ErdosRényinäytteitäs

ErdosRenyi samples, often denoted as G(n,p) or G(n,M), are a fundamental class of random graphs used in the study of network structures and properties. The Erdos-Renyi model, introduced by Paul Erdos and Alfréd Renyi, provides a probabilistic framework for constructing graphs. There are two primary variations. The G(n,p) model generates a graph with n vertices, where each possible edge between any pair of vertices exists independently with probability p. The G(n,M) model, on the other hand, selects exactly M edges uniformly at random from the set of all possible n(n-1)/2 edges.

These random graphs are crucial for understanding the emergence of various graph properties as the number of vertices or edges changes. For instance, they are used to study phenomena like the emergence of giant connected components, the distribution of path lengths, and the presence of specific subgraphs. The behavior of Erdos-Renyi graphs often exhibits sharp transitions, meaning certain properties appear or disappear abruptly as parameters like n or p cross a critical threshold. This makes them powerful tools for theoretical analysis and for forming baseline expectations against which real-world networks can be compared. Despite their simplicity, Erdos-Renyi samples have shed light on many complex network behaviors.