Research Article On the Isolated Vertices and Connectivity in Random Intersection Graphs

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1 International Cobinatorics Volue 2011, Article ID , 9 pages doi: /2011/ Research Article On the Isolated Vertices and Connectivity in Rando Intersection Graphs Yilun Shang Institute for Cyber Security, The University of Texas at San Antonio, San Antonio, TX 78249, USA Correspondence should be addressed to Yilun Shang, shylath@hotail.co Received 10 January 2011; Accepted 5 April 2011 Acadeic Editor: Liying Kang Copyright q 2011 Yilun Shang. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We study isolated vertices and connectivity in the rando intersection graph G n,, p. A Poisson convergence for the nuber of isolated vertices is deterined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When n α and α>6, we give the asyptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi rando graphs. 1. Introduction The classical rando graph G n, p, introduced by Erdős and Rényi in the late 1950s, consists of a fixed set of n vertices and edges that exist with a certain probability p, independently fro each other. Since then any other rando graph odels with dependent edges have been developed. Aong the, rando intersection graph 1, 2 is defined as follows. Consider a set V with n vertices and another universal set W with eleents. Define a bipartite graph B n,, p with independent vertex sets V and W. Edges between v V and w W exist independently with probability p. The rando intersection graph G n,, p derived fro B n,, p is defined on the vertex set V with vertices v 1,v 2 V adjacent if and only if there exists soe w W such that both v 1 and v 2 are adjacent to w in B n,, p. Appropriately scaling the paraeter as n α with soe α>0, Singer-Cohen 1 establishes connectivity thresholds for G n,, p : the threshold lies at p ln n / and ln n /n for α 1andα > 1, respectively. The result also reveals an asyptotic equivalence of graph connectivity and absence of isolated vertices in G n,, p, thatis,the zero-one law for the absence of isolated vertices is equal to that for connectivity. This is failiar in Erdős-Rényi odel; see 3, 4 for ore details. The study in the present paper is in continuation of Chapter 3 in 1. Taking our cue fro existing results for Erdős-Rényi

2 2 International Cobinatorics graphs e.g., 4, Corollary 3.31 and 3, Theore7.3, we ai to explore siilar results for the properties of isolated vertices and connectivity in G n,, p. The connectivity thresholds of another class of rando intersection graphs G n,, k, called rando key graphs or unifor rando intersection graphs, have been investigated recently 5, 6. BothG n,, p and G n,, k can be viewed as subclasses of a general odel 7. In 8, the authors deterine a zero-one law for the absence of isolated vertices in G n,, k, which again turns out to be equivalent to that for graph connectivity 6. Moreover, they show a Poisson convergence for the nuber of isolated vertices, which refines the corresponding zero-one law and leads to a double exponential result. In this paper, we deal with the asyptotic distribution of the nuber of isolated vertices and address the connectivity probability in G n,, p with n α, α > 6. A Poisson approxiation result Theore 2.1 for the nuber of isolated vertices is obtained by utilizing the Stein-Chen ethod, which yields convergence to a Poisson rando variable. The isolated vertices threshold 1,Proposition3.2 now readily follows fro our Theore 2.1 by an easy onotonicity arguent. In addition, based on a strong equivalence theore 9 relating the G n,, p and G n, p odels we derive an approxiation of the probability of connectivity at the threshold when α>6 see Theore 2.3, which is analogous to the wellknown double exponential result of Erdős and Rényi 10. Other related works regarding G n,, p odel have been reported. For exaple, 11, 12 exaines the liiting distribution of the degree of a typical vertex, 13 treats the evolution of the order of the largest coponent, and rando weights are assigned to the vertices in 14 to get general degree distributions. The rest of the paper is organized as follows. Our ain results are presented in Section 2. Sections3 and 4 contain technical proofs of Theores 2.1 and 2.3, respectively. Throughout the paper we set n α for soe α>0. 2. Main Results In this section we provide our ain results. Let X denote the nuber of isolated vertices in G n,, p and let Poi λ be a Poisson rando variable with paraeter λ. DenotebyE X and Var X the ean and variance of rando variable X, respectively. Recall that the Poisson rando variable has the unusual property that the ean and variance are both equal to the paraeter λ. Theore 2.1. In the odel G n,, p, let p, α 1, n, α > 1, where β n R. Ifli n β n β R, then one has X D Poi e β) as n,where D represents convergence in distribution.

3 International Cobinatorics 3 The upcoing corollary is iediate fro Theore 2.1. Corollary 2.2. In the odel G n,, p with p deterined through 2.1, suppose li n β n β R. Then one gets li P G n,, p ) contains no isolated vertices ) e e β. 2.3 n For a parallel double exponential result for connectivity when α is large, we have the following. Theore 2.3. In the odel G n,, p with α > 6 and p deterined through 2.1 i.e., p ln n βn /n), assue that li n β n β R. Then one has li P G n,, p ) is connected ) e e β. 2.4 n These results copleent those presented in 1 and get further insight into the evolutionary siilarities and differences between G n,, p and G n, p odels. A natural question would be to ask what happens for connectivity probability when α is sall. This is currently under investigation. 3. Proof of Theore 2.1 For i 1,...,n,letX i 1 vertex i is isolated in G n,,p and X n i 1 X i. Therefore, X counts the nuber of isolated vertices in G n,, p as defined in Section 2. We will deonstrate the asyptotic Poisson distribution of X by eploying the Stein-Chen ethod 15. Before proceeding, we first introduce soe definitions and notations. Let q 1 p and S denote the cardinality of a set S. For two integer-valued rando variables X and Y, the total variation distance between the ore correctly, between their distributions L X and L Y is given by d TV X, Y d TV L X, L Y sup P X A P Y A. 3.1 A Z Let Γ be a finite set of indices and let I a a Γ be a faily of rando indicator variables. We say I a a Γ are positively related c.f. 15 if, for each a Γ, there exist rando indicator variables J ba b Γ\{a} with the distributions ) ) L J ba b Γ\{a} L I b b Γ\{a} I a 1, 3.2 such that J ba I b for every b / a. It is notable that positively related is uch stronger than positively correlated. Suppose a n n 1 and b n n 1 are sequences of positive real nubers. We write a n b n if li n a n /b n 1. A useful result obtained by the Stein-Chen ethod is the following.

4 4 International Cobinatorics Lea 3.1 see 4, 15. Suppose that Y a Γ I a,wherethe I a a Γ are positively related rando indicator variables. Then one has d TV Y, Poi EY 1 e EY EY Var Y EY 2 EI a ] a Γ The next lea collects soe well-known approxiations that are used in this paper. Lea 3.2. If p 0,then 1 p 1 p; andifp 2 0,then 1 p e p. In the sequel, we estiate the expectation of rando variable X. Lea 3.3. Suppose α>0. Under the assuptions of Theore 2.1, onegets li EX n e β. 3.4 Proof. The probability that a vertex i is isolated can be coputed as EX i ) p s 1 p ) s ) n 1 s 1 p s s 0 1 p p 1 p ) n 1 ], 3.5 where the index s represents the nuber of vertices in W which are adjacent to i in B n,, p. Hence EX n 1 p p 1 p ) ] n For α 1, we have ) 2 p 2 1 q n 1) 2 ln n βn p 2 0, pq n 1 ) 1 ln n β ) n 1 n ) e n 1 / ln n β n as n.thusbylea 3.2,weobtain EX ne p 1 qn 1 ne p e pqn 1 ne p ne ln n β n e β, 3.8 as n.

5 International Cobinatorics 5 For α>1, note that np n n n α 1 0, np 2) ) 2 ln n 2 ) 2 βn ln n βn as n.byusinglea 3.2,wehave EX n 1 p p 1 np )] n 1 np 2) ne np2 ne ln n β n e β, 3.10 as n. The proof is then coplete. Proof of Theore 2.1. The triangular inequality for the total variation distance iplies d TV X, Poi e β)) d TV X, Poi EX d TV Poi EX, Poi e β)) By a coupling arguent 16,page58 and Lea 3.3, wehave d TV Poi EX, Poi e β)) EX e β as n. Cobining this with 3.11, we now only need to prove li n d TV X, Poi EX First, we clai that X i n i 1 are positively related. To see this, define X ji 1 vertex j is isolated in G n, Si,p 3.14 for every j / i,wheres i W represents the eleents in W which are adjacent to i in B n,, p S i is possibly epty. The rando graphs G n, S i,p and G n,, p are coupled in a natural way. Conditional on the isolation of vertex i in G n,, p, anyvertexj j / i is not adjacent to vertices of S i in B n,, p. Hence,wehave Xji ) n ) Xj ) n ) L L j 1,j / i j 1,j / i X i For every j / i, ifx j 1thenX ji 1. Consequently, we get X ji X i.

6 6 International Cobinatorics By Lea 3.1, the binary nature and exchangeability of the rando variables involved, we find that d TV X, Poi EX 1 e EX Var X EX 2 EX 1 Var X EX 2 EX 1 EX 1 EX E The cross ter E X 1 X 2 in 3.16 is shown to be given by ] n EX i 2 i 1 ] n EX i 2 i 1 X 2) EX 2 EX 2n EX 1 2] n n 1 E X 1 X 2 n n 2 EX 1 2] ) E X 1 X 2 s 02 s 1 ) 2s ) s n 2 )] s p 1 p p 1 p s ) 2 ) ] n 1, p 2p 1 p where s counts the nuber of vertices in W adjacent to neither 1 or 2 in B n,, p, leaving s vertices in W adjacent to exactly one of 1, 2. Cobining 3.5, 3.16, and 3.17 readily gives d TV X, Poi EX 1 ) 2 n 1 p 2p 1 p ) ] n 1 n 2 1 p p 1 p ) ] n p p 1 p ) ] n For α 1, we have siilarly as in the proof of Lea 3.3, p 2 2 p 2q n 1) 2 4p 2 0, pq n as n.thereby,itfollowsfrolea 3.2 that 1 ) 2 n 1 p 2p 1 p ) ] n 1 n 1 1 p 2 p 2q n 1)] ne p 2 p 2qn 1 ne 2p e p p 2qn ne 2p.

7 International Cobinatorics 7 Applying this to 3.18,weobtain d TV X, Poi EX 1 o 1 ne 2p 1 o 1 ne 2p 1 o 1 e p o 1 ne p o 1 e β n 0, 3.21 as n. For α>1, we get as in the proof of Lea 3.3, np 0, np 2) as n.hence,frolea 3.2 we have 1 ) 2 ) ] n 1 n 1 p 2p 1 p n 1 1 2p p 2 2p 1 p ) ] n 1 n 1 2p p 2 2p 1 np )] n 1 2np 2) 3.23 ne 2np2. Applying this to 3.18,wehave d TV X, Poi EX 1 o 1 ne 2np2 1 o 1 ne 2np2 1 o 1 e np2 o 1 ne np2 o 1 e β n 0, 3.24 as n, which concludes the proof. 4. Proof of Theore 2.3 Let ) p 2 p 1 1 q 2 npq n p 2 The following lea drawn fro 9 states an equivalence of G n,, p and G n, p odels. Lea 4.1 see 9. Let α>6and p be such that ω n p 2lnn ω 4.2 for soe ω. For any a 0, 1 and any graph property A,asn if it follows that P G n,, p ) A ) a if and only if P G n, p ) A ) a. 4.3

8 8 International Cobinatorics We recall the following classical result for connectivity threshold of G n, p. Lea 4.2 see 10. Let c R be fixed and p ln n c o 1 /n. Then P G n, p ) is connected ) e e c, 4.4 as n. Proof of Theore 2.3. In view of Leas 4.1 and 4.2,itsuffices to prove that n p ln n β, 4.5 as n. By the assuptions, we find that n p ln n n 1 1 n 1 1 D 1 D ) 2 n D n n 1 D ) n 2 /2 D ln n n ) ) ln n βn 1 n n2 /2 ln n, 4.6 where D denotes /n. Since n n n ) ) ln n βn 1 n n2 / as n,bylea 3.2, the right-hand side of 4.6 n ) n n n ) ) ln n ln n βn 1 n n2 /2 β n ln n n ) ) / ln n ln n βn 1/n 1/ n/2 1 n ) ) / ln n βn 1/n 1/ n/2 4.8 β n o 1 1 o 1 β, which concludes the proof.

9 International Cobinatorics 9 Acknowledgent The author would like to thank Karen Singer-Cohen for her war encourageent and supplying a copy of 1. References 1 K. B. Singer, Rando intersection graphs, Ph.D. thesis, The Johns Hopkins University, Baltiore, Md, USA, M. Karoński, E. R. Scheineran, and K. B. Singer-Cohen, On rando intersection graphs: the subgraph proble, Cobinatorics, Probability and Coputing, vol. 8, no. 1-2, pp , B. Bollobás, Rando Graphs, vol. 73 of Cabridge Studies in Advanced Matheatics, Cabridge University Press, Cabridge, UK, 2nd edition, S. Janson, T. Łuczak, and A. Rucinski, Rando Graphs, Wiley-Interscience Series in Discrete Matheatics and Optiization, Wiley-Interscience, New York, NY, USA, S. R. Blackburn and S. Gerke, Connectivity of the unifor rando intersection graph, Discrete Matheatics, vol. 309, no. 16, pp , O. Yağan and A. M. Makowski, Zero-one laws for connectivity in rando key graphs, ISR Technical Report , 2009, 7 E. Godehardt and J. Jaworski, Two odels of rando intersection graphs for classification, in Exploratory Data Analysis in Epirical Research, M. Schwaiger and O. Opitz, Eds., pp , Springer, Berlin, Gerany, O. Yağan and A. M. Makowski, On the rando graph induced by a rando key predistribution schee under full visibility, in Proceedings of the IEEE International Syposiu on Inforation Theory, pp , Toronto, Canada, J. A. Fill, E. R. Scheineran, and K. B. Singer-Cohen, Rando intersection graphs when ω n :an equivalence theore relating the evolution of the G n,, p and G n, p odels, Rando Structures & Algoriths, vol. 16, no. 2, pp , P. Erdős and A. Rényi, On the evolution of rando graphs, Publication of the Matheatical Institute of the Hungarian Acadey of Sciences, vol. 5, pp , Y. Shang, Typical vertex degrees in dense generalized rando intersection graphs, Matheatica Applicata, vol. 23, no. 4, pp , D. Stark, The vertex degree distribution of rando intersection graphs, Rando Structures & Algoriths, vol. 24, no. 3, pp , M. Behrisch, Coponent evolution in rando intersection graphs, Electronic Cobinatorics, vol. 14, no. 1, p. R17, Y. Shang, Degree distributions in general rando intersection graphs, Electronic Cobinatorics, vol. 17, no. 1, p. R23, A. D. Barbour, L. Holst, and S. Janson, Poisson Approxiation, vol. 2 of Oxford Studies in Probability, Oxford University Press, New York, NY, USA, T. Lindvall, Lectures on the Coupling Method, Dover, Mineola, NY, USA, 2002.

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