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It is now a well known observation that many graphical structures that arise in natural settings tend to obey scale-free properties, such as the power law of degree distribution.

Are there any good example of natural graphs that are fairly random and not do not necessarily obey scale-free properties?

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The connections between neurons do not form scale-free networks, I think. See e.g. Figure 1 on p.181 of http://www.indiana.edu/~cortex/Koetter_chapter_modified.pdf and observe that the vertex degrees only span a single order of magnitude.

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The network topology of wireless ad hoc networks and wireless sensor networks are often captured by a random geometric graph. This means, picking random points in a planar domain, and connecting any two that are within a certain distance threshold. These graphs have Poisson degree distributions, even though they are quite different from the Erdos-Renyi random graphs, and they arise naturally as a network model.

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Although I am not sure what you mean by "random", obvious natural non-scale-free graphs are the road networks.

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  • $\begingroup$ Well, random in the sense that the vertices would come at random and attach to others following some (possibly hidden) probability distribution. $\endgroup$ – Arnab Dec 6 '12 at 13:59
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    $\begingroup$ Actually, road networks are scale-free, when viewed appropriately. Obviously the natural representation isn't, since most intersection have degree at most 4 or sometimes 5 or rarely 6. But if you take the dual representation - a node for each road, and edge between them if the roads ever meet - you do get a scale-free network: cs.unm.edu/~treport/tr/05-10/RoadNetworks.pdf. $\endgroup$ – Joshua Grochow Dec 6 '12 at 18:28

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