The structure of a random graph at the point of the phase transition

Abstract:

Consider the random graph models $G(n,\# \;{\text {edges}} = M)$ and $G(n,\operatorname {Prob}({\text {edge}}) = p)$ with $M = M(n) = (1 + \lambda {n^{ - 1/3}})n/2$ and $p = p(n) = (1 + \lambda {n^{ - 1/3}})/n$. For $l \geq - 1$ define an $l$-component of a random graph as a component which has exactly $l$ more edges than vertices. Call an $l$-component with $l \geq 1$ a complex component. For both models, we show that when $\lambda$ is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order ${n^{2/3}}$, and the maximum value of $l$ is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order ${n^{2/3}}$, and deletion of all suspended trees results in a "smoothed" graph of size of order ${n^{1/3}}$, with the maximum vertex degree $3$. The total number of branching vertices, i.e., of degree $3$, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a $3$-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order ${n^{1/3}}$. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960. It is proven that there exists $p(\lambda )$, the limiting planarity probability, with $0 < p(\lambda ) < 1$, $p( - \infty ) = 1$, $p(\infty ) = 0$. In particular, $G(n,M)\quad (G(n,p),{\text {resp}}.)$ is almost surely nonplanar iff $(M - n/2){n^{ - 2/3}} \to \infty \;((np - 1){n^{ - 1/3}}) \to \infty ,{\text {resp}}.)$.