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arXiv · 0804.2575

Structural properties of spatially embedded networks

Kosmas Kosmidis·Shlomo Havlin·Armin Bunde
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Abstract

We study the effects of spatial constraints on the structural properties of networks embedded in one or two dimensional space. When nodes are embedded in space, they have a well defined Euclidean distance $r$ between any pair. We assume that nodes at distance $r$ have a link with probability $p(r) \sim r^{- \delta}$. We study the mean topological distance $l$ and the clustering coefficient $C$ of these networks and find that they both exhibit phase transitions for some critical value of the control parameter $\delta$ depending on the dimensionality $d$ of the embedding space. We have identified three regimes. When $\delta 2d$ the networks are ``large'' worlds $l \sim N^{1/d}$ with high clustering. Our results indicate that spatial constrains have a significant impact on the network properties, a fact that should be taken into account when modeling complex networks.

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Kosmas Kosmidis, Shlomo Havlin, Armin Bunde. 2008-04-16. Structural properties of spatially embedded networks. https://doi.org/10.1209/0295-5075/82/48005

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