Treewidth of Erdős-Rényi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs

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1 Treewidth of Erdős-Rényi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs Yong Gao Department of Computer Science, Irving K. Barber School of Arts and Sciences University of British Columbia Okanagan, Kelowna, Canada VV V7 Abstract We study conditions under which the treewidth of three different classes of random graphs is linear in the number of vertices. For the Erdős-Rényi random graph Gn, m), our result improves a previous lower bound obtained by Kloks and Bodlaender [3]. For random intersection graphs, our result strengthens a previous observation on the treewidth by Karoński et al. [9]. For scale-free random graphs based on the Barabási-Albert preferential-attachment model, it is shown that if more than vertices are attached to a new vertex, then the treewidth of the obtained network is linear in the size of the network with high probability. Key words: Treewidth, random graphs, random intersection graphs, scale-free random graphs. Introduction The notion of treewidth introduced by Robertson and Seymour [6] plays an important role in characterizing the structural properties of a graph see, e.g., [, 8, 3]) and the complexity of a variety of algorithmic problems of practical importance. Many NP-hard problems can be solved efficiently if the treewidth of the underlying graph is bounded or small see, e.g. [6, 8, 7]). In fact, the treewidth of many interesting graph classes has been shown to be bounded [7]. On the other hand, graphs with treewidth linear in the number of vertices are also interesting. For example, it has been shown recently that the property of having a linear treewidth is closely related to the existence of linear-sized subgraphs with positive vertex expansion [8]. The research is supported by National Science and Engineering Research Council of Canada NSERC) RGPIN address: yong.gao@ubc.ca Yong Gao) Preprint submitted to Elsevier May 4, 0

2 The theory of random graphs pioneered by the work of Erdős and Rényi [3] deals with the probabilistic behavior of various graph properties such as connectivity, colorability, and the size of connected) components [, 9, 3, 4]. In addition to the Erdős-Rényi random graph, other models of random graphs have been proposed and studied in recent years in an effort to better capture the characteristics observed in large-scale complex networks arising in real-world domains such as communication networks Internet, WWW, Wireless and PP networks), computational biology protein networks), and sociology social networks). An intersection model for random graphs was introduced by Karoński, et al. [9] and has drawn much recent interest see [4] and references therein). The Barabási-Albert scale-free model for random graphs was proposed in [3] and has been shown to have a power law degree distribution and other interesting characteristics similar to those observed in many real-world networks [0]. This paper is concerned with conditions under which the treewidth of the above three classes of random graphs is linear in the number of vertices. We prove that for an edge-to-vertex ratio greater than or equal to.073, the treewidth of the Erdős-Rényi random graph is linear in the number of vertices with high probability., which improves a previous result by Kloks and Bodlaender [3, Theorem 5.3.] requiring that the edge-to-vertex ratio be greater than.8. Our result on the treewidth of random intersection graphs strengthens an observation by Karoński et al. [9, Corollary 6]. For scale-free random graphs, we show that if more than vertices are attached to a new vertex, then the treewidth of the obtained network is linear in the number of vertices of the network with high probability. The next section fixes the notation and contains preliminaries. Also discussed in the next section is a variant of the Erdős-Rényi model for random graphs which we will be using in our proof. Sections 3-5 present and prove our results on the treewidth of the Erdős-Rényi random graph, random intersection graphs, and scale-free random graphs respectively. Section 6 concludes the paper with a discussion on some recent progress and some unsolved problems.. Notation and Preliminaries Throughout this paper, all logarithms are natural logarithms, i.e., to the base e. The cardinality of a set U is denoted by U. All graphs are simple and undirected. Standard terminologies in graph theory [7] are used. Given a graph GV, E), the neighborhood of a vertex v V is denoted by Nv) = {u V u v and u, v) E} and the neighborhood of a vertex subset U V is denoted by NU) = {w V \ U w, u) E for some u U}. The induced When revising this paper, we learned from one of the referees that a more recent work has improved the lower bound to 0.5. See Section 6 for a more detailed discussion.

3 subgraph on a subset of vertices U is denoted by G[U]. By a component of a graph, we mean a maximal connected subgraph. In the proofs, we will be using the following upper bound on n βn) that can be derived from Stirling s formula. Lemma.. For any constants 0 < β <, ) n βn where θ > 0 is a constant. ) n θ β β)n β β β) β, We also need the following three lemmas on the properties of some useful functions. The proof of these lemmas are routine and can be found in the technical report version of this paper [6]. Lemma.. On the internal 0, ), the function ft) = t t t) t attains its minimum at t = and lim ft) =. Furthermore, ft) is decreasing t 0 on the interval 0, ] and decreasing on the interval [, ). Lemma.3. For any c > and sufficiently small β > 0, the function rt) = t + ɛ) c is decreasing on the interval [ β, 3 ]. Lemma.4. For sufficiently small β, the function ) 4ct t t), where c > 0 is a constant,.) e gt) = t + t + βt) c t t t) t, where c > and β > 0 are constants,.) is increasing on [ β, 3 ]... Treewidth and Random Graphs Several equivalent definitions of treewidth exist and the one based on k- trees is probably the easiest to explain. The graph class of k-trees is defined recursively as follows see, e.g., [3, Definition..8]): 3

4 . A clique with k+ vertices is a k-tree;. Given a k-tree T n with n vertices, a k-tree with n+ vertices is constructed by adding to T n a new vertex and connecting it to a k-clique in T n. A graph is called a partial k-tree if it is a subgraph of a k-tree. The treewidth of a graph G, denoted by twg), is the minimum k such that G is a partial k-tree. We use Gn, m) to denote the Erdős-Rényi random graph [9] on n vertices with m edges selected from the N = n ) possible edges uniformly at random and without replacement. Throughout this paper by with high probability, abbreviated as whp, we mean that the probability of the event under consideration is o) as n goes to infinity. In the proof of our result on the treewith of the Erdős-Rényi random graph, we will be working with a random graph model Gn, m) that is slightly different from Gn, m) in that the m edges are selected independently and uniformly at random, but with replacement. There is a one-to-one correspondence between the random graph Gn, m) and the product probability space Ω, A, P Gn,m) { }) defined as follows:. Ω = m i= E i where each E i is the set of all n ) possible edges.. A is the σ-field consisting of all subsets of Ω. 3. The probability measure P Gn,m) { } is m P Gn,m) {ω} = n, ω Ω. )) A sample point ω Ω is interpreted as an outcome of a random experiment that selects m edges independently, uniformly at random with replacement from the set of all possible edges. Note that the graph corresponding to a sample point ω Ω is actually a multi-graph, i.e., a graph in which parallel edges are allowed. The existence of parallel edges does not have any impact on our analysis. It turns out that as far as the property of having a treewidth linear in the number of vertices is concerned, it is sufficient to work on the random graph Gn, m), as indicated in the following proposition. Proposition.. If there exists a constant β > 0 such that lim n P { twgn, m)) βn } =, then lim P {twgn, m)) βn} =. n 4

5 An observation similar to the above proposition, but on general monotone increasing combinatorial properties of random discrete structures, has been made in [0] and [, Section 4.6] and formally proved in []. We therefore omit the proof of the above proposition. Due to Proposition., we will continue to use the notation Gn, m) instead of Gn, m) throughout this paper, but with the understanding that the m edges are selected independently and uniformly at random with replacement... Random Intersection Graphs The intersection model for random graphs was introduced by Karoński, et al. [9]. A random intersection graph G I n, m, p) over a vertex set V is defined by a universe M and three parameters: n the number of vertices), m = M, and 0 p. Associated with a vertex v V is a random subset S v M formed by selecting each element in M independently with probability p. A pair of vertices u and v is an edge in G I n, m, p) if and only if S u S v. It is shown in [9] that for a fixed value k, G I n, m, p) contains a clique of size k whp if the following holds { /nm p /k ), if α k/k ) /n /k )) m / ), if α k/k ) where m = n α. As a consequence, the treewidth of G I n, m, p) is greater than k ) if p is in that range. Our result in Section 4 Theorem 3) indicates that for p /m with m = n α, the treewidth of G I n, m, p) is linear in the number of vertices..3. The Barabási-Albert Scale-Free Random Graph Following the formal definition given in [0], the Barabási-Albert random graph G S n, m) on a set of n vertices V = {v,, v n } is defined by a graph evolution process in which vertices are added to the graph one at a time. In each step, the newly-added vertex is made adjacent to m existing vertices selected according to the preferential attachment mechanism, i.e. the probability that an existing vertex is selected as a neighbor is in proportion to its degree. To be more precise, let v i be the vertex to be added and let G i be the graph obtained after vertex v i is added. The m neighbors of v i are selected in m steps. In step j m, the probability that an existing vertex w is selected as the neighbor of the new vertex v i is where deg Gi w) + d w j) i )m + j ),.3) 5

6 . i )m = k i deg Gi v k ) is the total degree of G i,. d w j) is the number of times w has been picked as the neighbor of v in the first j ) trials, and 3. the term j ) is the increase in the total degree as a result of selecting the first j neighbors. 3. Treewidth of the Erdős-Rényi Random Graph In this section, we prove the following theorem showing whenever m n.073, the treewidth of the Erdős-Rényi random graph Gn, m) is whp greater than βn for some constant β > 0, improving the previous result of Kloks and Bodlaender [] Also in [3, Theorem 5.3.]). Theorem. For any m n.073, there is a constant β > 0 such that lim P Gn,m) {twgn, m)) > βn} =. 3.4) n The following notion of balanced l-partition is used by Kloks and Bodlaender [3, ] in their study on the treewidth of random graphs. Definition 3. [3]). Let G = GV, E) be a graph with V = n. Let W = S, A, B) be a triple of disjoint vertex subsets such that V = S A B and S = l +. Without loss of generality, we will always assume that B A. W is said to be balanced if 3 n l ) A, B 3 n l ). W is said to be an l-partition if S separates A and B, i.e., there are no edges between vertices in A and vertices in B. Theorem is proved by an application of the first-moment method to the random variable that counts in Gn, m) the total number of d-rigid and balanced partitions defined as follows. Definition 3.. Let d > 0 be an integer. A triple W = S, A, B) with B > A + d is said to be d-rigid if there is no subset of vertices U B with U d that induces a connected component of G[B]. The notion of d-rigid and balanced partition generalizes that of balanced partition by requiring that any vertex set of size at most d in the larger subset of a partition cannot be moved to the other subset of the partition, and hence the word rigid. As we will have to consider all the vertex sets of size at most d to get the best possible estimation, the requirement of connectivity is a kind 6

7 of maximality condition in order to avoid repeated counting of vertex sets of different sizes. For the case of d =, being d-rigid means that G[B] has no isolated vertices. The motivation is that by considering the expected number of these more restricted partitions, we will be able to get a more accurate estimation when applying Markov s inequality. We note that the idea of imposing various restrictions on the combinatorial objects under consideration has been used in recent years to increase the power of the first moment method when estimating the threshold of the satisfiability of random CNF formulas [0, ] and the colorability of random graphs [, Section 4.7]. The difficulty we have to overcome is that to estimate the expected number of d-rigid and balanced partitions S, A, B) in Gn, m), an exponentially-small upper bound is required on the probability that the induced subgraph G[B] of Gn, m) does not have small-sized tree components. We managed to obtain such an exponentially-small upper bound in a conditional probability space. To achieve the best possible Lipschitz constant in our application of the Hoeffding- Azuma inequality in the conditional probability space, we use a weighted count on the number of tree components of size up to a fixed constant d. We are not aware of any other application of the Hoeffding-Azuma inequality in the study of random discrete structures where this idea of using a weighted count is beneficial. Lemma 3.. Let d be an integer. Any graph with treewidth at most l > 4 must have a balanced l-partition W = S, A, B) such that either B A + d or W is d-rigid. Proof. From [3], any graph with treewidth at most l > 4 must have a balanced l-partition W = S, A, B). If B A +d, we are done. Otherwise, if the triple W is not d-rigid, then there must be a vertex subset U B that induces a component of G[B] and consequently NU) B \ U) =. Therefore, we can move U from B to A and create a new balanced l-partition with the size of B decreased by U. We continue this process until either B A + d or the partition becomes d-rigid. 3.. Conditional Probability of a d-rigid and balanced l-partition We bound the conditional probability that a balanced triple W = S, A, B) with S = l + and B A + d is d-rigid given that it is an l-partition for Gn, m). The following variate of the Hoeffding-Azuma inequality will be used in the proof. Lemma 3. Lemma. [5] and Theorem.9 [9]). Let Ω = m Ω i be 7 i=

8 an independent product probability space where each Ω i is a finite set, and f : Ω R be a random variable satisfying the following Lipschitz condition fω) fω ) c f 3.5) for every pair ω, ω Ω that only differ in one coordinate. Then, for any t > 0, P {fω) E [fω)] t} e t c f m. To ease the presentation, let d > 0 be a constant and write xt, c) = gt, c) = rt, c) = ct t t +, d ) d i i i d i! i= xt, c)e xt,c)) i, t + /d )) c e xt,c) 3.6) Theorem. Let Gn, m), c = m n, be a random graph and let W = S, A, B) be a balanced triple such that S = l +, A = a, and B = b = tn. Let d > 0 be a constant integer less than l +. Then for n sufficiently large, P Gn,m) {W is d-rigid W is an l-partition} ) r+g) n e 3.7) where r = rt, c) and g = gt, c) are defined in Equation 3.6). Proof. Conditional on that W is an l-partition of Gn, m), each of the m edges can only be selected from the set of edges E W = V \ {u, v) : u A, v B}, where V denotes the set of unordered pair of vertices. The size s of E W is s = E W = nn ) ba = nn ) tnn tn l + )). In the rest of the proof, we will work on the conditional probability space P = Ω, P P { }) where Ω = Ω Ω Ω m and Ω i = E W for each i m. A sample point ω = ω,, ω m ) Ω corresponds to an outcome of selecting m edges from E W uniformly at random and with replacement. Note that W is a balanced l-partition for the graph determined by ω Ω. The probability measure P P { } is defined as P P {ω} = /s) m. The following lemma guarantees that we can obtain Equation 3.7) by studying the probability P P {W is d-rigid}. 8

9 Lemma 3.3. Proof. P Gn,m) {W is d-rigid W is an l-partition} = P P {W is d-rigid}. Recall that P Gn,m) { } is the probability measure for the probability space Ω, P Gn,m) { }) and P P { } is the probability measure for the probability space P = Ω, P P { }). Note that Ω is the set of sample points ω in Ω such that W is an l-partition in the graph determined by ω. Let Q Ω be the set of sample points ω such that W is d-rigid in the graph determined by ω. We have P Gn,m) {W is d-rigid W is an l-partition} = Q Ω Ω definition of conditional probability) = P P {W is d-rigid}. definition of the two probability spaces) This proves the lemma. Continuing the proof of Theorem, we bound P P {W is d-rigid} by using the Hoeffding-Azuma inequality. To make things simpler, we will bound the probability that there exist tree components, instead of general connected components, of size at most d in the subgraph of Gn, m) induced on the vertex subset B. To apply the Hoeffding-Azuma inequality see Lemma 3.) to our case, the probability space is P = Ω, P P { }), and we can use any Lipschitz function f : Ω R such that fω) = 0 whenever the total number of tree components of size at most d in the graph determined by ω is zero. To achieve the best possible Lipschitz constant c f in Equation 3.5), we consider a weighted sum I : Ω R of all tree components of size at most d defined as follows. For any i d, let U i = {U B : U = i} be the collection of size-i vertex subsets in B and let d U = U i. i= For a vertex subset U U, we use I U to denote the indicator function of the event that G[U] is a tree component of G[B], i.e., G[U] is a tree and NU) B \ U) =. Define I = U ) I U 3.8) d U U The idea is that, instead of counting the total number of tree components of size at most d, we use the random variable I as a weighted count to which the contribution of a tree component on a vertex subset of size i is i d ). The purpose is to make Iω) Iω ) as close to as possible for every pair ω and ω that differ only on one coordinate. Note that if we had used the unweighted sum I = U U I U, the best we can have is max Iω) Iω ). 9

10 Since U U I U is the number of tree components of size at most d, and that for any d > 0, I = U ) I U I U, d U U U U we have, by the definition of a d-rigid triple, that P P {W is d-rigid} P P { U U I U = 0 } P P {I = 0}. By Lemma 3.3 and Lemma 3., we have P Gn,m) {W is d-rigid W is an l-partition} P P {I = 0} P P {I E P [I] E P [I]} ) E P [I]) c f cn 3.9) e where c f = max Iω) Iω ) with the maximum taken over all pairs of ω and ω in Ω that differ only on one coordinate. The following lemma bounds max Iω) Iω ). Lemma 3.4. For any ω, ω Ω that differ only in one coordinate, Iω) Iω ) + d. Proof. Note that ω and ω represent two possible outcomes of the independent random experiments that select the m edges of a random graph. If ω, ω Ω differ only in one coordinate, say the i-th coordinate, then the edge sets of the corresponding graphs G ω and G ω only differ in the i-th edge. Let us consider the change of the value of I when we modify G ω to G ω by removing the i-th edge of G ω and adding the i-th edge of G ω. First, removing the i-th edge can only increase I. Let the amount of the increase be δ + I. The maximum increase occurs in situations where a tree component T is broken up into two smaller tree components T and T. Suppose that there are i vertices in T and j vertices in T, we have δ + I = i ) + j ) i + j ) I i+j d d d d + d 0

11 where I i+j d = if i + j d and I i+j d = 0 otherwise. Secondly, adding the i-th edge can only decrease I. Let the amount of the decrease be δ I. The maximum decrease occurs in situations where two tree components are merged into a larger one, and similar argument as in the above shows that δ I + d as well. Therefore, the maximum net change of I is + d and is achieved when δ + I = + d and δ I = 0, or δ+ I = 0 and δ I = + d. Consequently, Iω) Iω ) + d. This proves the lemma. We now estimate the expectation E P [I] of the function I defined in Equation 3.8). Let U, U = i, be a vertex subset in U and recall that in Gn, m), the m = cn edges are selected uniformly at random and with replacement. Conditional on the event that W = S, A, B) is a balanced l-partition, the m edges are selected from the set E W uniformly at random with replacement. Therefore for i, the probability that G[U] is an induced tree component in G[B] is P P {I U = } = ) cn i i i i s = c i n i i i s i s ) itn i) + ) i s s ) i itn i) + ) i s ) cn i+ ) cn i+ For the case of U =, P P {I U = } is the probability that the single vertex in U is isolated in G[B], and thus P P {I U = } = ) cn tn ). s Since there are ) tn i vertex subsets of size i in B, the expected number of tree components of size at most d in G[B] is E P [I] = P P {I U = } U ) d U U = tn d i= ) cn tn ) + s d i d ) ) tn i i i cn s ) i itn i) + i s ) ) cn i+

12 Since s = nn ) tnn tn l + )) = t t))n +tnl+) n, we have that for sufficiently large n E P [I] tn e ct t t) = te xt,c) + + d d i= i= ) d i ct ) i i i d t t + i! ) d i i i d i! e ict t t+ ) ) xt, c)e xt,c)) i n. 3.0) To complete the proof of Theorem, we see that Equation 3.7) follows from Lemma 3.4, Equation 3.9), and Equation 3.0). 3.. Proof of Theorem We prove Theorem by applying Markov s inequality and using the upper bound obtained in Section 3. on the conditional probability of a d-rigid and balanced l-partition. Let l + = βn where β > 0 is a sufficiently small number to be determined at the end of the proof. Consider the following two random variables J : the total number of balanced βn-partitions W = S, A, B) such that A B A + d, and J : the total number of balanced βn-partitions W = S, A, B) such that B > A + d and W is d-rigid. By Lemma 3., if the treewidth of Gn, m) is at most βn, then either J > 0 or J > 0. It follows that P Gn,m) {twgn, m)) βn} P Gn,m) {J + J > 0}. 3.) If we can show that E Gn,m) [J + J ] tends to zero as n goes to infinity, Theorem follows from Markov s inequality. To simplify the presentation, define φ t) = t + t + tβ + O/n) ) c, φ t) = e rt,c)+gt,c))) c c, φt) = φ t)φ t). For the expectation of J, we have Lemma 3.5. For any c >, there is a constant β β < β, lim E Gn,m) [J ] = 0. n > 0 such that for any

13 Proof. Consider a partition W = S, A, B) of the vertices of Gn, m) such that B A and write B = b = tn. Since A + B = β)n, we see that B A + d if and only if B β)n+d. The probability for W to be a balanced βn-partition is P Gn,m) {W is an βn-partition} = ) cn tnn tn βn) nn )/ = t + t + tβ + O/n) ) cn = φ t)) n. For a fixed vertex subset S, there are ) β)n b ways n b = B 3n) to choose the pair A, B) such that one of them has size b. It follows that E Gn,m) [J ] = ) n βn ) n βn β)n b β)n +d β)n b β)n +d ) n βn φ bn ) n ) b ) n φ b ) n b n ). Since ) n b attains its maximum at b = n and the function φ t) is increasing in the interval [ β, ], we have by Stirling s formula Lemma.) that ) n n E Gn,m) [J ] d n φ βn) ) n ) ) cn n d ) n + β βn d β β β) β ) n + β)c ) n. For any c >, there is a constant β > 0 such that + β)c < for any β < β. Since lim =, there exists a constant β β 0 β β β) β > 0 such that β β β) β β < β, + β ) c). Taking β = minβ, β ), we see that for any E Gn,m) [J ] d β β β) β ) n + β)c ) n dγ n where 0 < γ <. Lemma 3.5 follows. For the expectation of J, we have the following 3

14 Lemma 3.6. For c =.073, there is a constant β β < β, lim E Gn,m) [J ] = 0. n > 0 such that for any Proof. Consider a partition W = S, A, B) of the vertices of Gn, m) such that S = l + = βn, B A + d, B = b = tn, with β t β) 3. Let I W be the indicator function of the event that W is a d-rigid and balanced βn-partition. We have E Gn,m) [I W ] = P Gn,m) {W is a d-rigid and balanced βn partition} = P Gn,m) {W is a balanced βn partition} From Theorem, we know that P Gn,m) {W is d-rigid W is a balanced βn partition}. P Gn,m) {W is d-rigid W is a balanced βn partition} e r+g) n By the definition of a balanced partition, P Gn,m) {W is a balanced βn partition} = = φ t)) n. ) cn tnn tn βn) nn )/ = φ t)) n. For a fixed vertex subset S with S = βn, there are ) n βn b ways n b 3 n) to choose the pair A, B) such that B = b. Therefore, E Gn,m) [J ] = W E Gn,m) [I W ] ) n βn ) n βn n b 3 n n b 3 n ) n βn φ bn )φ bn ) n )) b ) n φ b b n )φ b ) n n )). By Lemma., we have for n large enough E Gn,m) [J ] ) n β β β) β n b 3 n φ b n )φ b n ) b n b n ) b n b n n Recall that φ t) = e c rt,c)+gt,c))) c, 4

15 and see Equation 3.6) for the definition of rt, c) and gt, c). By Lemma.3, rt, c) and gt, c) are decreasing on [ β, 3 ] for any fixed c. Consequently φ t) is increasing on [ β, 3 ]. It follows that φ b n ) φ 3 ). By Lemma.4, Therefore, b n φ b n ) b n b n ) b n E Gn,m) [J ] On) Consider the function φ 3 ) 3 ) 3 3 ) 3 β β β) β = β)c 3 ) 3 3 ). 3 ) n β)φ 3 )) c 5 9 zβ, d, c) = β)φ 3 )) c 3 ) 3 3 ). 3 3 ) 3 3 ) 3 ) n. Numerical calculations using MATLAB shows that for c =.073, β = 0, and d = 70, we have z0, 70,.073) <. Since zβ, d, c) is continuous in β on [0, ], there exist constants β > 0 such that zβ, 70,.073) <. By Lemma., there exists a constant β > 0 such that for any β β, β β β) β < zβ, 70,.073). Let β = minβ, β ). We have that for any β < β, E Gn,m) [J ] On) β β zβ, 70,.073) β) β On) β β β) β zβ, 70,.073) On)γ n for some constant 0 < γ <. This proves Lemma 3.6. It follows from Equation 3.) that for any β β, lim n P Gn,m) {twgn, m)) βn} = 0, if m n =.073. Since the property that the treewidth of a graph is greater βn is a monotone increasing graph property, we have that for any c.073, lim n P Gn,m) {twgn, cn)) βn} = 0. Theorem follows. 5

16 4. Treewidth of Random Intersection Graphs In this section, we prove the following theorem on the treewidth of random intersection graphs by applying Markov s inequality to the number of balanced l-partitions in a random intersection graph. Theorem 3. Let G I n, m, p) be a random intersection graph with the universe M = {,, m} and m = n α. For any p m and α > 0, there exists a constant β > 0 such that lim P G n I n,m,p) {twg I n, m, p)) > βn} =. 4.) Proof. Let p = c m. Consider a balanced triple W = S, A, B) with S = βn, A = an, and B = bn. We upper bound the probability that W is a balanced βn-partition and then use Markov s inequality. By the definition of random intersection graphs, there is no edge between the two vertex subsets A and B if and only if for every element e M ) ) e S v S v, v A v B which in turn is equivalent to the following condition: for every e M, either e S v, v A, or e S v, v B. 4.3) Since S v s are formed independently and since P {e S v } = p for every e M and v V, the probability for the event in Equation 4.3) to occur is It follows that p) an + p) bn p) a+b)n) m. P GI n,m,p) {W is a balanced βn-partition} = p) an + p) bn p) a+b)n) m = p) amn + p) b a)n p) bn) m. There are ) n βn ways to choose S and for each fixed S, there are n βn ) an ways to choose A. Since the treewidth of G I n, m, p) is at most βn only if there is a 6

17 balanced βn-partition, we have by Markov s inequality that for p c m, c >, P GI n,m,p) {twg I n, m, p)) βn} P GI n,m,p) {There exsits a balanced βn-partition} ) n ) n p) amn + p) b a)n p) bn) m βn an n O) βn 3 a ) 3 a ) n e O)n ) 3 βn 3 ) 3 3 ) 3 e )ac a a a) a ) n. where last inequality is because the function for any c >. Note that have This proves Theorem 3. e ) 3 3 ) 3 3 ) 3 )n e )tc t t t) t is decreasing on [ 3, ] <. Therefore, for sufficiently small β, we lim P G n I n,m,p) {twg I n, m, p)) βn} = The Treewidth of Scale-Free Random Graphs In this section, we prove a theorem on the treewidth of the Barabási-Albert scale-free random graph by applying Markov s inequality to the number of balanced l-partitions. It turns out that for the case of the Barabási-Albert random graph, calculating the probability that a set of three disjoint vertex sets is a balanced partition is not as easy as for the case of the Erdős-Rényi random graph and the random intersection graph. We introduce the following definition. Definition 5.. Let G = GV, E) be a graph with V = n. Let W = S, A, B) be a balanced triple of disjoint vertex subsets such that V = S A B, S = l+, and A B. Let I and I be two nonempty disjoint vertex sets such that I I = V. We say that W is a balanced l-partition with respect to I if the following two conditions are true. Nv) I B) = for every v I A, and. Nv) I A) = for every v I B. It is not hard to see that a balanced l-partition is a balanced l-partition with respect to any subset of V. We will upper bound the probability of being a 7

18 balanced l-partition by the probability of being a balanced l-partition with respect to some special subset I of V. For the Barabási-Albert scale-free random graph, we are able to pick a special subset I so that the latter probability can be estimated analytically. Theorem 4. Let G S n, m) be the Barabási-Albert random graph. For any m, there is a constant β > 0 such that lim P G n S n,m) {twg S n, m)) > βn} =. 5.4) Proof. Let V = {v, v,, v n } be the set of vertices in G S n, m) and V i = {v,, v i }. Without loss of generality, assume that the vertices are added to G S n, m) in this order in the iterative construction of G S n, m). Let I be the first half of the vertices, i.e, I = {v, v,, v n }, and I be the second half {v n +,, v n }. Let W = S, A, B) be a balanced triple of disjoint vertex subsets with S = βn, A = an, and B = bn. Assume, without loss of generality, that A B so that β 3 a β. Considering the way in which A and B intersect with I and I, let us write I A = sn, I A = a s)n; I B = tn, I B = b t)n; where s and t shall satisfy 0 s β, s + t = β. We upper bound the probability P GS n,m) {W is a balanced βn-partition}. Let E be the event that W is a balanced βn-partition and E be the event that W is a balanced βn-partition with respect to I. Define the following events E i = { {Nvi ) I B) = }, if v i I A {Nv i ) I A) = }, if v i I B By Definition 5., we have E E E n + E n. Therefore, P GS n,m) {E} P GS n,m) { E n + E n }. The following lemma bounds the conditional probability of E i given G S n, m)[v i ]. Lemma 5.. s/) m, if v i I B P GS n,m) {E i G S n, m)[v i ]} t/) m, if v i I A 8

19 Proof. Consider a vertex v i I B The case for v i I A is similar). The total vertex degree of G S n, m)[v i ] is i )m nm. The total vertex degree of the vertices in I A is at least snm. Note that the event E i implies that none of the vertices in I A is selected as the neighbor of v i in the m-step procedure to decide v i s neighbors. By the definition of preferential attachment mechanism in the Barabási-Albert model Equation.3)), we have that P GS n,m) {E i G S n, m)[v i ]} snm i )m ) snm i )m + ) snm i )m + m ) ) snm nm )m = s )m. This proves the lemma. We continue the proof of Theorem 4. From Lemma 5., we have P GS n,m) {E} { } P GS n,m) En/+ E n n = P GS n,m) {E i G S n, m)[v i ]} i=n/+ s/) m ) I B t/) m ) I A = s/) b t t/) a s) mn. Taking into consideration the facts that a + b = β)n and s + t = β, we see P GS n,m) {E} s/) b t t/) a s) mn = s/) b+s β)/ 3/4 + s/) a s) mn = ) β mn s a+ s/) 3/4 + s/) a s. Consider the behavior of the function β s a+ fs, β) = s/) 3/4 + s/) a s = s/ 3/4 + s/ ) s a s/) s/) β/, 5.5) for 0 s and β 3 a β). We have Lemma 5.. There is a constant β > 0 such that for any β < β, f max = max{fs, β) : s [0, /], a [ β)/3, β)/]} < ) 9

20 Proof. Note that the last term s/) β/ of fs, β) can be made arbitrarily close to by requiring that β is less than a sufficiently small number, say β 0. We, therefore, only need to consider the function ) s a s/ fs) = s/). 3/4 + s/ First, we note that basic calculus shows that fs) ) 7 8 for any s [ 4, ]. Now consider the interval [0, 4 ]. Let β be a constant such that for any β < β, 4 β 3 < 0. Split [0, 4 ] into d + segments and consider the d + ) intervals [s i, s i+ ] where s i = i 4d, 0 i d. Since gs) = s/ 3/4+s/) s a is increasing in [0, /], s a < s 3 < 0 for any s [0, /4] and a [ β)/3, β)/], and hs) = s/) is decreasing in [0, /4], we have max fs) = max s [0,/4] 0 i d { max fs)} s [s i,s i+] max gs i+)hs i )). 0 i d Numerical calculations using d = 0 gives us max 0 i d gs i+ )hs i )) < Taking β = min{β 0, β }, we get Equation 5.6). To complete the proof of Theorem 4, we see from Markov s inequality that the expected number of balanced βn-partitions is at most ) ) n n s βn a )s a s ) mn ) ) n n )a s mn. βn an Numerical calculation shows that <. Since an n, we have by Lemma. and Lemma. that there is a constant β such that for any β < β and m ) ) n n lim mn = 0. n βn an Let β = min{β, β } where β is the constant required in Equation 5.6). It follows that for any m, the expected number of balanced βn-partitions in G S n, m) tends to zero, and consequently lim P G n S n,m) {twg S n, m)) > βn} =. This completes the proof of Theorem Conclusions After the submission of the current paper, C. Lee, J. Lee and S. Oum proved in their recent manuscript [4] that the Erdős-Rényi random graph has a linear 0

21 treewidth with high probability if the edge-to-vertex ratio is greater than /, using a different approach that is based on a theorem proved in a manuscript of I. Benjamini, G. Kozma, and N. Wormald [5] on the structure of the giant component in the the Erdős-Rényi random graph. This, together with the wellknown observation that the Erdős-Rényi random graph has treewidth at most if the edge-to-vertex ratio is less than /, completely settles the exact threshold of the edge-to-vertex ration for the property of having a linear treewidth in the Erdős-Rényi random graph. The results presented in this paper on the treewidth of the random intersection graph and the Barabási-Albert random graph may be further strengthened. However, it is likely that neither our approach nor the approach based on the result of Benjamini et al is sufficient to resolve the question of linear treewidth in these random graphs completely. The major obstacle is the fact that in these random graph models, the edges are highly correlated, rendering it hard to apply those techniques that are effective for the Erdős-Rényi random graph. As we have shown in Section 5, the treewidth of the Barabási-Albert scalefree random graph is linear in the number of vertices if m, the number of the previous vertices attached to a new vertex, is greater than. It can be seen that if m < 3, then the treewidth of the Barabási-Albert random graph is at most. It is reasonable to conjecture that the treewith of the Barabási-Albert random graph becomes linear when m 3, and proving this conjecture is an interesting open problem. Acknowledgment This paper was submitted in August, 009 and the result on the treewidth of the Erdős-Rényi random graph is an improved version of the author s earlier conference paper [5] in 006. We thank one of the referees for bringing the recent work of C. Lee, J. Lee and S. Oum [Rank-width of random graphs, arxiv:00.046, January, 00] to our attention. The conjecture stated in Section 6 on the linear treewidth of scale-free graphs is due to one of the referees. We also thank the referees for their careful reading of the paper and for their thoughtful feedback. Their detailed suggestions and criticisms have helped improve the presentation of the paper significantly. References [] D. Achlioptas. Threshold Phenomena in Random Graph Colouring and Satisfiability. PhD thesis, Department of Computer Science, University of Toronto, Toronton, Canada, 999.

22 [] D. Achlioptas and E. Friedgut. A sharp threshold for k-colorability. Random Structures and Algorithms, 4):63 70, 999. [3] R. Albert and A. Barabási. Statistical mechanics of complex networks. Reviews of Modern Physics, 7447):47 97, 00. [4] M. Behrisch, A. Taraz, and M. Ueckerdt. Coloring random intersection graphs and complex networks. SIAM Journal on Discrete Mathematics, 3):88 99, 008. [5] I. Benjamini, G. Kozma, and N. Wormald. The mixing time of the giant component of a random graph. Technical report, arxiv:060459, 006. [6] H. Bodlaender. A tourist guide through treewidth. Acta Cybernetica, - ):, 993. [7] H. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 09-): 45, 998. [8] H. Bodlaender and A. Koster. Combinatorial optimization on graphs of bounded treewidth. The Computer Journal, 53):55 69, 008. [9] B. Bollobás. Random Graphs. Cambridge University Press, 00. [0] B. Bollobás, O. Riordan, J. Spencer, and G. Tusnady. The degree sequence of a scale-free random graph process. Random Structures and Algorithms, 8:79 90, 00. [] J. Böttcher, K. Pruessmann, A. Taraz, and A. Würfl. Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. European Journal of Combinatorics, 3:7 7, 00. [] J. Díaz, L. Kirousis, D. Mitsche, and X. Pérez-Gimńez. On the satisfiability threshold of formulas with three literals per clause. Theoretical Computer Science, 40:90 934, 009. [3] P. Erdős and A. Rényi. On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci., 5:7 6, 960. [4] E. Friedgut. Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc., :07 054, 999. [5] Y. Gao. On the threshold of having a linear treewidth in random graphs. In Proceedings of th Annual International Conference on Computing and Combinatorics COCOON 06), pages 6 34, 006. [6] Y. Gao. Treewidth of Erdős-Rényi random graphs, random intersection graphs, and scale-free random graphs. Technical report, arxiv: , 009.

23 [7] G. Gottlob and S. Szeider. Fixed-parameter algorithms for artificial intelligence, constraint satisfaction, and database problems. The Computer Journal, 53):303 35, 008. [8] M. Grohe and D. Marx. On tree width, bramble size, and expansion. Journal of Combinatorial Theory Series B), 99:8 8, 009. [9] M. Karoński, E. Scheinerman, and K. Singer-Cohen. On random intersection graphs: The subgraph problem. Combinatorics, Probability, and Computing, pages 3 59, 999. [0] L. Kirousis, P. Kranakis, D. Krizanc, and Y. Stamation. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 3):53 69, 994. [] L. Kirousis and Y. Stamatiou. An inequality for reducible, increasing properties of randomly generated words. Technical Report TR , Computer Technology Institute, University of Patras, Patras, Greece, 996. [] Kloks and H. Bodlaender. Only few graphs have bounded treewidth. Technical report, Technical Report RUU-CS-9-35, Department of Computer Science, Utrecht University, 99. [3] T. Kloks. Treewidth: Computations and Approximations. Springer-Verlag, 994. [4] C. Lee, J. Lee, and S. Oum. Rank-width of random graphs. Technical report, arxiv:00.046, 00. [5] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, London Mathematical Society Lecture Note Series, vol. 4, pages Cambridge Univ. Press, 989. [6] N. Robertson and P. Seymour. Graph minors. ii. algorithmic aspect of tree-width. Journal of Algorithms, 7:309 3, 986. [7] D. West. Introduction to Graph Theory. Prentice Hall, 00. 3