Random Graph Models with Non-Independent Edges

Random graph models play an important role in describing networks with random structural features. The most classical model with the largest number of existing results is the Erd˝os-R´enyi random graph, in which the edges are chosen interdependently at random, with the same probability. In many real-life situations, however, the independence assumption is not realistic. We consider random graphs in which the edges are allowed to be dependent. In our model the edge dependence is quite general, we call it p-robust random graph. Our main result is that for any monotone graph property, the p-robust random graph has at least as high probability to have the property as an Erd˝os-R´enyi random graph with edge probability p. This is very useful, as it allows the adaptation of many results from Erd˝os-R´enyi random graphs to a non-independent setting, as lower bounds. Random networks occur in many practical scenarios. Some examples are wireless ad-hoc networks, various social networks, the web graph describing the World Wide Web, and a multitude of others. Random graph models are often used to describe and analyze such networks. The oldest and most researched random graph model is the Erd˝os-R´enyi random graph G n, p . This denotes a random graph on n nodes, such that each edge is added with probability p, and it is done independently for each edge. A large number of deep results are available about such random graphs, see expositions in the books [1-4]. Below we list some examples. They are asymptotic results, and for simplicity we ignore rounding issues (i.e., an asymptotic formula may provide a non-integer value for a parameter which is defined as integer for finite graphs).

The size of a maximum clique in , n p G is asymptotically 2 1/ log p n .
If , n p G has average degree d, then its maximum independent set has asymptotic size (2 ) / n In d d .
The chromatic number of  However, the requirement that the edges are independent is often a severe restriction in modelling real-life networks. Therefore, numerous attempts have been made to develop models with various dependencies among the edges, see a survey in [2]. Here we consider a general form of edge dependency. We call a random graph with this type of dependency a p-robust random graph.
probability never drops below p. Let us show some examples of p-robust random graphs.
1. First note that the classical Erd˝os-R´enyi random graph , n p G is a special case of our model, since our model also allows taking all edges independently with probability p.
2. However, we can also allow (possibly messy) dependencies. For example, let P(e) denote the probability that a given edge e is present in the graph, and let us condition on k, the number of other edges in the whole graph. Let P e (k) denote the probability that there are k edges in the graph, other than e. For any 4. Consider the model described above in 3, but with the additional condition that each potential edge e has at least 3 adjacent edges, whether or not e is in the graph. What can we say about this conditional random graph? The same derivation as in 3, but with 3 k ≥ , gives us that the new random graph will remain p-robust, but now with If we have a random graph like the examples 2,3,4 above (and many possible others with dependent edges), then how can we estimate some parameter of the random graph, like the size of the maximum clique? We show that at least for so called monotone properties we can use the existing results about Erd˝os-R´enyi random graphs as lower bounds. Let Q be a set of graphs. We use it to represent a graph property: a graph G has property Q if and only if G Q ∈ . Therefore, we identify the property with Q. We are going to consider monotone graph properties, as defined below.

Definition 2 (Monotone Graph Property)
A graph property Q is called monotone, if it is closed with Note that many important graph properties are monotone.
Examples: having a clique of size at least k, having a Hamiltonian circuit, having k disjoint spanning trees, having chromatic number at least k, having diameter at most k, having a dominating set of size at most k, having a matching of size at least k, and numerous others.
In fact, almost all interesting graph properties have a monotone version. Our result is that for any monotone graph property, and for any n, p, it always holds that where the "∼" relation between random graphs means that they have the same distribution, i.e., they are statistically indistinguishable. If this can be accomplished, then the claim will directly follow, since then a random graph distributed as . Now let us generate the random graphs , n p G and G 2 , as follows.
Step 2: If i = m, then let Step 5: If i > 1, then decrease i by one, and go to Step 2: else HALT.
First note that the value