Clustering in random line graphs

We investigate the degree distribution $P(k)$ and the clustering coefficient $C$ of the line graphs constructed on the Erd"os-R'enyi networks, the exponential and the scale-free growing networks. We show that the character of the degree distribution in these graphs remains Poissonian, exponential and power law, respectively, i.e. the same as in the original networks. When the mean degree $$ increases, the obtained clustering coefficient $C$ tends to 0.50 for the transformed Erd"os-R'enyi networks, to 0.53 for the transformed exponential networks and to 0.61 for the transformed scale-free networks. These results are close to theoretical values, obtained with the model assumption that the degree-degree correlations in the initial networks are negligible.

💡 Research Summary

The paper investigates how the line‑graph transformation reshapes three canonical families of random networks – Erdős‑Rényi (ER) graphs, exponential (growth‑with‑constant‑average‑degree) networks, and scale‑free (Barabási‑Albert‑type) networks – focusing on two fundamental structural descriptors: the degree distribution (P(k)) and the clustering coefficient (C). A line graph (L(G)) is constructed by turning every edge of the original graph (G) into a node; two nodes in (L(G)) are linked if the corresponding edges in (G) share a common endpoint. This edge‑centric representation is relevant for processes that naturally live on links (e.g., traffic flow, disease transmission across contacts, or information diffusion along communication channels).

Degree distribution preservation
Under the standard assumption that the original network is degree‑uncorrelated, the probability that an edge connects vertices of degrees (k_i) and (k_j) is proportional to (k_i P(k_i) k_j P(k_j) / \langle k\rangle^2). In the line graph this edge becomes a node whose degree equals (k_i + k_j - 2) (the two incident vertices each contribute all their other incident edges). Consequently the line‑graph degree distribution can be expressed as a convolution:

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