Motivation

In this lecture, we will learn the definitions and properties of four graphical models:

1. The definition of the Erdos-Renyi model.
   • Number of edges the graph has
   • Random realization be a complete graph
   • The edge probability and degree distributions of the Erdos-Renyi model.
   • Graph structure: the phase transitions of the Erdos-Renyi model.
2. The definition of the Configuration model.
   • The edge probability of the Configuration model.
3. The definition of Price's preferential attachment model.
4. The definition of the Small-World model.

I. Erdos-Renyi Model

The Erdos-Renyi (ER) model is a fundamental model for generating random graphs. It is defined in two primary ways: $G(n, M)$ and $G(n, p)$.

1. Definitions:
   • $G(n, M)$ Model: A graph is constructed by selecting $n$ nodes and randomly adding $M$edges between them.
   • $G(n, p)$ Model: A graph is constructed by selecting $n$ nodes, and each possible edge between any pair of nodes is included with probability $p$.
2. Number of Edges:
   • In the $G(n, M)$ model, the graph has exactly $M$ edges.
   • In the $G(n, p)$ model, the expected number of edges is $\binom{n}{2} p$, where $\binom{n}{2}$ is the number of possible edges in a graph with $n$ nodes.
3. Random Realization Be a Complete Graph:
   • In the $G(n, p)$ model, if $p = 1$, every possible edge is included, resulting in a complete graph.
   • As $p$ decreases from 1 to 0, the graph transitions from being a complete graph to an empty graph.

Edge Probability

In the $G(n, p)$ model, the probability that an edge exists between any two given nodes is $p$. MLE of $p$:

• The number of edges $M$ in a realization of the graph follows a binomial distribution: $M \sim \text{Binomial}\left(\binom{n}{2}, p\right)$.
• The likelihood function for observing $m$ edges in the graph is given by the probability mass function of the binomial distribution: $P(M = m \mid p) = \binom{\binom{n}{2}}{m} p^m (1 - p)^{\binom{n}{2} - m}$.
• To find the MLE of $p$, we need to maximize the likelihood function with respect to $p$. This is often done by maximizing the log-likelihood function instead: $\log L(p) = \log \left( \binom{\binom{n}{2}}{m} \right) + m \log p + \left( \binom{n}{2} - m \right) \log (1 - p)$ .
• The binomial coefficient $\binom{\binom{n}{2}}{m}$ is a constant with respect to $p$ and can be ignored for the purpose of maximization: $\log L(p) = m \log p + \left( \binom{n}{2} - m \right) \log (1 - p)$ .
• To find the MLE, we take the derivative of the log-likelihood function with respect to $p$ and set it to zero: $\frac{d}{dp} \log L(p) = \frac{m}{p} - \frac{\binom{n}{2} - m}{1 - p} = 0$ .
• Solving for $p$: