arXiv subjects
Publications and source records attributed to Martin Nehez.
The random networks enriched with additional structures as metric and group-symmetry in background metric space are investigated. The important quantities like he clustering coefficient as well as the mean degree of separation in such networks are effectively computed with help of additional structures. Representative models are discussed in details.
Emergence of dominating cliques in Erdös-Rényi random graph model ${\bbbg(n,p)}$ is investigated in this paper. It is shown this phenomenon possesses a phase transition. Namely, we have argued that, given a constant probability $p$, an $n$-node random graph $G$ from ${\bbbg(n,p)}$ and for $r= c \log_{1/p} n$ with $1 \leq c \leq 2$, it holds: (1) if $p > 1/2$ then an $r$-node clique is dominating in $G$ almost surely and, (2) if $p \leq (3 - \sqrt{5})/2$ then an $r$-node clique is not dominating in $G$ almost surely. The remaining range of probability $p$ is discussed with more attention. A detailed study shows that this problem is answered by examination of sub-logarithmic growth of $r$ upon $n$.