In the mathematical field of graph theory, the Erdà ÂsâÂÂRényi model refers to one of two closely related models for generating random graphs or the evolution of a random network. These models are named after Hungarian mathematicians Paul Erdà Âs and Alfréd Rényi, who introduced one of the models in 1959. Edgar Gilbert introduced the other model contemporaneously with and independently of Erdà Âs and Rényi. In the model of Erdà Âs and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely. In the model introduced by Gilbert, also called the Erdà ÂsâÂÂRényiâÂÂGilbert model, each edge has a fixed probability of being present or absent, independently of the other edges. These models can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.
There are two closely related variants of the Erdà ÂsâÂÂRényi random graph model.
The behavior of random graphs are often studied in the case where , the number of vertices, tends to infinity. Although and can be fixed in this case, they can also be functions depending on . For example, the statement that almost every graph in is connected means that, as tends to infinity, the probability that a graph on vertices with edge probability is connected tends to .
The expected number of edges in G(n, p) is , with a standard deviation asymptotic to . Therefore, a rough heuristic is that if some property of G(n, M) with does not significantly change in behavior if M is changed by up to s(n), then G(n, p) should share that behavior.
This is formalized in a result of à Âuczak. Suppose that P is a graph property such that for every sequence M = M(n) with , the probability that a graph sampled from G(n, M) has property P tends to a as n â âÂÂ. Then the probability that G(n, p) has property P also tends to a.
Implications in the other direction are less reliable, but a partial converse (also shown by à Âuczak) is known when P is monotone with respect to the subgraph ordering (meaning that if A is a subgraph of B and B satisfies P, then A will satisfy P as well). Let , and suppose that a monotone property P is true of both G(n, p â õ) and G(n, p + õ) with a probability tending to the same constant a as n â âÂÂ. Then the probability that has property P also tends to a.
For example, both directions of equivalency hold if P is the property of being connected, or if P is the property of containing a Hamiltonian cycle. However, properties that are not monotone (e.g. the property of having an even number of edges) or that change too rapidly (e.g. the property of having at least edges) may behave differently in the two models.
In practice, the G(n, p) model is the one more commonly used today, in part due to the ease of analysis allowed by the independence of the edges.
With the notation above, a graph in G(n, p) has on average edges. The distribution of the degree of any particular vertex is binomial:
where n is the total number of vertices in the graph. Since
this distribution is Poisson for large n and np = const.
In a 1960 paper, Erdà Âs and Rényi described the behavior of G(n, p) very precisely for various values of p. Their results included that:
Thus is a sharp threshold for the connectedness of G(n, p).
Further properties of the graph can be described almost precisely as n tends to infinity. For example, there is a k(n) (approximately equal to 2log<sub>2</sub>(n)) such that the largest clique in G(n, 0.5) has almost surely either size k(n) or k(n) + 1.
Thus, even though finding the size of the largest clique in a graph is NP-complete, the size of the largest clique in a "typical" graph (according to this model) is very well understood.
Edge-dual graphs of Erdos-Renyi graphs are graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.
In percolation theory one examines a finite or infinite graph and removes edges (or links) randomly. Thus the Erdà ÂsâÂÂRényi process is in fact unweighted link percolation on the complete graph. (One refers to percolation in which nodes and/or links are removed with heterogeneous weights as weighted percolation). As percolation theory has much of its roots in physics, much of the research done was on the lattices in Euclidean spaces. The transition at np = 1 from giant component to small component has analogs for these graphs, but for lattices the transition point is difficult to determine. Physicists often refer to study of the complete graph as a mean field theory. Thus the Erdà ÂsâÂÂRényi process is the mean-field case of percolation.
Some significant work was also done on percolation on random graphs. From a physicist's point of view this would still be a mean-field model, so the justification of the research is often formulated in terms of the robustness of the graph, viewed as a communication network. Given a random graph of n â« 1 nodes with an average degree . Remove randomly a fraction of nodes and leave only a fraction from the network. There exists a critical percolation threshold below which the network becomes fragmented while above a giant connected component of order n exists. The relative size of the giant component, P<sub>âÂÂ</sub>, is given by
Both of the two major assumptions of the G(n, p) model (that edges are independent and that each edge is equally likely) may be inappropriate for modeling certain real-life phenomena. Erdà ÂsâÂÂRényi graphs have low clustering, unlike many social networks. Some modeling alternatives include BarabásiâÂÂAlbert model and Watts and Strogatz model. These alternative models are not percolation processes, but instead represent a growth and rewiring model, respectively. Another alternative family of random graph models, capable of reproducing many real-life phenomena, are exponential random graph models.
The G(n, p) model was first introduced by Edgar Gilbert in a 1959 paper studying the connectivity threshold mentioned above. The G(n, M) model was introduced by Erdà Âs and Rényi in their 1959 paper. As with Gilbert, their first investigations were as to the connectivity of G(n, M), with the more detailed analysis following in 1960.
A continuum limit of the graph was obtained when is of order . Specifically, consider the sequence of graphs for . The limit object can be constructed as follows:
Applying this procedure, one obtains a sequence of random infinite graphs of decreasing sizes: . The theorem states that this graph corresponds in a certain sense to the limit object of as .