Dense subgraphs of power-law random graphs

Abstract

The problem of finding a maximal dense subgraph of a power-law random graph $G\left(n,\alpha \right)$ is considered for every value of density $c\in \left(0,1\right)$ and for every $\alpha \in \left(0,+\infty \right)$. It is shown that in case $\alpha <2$ a maximal $c$-dense subgraph has size ${\mathrm{\Theta }}_p\left(n^{1-\alpha ∕2}\right)$, in case $\alpha =2$ it is limited whp, and in case $\alpha >2$ it is whp less than $\frac{4}{c}$.