Sharp Thresholds of Graph properties and the k sat Problem

Given a monotone graph property P consider p P the proba bility that a random graph with edge probability p will have P The function d p P dp is the key to understanding the threshold behavior of the property P We show that if d p P dp is small corresponding to a non sharp threshold then there is a list of graphs of bounded size such that P can be approximated by the property of having one of the graphs as a subgraph One striking consequences of this result is that a coarse threshold for a random graph property can only happen when the value of the critical edge probability is a rational power of n As an application of the main theorem we settle the question of the existence of a sharp threshold for the satis ability of a random k CNF formula An appendix by Jean Bourgain was added after the rst version of this paper was written In this appendix some of the conjectures raised in this paper are proven along with more general results Institute of Mathematics The Hebrew University of Jerusalem Givat Ram Jerusalem Israel Email ehudf math huji ac il This Paper is part of a Ph D thesis prepared under the supervision of Prof Gil Kalai Mathematics Subject Classi cation C A Introduction De nitions Consider G n p the probability space of random graphs on n vertices with edge probability p We will be considering subsets of this space de ned by monotone graph properties A monotone graph property P is a property of graphs such that a P is invariant under graph automorphisims b If graph H has property P than so does any graph G having H as a sub graph A monotone symmetric family of graphs is a family de ned by such a prop erty One of the rst observations made about random graphs by Erd os and R enyi in their seminal work on random graph theory was the existence of threshold phenomena the fact that for many interesting properties P the probability of P appearing in G n p exhibits a sharp increase at a certain critical value of the parameter p Bollob as and Thomason proved the exis tence of threshold functions for all monotone set properties and in it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appear ance increases from values very close to to values close to in a very small interval More precise analysis of the size of the threshold interval is done in This threshold behavior which occurs in various settings which arise in com binatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics One of the main questions that arise in studying phase transitions is how sharp is the transition For example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satis ability of a random k CNF Boolean formula Nati Linial who intro duced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to nd gen eral conditions for sharpness of the phase transition answering the question posed in as to the relation between the length of the threshold interval and the value of the critical probability In this paper we indeed introduce a simple condition and prove it is su cient Stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph This condition can be applied in a more general setting such as that of the k sat problem where indeed it can be used to demonstrate the sharpness of the threshold Let us now de ne precisely the question with which we wish to deal Consider An a family of graphs on n vertices de ned by a monotone graph property Pn Let us de ne what we mean by a sharp threshold vs a coarse one for a series of such properties Recall that G n p is actually a product space of n copies of the point space endowed with the product measure and p A the measure of A is the probability that a random graph with edge probability p will belong to A and is a monotone function of p Fix and for a property P and the family A de ned by it let p be such that p A and p be de ned by p A De ne the threshold length to be p p There exists pc p p the critical p such that pc A Now for a series of properties P n we will say that the properties have a sharp threshold if lim n pc n where pc n is the critical p for P n If the ratio pc is bounded away from zero we will say that properties have a coarse threshold Bollob as and Thomason showed that this ratio is bounded from above From a coarse threshold for a graph property can only happen for small enough p i e p bounded from above by a negative power of n The question of understanding coarse thresholds for non symmetric properties at values of p that are bounded from is also interesting see Example Connectivity has a sharp threshold since the critical p is approxi mately log n n where as n On the other hand the property of having a triangle in the graph has a coarse threshold since both the critical p and the length of the threshold interval are of magnitude n The rst naive conjecture that one might raise is that a coarse threshold happens only for such properties i e having a certain graph as a subgraph The following example shows however that this conjecture must be slightly modi ed Consider the property G is a graph on n vertices with a triangle as a sub graph and at least log n edges A moment s re ection shows that this property is probabilisticly equivalent to the previous one and di ers from it by a set of graphs with total probability which is negligible What we suggest in this paper is that the naive conjecture is correct except for such arti cial examples Before presenting the main theorems here are a few de nitions and nota tions A balanced graph is a graph with average degree no smaller than that of any of its subgraphs A strictly balanced graph is one where the average degree is strictly larger than that of any proper subgraph For example any cycle is a strictly balanced graph where as two disjoint copies of a cycle make up a balanced but not strictly balanced graph For a family of graphs A we will call a graph H minimal if H belongs to A but no subgraph of H does Let kAk denote the number of edges of the largest minimal graph in A when A is non empty and de ne kAk when A is the empty family Throughout this paper c will denote a constant not necessarily the same one each time it appears When dealing with graphs n will denote the number of vertices and N n the number of edges in the complete graph We will be interested in p p n such that p tends to zero as n tends to in nity Let q p For a graph H jHj will denote the number of edges in H and v H the number of vertices E H will denote the expected number of copies of H in G n p E H p n v H v H jAut H j For graphs H S we will denote the fact that they are isomorphic by H S For a graph H let H the orbit of H be the set of all subgraphs of the complete graph on n vertices which are isomorphic to H So E H j H jpjHj We de ne also another function of H which is more convenient to work with D H n H p Note that for H of bounded size D H E H cD H Obviously for a property to have a coarse threshold there must be points within the critical interval for which the derivative of the function p A with respect to p is small More precisely Remark if fAig is a series of properties with a coarse threshold i e An pc An C for all n then for each n there exists p p n such that p is in the critical interval for An and p d dp jp p C We will attack this aspect of the problem denoting the slope at a point p by I for reasons to be explained give a condition on the family A such that p I is bounded from above We now come to our main theorem Theorem There exists a function k c such that for all c any n and any monotone symmetric family of graphs A on n vertices such that p I c for every there exists a monotone symmetric family B such that kBk k c and p A B Furthermore the minimal graphs in B are all balanced What the theorem essentially means is that a family with a coarse threshold can be approximated by a family whose minimal graphs are all small Notice that any monotone family is characterized by its minimal graphs The following theorem seems at rst sight to be slightly less informative than the previous one it is however more suitable for applications i e proving certain properties have a coarse threshold Theorem Let There exist functions B c b c b c such that for all c any n and any monotone symmetric family of graphs A on n vertices such that p I c and p A for every there exists a graph G with the following properties