A General Dependency Structure for Random Graphs and Its Effect on Monotone Properties
aa r X i v : . [ c s . D M ] D ec A General Dependency Structure forRandom Graphs and Its Effect on MonotoneProperties
Zohre Ranjbar-Mojaveri and Andr´as Farag´oDepartment of Computer ScienceThe University of Texas at DallasRichardson, Texas
Abstract
We consider random graphs in which the edges are allowed to be dependent . In ourmodel the edge dependence is quite general, we call it p -robust random graph. It meansthat every edge is present with probability at least p , regardless of the presence/absenceof other edges. This is more general than independent edges with probability p , as weillustrate with examples. 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 anErd˝os-R´enyi random graph with edge probability p . This is very useful, as it allowsthe adaptation of many results from classical 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-hocnetworks, various social networks, the web graph describing the World Wide Web, and amultitude of others. Random graph models are often used to describe and analyze suchnetworks.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 availableabout such random graphs, see expositions in the books [2, 4, 5]. Below we list some exam-ples. They are asymptotic results, and for simplicity/clarity we omit potential restrictionsfor the range of p , as well as ignore rounding issues (i.e., an asymptotic formula may providea non-integer value for a parameter which is defined as integer for finite graphs). • The size of a maximum clique in G n,p is asymptotically 2 log /p n . • If G n,p has average degree d , then its maximum independent set has asymptotic size n ln dd . 1 The chromatic number of G n,p is asymptotically n log b n , where b = − p . • The size of a minimum dominating set in G n,p is asymptotically log b n , where b = − p . • The length of the longest cycle in G n,p , when the graph has a constant average degree d , is asymptotically n (1 − d e − d ). • The diameter of G n,p is asymptotically log n log( np ) , when np → ∞ . (If the graph is notconnected, then the diameter is defined as the largest diameter of its connected com-ponents.) • If G n,p has average degree d , then the number of nodes of degree k in the graph isasymptotically d k e − d k ! n .These results (and many others) make it possible that for random graphs that one canfind good and directly computable estimates of graph parameters that are hard to computefor deterministic graphs. Moreover, the parameters often show very strong concentration.For example, as listed above, the chromatic number of G n,p is asymptotically n log b n , where b = − p . However, we can say more: the chromatic number of a random graph is so stronglyconcentrated that with probability approaching one, as n → ∞ , it falls on one of twoconsecutive integers (see Alon and Krivelevich [1]). The requirement that the edges are independent is often a severe restriction in modelingreal-life networks. Therefore, numerous attempts have been made to develop models withvarious dependencies among the edges, see, e.g., a survey in [3]. Here we consider a generalform of edge dependency. We call a random graph with this type of dependency a p -robustrandom graph. Definition 1 ( p -robust random graph) A random graph on n vertices is called p -robust ,if every edge is present with probability at least p , regardless of the status (present or not) ofother edges. Such a random graph is denoted by e G n,p . Note that p -robustness does not imply independence. It allows that the existence proba-bility of an edge may depend on other edges, possibly in a complicated way, it only requiresthat the probability never drops below p . Let us show some examples of p -robust randomgraphs. Example 1
First note that the classical Erd˝os-R´enyi random graph G n,p is a special caseof e G n,p , since our model also allows adding all edges independently with probability p . Example 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 uscondition on k , the number of other edges in the whole graph. Let P e ( k ) denote the2robability that there are k edges in the graph, other than e . For any fixed k , set P ( e | k ) = 1 − ( k + 1) /n ; let this be the probability that edge e exists, given that thereare k other edges in the graph. Using that the total number of edges cannot be morethan n ( n − /
2, we have that k ≤ n ( n − / − P ( e | k ) ≥ − n ( n − n = 1 − n − n ≥ , for any k , implying P ( e ) = P n ( n − / − k =0 P ( e | k ) P e ( k ) ≥ / p = 1 /
2, this random graph is p -robust. At the same time, the edges arenot independent, since the probability that e is present depends on how many otheredges are present. Example 3
For a given edge e , let r ( e ) denote the number of edges that are adjacent with e (not including e itself). If e does not exist, then let r ( e ) = 0. Let the conditionalprobability that edge e exists, given that it has k adjacent edges, be P ( e | r ( e ) = k ) = − k +5 . Note that the possible range of k is 0 ≤ k ≤ n − P ( e | r ( e ) = k ) ≥ − = . This implies P ( e ) = P n − k =0 P ( e | r ( e ) = k ) P ( r ( e ) = k ) ≥ P n − k =0 P ( r ( e ) = k ) = . Thus, with p = , this random graph is p -robust. Atthe same time, the edges are not independent, since the probability that e is presentis influenced by the number of adjacent edges. Example 4
Consider the model described above in 3, but with the additional conditionthat each potential edge e has at least 3 adjacent edges, whether or not e is in thegraph. What can we say about this conditional random graph? The same derivationas in 3, but with k ≥
3, gives us that the new random graph will remain p -robust, butnow with p = .If we have a random graph like the ones in Examples 2,3,4 above (and many possibleothers with dependent edges), then how can we estimate some parameter of the randomgraph, like the size of the maximum clique? We show that at least for so called monotoneproperties 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 itis closed with respect to adding new edges. That is, if G ∈ Q and G ⊆ G ′ , then G ′ ∈ Q . Note that many important graph properties are monotone. Examples: having a cliqueof size at least k , having a Hamiltonian circuit, having k disjoint spanning trees, havingchromatic number at least k , having diameter at most k , having a dominating set of sizeat most k , having a matching of size at least k , and numerous others. In fact, almost allinteresting graph properties have a monotone version. Our result is that for any monotonegraph property, and for any n, p , it always holds that e G n,p is more likely to have the propertythan G n,p (or at least as likely). This is very useful, as it allows the application of the richtreasury of results on Erd˝os-R´enyi random graphs to the non-independent setting, as lowerbounds on the probability of having a monotone property. Below we state and prove ourgeneral result. 3 heorem 1 Let Q be any monotone graph property. Then the following holds: Pr( G n,p ∈ Q ) ≤ Pr( e G n,p ∈ Q ) . Proof.
We are going to generate e G n,p as the union of two random graphs, G n,p and G , bothon the same vertex set V . G n,p is the usual Erd˝os-R´enyi random graph, G will be definedlater. The union G n,p ∪ G is meant with the understanding that if the same edge occurs inboth graphs, then we merge them into a single edge. We plan to chose the edge probabilitiesin G , such that G n,p ∪ G ∼ e G n,p , where the “ ∼ ” relation between random graphs meansthat they have the same distribution, i.e., they are statistically indistinguishable. If this canbe accomplished, then the claim will directly follow, since then a random graph distributedas e G n,p can be obtained by adding edges to G n,p , which cannot destroy a monotone property,once G n,p has it. This will imply the claim.We introduce some notations. Let e , . . . , e m denote the (potential) edges. For every i ,let h i be the indicator of the event that the edge e i is included in e G n,p . Further, let us usethe abbreviation h mi = ( h i , . . . , h m ). For any a = ( a , . . . , a m ) ∈ { , } m , the event { h m = a } means that e G n,p takes a realization in which edge e i is included if and only a i = 1. Similarly, { h mi = a mi } means { h i = a i , . . . , h m = a m } . We also use the abbreviation a mi = ( a i , . . . , a m ).Now let us generate the random graphs G n,p and G , as follows.Step 1. Let i = m .Step 2. If i = m , then let q m = Pr( h m = 1). If i < m , then set q i = Pr( h i = 1 | h mi +1 = a mi +1 ), where a mi +1 indicates the already generated edges of G n,p ∪ G .Step 3. Compute p ′ i = p (1 − q i )1 − p . (1)Step 4. Put e i into G n,p with probability p , and put e i into G with probability q i − p ′ i .Step 5. If i >
1, then decrease i by one, and go to Step 2; else halt. First note that the value q i − p ′ i in Step 4 can indeed be used as a probability. Clearly, q i − p ′ i ≤ q i is a probability and p ′ i ≥
0. To show q i − p ′ i ≥
0, observe that p ′ i = p (1 − q i )1 − p ≤ q i , since the inequality can be rearranged into p (1 − q i ) ≤ q i (1 − p ), whichsimplifies to p ≤ q i . The latter is indeed true, due to q i = Pr( h i = 1 | h mi +1 = a mi +1 ) ≥ p ,which follows from the p -robust property.Next we show that the algorithm generates the random graphs G n,p and G in a waythat they satisfy G n,p ∪ G ∼ e G n,p . We prove it by induction, starting from i = m andprogressing downward to i = 1. For any i , let G in,p , G i denote the already generated partsof G n,p , G , respectively, after executing Step 4 m − i + 1 times, so they can only containedges with index ≥ i . Further, let e G in,p be the subgraph of e G n,p in which we only keep theedges with index ≥ i , that is, e G in,p = e G n,p − { e i − , . . . , e } . The inductive proof will showthat G in,p ∪ G i ∼ e G in,p holds for every i . At the end of the induction, having reached i = 1,we are going to get G n,p ∪ G ∼ e G n,p , which is the same as G n,p ∪ G ∼ e G n,p .4et us consider first the base case i = m . Then we have Pr( e m ∈ G n,p ) = Pr( e m ∈ G mn,p ) = p by Step 4. Then in Step 4, edge e m is put into G with probability q m − p ′ m ,yielding Pr( e m ∈ G m ) = q m − p ′ m . Now observe that the formula (1) is chosen such that p ′ i is precisely the solution of the equation p + q i − p ′ i − ( q i − p ′ i ) p = q i (2)for p ′ i . For i = m the equation becomes p + q m − p ′ m − ( q m − p ′ m ) p = q m , (3)and p ′ m = p (1 − q m )1 − p is the solution of this equation. Since by Step 4 we have Pr( e m ∈ G mn,p ) = p and Pr( e m ∈ G m ) = q m − p ′ m , therefore, we get that the left-hand side of (3) is precisely theprobability of the event { e m ∈ G mn,p ∪ G m } . By (3), this probability is equal to q m , which isset to q m = Pr( h m = 1) = Pr( e m ∈ e G mn,p ) in Step 2. This means that G mn,p ∪ G m ∼ e G mn,p , asdesired.For the induction step, assume that the claim is true for i + 1, i.e., G i +1 n,p ∪ G i +12 ∼ e G i +1 n,p holds. In Step 4, edge e i is added to G i +1 n,p with probability p . It is also added to G i +12 withprobability q i − p ′ i . Therefore, just like in the base case, we get that p + q i − p ′ i − ( q i − p ′ i ) p =Pr( e i ∈ G in,p ∪ G i ) . We already know that p ′ i satisfies the equation (2), so e i is added to e G i +1 n,p with probability q i = Pr( h i = 1 | h mi +1 = a mi +1 ), given the already generated part, representedby a mi +1 . By the inductive assumption, h mi +1 is distributed as e G i +1 n,p , which is the truncatedversion of e G n,p , keeping only the ≥ i + 1 indexed edges. Hence, for h mi +1 , we can write by thechain rule of conditional probabilities:Pr( h mi +1 = a mi +1 ) = Pr( h m = a m ) m − Y j = i +1 Pr( h j = a j | h mj +1 = a mj +1 ) . After processing e i (i.e., adding it with probability q i ), we getPr( h mi = a mi ) = Pr( h i = a i | h mi +1 = a mi +1 ) Pr( h mi +1 = a mi +1 )= Pr( h i = a i | h mi +1 = a mi +1 ) Pr( h m = a m ) m − Y j = i +1 Pr( h j = a j | h mj +1 = a mj +1 ) | {z } Pr( h mi +1 = a mi +1 ) = Pr( h m = a m ) m − Y j = i Pr( h j = a j | h mj +1 = a mj +1 ) , which, by the chain rule, is indeed the distribution of e G in,p , completing the induction.Thus, at the end, a realization a = a m ∈ { , } m of e G n,p is generated with probabilityPr( h m = a ) = Pr( h m = a m ) m − Y j =1 Pr( h j = a j | h mj +1 = a mj +1 ) , e G n,p with its correct probability. Therefore, we get G n,p ∪ G ∼ e G n,p , so e G n,p arises by adding edges to G n,p , which cannot destroy a monotone property. This implies thestatement of the Theorem, completing the proof. ♠ 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 e , let r ( e ) denote the number of edges that areadjacent with e (not including e itself). If e does not exist, then let r ( e ) = 0. Letthe conditional probability that edge e exists, given that it has k adjacent edges,be P ( e | r ( e ) = k ) = 12 − k + 5 . Note that the possible range of k is 0 ≤ k ≤ n − P ( e | r ( e ) = k ) ≥ −
15 = 310 . This implies P ( e ) = n − X k =0 P ( e | r ( e ) = k ) P ( r ( e ) = k ) ≥ n − X k =0 P ( r ( e ) = k ) = 310 , Thus, with p = , this random graph is p -robust. At the same time, the edgesare not independent, since the probability that e is present is influenced by thenumber of adjacent edges.Now we can claim that this random graph asymptotically has a maximum clique of sizeat least 2 log / n , chromatic number at least n log b n with b = − p = , minimum dominatingset at most log b n , also with b = 10 /
7, and diameter at most log n log(3 n/ . All these follow fromthe known results about the Erd˝os-R´enyi random graph G n,p (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. References [1] N. Alon and M. Krivelevich, “The Concentration of the Chromatic Number of RandomGraphs,”
Combinatorica , vol. 17, 1977, pp. 303–313.[2] B. Bollob´as,
Random Graphs,
Cambridge University Press, 2001.63] A. Farag´o, “Network Topology Models for Multihop Wireless Networks,”
ISRN Com-munications and Networking,
Vol. 2012, Article ID 362603, doi:10.5402/2012/362603[4] A. Frieze and M. Karo´nski,
Introduction to Random Graphs,
Cambridge Univ. Press,2016.[5] S. Janson, T. Luczak, and A. Rucinski,