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 Erdős-Rényi random graph . This denotes a random graph on nodes, such that each edge is added with probability , 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 [2, 4, 5]. Below we list some examples. They are asymptotic results, and for simplicity/clarity we omit potential restrictions for the range of , as well as 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 is asymptotically .
If has average degree , then its maximum independent set has asymptotic size .
The chromatic number of is asymptotically , where .
The size of a minimum dominating set in is asymptotically , where .
The length of the longest cycle in , when the graph has a constant average degree , is asymptotically .
The diameter of is asymptotically , when . (If the graph is not connected, then the diameter is defined as the largest diameter of its connected components.)
If has average degree , then the number of nodes of degree in the graph is asymptotically .
These results (and many others) make it possible that for random graphs that one can find good and directly computable estimates of graph parameters that are hard to compute for deterministic graphs. Moreover, the parameters often show very strong concentration. For example, as listed above, the chromatic number ofis asymptotically , where . However, we can say more: the chromatic number of a random graph is so strongly concentrated that with probability approaching one, as , it falls on one of two consecutive integers (see Alon and Krivelevich ).
2 Random Graphs With Dependent Edges
The requirement that the edges are independent is often a severe restriction in modeling real-life networks. Therefore, numerous attempts have been made to develop models with various dependencies among the edges, see, e.g., a survey in . Here we consider a general form of edge dependency. We call a random graph with this type of dependency a -robust random graph.
(-robust random graph) A random graph on vertices is called -robust, if every edge is present with probability at least , regardless of the status (present or not) of other edges. Such a random graph is denoted by .
Note that -robustness does not imply independence. It allows that the existence probability of an edge may depend on other edges, possibly in a complicated way, it only requires that the probability never drops below . Let us show some examples of -robust random graphs.
- Example 1
First note that the classical Erdős-Rényi random graph is a special case of , since our model also allows adding all edges independently with probability .
- Example 2
However, we can also allow (possibly messy) dependencies. For example, let denote the probability that a given edge is present in the graph, and let us condition on , the number of other edges in the whole graph. Let denote the probability that there are edges in the graph, other than . For any fixed , set ; let this be the probability that edge exists, given that there are other edges in the graph. Using that the total number of edges cannot be more than , we have that always holds. Therefore, , for any , implying . Thus, with , this random graph is -robust. At the same time, the edges are not independent, since the probability that is present depends on how many other edges are present.
- Example 3
For a given edge , let denote the number of edges that are adjacent with (not including itself). If does not exist, then let . Let the conditional probability that edge exists, given that it has adjacent edges, be . Note that the possible range of is . Then we have . This implies . Thus, with , this random graph is -robust. At the same time, the edges are not independent, since the probability that is present is influenced by the number of adjacent edges.
- Example 4
Consider the model described above in 3, but with the additional condition that each potential edge has at least 3 adjacent edges, whether or not is in the graph. What can we say about this conditional random graph? The same derivation as in 3, but with , gives us that the new random graph will remain -robust, but now with .
If we have a random graph like the ones in 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 Erdős-Rényi random graphs as lower bounds.
Let be a set of graphs. We use it to represent a graph property: a graph has property if and only if . Therefore, we identify the property with . We are going to consider monotone graph properties, as defined below.
(Monotone graph property) A graph property is called monotone, if it is closed with respect to adding new edges. That is, if and , then .
Note that many important graph properties are monotone. Examples: having a clique of size at least , having a Hamiltonian circuit, having disjoint spanning trees, having chromatic number at least , having diameter at most , having a dominating set of size at most , having a matching of size at least , 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 , it always holds that is more likely to have the property than (or at least as likely). This is very useful, as it allows the application of the rich treasury of results on Erdős-Rényi random graphs to the non-independent setting, as lower bounds on the probability of having a monotone property. Below we state and prove our general result.
Let be any monotone graph property. Then the following holds:
Proof. We are going to generate as the union of two random graphs, and , both on the same vertex set . is the usual Erdős-Rényi random graph, will be defined later. The union is meant with the understanding that if the same edge occurs in both graphs, then we merge them into a single edge. We plan to chose the edge probabilities in , such 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 can be obtained by adding edges to , which cannot destroy a monotone property, once has it. This will imply the claim.
We introduce some notations. Let denote the (potential) edges. For every , let be the indicator of the event that the edge is included in . Further, let us use the abbreviation . For any , the event means that takes a realization in which edge is included if and only . Similarly, means . We also use the abbreviation . Now let us generate the random graphs and , as follows.
- Step 1.
- Step 2.
If , then let . If , then set , where indicates the already generated edges of .
- Step 3.
- Step 4.
Put into with probability , and put into with probability .
- Step 5.
If , then decrease by one, and go to Step 2; else halt.
First note that the value in Step 4 can indeed be used as a probability. Clearly, holds, as is a probability and . To show , observe that since the inequality can be rearranged into , which simplifies to . The latter is indeed true, due to , which follows from the -robust property.
Next we show that the algorithm generates the random graphs and in a way that they satisfy . We prove it by induction, starting from and progressing downward to . For any , let , denote the already generated parts of , respectively, after executing Step 4 times, so they can only contain edges with index . Further, let be the subgraph of in which we only keep the edges with index , that is, . The inductive proof will show that holds for every . At the end of the induction, having reached , we are going to get , which is the same as .
Let us consider first the base case . Then we have by Step 4. Then in Step 4, edge is put into with probability , yielding . Now observe that the formula (1) is chosen such that is precisely the solution of the equation
for . For the equation becomes
and is the solution of this equation. Since by Step 4 we have and , therefore, we get that the left-hand side of (3) is precisely the probability of the event . By (3), this probability is equal to , which is set to in Step 2. This means that , as desired.
For the induction step, assume that the claim is true for , i.e., holds. In Step 4, edge is added to with probability . It is also added to with probability . Therefore, just like in the base case, we get that We already know that satisfies the equation (2), so is added to with probability , given the already generated part, represented by . By the inductive assumption, is distributed as , which is the truncated version of , keeping only the indexed edges. Hence, for
, we can write by the chain rule of conditional probabilities:
After processing (i.e., adding it with probability ), we get
which, by the chain rule, is indeed the distribution of , completing the induction.
Thus, at the end, a realization of is generated with probability
indeed creating with its correct probability. Therefore, we get , so arises by adding edges to , which cannot destroy a monotone property. This implies the statement of the Theorem, completing the proof.
3 An Example
As a sample application of the result, consider the random graph described in Example 3. For handy access, let us repeat the example here:
Example 3. For a given edge , let denote the number of edges that are adjacent with (not including itself). If does not exist, then let . Let the conditional probability that edge exists, given that it has adjacent edges, be
Note that the possible range of is . Then we have
Thus, with , this random graph is -robust. At the same time, the edges are not independent, since the probability that is present is influenced by the number of adjacent edges.
Now we can claim that this random graph asymptotically has a maximum clique of size at least , chromatic number at least with , minimum dominating set at most , also with , and diameter at most . All these follow from the known results about the Erdős-Rényi random graph (listed in the Introduction), complemented with our Theorem 1. Note that such bounds would be very hard to prove directly from the definition of the edge dependent random graph model.
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