Diclique clustering in a directed network

Transcription

1 Diclique clustering in a directed network Mindaugas Bloznelis and Lasse Leskelä Vilnius University and Aalto University October 17, 016 Abstract We discuss a notion of clustering for directed graphs, which describes how likely two followers of a node are to follow a common target. The associated network motifs, called dicliques or bi-fans, have been found to be key structural components in various real-world networks. We introduce a two-mode statistical network model consisting of actors and auxiliary attributes, where an actor i decides to follow an actor j whenever i demands an attribute supplied by j. We show that the digraph admits nontrivial clustering properties of the aforementioned type, as well as power-law indegree and outdegree distributions. Keywords:intersection graph, two-mode network, affiliation network, digraph, diclique, bi-fan, complex network 1 Introduction 1.1 Clustering in directed networks Many real networks display a tendency to cluster, that is, to form dense local neighborhoods in a globally sparse graph. In an undirected social network this may be phrased as: your friends are likely to be friends. This feature is typically quantified in terms of local and global clustering coefficients measuring how likely two neighbors of a node are neighbors [11, 13, 14, 16]. In directed networks there are many ways to define the concept of clustering, for example by considering the thirteen different ways that a set of three nodes may form a weakly connected directed graph [5]. In this paper we discuss a new type of clustering concept which is motivated by directed online social networks, where a directed link i j means that an actor i follows actor j. In such networks a natural way to describe clustering is to say that your followers are likely to follow common targets. When the topology of the network is unknown and modeled as a random graph distributed according to a probability measure P, the above statement 1

2 can be expressed as Pi i 4 i 1 i 3, i i 3, i 1 i 4 ) > Pi i 4 ), 1) where you corresponds to actor i 3. Interestingly, the conditional probability on the left can stay bounded away from zero even for sparse random digraphs [9]. The associated subgraph Fig. 1) is called a diclique. Earlier experimental studies have observed that dicliques a.k.a. bi-fans) constitute a key structural motif in gene regulation networks [10], citation networks, and several types of online social networks [17]. Figure 1: Forming a diclique by adding a link i i 4. Motivated by the above discussion, we define a global diclique clustering coefficient of a finite directed graph D with an adjacency matrix D ij ) by i C di D) = 1,i,i 3,i 4 ) D i 1,i 3 D i1,i 4 D i,i 3 D i,i 4 i 1,i,i 3,i 4 ) D, ) i 1,i 3 D i1,i 4 D i,i 3 where the sums are computed over all ordered quadruples of distinct nodes. It provides an empirical counterpart to the conditional probability 1) in the sense that the ratio in ) defines the conditional probability P D I I 4 I 1 I 3, I 1 I 4, I I 3 ), 3) where P D refers to the distribution of the random quadruple I 1, I, I 3, I 4 ) sampled uniformly at random among all ordered quadruples of distinct nodes in D. To quantify diclique clustering among the followers of a selected actor i, we may define a local diclique clustering coefficient by i C di D, i) = 1,i,i 4 ) D i 1,iD i1,i 4 D i,id i,i 4 i 1,i,i 4 ) D, 4) i 1,iD i1,i 4 D i,i where the sums are computed over all ordered triples of distinct nodes excluding i. We remark that C di D, i) = P D I I 4 I 1 I 3, I 1 I 4, I I 3, I 3 = i ). Remark 1. By replacing by in 3), we see that the analogue of the above notion for undirected graphs corresponds to predicting how likely the endpoints of the 3-path I I 3 I 1 I 4 are linked together.

3 1. A directed random graph model Our goal is to define a parsimonious yet powerful statistical model of a directed social network which displays diclique clustering properties as discussed in the previous section. Clustering properties in many social networks, such as movie actor networks or scientific collaboration networks, are explained by underlying bipartite structures relating actors to movies and scientists to papers [7, 1]. Such networks are naturally modeled using directed or undirected random intersection graphs [1, 3, 4, 6, 8]. A directed intersection graph on a node set V = {1,..., n} is constructed with the help of an auxiliary set of attributes W = {w 1,..., w m } and a directed bipartite graph H with bipartition V W, which models how nodes or actors) relate to attributes. We say that actor i demands or follows) attribute w k when i w k, and supplies it when i w k. The directed intersection graph D induced by H is the directed graph on V such that i j if and only if H contains a path i w k j, or equivalently, i demands one or more attributes supplied by j see Fig. ). For example, in a citation network the fact that an author i cites a paper w k coauthored by j, corresponds to i w k j. Figure : Node 1 is follower of node, because 1 demands an attribute a supplied by. We consider a random bipartite digraph H where the pairs i, w k ), i V, w k W establish adjacency relations independently of each other. That is, the bivariate binary random vectors I i k, I k i ), 1 i n, 1 k m, are stochastically independent. Here I i k and I k i stand for the indicators of the events that links i w k and w k i are present in H. We assume that every pair i, w k ) is assigned a triple of probabilities p ik = Pi w k ), q ik = Pw k i), r ik = Pi w k, w k i). 5) Note that, by definition, r ik satisfies the inequalities max{p ik + q ik 1, 0} r ik min{p ik, q ik }. 6) A collection of triples {p ik, q ik, r ik ), 1 i n, 1 k m} defines the distribution of a random bipartite digraph H. 3

4 We will focus on a fitness model where every node i is prescribed a pair of weights x i, y i 0 modelling the demand and supply intensities of i. Similarly, every attribute w k is prescribed a weight z k > 0 modelling its relative popularity. Denoting a b = min{a, b} and letting p ik = γx i z k ) 1 and q ik = γy i z k ) 1 i, k 1, 7) we obtain link probabilities proportional to respective weights. Furthermore, we assume that r ik = rx i, y i, z k, γ), i, k 1, 8) for some function r 0 satisfying 6). Here γ > 0 is a parameter, defining the link density in H, which generally depends on m and n. Note that r defines the correlation between reciprocal links i w k and w k i. For example, by letting rx, y, z, γ) = γxz 1)γyz 1), we obtain a random bipartite digraph with independent links. We will consider weight sequences having desired statistical properties for complex network modelling. For this purpose we assume that the node and attributes weights are realizations of random sequences X = X i ) i 1, Y = Y i ) i 1, and Z = Z k ) k 1, such that the sequences {X i, Y i ), i 1} and {Z k, k 1} are mutually independent and consist of independent and identically distributed terms. However, the attribute weights X i and Y i of any given node i are allowed to be correlated with each other.) The resulting random bipartite digraph is denoted by H, and the resulting random intersection digraph by D. We remark that D extends the random intersection digraph model introduced in [1]. 1.3 Degree distributions When γ = mn) 1/ and m, n, the random digraph D defined in Sec. 1. becomes sparse, having the number of links proportional to the number of nodes. Theorem 1 below describes the class of limiting distributions of the outdegree of a typical vertex i. We remark that for each n the outdegrees d + 1),..., d + n) are identically distributed. To state the theorem, we let Λ 1, Λ, Λ 3 be mixed-poisson random variables distributed according to PΛ i = r) = E e λ i λr i r!, r 0, where λ 1 = X 1 β 1/ E Z 1, λ = Z 1 β 1/ E Y 1, and λ 3 = X 1 E Y 1 )E Z1 ). We also denote by Λ i a downshifted size-biased version of Λ i, distributed according to PΛ i = r) = r + 1 E Λ i PΛ i = r + 1), r 0. Below d refers to convergence in distribution. 4

5 Theorem 1. Consider a model with n, m and γ = nm) 1/, and assume that E Y 1, E Z 1 <. 1) If m/n 0 then d + 1) d 0. ) If m/n β for some β 0, ), then d + 1) d Λ 1 j=1 Λ,j, where Λ,1, Λ,,... are independent copies of Λ, and independent of Λ 1. 3) If m/n, then d + 1) d Λ 3. Remark. By symmetry, the results of Theorem 1 extend to the indegree d 1) when we redefine λ 1 = Y 1 β 1/ E Z 1, λ = Z 1 β 1/ E X 1, and λ 3 = Y 1 E X 1 )E Z 1 ). Remark 3. The limiting distributions appearing in Theorem 1:1) 1) admit heavy tails. This random digraph model is rich enough to model powerlaw indegree and outdegree distributions, or power-law indegree and lighttailed outdegree distributions []. Remark 4. In Theorem 1:1) it is sufficient to assume that E Z 1 <. We note that a related result for simple undirected) random intersection graph has been shown in []. Theorem 1 extends the result of [] to digraphs. 1.4 Diclique clustering We investigate clustering in the random digraph D defined in Sec. 1. by approximating the random) diclique clustering coefficient C di D) defined in ) by a related nonrandom quantity c di := PI I 4 I 1 I 3, I 1 I 4, I I 3 ), where I 1, I, I 3, I 4 ) is a random ordered quadruple of distinct nodes chosen uniformly at random. Note that here P refers to two independent sources of randomness: the random digraph generation mechanism and the sampling of the nodes. Because the distribution of D is invariant with respect to a relabeling of the nodes, the above quantity can also be written as c di = P 4 1 3, 1 4, 3 ). We believe that under mild regularity conditions C di D) c di, provided that m and n are sufficiently large. Proving this is left for future work. Theorem below shows that the random digraph D admits a nonvanishing clustering coefficient c di when the intensity γ is inversely proportional to the number of attributes. For example, by choosing γ = nm) 1/ and letting m, n so that m/n β > 0, we obtain a sparse random digraph with tunable clustering coefficient c di and limiting degree distributions defined by Theorem 1 and Remark. 5

6 Theorem. Assume that m and γm α for some constant α 0, ), and that E X1 3, E Y 1 3, E Z4 1 <. Then E X c di 1 + α 1 + E Y ) 1 E Z 1 )E Z1 3) E X 1 E Y 1 E Z1 4 + α E X 1 E Y1 E Z ) 1 1 )3 E X 1 E Y 1 E Z1 4. 9) Remark 5. When E X1 4, E Y 1 4, E Z4 1 <, the argument in the proof of Theorem allows to conclude that c di 0 when γm. To investigate clustering among the followers of a particular node i, we study a theoretical analogue of the local diclique clustering coefficient C di D, i) defined in 4). By symmetry, we may relabel the nodes so that i = 3. We will consider the weights of node 3 as known and analyze the conditional probability c di X 3, Y 3 ) = P X3,Y , 1 4, 3 ), where P X3,Y 3 refers to the conditional probability given X 3, Y 3 ). Actually, we may replace P X3,Y 3 by P Y3 above, because all events appearing on the right are independent of X 3. One may also be interested in analyzing the conditional probability c di X, Y ) = P X,Y 4 1 3, 1 4, 3 ), where P X,Y refers to the conditional probability given the values of all node weights X = X i ) and Y = Y i ). Again, we may replace P X,Y by P X1,X,Y 3,Y 4 above, because the events on the right are independent of the other nodes weights. More interestingly, c di X, Y ) turns out to be asymptotically independent of X and Y 4 as well in the sparse regime. Below P refers to convergence in probability. Theorem 3. Assume that m and γm α for some constant α 0, ). 1) If E X1 3, E Y 1 3, E Z4 1 <, then c di X 3, Y 3 ) P 1 + α ) If E Z1 4 <, then c di X, Y ) P E X 1 E X 1 + Y 3 ) E Z 3 1 )E Z 1 ) E Z α E X1 E Z ) 1 1 Y )3 3 E X 1 E Z αx 1 + Y 3 ) E Z3 1 )E Z 1 ) E Z1 4 + α E Z1 X 1 Y )3 3 E Z1 4 ) 1. Note that for large Y 3, the clustering coefficient c di X 3, Y 3 ) = c di Y 3 ) scales as Y3 1. Similarly, for large X 1 and Y 3, the probability c di X, Y ) scales as X1 1 Y 3 1. We remark that similar scaling of a related clustering coefficient in an undirected random intersection graph has been observed in [4]. 6

7 Remark 6. When all attribute weights are equal to a constant z > 0, the statement in Theorem 3:3) simplifies into c di X, Y ) P 1 + αzx 1 ) αzy 3 ) 1, a result reported in [9]. Remark 7. Theorems 1,, and 3 do not impose any restrictions on the correlation structure of the supply and demand indicators defined by 8). 1.5 Diclique versus transitivity clustering An interesting question is to compare the diclique clustering coefficient c di with the commonly used transitive closure clustering coefficient c tr = P ), see e.g. [5, 15]. The next result illustrates that c tr depends heavily on the correlation between the supply and demand indicators characterized by the function rx, y, z, γ) in 8). A similar finding for a related random intersection graph has been discussed in [1]. Theorem 4. Assume that m and γm α for some constant α 0, ). Suppose also that E X1, E Y 1, E Z3 1 <. 1) If rx, y, z, γ) = γxz) 1)γyz) 1), then c tr 0. ) If rx, y, z, γ) = εγxz) γyz) 1) for some 0 < ε 1 and E X 1 Y 1 ) > 0, then c tr 1 + α E X 1 Y 1 ) E Z ) 1 1 ) ε E X 1 Y 1 ) E Z ) The assumption in 4) means that the supply and demand indicators of any particular node attribute pair are conditionally independent given the weights. In contrast, the assumption in 4) forces a strong correlation between the supply and demand indicators. We note that condition 6) is satisfied in case 4) for all i n and k m with high probability as n, m, because n 1/ max i n X i +Y i ) P 0 and m 1/ max k m Z k P 0 imply that γx i Z k +γy i Z k 1 for all i n and k m with high probability. We remark that in case 4), and in case 4) with a very small ε, the transitive closure clustering coefficient c tr becomes negligibly small, whereas the diclique clustering coefficient c di remains bounded away from zero. Hence, it makes sense to consider the event {1 3, 1 4, 3} as a more robust predictor of the link 4 than the event { 3 4}. This conclusion has been empirically confirmed for various real-world networks in [10, 17]. Proofs The proof of Theorem 1 goes along similar lines as that of Theorem 1 in []. It is omitted. 7

8 We assume for notational convenience that γ = αm 1. For γ = α1 + o1))m 1 proofs remain unchanged. Denote the events A = {1 3, 1 4, 3}, B = { 4}, E = { 3, 3 4} and introduce random variables p ik = α X iz k m, q ik = α Y iz k m. Note that p ik = PI i k = 1), q ik = PI k i = 1), and p ik = 1 p ik, q ik = 1 q ik. 11) Here P and Ẽ denote the conditional probability and expectation given X, Y, Z. We denote by I M the indicator of an event M. We use the shorthand notation and a r = E X1 r, b r = E Y1 r, h r = E Z1 r. Proof of Theorem. We observe that A = i [4] A i, where { A 1 = A 1.k, A 1.k = I1 k I k I k 3 I k 4 = 1 }, k C 1 { A = A.kl, A.kl = I1 k I l I k 3 I k 4 I l 3 = 1 }, k,l) C { A 3 = A 3.kl, A 3.kl = I1 k I 1 l I k I k 3 I l 4 = 1 }, k,l) C 3 A 4 = { A 4.jkl, A 4.jkl = I1 j I 1 k I l I j 3 I k 4 I l 3 = 1 }. j,k,l) C 4 Here C 1 = [m], C = C 3 = {k, l) : k l; k, l [m]}, and C 4 = {j, k, l) : j k l j; j, k, l [m]}. Hence, by inclusion exclusion, PA i ) PA i ). i [4] {i,j} [4] PA i A j ) PA) i [4] We prove the theorem in Claims 1 3 below. Claim implies that PA) = i [4] PA i) + Om 4 ). Claim 3 implies that PA B) = PA 1 ) + Om 4 ). Finally, Claim 1 establishes the approximation 9) to the ratio C di = PA B)/PA). Claim 1. We have PA 1 ) = α 4 m 3 A o1)), 1) PA ) = α 5 m 3 A 1 + o1)), 13) PA 3 ) = α 5 m 3 A o1)), 14) PA 4 ) = α 6 m 3 A o1)). 15) 8

9 Here we denote A 1 = a 1b 1h 4, A = a 1b 1 b h h 3, A 3 = a 1 a b 1h h 3, A 4 = a 1 a b 1 b h 3. Claim. For 1 i < j 4 we have Claim 3. We have PA i A j ) = Om 4 ). 16) PB A) = PA 1 ) + Om 4 ). 17) Proof of Claim 1. We estimate every PA r ) using inclusion-exclusion I 1 I PA r ) I 1. Here I 1 = I 1 r) = x C r PA r.x ), I = I r) = Now 1 15) follow from the approximations {x,y} C r PA r.x A r.y ). I 1 1) = α 4 m 3 A o1)), I 1 3) = α 5 m 3 A o1)), I 1 ) = α 5 m 3 A 1 + o1)), 18) I 1 4) = α 6 m 3 A o1)) and bounds I r) = om 3 ), for 1 r 4. Firstly we show 18). We only prove the first relation. The remaining cases are treated in much the same way. From the inequalities, see 11), p 1k p k q 3k q 4k p 1k p k q 3k q 4k p 1k p k q 3k q 4k I k p 1k p k q 3k q 4k p 1k p k q 3k q 4k I k, I k := I p 1k 1I pk 1I q3k 1I q4k 1, I k := I p 1k >1 + I pk >1 + I q3k >1 + I q4k >1, we obtain where PA 1.k ) = E p 1k p k q 3k q 4k = E p 1k p k q 3k q 4k + R, 19) E p 1k p k q 3k q 4k = α 4 m 4 A 1 and R E p 1k p k q 3k q 4k I k = om 4 ). The latter limit follows by dominated convergence because I k is bounded by 4 and tends to zero a.s. as m. Hence I 1 1) = mpa 1.k ) = α 4 m 3 A o1)). Secondly we show that I r) = om 3 ), for 1 r 4. For r = 1 the bound I 1) = m ) PA1.k A 1.l ) = om 3 ) follows from the inequalities PA 1.k A 1.l ) E p 1k p k q 3k q 4k p 1l p l q 3l q 4l = Om 8 ). 9

10 For r =, 3 we split I r) = J J 5, where J 1 = PA r.kl A r.kl ), J = PA r.kl A r.k l), J 3 = J 5 = {k,l),k,l )} C r {k,l),k,l )} C r k,l) C r PA r.kl A r.lk ). {k,l),k,l)} C r PA r.kl A r.k l ), J 4 = {k,l),k,k)} C r, k l PA r.kl A r.k k), In the first second) sum distinct pairs x = k, l) and y = k, l ) share the first second) coordinate. In the third sum all coordinates of the pairs k, l), k, l ) are different. In the fourth sum the pairs k, l), k, k) only share one common element, but it appears in different coordinates. We show that each J i = om 3 ). Next we only consider the case of r =. The case of r = 3 is treated in a similar way. We have ) m 1 J 1 = m PA.kl A.kl ) m 3 E H 1, H 1 = p 1k p l p l q 3k q 4k q 3l q 3l, ) m 1 J = m PA.kl A.k l) m 3 E H, H = p 1k p 1k p l q 3k q 4k q 3k q 4k q 3l, J 3 = m ) m ) PA.kl A.k l ) m4 E H 3, H 3 = p 1k p 1k p l p l q 3k q 4k q 3k q 4k q 3l q 3l, J 4 = mm 1)m )PA.kl A.k k) m 3 E H 4, H 4 = p 1k p 1k p l p k q 3k q 4k q 3k q 4k q 3l, ) m J 5 = PA.kl A.lk ) m E H 5, H 5 = p 1k p 1l p l p k q 3k q 3l q 4k q 4l. For H 1 we estimate the typical factors p ij p ij and q ij q ij, but We obtain q 3l q 3l I Y3 m + I Y3 > m αm 1/ Z l + I Y3 > m. 0) E H 1 α 6 m 6 a 1 a b 1 h h 3 b h αm 1/ + h 1 E Y 3 I Y3 > m) = om 6 ). 1) Hence J 1 = om 3 ). Similarly, we show that J = om 4 ). Furthermore, while estimating H 3 we apply 0) to q 3l and q 3l and apply p ij p ij and q ij q ij to remaining factors. We obtain H 3 p 1k p 1k p l p l q 3k q 4k q 3k q 4k αm 1/ Z l +I Y3 > m)αm 1/ Z l +I Y3 > m). ) Since the expected value of the product on the right is om 8 ), we conclude that E H 3 = om 8 ). Hence J 3 = om 4 ). Proceeding in a similar way we establish the bounds J 4 = om 5 ) and J 5 = Om 6 ). 10

11 We explain the truncation step 0) in some more detail. A simple upper bound for H 1 is the product p 1k p l p l q 3k q 4k q 3l q 3l = α 7 m 7 X 1 X Y 3 3 Y 4 Z 3 k Z l Z l. It contains an undesirable high power Y3 3. Using 0) instead of the simple upper bound q 3l q 3l we have reduced in 1) the power of Y 3 down to. Similarly, in ) we have reduced the power of Y 3 from 4 to. Using the truncation argument we obtain the upper bound I 4) = om 3 ) under moment conditions E X1 3, E Y 1 3, E Z4 1 <. The proof is similar to that of the bound I ) = om 3 ) above. We omit routine, but tedious calculation. Proof of Claim. We only prove that q := PA 3 A 4 ) = Om 4 ). The remaining cases are treated in a similar way. For x = j, k, l) C 4 and y = r, t) C 3 we denote, for short, I A4.x = I x = I jkl and I A 3.y = I y = I rt. For q = E I A4 I A3, we write, by symmetry, ) q E I A3 = mm 1)m )E I 13I A3 and Here J 1 = r,t [m]\[3], r t x C 4 I x E I 13I A3 E I 13 y C 3 I y I rt, J = r [m]\[3] ) = E I 13J 1 + J + J 3 ). ) Isr +I rs, J3 = s [3] r,t [3], r t Finally, we show that E I 13 J i = Om 7 ), i [3]. For i = 1 we have, by symmetry, Invoking the inequalities E I 13J 1 = m 3)m 4)E I 13I 45. 3) E I 13I 45 = E Ẽ I 13I 45 E p 11 p 1 p 15 p 3 p 4 q 31 q 33 q 34 q 4 q 45 = Om 10 ) 4) we obtain E I 13 J 1 = Om 8 ). The bound E I 13 J = Om 7 ) is obtained from the identity which follows by symmetry) E I 13J = m 3) s [3] E I 13 I s4 + E I 13I 4s ), I rt. 11

12 combined with the bounds E I 13 I s4 + E I 13 I 4s = Om 8 ), s [3]. We only show the latter bound for s = 3. The cases s = 1, are treated in a similar way. We have E I 13I 34 E p 11 p 1 p 13 p 3 q 31 q 33 q 4 q 44 = Om 8 ), E I 13I 43 E p 11 p 1 p 13 p 3 p 4 q 31 q 33 q 34 q 4 q 43 = Om 10 ). The proof of E I 13 J 3 = Om 7 ) is similar. It is omitted. Proof of Claim 3. We use the notation I Aj = 1 I Aj for the indicator of the event A j complement to A j. For i 4 we denote H i = A i B ) \ 1 j i 1 A j. We have PA B) = P i [4] A i B) = PA 1 B) + R, 0 R P i 4 H i ). Note that A 1 B = A 1. It remains to show that PH i ) = Om 4 ), i 4. We have, by symmetry, PH ) = E I A I B I A1 E x C I A.x I B I A1 = mm 1)E I A.1 I B I A1. 5) Note that A 1 B = A 1. It remains to show that PH i ) = Om 4 ), i 4. We have, by symmetry, PH ) = E I A I B I A1 E x C I A.x I B I A1 = mm 1)E I A.1 I B I A1. 6) Since on the event A.1 Ā1 the link { 4} can be witnessed by any attribute, but w 1, we have I A.1 I B I A1 I A.1 I j I j 4 = I A.1 I 4 + ) I j I j 4 j m Furthermore, by symmetry, 3 j m E I A.1 I B I A1 E I A.1 I 4 + m )E I A.1 I 3 I 3 4. Here E I A.1 I 4 E p 11 p q 31 q 3 q 41 q 4 = Om 6 ) and similarly E I A.1 I 3 I 3 4 = Om 7 ). Therefore, E I A.1 I B I A1 = Om 6 ). Now 6) implies PH ) = Om 4 ). The bounds PH j ) = Om 4 ), j = 3, 4 are obtained in a similar way. Proof of Remark 5. Proceeding as in the proof of Theorem, but without using truncation 0), we obtain 9) with α = α m = γm +. Proof of Theorem 3. The proof is the same as that of Theorem, but while evaluating the probabilities of events A and A B we treat X 3, Y 3 and X 1, X, Y 3, Y 4 as constants in the cases i) and ii) respectively. 1

13 Proof of Theorem 4. We write C T = PB E)/PE) and evaluate the probabilities PB E) and PE). We firstly observe that E = E 1 E, where E 1 = { E 1.k, E 1.k = I k I k 3 I 3 k I k 4 = 1 }, k C 1 E = { E.kl, E.kl = I k I k 3 I 3 l I l 4 = 1 }. k,l) C For j = 1, we obtain by inclusion-exclusion that PE j ) = p j R j, where p j = x C j PE j.x ) and 0 R j Furthermore, denoting R 0 = PE 1 E ) we have {x,y} C j PE j,x E j.y ). PE) = PE 1 ) + PE ) R 0 = p 1 + p R 0 R 1 R. 7) We secondly observe that B E 1 = E 1. Hence PB E) = PE 1 ) + R 3, where R 3 = PB E ) \ E 1 ). 8) In the remaining part of the proof we establish a first order asymptotics for p 1 and p { om ) in the case i), p 1 = ε1 + o1))α 3 m a 1 b 1 h 3 E X 3 Y 3 ) in the case ii), p = α 4 m a 1 b 1 h E X 3 Y 3 ) + om ) and bound the remainders R i = om ), 0 i 3. Now, the result of the theorem follows from 7), 8). Evaluation of p 1. We obtain, by symmetry, p 1 = me I k I k 3 I 3 k I k 4 = me p k r 3k q 4k. 9) In the case i) we have r 3k = p 3k q 3k. Invoking the inqualities q 3k = q 3k IZk m + I Zk > m) αy3 / m + I Zk > m and p k p 3k q 4k p k p 3k q 4k we obtain E p k r 3k q 4k E p k p 3k αy3 / m + I Zk > m) q4k = om 3 ). Hence p 1 = om ). In the case ii) we have r 3k = ε p 3k q 3k ). Now E p k q 4k r 3k = εe p k q 4k p 3k q 3k ) = ε1 + o1))e p k p 3k q 3k ) q 4k. 13

14 In the last step we used the same argument as in 19) above. Finally, we evaluate the expectation E p k p 3k q 3k ) q 4k = α/m) 3 E X Y 4 Z 3 k X 3 Y 3 ) and invoke the result in 9). Evaluation of p. We obtain, by symmetry, p = mm 1)E I k I k 3 I 3 l I l 4 = mm 1)E p k q 3k p 3l q 4l. Then proceeding as in 19) above we evaluate the expectation E p k q 3k p 3l q 4l = 1+o1))E p k q 3k p 3l q 4l = 1+o1))α/m) 4 a 1 b 1 h E X 3 Y 3 ). Estimation of R i. The bound R 0 = Om 3 ) follows from the union bound R 0 k,l) C j C 1 P E.kl E 1.j ) combined with the following simple inequalities PE.kl E 1.j ) E p k p 3l p j q 3k q 4l q 4j = Om 6 ), for j {k, l}, PE.kl E 1.k ) E p k p 3l q 3k q 4k q 4l = Om 5 ), PE.kl E 1.l ) E p k p l p 3l q 3k q 4l = Om 5 ). The bound R 1 = Om 4 ) follows from the inequalities PE 1.k E 1.l ) E p k p l p 3l q 3k q 4k q 4l = Om 6 ), for k l. The bound R = om ) is obtained similarly as the upper bound for I ) in the proof of Theorem. Note that here we use the truncation argument as in ) above. Finally, we show that R 3 = Om 3 ). We decompose apply the union bound B = { B j, B j = I j I j 4 = 1 }, j C 1 R 3 PB E ) and evaluate the summands k,l) C j C 1 PE.kl B j ) PE.kl B j ) = E p k p j p 3l q 3k q 4j q 4l = Om 6 ), for j / {k, l}, PE.kl B k ) = E p k p 3l q 3k q 4k q 4l = Om 5 ), PE.kl B l ) = E p k p l p 3l q 3k q 4l = Om 5 ). 14

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