Modulated Scale-free Network in the Euclidean Space
S. S. Manna, Parongama Sen

TL;DR
This paper introduces a model for growing Euclidean space networks with modulated scale-free properties, analyzing how connection probabilities based on degree and distance influence network topology and degree distributions.
Contribution
It presents a novel network growth model incorporating Euclidean distances and degree-based attachment, deriving exact link length distribution and identifying conditions for scale-free behavior.
Findings
Network is scale-free for all $\alpha > \alpha_c$
Degree distribution decays stretched exponentially for $\alpha \\leq \\alpha_c$
Link length distribution follows a power law $D(\\ell) \\sim \\ell^{\\delta}$ with exact $\\delta$
Abstract
A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its -th predecessor of degree with a directed link of length using a probability proportional to . Our numerical study indicates that the network is Scale-free for all values of and the degree distribution decays stretched exponentially for the other values of . The link length distribution follows a power law: where is calculated exactly for the whole range of values of .
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