Session 13

Apuntes

Session 13 🔹 What Are Random Networks? Random networks are mathematical models of networks where edges (links) between nodes are placed randomly . They lie on the "disorder" end of the order-disorder spectrum, opposite to regular networks (where every node has a fixed number of neighbors). The most famous model of random networks is the Erdős-Rényi (ER) model , particularly the G(n, p) model. 🔹 Erdős–Rényi (G(n, p)) Model: ● N (or n) = number of nodes ● p = probability that any given pair of nodes is connected by an edge In this model: ● Every possible link between two nodes exists independently with probability p . ● The degree distribution (i.e., how many connections each node has) follows a binomial distribution , which becomes Poisson for large N and small p. Binomial Degree Distribution: P(k)=(N−1k)pk(1−p)N−1−kP(k) = \binom{N-1}{k} p^k (1-p)^{N-1-k} Poisson Approximation (for sparse networks): P(k)=e− ⟨ k ⟩⟨ k ⟩ kk!P(k) = \frac{e^{-\langle k \rangle} \langle k \rangle^k}{k!} Where ⟨ k ⟩ =(N−1)p\langle k \rangle = (N - 1) p is the average degree. 🔹 Key Properties of Random Networks 1. Clustering Coefficient (C) ● Measures how l...