In the mathematical field of graph theory, the Erdős–Rényi model is either of two closely related models for generating random graphs. They are named after mathematicians Paul Erdős and Alfréd Rényi, who first introduced one of the models in 1959, while Edgar Gilbert introduced the other model contemporaneously and independently of Erdős and Rényi. In the model of Erdős and Rényi, all graphs on a fixed vertex set with a fixed number of edges are equally likely; in the model introduced by Gilbert, 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.
Definition
There are two closely related variants of the Erdős–Rényi random graph model.
In the G model, a graph is chosen uniformly at random from the collection of all graphs which have n nodes and M edges. For example, in the G model, each of the three possible graphs on three vertices and two edges are included with probability 1/3.
In the G model, a graph is constructed by connecting nodes randomly. Each edge is included in the graph with probability p independent from every other edge. Equivalently, all graphs with n nodes and M edges have equal probability of
The behavior of random graphs are often studied in the case where n, the number of vertices, tends to infinity. Although p and M can be fixed in this case, they can also be functions depending on n. For example, the statement means
Comparison between the two models
The expected number of edges in G is, and by the law of large numbers any graph in G will almost surely have approximately this many edges. Therefore, a rough heuristic is that if pn2 → ∞, then G should behave similarly to G with as n increases. For many graph properties, this is the case. If P is any graph property which is monotonewith respect to the subgraph ordering, then the statements "P holds for almost all graphs in G" and "P holds for almost all graphs in " are equivalent. For example, this holds if P is the property of being connected, or if P is the property of containing a Hamiltonian cycle. However, this will not necessarily hold for non-monotone properties. In practice, the G model is the one more commonly used today, in part due to the ease of analysis allowed by the independence of the edges.
Properties of ''G''(''n'', ''p'')
With the notation above, a graph in G 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, Erdős and Rényi described the behavior of G very precisely for various values of p. Their results included that:
If np < 1, then a graph in G will almost surely have no connected components of size larger than O.
If np = 1, then a graph in G will almost surely have a largest component whose size is of order n2/3.
If np → c > 1, where c is a constant, then a graph in G will almost surely have a unique giant component containing a positive fraction of the vertices. No other component will contain more than O vertices.
If, then a graph in G will almost surely contain isolated vertices, and thus be disconnected.
If, then a graph in G will almost surely be connected.
Thus is a sharp threshold for the connectedness of G. Further properties of the graph can be described almost precisely as n tends to infinity. For example, there is a k such that the largest clique in G has almost surely either size k or k + 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 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.
Relation to percolation
In percolation theory one examines a finite or infinite graph and removes edges randomly. Thus the Erdős–Rényi process is in fact unweighted link percolation on the complete graph.. 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 Erdős–Rényi 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 ≫ 1 nodes with an average degree <k>. Remove randomly a fraction 1 − p′ of nodes and leave only a fraction p′ 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∞, is given by
Caveats
Both of the two major assumptions of the G model may be inappropriate for modeling certain real-life phenomena. Erdős–Rényi graphs have low clustering, unlike many social networks. Some modeling alternatives include Barabási–Albert model and Watts and Strogatz model. These alternative models are not percolation processes, but instead represent a growth and rewiring model, respectively. A model for interacting Erdős–Rényi networks was developed recently by Buldyrev et al.
History
The G model was first introduced by Edgar Gilbert in a 1959 paper studying the connectivity threshold mentioned above. The G model was introduced by Erdős and Rényi in their 1959 paper. As with Gilbert, their first investigations were as to the connectivity of G, with the more detailed analysis following in 1960.
