construction is found to apply, and conditions are derived for a simple linear dependence. Together, these measures (i) reveal patterns that are common to close-packed clusters of atoms/molecules, (ii) identify the type of surface effects prominent in different systems, and (iii) associate fingerprints that may be used to classify networks with varying types of correlations.

💡 Research Summary

The paper presents a unified theoretical framework and empirical analysis for understanding degree–degree correlations in densely packed spatial networks that arise from atomic and molecular systems. Starting from the well‑known nearest‑neighbor degree function ⟨k_nn(k)⟩, the authors derive a general expression that links this correlation directly to the degree dependence of the local clustering coefficient C(k). By assuming a power‑law scaling C(k) ∝ k^−α, they obtain a linear approximation
⟨k_nn(k)⟩ ≈ (⟨k²⟩/⟨k⟩)