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What is the difference between scale-free network and small-world network?

I can't understand from the definitions around the web if they are both the same name for one thing. Do both follow a power-law distribution of the nodes degree?

In particular is it true that scale-free networks are a subset of small-world networks?

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    $\begingroup$ This question appears to be off-topic because it does not seem directly relevant to physics $\endgroup$
    – akhmeteli
    Oct 24, 2013 at 15:41
  • $\begingroup$ This question seems closable since a straightforward Google search would immediately provide the answer, cf. e.g. this and this meta post. $\endgroup$
    – Qmechanic
    Oct 26, 2013 at 0:30
  • $\begingroup$ @akhmeteli lots of physicists study scalefree and/or small-worlds networks. Have a read through PRE or PRL. $\endgroup$
    – Joel
    Dec 16, 2014 at 14:54
  • $\begingroup$ @Joel: With all due respect, I am not sure I have to "read through PRE or PRL" to decide if a question should be closed. If the OP does not want his/her question to be closed, it's (s)he who should care about that. The question and the links offered by the OP did not impress me as directly relevant to physics. $\endgroup$
    – akhmeteli
    Dec 16, 2014 at 22:02

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They're different, but in many cases they present themselves together.

I don't remember the formal definitions, but a scale-free network should certainly have a power law distribution of nodes degree. And a small-world has to do with a low average average path-length.

Starting from a scale-free network and adding a few randomized links between nodes you obtain a nice small-world network.

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    $\begingroup$ Scale-free = powerlaw distribution of degrees. Small-worlds = high clustering, but short path lengths. Adding randomized links in a scale-free network does NOT generate a small-world network. $\endgroup$
    – Joel
    Dec 16, 2014 at 14:55
  • $\begingroup$ Are you thinking about what happens when you randomly rewire a lattice? $\endgroup$ May 4, 2021 at 1:53

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