We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every ε>0, whp O(logn) rounds are
sufficient to inform all but an ε-fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp nΩ(1) rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with β>3, the push-pull protocol needs Ω(logn)
to inform all but εn vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with 2<β<3, push-pull is exponentially faster than pull