Harvard linguist George Kingsley Zipf identified a distribution of how frequent words are in a corpus. Most of you should be familiar with that law in the social realm: given a “relation” criterion, most social networks lead to a zipfian distribution of degree, the number of contacts.
Looking into a database of mobile calls (frequent & reciprocal to filter out weak ties; so far the database is confidential information, but I have access to a hashed version) I came across a similar law, but about the cross-distribution of degrees at the two ends of relations.
- let f (n) be the zipfian distribution of degrees:
f (n) = 1/z(a) n^(- a)
- z(.) is the Rieman’s zeta function; and
- a is a parameter, generally between two and three.
- let f (n, m) be the distribution of ties between users of degree n and m; it seems that:
log f (n, m) = – log f (n) . log f (m)
This is the simplest way to put it; the actual formula for f (n, m) is therefore:
f (n, m) = exp (- [log z(a) + a log n] . [log z(a) + a log m] )
Actually, f (1, 1) is much lower then expected (but still higher then most other case).
What I need now it to check or have specialists check such a law on as many complex graph as possible: social, assortative ones might boast the same result. This result appears more precise then over-all assortativity. I am assuming it wouldn’t be true on bipartite-based networks, as f (n, ñ) — with n and ñ close — would then be higher.
Comments are more then welcome.