Large Graphs

We consider a particular model for random graphs called the Erdos-Renyi. An undirected random graph $G(n,p)$ is defined on a set of vertices $V = \{1,2,\ldots,n\}$ such that an edge in the graph is present with probability $p$ and the occurence of these edges are statistically independant. An example would be $G(n,\frac{1}{2})$. In this graph the probability of vertex with degree $d$ would be

$P(d) = {n\choose d} \left(\frac{1}{2}\right)^d\left(\frac{1}{2}\right)^{n-d} = {n\choose d} \left(\frac{1}{2}\right)^n$

The expected degree (i.e. mean) of any vertex is $\frac{n}{2}$ with variance $\sqrt{n}$.

Theorem 1.1   In $G(n,\frac{1}{2})$ graph, almost all vertices of degree $\frac{n}{2}+5\sqrt{n}$ or $(1+\epsilon)\frac{n}{2}$.

In the asymptotic case as $n$ tends to infinity by the central limit theorem we get the distribution for the degree to be

$P(d) = \frac{1}{\sqrt{\frac{\pi n}{2}}} e^{-\frac{(d-\frac{n}{2})^2}{\frac{n}{2}}}.$

Let us consider $G(n,\frac{1}{n})$. We shall prove that we are very likely to find a vertex of degree $\frac{\log n}{\log\log n}$. Note that $\log {d^d} \approx \log n$ and hence $d^d \approx n$.

$$\begin{array}{rl} P(d) &= {n\choose d} \left(\frac{1}{n}\rig... ...}{n}\right)^{n}\left(1 - \frac{1}{n}\right)^{-d}\\ \end{array}$$

As $n$ grows bigger we see that $\left(1 - \frac{1}{n}\right)^{-d}$ becomes 1 and $(n)(n-1)\ldots(n-d+1)$ gets close to $n^d$. Further $\left(1 - \frac{1}{n}\right)^{n}$ tends to $e{-1}$. Therefore as $n$ tends to infinity we get $P(d) \approx \frac{e^{-1}}{d!} > \frac{e^{-1}}{d^d} \approx \frac{e^{-1}}{n}$ when $d=\frac{\log n}{\log\log n}$. Hence with probability $e^{-1}$ there exists some vertex with degree $d$.

Giant Components database of protein interaction. Vertices are proteins, edges mark interactions. 2730 vertices, 3602 edges.

Size of component 1  2   3  4  5  6 7 8 9 10 11 12 13 14 15 16 ... 1851
No of component 48 174 50 25 14 6 4 6 1 1  1  1  0  0  0  1       1

We know discuss and alternative method for generating random graphs. Consider a graph on $n$ vertices and pick two random numbers between 1 and n and create an edge between them. After adding $n$ edges, a giant component appears. After adding $n\log n$ edges the graph becomes connected.

Let us get back to our original model of graphs. Consider $G(n,\frac{d}{n})$.

• $d \geq 1$ : gaint component
• $d < 1$ : no giant component
• $d \geq \log n$ connected.

Three alternatives are possible in the asymptotic case.

1. The graph becomes connected.
2. There exists an isolated vertex.
3. The graph is not connected and there are no isolated vertices.

It can be shown that a shift occurs for the first two alternatives when $p = \frac{\log n + c}{n}$. Further the probability of alternative three occuring is small compared to two and hence can be ignored.

Let us compute the probability that a specific vertex is isolated, i.e. $(1-p)^{n-1} = \frac{e^{-c}}{n}$. Hence the probability that there are no isolated vertices is $\left(1 - \frac{e^{-c}}{n}\right)^n = e^{-e^{-c}}$. We shall compute this probability for a few values of $c$.

When $c=0$ the probability is $\approx 0.3$

When $c=1$ the probability is $e^{-e^{-1}}$

When $c=10$ the probability is $\approx 0.999$

The probability that a pair of vertices connected to each other but isolated otherwise is $(1 - \frac{\log n +c}{n})^{2(n-2)}$ = $\frac{1}{n^2}e^{-2c}$. Probability of no isolated connected pair is therefore $\frac{n \log n}{n}e^{-2c}$