Where do power laws come from?

Abstract

What distribution of graphical degree sequence is invariant under ``scaling''? Are these graphs always power-law graphs? We show the answer is a surprising ``yes'' for sparse graphs if we ignore isolated vertices, or more generally, the vertices with degrees less than a fixed constant $k$. We obtain a concentration result on the degree sequence of a random induced subgraph. The case of hypergraphs (or set-systems) is also examined.