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Phase transition in long-range percolation on bipartite hierarchical lattices.

1. Introduction

For an integer N [greater than or equal to] 2, the hierarchical lattice of order N is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The hierarchical distance d on [[OMEGA].sub.N] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

which satisfies the strong (non-Archimedean) triangle inequality:

d(x, y) [less than or equal to] max {d(x, z), d(z, y)}, (3)

for any x, y, z [member of] [[OMEGA].sub.N]. This means that ([[OMEGA].sub.N], d) is an ultrametric space. Roughly speaking, this corresponds to the leaves of an infinite N-ary tree, with metric distance half the graph distance.

Some stochastic models based on hierarchical lattices have been studied. The asymptotic long-range percolation on [[OMEGA].sub.N] is analyzed in [1] for N [right arrow] [infinity]. To the best of our knowledge, this is the first paper devoted to ([[OMEGA].sub.N], d). For different purpose, the works [2-4] study the long-range percolation on [[OMEGA].sub.N] for fixed N by using different connection probabilities. The contact process and perturbation analysis on [[OMEGA].sub.N] for finite N have been studied in [5, 6], respectively. Random walks on hierarchical lattices have been examined in [7, 8].

In this paper, we study percolation on a class of bipartite hierarchical lattices, where edges always run between vertices of unlike type. Bipartite graphs have been studied intensively in the literature (see e.g., [9, 10]) and bipartite structure is popular in many social networks including sexual-contact networks [11] and affiliation networks [12], but we have not seen the setup that we consider here. For two integers l [greater than or equal to] 1 and 1 [less than or equal to] [gamma] [less than or equal to] N - 1, consider a partition of [[OMEGA].sub.N] into two sets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Vertices in [[OMEGA].sup.1.sub.N] and [[OMEGA].sup.2.sub.N] are said to have types 1 and 2, respectively. For each k [greater than or equal to] 1, the probability of connection between two vertices x and y of unlike type such that d(x, y) = k is given by

[P.sub.k] = 1 - exp (- [alpha]/[[beta].sup.k]), (5)

where 0 [less than or equal to] [alpha] < [infinity] and 0 < [beta] < [infinity], all connections being independent. Vertices of the same type cannot be connected with each other, and hence the resulting graph is a class of random bipartite graph.

In the above bipartite hierarchical lattice, denoted by ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d), vertices of both types are countable and the shortest distance between vertices in [[OMEGA].sup.1.sub.N] and [[OMEGA].sup.2.sub.N] is l. The vertices in ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d) can be represented by the leaves at the bottom of an infinite regular tree, where N branches emerge from each inner node, see Figure 1. The distance between two vertices (leaves at level 0) is the number of levels from the bottom to their most recent common ancestor. The partition of types for vertices is determined by their ancestors at level l; in other words, we need to track back at least l levels to find the most recent common ancestor of two vertices of unlike type.

Two vertices x, y [member of] ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d) are in the same component if there exists a finite sequence x = [x.sub.0], [x.sub.1], ..., [x.sub.n] = y such that each pair of vertices [x.sub.i - 1] and x; has different types and shares an edge for i = 1, ..., n. In our model, the parameter [beta] > 0 describes the long-range nature, while we think of [alpha] [greater than or equal to] 0 as a percolation parameter. We are interested in studying when there is a nontrivial percolation threshold in ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d), namely, the critical percolation value [[alpha].sub.c] [member of] (0, [infinity]). Our results for phase transition are analogous to those in the monopartite counterpart ([[OMEGA].sub.N], d) (see [3]). The similar (comparable) behavior of phase transitions in bipartite and corresponding monopartite networks has also been observed in other percolation contexts (see the discussion in Section 2).

The rest of the paper is organized as follows. The results are stated and discussed in Section 2, and the proofs are given in Section 3.

2. Results

Let [absolute value of (S)] be the size of a vertex set S. The connected component containing the vertex x is denoted by C(x). By definition, the origin 0 [member of] [[OMEGA].sup.1.sub.N](l, y) for all l [greater than or equal to] 1 and 1 [less than or equal to] [gamma] [less than or equal to] N - 1. Since, for all x [member of] [[OMEGA].sup.1.sub.N](l, y) and y [member of] [[OMEGA].sup.1.sub.N](l,N - [gamma]), [absolute value of (C(x))] and [absolute value of (C(y))] have the same distribution, it suffices to consider only [absolute value of (C(0))] without loss of generality. The percolation probability is defined as

[theta](l, [gamma], [alpha], [beta]) = P([absolute value of (C(0))] = [infinity]), (6)

and the critical percolation value is defined as

[[alpha].sub.c] = [[alpha].sub.c] (l, [gamma], [beta]) = inf [[alpha] [greater than or equal to] 0 : [theta] (l, [gamma], [alpha], [beta]) > 0}, (7)

which is nondecreasing in [beta] for any given l and [gamma].

Theorem 1. Assume that l [greater than or equal to] 1, 1 [less than or equal to] [gamma] [less than or equal to] N - 1 and that 0 < [beta] < [infinity]. One has the following:

(i) If [beta] [less than or equal to] N, then [[alpha].sub.c] = 0.

(ii) If [beta] [greater than or equal to] [N.sup.2], then [[alpha].sub.c] = [infinity].

(iii) If N < [beta] < [N.sup.2], then 0 < [[alpha].sub.c] < [infinity].

Moreover, there is almost surely at most one infinite component when [alpha] > [[alpha].sub.c] .

Remark 2. The critical value [[alpha].sub.c] = [[alpha].sub.c] (l, [gamma], [beta]) turns out to be a function of only p irrespective of the values of l and [gamma]. Koval et al. [3] showed the same behavior of [[alpha].sub.c] for percolation in the monopartite lattice [[OMEGA].sub.N]. This analogy of phase transition has been recognized in other percolation problems in statistical physics. An example is the AB percolation introduced by Mai and Halley [13] for the study of gelation processes. In this model, each vertex of an infinite connected graph G is assigned one of two states, say A and B, with probability p and 1 - p, respectively, independently of all other vertices. Edges with two end-vertices having unlike states (called AB bonds) are occupied. Thus, the AB percolation can be viewed as a bond percolation with occupation probability 2p(1-p) (although some dependence is involved, namely, no odd path of AB bonds exists). Appel and Wierman [14] proved that AB percolation does not occur for any value of p [member of] [0,1] on a bipartite square lattice with bipartition [V.sub.1] = {v = ([v.sub.x], [v.sub.y]) [member of] [Z.sup.2] : [v.sub.x] - [v.sub.y] is odd} and [V.sub.2] = {v = ([v.sub.x], [v.sub.y]) [member of] [Z.sup.2] : [v.sub.x] - [v.sub.y] is even} such that [Z.sup.2] = [V.sub.1] [union] [V.sub.2]. In other words, the bond percolation cannot occur on the above bipartite square lattice for occupation probability 2p(1 - p) [less than or equal to] 1/2. This is consistent with the classical result which says that bond percolation on [Z.sup.2] does not occur when occupation probability [less than or equal to] 1/2 (see, e.g., [15, 16]). Other comparable AB percolation thresholds for monopartite and bipartite high-dimensional lattices can be found in [17].

Another example is the biased percolation [18,19] on infinite scale-free networks with a power-law degree distribution P(k) [varies] [k.sup.-[gamma]]. In this model, an edge between vertices with degrees [k.sub.1] and [k.sub.2] is occupied with probability proportional to [([k.sub.1][k.sub.2]).sup.-[alpha]]. By using generating function method, Hooyberghs et al. [9] showed that biased percolation on a bipartite scale-free network with two bipartition sets following degree distributions [P.sup.A](k) [varies] [k.sup.-[gamma]A] and [P.sup.B](k) [varies] [k.sup.-[gamma]B], respectively, has the same critical behaviors with biased percolation on a monopartite scale-free network when [[gamma].sub.A] = [[gamma].sub.B] = [gamma].

Remark 3. The uniqueness of the infinite component holds here for the same reason as the uniqueness result for the percolation graph of [[OMEGA].sub.N] (see [3, Theorem 2]). Note that our graph resulting from ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.1.sub.N], d) can be viewed as a spanning subgraph of that from ([[OMEGA].sub.N], d).

We consider Theorem 1 as an intermediate step towards the study of percolation on bipartite hierarchical lattices. In particular, one may explore the connectivity at the critical regime [beta] = [N.sup.2] and the graph distance (chemical distance) between 0 and a vertex x. It is also interesting to study the mean field percolation (N [right arrow] [infinity]) and compare it with that on [Q.sub.N] [1]. Directed percolation [20] and other meaningful colorings on [[OMEGA].sub.N] (other than the 2-coloring addressed in this paper) are possible.

3. Proofs

We start with some notations. Then we prove Theorem 1.

For a vertex x [member of] ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d), define Br(x) as the ball of radius r around x; that is, [B.sub.r](x) = {y : d(x, y) [less than or equal to] r}. We make the following observations. Firstly, for any vertex x, [B.sub.r](x) contains [N.sup.r] vertices. In particular, if r < l, all vertices in the ball have the same type. Secondly, [B.sub.r](x) = [B.sub.r](y) if d(x, y) [less than or equal to] r. Finally, for any x, y, and r, we have either Br(x) = Br(y) or [B.sub.r](x) [intersection] [B.sub.r](y) = 0.

For a set S of vertices, denote by [bar.S] = [[OMEGA].sub.N]\S its complement. Let [C.sub.n](x) be the component of vertices that are connected to x by a path using only vertices within [B.sub.n](x). For disjoint sets [S.sub.1], [S.sub.2] [subset or equal to] [[OMEGA].sub.N], we denote by [S.sub.1] [left/right arrow] [S.sub.2] the event that at least one edge joins a vertex in S1 to a vertex in [S.sub.2]. [S.sub.1] [??] [S.sub.2] means the event that such an edge does not exist. By definition, if [S.sub.1], [S.sub.2] [subset or equal to] [[OMEGA].sup.1.sub.N] for i = 1 or 2, then [S.sub.1] [??] [S.sub.2] occurs with probability 1. Let [C.sup.m.sub.n] (x) be the largest component in [B.sub.n](x). If there are more than one such components, just take any one of them as [C.sup.m.sub.n](x). It is clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][3].

Proof of (i). Let [A.sub.k] be the event that the origin 0 [member of] [[OMEGA].sup.1.sub.N] connects by an edge to at least one vertex at distance k in [[OMEGA].sup.2.sub.N]. By construction, for k < l, P([A.sub.k]) = 0. For k = l, there are ((N - [gamma])/(N - 1))(N - 1)[N.sup.k-1] vertices in [[OMEGA].sup.2.sub.N] at distance k from 0. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

by using (5). For k > l, there are ((N - [gamma])/N)(N - 1)[N.sup.k-1] vertices in [[OMEGA].sup.2.sub.N] at distance k from 0. Similarly, we obtain

P([A.sub.k]) = 1 - exp (- [alpha](N - [gamma])(N - 1)/[[beta].sup.k]N [N.sup.k- 1]), (9)

For k > l.

Since all the events [{[A.sub.k]}.sub.k[greater than or equal to] 1] are independent and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

diverges for any 0 < [beta] [less than or equal to] N, 1 [less than or equal to] [gamma] [less than or equal to] N - 1, and [alpha] > 0, infinitely many of [A.sub.k] occur with probability 1 by the second Borel-Cantelli lemma. Thus, [theta](l, [gamma], [alpha], [beta]) = 1 for any l > 1, 1 [less than or equal to] y [less than or equal to] N - 1, [alpha] > 0, and 0 < [beta] [less than or equal to] N. The result then follows.

Proof of (ii). We only need to show [[alpha].sub.c](l, [gamma], [N.sup.2]) = [infinity] by virtue of the monotonicity. Note that, for j [greater than or equal to] l, there are [gamma]y/N)[N.sup.j] vertices in [B.sub.j](0) [intersection] [[OMEGA].sup.1.sub.N] and ((N - [gamma])/N)[N.sup.j] vertices in [B.sub.j](0) [intersectrion] [[OMEGA].sup.2.sub.N]. Hence, by the comments in the proof of (i) and taking [beta] = [N.sup.2], for any j [greater than or equal to] l, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

which is strictly less than 1 for any finite [alpha] [greater than or equal to] 0.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Since the events [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are independent and all have the same probability strictly less than 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Consequently, there exists an i such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with probability 1. It follows from (12) that [theta](l, [gamma], [alpha], [N.sup.2]) = 0 for all [alpha] [greater than or equal to] 0. This implies [[alpha].sub.c](l, [gamma], [N.sup.2]) = [infinity]

Proof of (iii). The positivity of ac is a direct consequence of the proof of Theorem 1(b) in [3]. Since the percolation graph of ([[OMEGA].sup.1.sub.N], [[OMEGA].sup.2.sub.N], d) can be viewed as a spanning subgraph of that of ([[OMEGA].sub.N], d), the percolation cluster C(0) is almost surely finite; namely, [theta](l, [gamma], [alpha], [beta]) = 0, for a small enough.

Now we turn to the proof of finiteness of [[alpha].sub.c]. The main technique to be used is an iteration involving the tail probability of binomial distributions [3, 21]. Since [beta] < [N.sup.2], we choose an integer K and a real number [delta] such that

[square root of ([beta])] < [delta] [less than or equal to] [([N.sup.K] - 1).sup.1/K]. (14)

Clearly, 1 < [delta] < N. For n [greater than or equal to] 1, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

and analogously,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Here, [a.sub.n] is the probability that the largest component of a ball of radius nK contains at least ([gamma]/N)[[delta].sup.nK] vertices in [[OMEGA].sup.1.sub.N] and at least ((N - [gamma])/N)[[delta].sup.nK] vertices in [[OMEGA].sup.2.sub.N]. Such a ball is said to be good. We set [a.sub.0] = [b.sub.0] = 1 by convention. It is clear that, for [alpha] > 0, all [a.sub.n] and [b.sub.n] are positive, since nK is a finite number and the connection probability in 5) is positive.

In what follows, we will prove [[alpha].sub.c] < [infinity] in two steps.

Step 1. We show that there exists some [alpha] > 0 such that an converges to 1 exponentially fast; namely, 1 - [a.sub.n] [less than or equal to] exp(-cn) for some c > 0.

Step 2. We show that there exists some [alpha] > 0 such that Lim [inf.sub.n[right arrow][infinity]] [b.sub.n] > 0.

We start with Step 1. To this end, denote by N the nonnegative integers. We can naturally label the vertices in [[OMEGA].sub.N] via the map f: [[OMEGA].sub.N] [right arrow] N as

f : x = ([x.sub.1], [x.sub.2], ...) [??] [[infinity].summation] over (i = 1) [x.sub.i] [N.sup.i - 1]. (17)

This order agrees with the depiction in Figure 1. A ball of radius nK is said to be very good if it is good and its largest component connects by an edge to the largest component of the first (as per the aforementioned order) good subball in the same ball of radius (n + 1)K. Clearly, the first good subball of radius nK in a ball of radius (n + 1)K is very good. Condition (14) implies that ([N.sup.K] - 1)[[delta].sup.nK] [greater than or equal to] [[delta].sup.(n + 1)K]. Thus we assert that the ball [B.sub.(n + 1)K](0) is good if (a) it contains[ .sup.NK] - 1 good subballs of radius nK, and (b) all these good subballs are very good.

The number of good subballs of radius nK in a ball of radius (n+ 1)K has a binomial distribution Bin([N.sup.K], an) with parameters [N.sup.K] and [a.sub.n]. Given the collection of good subballs, the probability that the first such good subball is very good equals to 1. Fix any of the other good subballs, say B; the probability that B is not very good is upper bounded by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where a and b are the number of vertices in the largest component of the first good subball in [[OMEGA].sup.1.sub.N] and [[OMEGA].sup.2.sub.N], respectively; likewise, c and d are the number of vertices in the largest component of the subball B in [[OMEGA].sup.1.sub.N] and [[OMEGA].sup.2.sub.N], respectively. By definition, we have a, c [greater than or equal to] ([gamma]/N)[[delta].sup.nK], b, d [less than or equal to] ((N - [gamma])/N)[[delta].sup.nK], and the distance between two vertices in a ball of radius (n+ 1)K is at most (n + 1)K.

Therefore, the probability for any of the other good subballs B to be very good is at leat 1 - [[epsilon].sub.n]. Thus, the number of very good subballs is stochastically larger than a random variable obeying a binomial distribution Bin([N.sup.K], [a.sub.n](1- [[epsilon].sub.n])). From the above comments (a) and (b) and the definition of [a.sub.n], it follows that

[a.sub.n + 1] [greater than or equal to] P (Bin ([N.sup.K], [a.sub.n] (1 - [[epsilon].sub.n])) [greater than or equal to] [N.sup.K] - 1). (19)

In general, we have the following inequality for the tail of binomial random variable:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

By (19), (20), and writing [[xi].sub.n] = 1 - [a.sub.n], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

We can choose c > 0 large enough so that 4([N.sup.K.sub.2]) [less than or equal to] exp(c), and then we choose a large enough so that(c) [[epsilon].sub.n] [less than or equal to] exp(-c(n + 1)) and (d) [[xi].sub.1] exp(-2c) hold. To see (c), note that [beta] < [[delta].sup.2] and then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

To see (d), note that [lim.sub.[alpha] [right arrow] [infinity]] [[epsilon].sub.0] = 0, [[xi].sub.0] = 0 and by (21) we obtain

[[xi].sub.1] = 1 - [a.sub.1] [less than or equal to] ([N.sup.K.sub.2])[([[xi].sub.0] + [[epsilon].sub.0]).sup.2], (23)

which also approaches 0.

According to our above choice of c and a, we have inductively, if [[xi].sub.n] [less than or equal to] exp(-c(n + 1)), then

[[xi].sub.n + 1] [less than or equal to] ([N.sup.K.sub.2])[([[xi].sub.n] + [[epsilon].sub.n]).sup.2] [less than or equal to] 4 ([N.sup.K.sub.2]) exp (-2c (n + 1)) [less than or equal to] exp (-c (2n + 1)) [less than or equal to] exp (-c (n + 2)), (24)

which implies that [[xi].sub.n] [less than or equal to] exp(-c(n + 1)) [less than or equal to] exp(-cn) for all n [member of] N. then finish the proof of Step 1.

For Step 2, recalling the definition of [b.sub.n], we claim that

[b.sub.n + 1] [less than or equal to] [b.sub.n] x P(Bin ([N.sup.K] - 1, [a.sub.n] (1- [[epsilon].sub.n]) [greater than or equal to] [N.sup.K] - 2). (25)

In deed, if [absolute value of ([C.sub.n](0) [intersectrion] [[OMEGA].sup.1.sub.N])] [greater than or equal to] ([gamma]/N)[[delta].sup.nK] and [absolute value of ([C.sub.nK](0) [intersectrion] [[OMEGA].sup.2.sub.N])] [greater than or equal to] ((N - [gamma])/N)[[delta].sup.nK], then [B.sub.nK] (0) is the first good subball in the derivation above. If this component is connected to at least [N.sup.K] - 2 other large components in [B.sub.(n + 1)K] (0) as above, then the component containing the origin in [B.sub.(n + 1)K] (0) has ([gamma]/N)[[delta].sup.nK]([N.sup.K] - 1) [greater than or equal to] ([gamma]/N)[[delta].sup.(n + 1)K] vertices in and [[OMEGA].sup.1.sub.N] ((N - [gamma])/N)[[delta].sup.nK]([N.sup.K] - 1) [greater than or equal to] ((N - [gamma])/N)[[delta].sup.(n + 1)K] vertices in [[OMEGA].sup.1.sub.N]. Thus, the inequality (25) follows.

A simple coupling gives

P (Bin ([N.sup.K] - 1, [a.sub.n] (1 - [[epsilon].sub.n])) [greater than or equal to] [N.sup.K] - 2) [greater than or equal to] P (Bin ([N.sup.K], [a.sub.n] (1 - [[epsilon].sub.n])) [greater than or equal to] [N.sup.K] - 1). (26)

Hence, we derive that the right-hand side of (26) converges to 1 exponentially fast by exploiting (20) and the fact that [a.sub.n](1 - [[epsilon].sub.n]) converges to 1 exponentially fast. It then follows from (25) that

[b.sub.n + 1] [greater than or equal to] [b.sub.n] (1- exp (-cn)), (27)

for some c > 0. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

It is direct to check that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

and hence [[PI].sup.n.sub.k = 1] (1 - exp(-ck)) > 0 for all n. Since [b.sub.1] > 0, inequality (28) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

as desired.

http://dx.doi.org/10.1155/2013/172393

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Yilun Shang

Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682

Correspondence should be addressed to Yilun Shang; shylmath@hotmail.com

Received 17 August 2013; Accepted 10 September 2013

Academic Editors: A. Iksanov, M. Kohl, and K. Saito
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