Sample (9)
Generating random graphs is an important primitive in graph theory. A simple way to generate undirected random graphs with n nodes
is to connect each pair of distinct nodes (self-loops ar

Question

Sample (9)
Generating random graphs is an important primitive in graph theory. A simple way to generate undirected random graphs with n nodes
is to connect each pair of distinct nodes (self-loops are not allowed) independently with probability p.
o For n nodes, if X is the r.v. equal to the number of edges in a randomly generated undirected graph, specify the PDF x.
© What is the expected number of edges in a random graph?
© The degree of a node is the number of other distinct nodes that it is connected to. Consider a node u € {1,2,...,n} and define
Y,, to be the r.v. equal to its degree in a random graph. When n is even, calculate Pr[Y; > n/2].
o A triangle in a graph is a set of 3 distinct nodes that are all connected to each other. For example, if there are edges from node
wu tov, v to w and w to u, then nodes {u,v,w} are said to form a triangle. If Z is the r.v. equal to the number of triangles in a
random graph, calculate E[Z].

VIPSolutions · Accepted Answer

# Random Graphs in Graph Theory

## Probability Density Function (PDF) of $ X $

For a random graph with $ n $ nodes where each pair of distinct nodes is connected with probability $ p $,  (the random variable denoting the number of edges) follows a Binomial distribution. The PDF is given by:

$$
\text{PDF}_X(k) = \binom{\binom{n}{2}}{k} p^k (1-p)^{\binom{n}{2} - k}
$$

where $\binom{n}{2}$ is the total number of possible edges.

## Expected Number of Edges

The expected number of edges, $ E[X] $, in the random graph is:

$$
E[X] = \binom{n}{2} \cdot p
$$

## Probability of Node Degree

For a node $ u $, the degree $ Y_u $ is a Binomial random variable with parameters $ n-1 $ (the number of other nodes) and $ p $. When $ n $ is even, the probability that $ Y_u \geq n/2 $ can be calculated using:

$$
\Pr[Y_u \geq n/2] = \sum_{k = n/2}^{n-1} \binom{n-1}{k} p^k (1-p)^{n-1-k}
$$

## Expected Number of Triangles

A triangle involves choosing 3 nodes and ensuring all three possible edges between them are present. The expected number of triangles $ E[Z] $ is given by:

$$
E[Z] = \binom{n}{3} \cdot p^3
$$

This accounts for choosing 3 nodes out of $ n $ and the probability that all three edges exist.

VIPSolutions · Answer

# Final Answers on Random Graphs

## 1. Probability Density Function (PDF) of $ X $

The PDF of the random variable $ X $, representing the number of edges in a random graph with $ n $ nodes and connection probability $ p $, is:

$$
\text{PDF}_X(k) = \binom{\binom{n}{2}}{k} p^k (1-p)^{\binom{n}{2} - k}
$$

## 2. Expected Number of Edges

The expected number of edges in the random graph is calculated as:

$$
E[X] = \binom{n}{2} \cdot p
$$

## 3. Probability of Node Degree

For a specific node $ u $, the degree $ Y_u $ is distributed as a Binomial random variable:

$$
Y_u \sim \text{Binomial}(n-1, p)
$$

The probability that the degree of node $ u $ is greater than or equal to $ n/2 $ (when $ n $ is even) is:

$$
\Pr[Y_u \geq n/2] = \sum_{k = n/2}^{n-1} \binom{n-1}{k} p^k (1-p)^{n-1-k}
$$

## 4. Expected Number of Triangles

The expected number of triangles $ Z $ in the graph is given by:

$$
E[Z] = \binom{n}{3} \cdot p^3
$$

This expression derives from selecting 3 nodes and ensuring that all three edges connecting them exist.

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