Consider a graph in Gn, p, with p=c lnn / n. Use the second...

Consider a graph in $G_{n, p}$, with $p=c \ln n / n$. Use the second moment method or the conditional expectation inequality to prove that if $c<1$ then, for any constant $\varepsilon>0$ and for $n$ sufficiently large, the graph has isolated vertices with probability at least $1-\varepsilon$.

Key Concepts

Random Graph Models

Random graph models, notably the Erd?s–Rényi model G?,p, study graphs where each edge between n vertices is independently included with a given probability p. This framework is essential for exploring how graph properties emerge and change as n grows and p varies.

Threshold Phenomena in Graphs

Threshold phenomena refer to the abrupt change in the structure of a random graph when a parameter, such as the edge probability p, crosses a critical value. In settings where p is proportional to (ln n)/n, this concept is key to understanding transitions like the emergence or disappearance of isolated vertices and overall connectivity.

Second Moment Method

The second moment method is a probabilistic technique used to show that a random variable is positive with high probability by comparing its first and second moments. In the context of random graphs, it is applied to demonstrate that the expected number of certain substructures (such as isolated vertices) concentrates around its mean, thus implying their firm existence with high probability.

Conditional Expectation Inequality

The conditional expectation inequality is a tool that bounds the probability of an event by analyzing the expected value conditioned on a related event. In random graph analysis, it is useful for estimating the occurrence of specific graph properties by reducing complex dependency structures, thereby providing rigorous high-probability guarantees for features like isolated vertices.

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