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The Erdős–Rényi model, named after mathematicians Paul Erdős and Alfred Rényi, is a fundamental probabilistic model in graph theory. Introduced in 1959, it provides a simple yet powerful framework for studying random graphs, which are graphs formed by connecting vertices with edges chosen randomly. The model is defined by two key parameters: the number of vertices *n* and a probability *p* that determines whether any given pair of vertices is connected by an edge.

In the Erdős–Rényi model, each possible edge in a graph with *n* vertices is included independently with probability *p*. This results in a random graph *G(n, p)*, where *n* represents the number of vertices and *p* the edge probability. The model assumes no restrictions on edge connections, meaning loops (edges connecting a vertex to itself) and multiple edges between the same pair of vertices are typically excluded unless specified otherwise.

The model has broad applications in network science, social sciences, and computer science, where it helps analyze properties like connectivity, clustering, and the emergence of giant components in large networks. Key phases of the model include the *disconnected phase* (when *p* is very low, the graph is sparse and fragmented), the *percolation threshold* (around *p ≈ 1/n*, where a giant connected component begins to form), and the *connected phase* (when *p* is sufficiently high, the graph becomes highly interconnected).

Variations of the model include the *Erdős–Rényi G(n, m)* model, where a fixed number of edges *m* is added to the graph, and the *configuration model*, which generalizes the approach to degree sequences. The Erdős–Rényi model remains a cornerstone in understanding randomness in complex systems, influencing both theoretical research and practical applications in network analysis.