A critical point for random graphs with a given degree sequence
Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
Random Structures & Algorithms
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Consider the set $\mathcal{SG}(Q_k)$ of all graphs whose vertices are labeled with nonidentity elements of the group $Q_k=\mathbb{Z}_2^k$ so that there is an edge between vertices with labels $a$ and $b$ if and only if the vertex labeled $a+b$ is also in the graph. Note that edges always appear in triangles since $a+b=c$, $b+c=a$, and $a+c=b$ are equivalent statements for $Q_k$. We define the random cubic sum graph $\mathcal{SG}(Q_k,p)$ to be the probability space over $\mathcal{SG}(Q_k)$ whose vertex sets are determined by $\Pr[x\in V]=p$ with these events mutually independent. As $p$ increases from 0 to 1, the expected structure of $\mathcal{SG}(Q_k,p)$ undergoes radical changes. We obtain thresholds for some graph properties of $\mathcal{SG}(Q_k,p)$ as $k\rightarrow\infty$. As with the classical random graph, the threshold for connectivity coincides with the disappearance of the last isolated vertex.