Bounded-hop percolation

Bounded-hop percolation

Christian Hirsch
Abstract.

Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate distribution of the minimum number of hops that are needed to reach some point of the second point process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the second point process as its intensity tends to infinity. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the second point process in a given number of hops.

Key words and phrases:
ad hoc network, chemical distance, connection probability, continuum percolation
2010 Mathematics Subject Classification:
Primary 60K35; Secondary 60D05
Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany; E-mail: hirsch@wias-berlin.de.

1. Introduction and main results

We consider a model for a wireless telecommunication network where users are scattered at random in the entire Euclidean plane. In order to meet the users’ communication demands, the operator sustains a network of base stations. In classical cellular networks, the base stations subdivide the plane into serving zones and all users inside a serving zone communicate directly with the associated base station. Although such networks exhibit a simple hierarchical topology, installation and upkeep are costly. Indeed, to guarantee good quality of service to all users, the operator either needs to install (and maintain) a relatively dense network of base stations, or the base stations’ transmission powers must be sufficiently high so that also distant users can be served.

Since the advent of LTE technology, operators have the possibility to reduce the number of required base stations substantially by using relays. As of today, this means installing fixed relays at locations that have been chosen in advance. For future generation networks it is desirable to extend this concept through the intelligent use of ad hoc technology. To be more precise, we assume that each user has a (comparatively small) transmission radius. A direct communication between users is possible if they are within each others communication radii. Additionally, by forwarding messages via chains of directly connected users, base stations can communicate with distant users, even if transmission radii are comparatively small.

Despite these virtues, having users act as relays entails a major drawback when it comes to quality of service for delay-sensitive applications. Indeed, the forwarding of messages via several hops induces substantial delay in message transmission. Hence, in network planning, it is crucial to have detailed knowledge of distributional properties of the minimum number of hops to a base station.

In the random-graphs community, the minimum number of hops that are needed to connect two vertices of a graph is known as chemical distance. In supercritical Bernoulli percolation on the lattice, chemical distance has been investigated in [antalPhD, antPiszt]. Loosely speaking, for distant points in the infinite connected component, the chemical distance is approximately proportional to the Euclidean distance, where the proportionality factor is called time constant. The extension of this result to the setting of continuum percolation [yao] will be the major tool for establishing the distributional limit of the rescaled minimum number of hops needed to connect a user to a base station.

Next, we provide a precise definition of the wireless spatial telecommunication network under consideration. It consists of two types of network components. The first component is formed by network users. They are modeled by a homogeneous Poisson point process in , with some intensity . The base stations constitute the second component. We assume that they are of the form , where is assumed to be a stationary and ergodic point process that is independent of and has a finite and positive intensity . Here, is some scaling parameter controlling the intensity of base stations. Since we only assume stationarity and ergodicity, our results are valid under quite weak conditions on the spatial distribution of base stations. For instance, they can be applied to homogeneous Poisson point processes as well as randomly shifted lattices. In other words, our results do not depend on the question whether the base stations are scattered at random in the Euclidean plane or are aligned according to a grid that is viewed from a random reference point.

The random network under consideration can be thought of as a model for a wireless telecommunication network, where users can connect to base stations indirectly via at most hops of Euclidean distance at most to other network users. To be more precise, we say that are -connectable if there exist (not necessarily distinct) such that for all , where and . Here, denotes the standard Euclidean norm in . We say that are connectable if they are -connectable for some . Figure 1 shows a realization of the network model, where the points of and are represented by dots and squares, respectively. Points of that are -connectable to some point of are shown in blue, while points of that are -connectable but not -connectable to some point in appear in green.

Figure 1. Realization of network model

In the following, we write for the smallest number such that is -connectable to some point of . The main object of investigation in this paper is the quantity

i.e., the normalized expected number of points in that are -connectable to some base station. In fact, we show that admits a more natural representation as limiting quantity of the average number of points in inside a large box that are -connectable to a point of .

Proposition 1.

Let and . Then, almost surely,

Provided that and are of the same order, the asymptotic behavior of depends sensitively on whether the intensity is below or above the critical intensity in continuum percolation. To be more precise, is the infimum over all intensities for which the union of balls of radius centered at points almost surely has an unbounded connected component.

In the sub-critical regime, decays polynomially in as .

Theorem 2.

Let and . Then,

where denotes the set of all that are connectable to the origin.

Note that for we have , see e.g. [Grim99, Theorem 12.35].

Next, consider the supercritical case, i.e., let . By a central result in continuum percolation [mRoy, Theorem 2.1], the set contains a unique unbounded connected component. In the following, denotes the subset of all elements of that are contained in this unbounded connected component. We write for the probability that there exists , with .

In order to describe the asymptotic behavior of for large and , it is important to understand that the chemical distance between two points of , i.e., the minimum number of hops needed to establish a connection, grows linearly in the Euclidean distance of the two points. This can be formalized in different ways.

First, fixing any point , there should exist an a.s. finite random variable such that for every the chemical distance between and is at most . As observed in [bk1, Lemma 5.2], when considering Bernoulli site percolation on the lattice, the corresponding result can be derived by adapting the bond percolation argument established in Lemma 2.4 in the thesis of Antal [antalPhD].

Additionally, when disregarding points in a small environment of , the random variable can be replaced by a deterministic quantity that does not depend on . To be more precise, we put if is the element of minimizing the distance to . Then, denotes the minimum integer such that and are -connectable, where is the first standard unit vector in . Using Kingman’s subadditive ergodic theorem, it is shown in [yao] that there exists a real number such that almost surely, ; see also [antPiszt] for the corresponding statement on the lattice.

With this background, we can now provide a heuristic explanation for the asymptotic behavior of if the speed at which and tend to infinity is chosen so that their quotient tends to some constant. To be more precise, by the Slivnyak-Mecke theorem [sWeil, Corollary 3.2.3], we have

Hence, it suffices to understand the asymptotic distribution of as . First, points of can only connect to points of that are contained in the unbounded connected component of continuum percolation and the probability that a given point of is contained in the unbounded connected component is given by . Hence, instead of we consider the process of relevant points , where is obtained from by independent thinning with survival probability . Then, for a given point of to be connectable to some point of , the former must also belong to the unbounded connected component, which occurs with probability . Moreover, the closest point of is at Euclidean distance and it can be reached in at most hops. This heuristic is made precise in the following result, where we use the convention .

Theorem 3.

Let . Then, converges in distribution to the random variable

where is a Bernoulli random variable that is independent of and which assumes the value with probability .

In other words, the asymptotic distribution of is a mixture between a Dirac measure at and the contact distribution of the point process . In particular, Theorem 3 can be used to compute .

Corollary 4.

Let and assume that for some . Then,

If is a homogeneous Poisson point process with intensity , then is again a homogeneous Poisson point process with intensity . In particular, we get the following result.

Corollary 5.

Let be a homogeneous Poisson point process with intensity . Then, under the assumptions of Corollary 4,

where denotes the volume of the unit ball in .

The limiting distribution provided in Theorem 3 depends on implicitly via and . In order to develop an intuition on the order of that is needed to achieve a given (high) connectivity probability, it is useful to have some information on the behavior of and as a function of . First, concerning , it is shown in [pcPerc, Corollary of Theorem 3] that converges exponentially fast to as tends to infinity. Second, we show that asymptotically tends to as and that the convergence occurs at least at a polynomial speed.

Theorem 6.

The present paper is organized as follows. In Section 2, we establish the ergodic representation of announced in Proposition 1 and investigate the asymptotic behavior of in the subcritical regime. That is, we prove Theorem 2. Section 3, is devoted to the proof of Theorem 3 which describes the distributional limit of the rescaled minimum number of hops in the supercritical regime. Finally, in Section 4, we prove Theorem 6, i.e., we show that the time constant tends to as the intensity tends to infinity. Additionally, we provide a lower bound for the speed of this convergence.

2. Proof of Proposition 1 and Theorem 2

The proof of Proposition 1 is based on the multidimensional ergodic theorem. To apply this result, it is important to note that the homogeneous Poisson point process is mixing [sWeil, Theorem 9.3.5], so that the pair of independent stationary point processes is again ergodic, see [krengel, Theorem 3.6].

Proof.

For let

denote the number of points in that are at most hops away from some point of . From the ergodic theorem for spatial processes (see, e.g. [krengel, Theorem 2.13]), we conclude that the random variable

converges almost surely to

Moreover, for sufficiently large the expression

is bounded below and above by and respectively. Hence, letting completes the proof. ∎

To prepare the proof of Theorem 2, we note that it is possible to express as the expected value of the suitably weighted size of the cluster at a typical point of . To be more precise, for , let denote the set of all such that is -connectable to . Additionally, put

Then, we show that .

Lemma 7.

Let and . Then,

Proof.

The claimed identity is a consequence of the mass-transport principle [groupPerc]. Indeed, define a function by mapping a pair of sites to

Then, clearly, . On the other hand,

By stationarity, we obtain that

which concludes the proof. ∎

Since for all , Lemma 7 gives rise to a simple upper bound for .

Proposition 8.

Let and . Then,

We note two corollaries of Proposition 8. First, must grow at least linearly in for to have a non-zero limit.

Corollary 9.

If , then .

Proof.

Since is contained in , we deduce that . In particular, applying the upper bound from Proposition 8 proves the claim. ∎

Moreover, Proposition 8 is also useful for proving Theorem 2.

Proof of Theorem 2.

Combining the trivial inequality with Proposition 8 yields the desired bound. ∎

3. Proof of Theorem 3

In this section, we prove Theorem 3. To this end, we fix throughout the entire section. Using the notation of Theorem 3, let . In order to show that converges to in distribution, we fix an arbitrary . Then, we proceed in three steps, namely

  1. ,

  2. ,

  3. .

As a first auxiliary result, we note that asymptotically the events that points in belong to the unbounded connected component become independent.

Lemma 10.

Let and be distinct points in . Furthermore, let denote the event that and for some . Then,

Proof.

Choose such that the cubes are pairwise disjoint. Furthermore, let denote the event that the connected component of at is not contained in . Since the events are independent, we can conclude that

if we can show that the probability that the connected component of at is finite, but not contained in tends to as . But this is a consequence of the uniqueness of the unbounded connected component in continuum percolation, see [mRoy, Theorem 2.1]. ∎

Lemma 10 allows us to compute .

Proof of .

First, we note that . For the reverse inequality, let be arbitrary. Uniqueness of the infinite connected component shows that if and for some , then . Hence, by Fatou’s lemma and Lemma 10,

Letting completes the proof of the lower bound. ∎

Lemma 11.

Let be arbitrary. Then,

where denotes the event that there exists such that , but is not -connectable to .

Proof.

The claim is an immediate consequence of [yao, Theorem 2.2]. ∎

After these preliminary results, we now proceed with the proof of .

Proof of .

Put , where denotes the event that there exists with . Then,

By Lemma 11, the second probability in the above expression is negligible as . Hence, by Lemma 10,

Letting completes the proof. ∎

In order to complete the proof of Theorem 3, it remains to show that . First, we derive an auxiliary result illustrating the close relationship between the Euclidean distance and the chemical distance in the unbounded connected component of continuum percolation [yao] to show that, asymptotically, users are not -connectable to base stations that are not within distance of . To be more precise, we use the following corollary to the shape theorem [yao, Theorem 2.2].

Lemma 12.

Let and . Then, for every ,

where is the event that the origin is -connectable to some point in .

Second, we note that, asymptotically, distinct points that are connectable must be contained in the unbounded connected component of continuum percolation.

Lemma 13.

Let and be distinct points in . Furthermore, let denote the event that is connectable to some with . Then, .

Proof.

Let be the minimum of the pairwise distances between elements of . By stationarity, is bounded above by two times the probability that , but the origin is connectable to some point with distance at least . By uniqueness of the unbounded connected component in continuum percolation, the probability of the latter event tends to as . ∎

Now, we can complete the proof of Theorem 3.

Proof of .

First, we see that is at most

where denotes the event that is connectable to some . We conclude from Lemma 12 that it suffices to investigate the first expression. Concerning , Lemma 13 shows that as this probability converges to the probability of the event that and for some . Hence, combining Lemma 10 with the dominated convergence theorem gives that

Repeating the final steps used in the derivation of the lower bound completes the proof. ∎

4. Proof of Theorem 6

Loosely speaking, in order to prove Corollary 6, we can proceed similarly as in [yao, Lemma 3.4] and modify the arguments used in the lattice setting [antPiszt]. The general construction presented in these papers is useful for the proof of Corollary 6, but the identification of the behavior of as requires a more refined analysis.

It is convenient to introduce a specific family of site percolation processes. For this purpose, we describe certain useful configurations in the unit cube. Let be arbitrary. First, we need to ensure that any two points of can be connected via hops of distance at most 1 to other points of . To be more precise, denotes the event consisting of all locally finite such that for all , where is a subdivision of into congruent subcubes of side length . In particular, if , then for all , .

Second, we demand that has a point close to the origin. This will allow us to pass through linear arrangements of adjacent cubes without deviating too much from the line segment connecting the centers of these cubes. More precisely, denotes the event consisting of all locally finite with . Note that for all and . Finally, for we say that is -good if .

To begin with, we show that we can traverse quickly linear arrangements of good sites.

Lemma 14.

Let and be such that the site is -good for all . Furthermore, let be such that and . Then, and are -connectable.

Proof.

Proceeding inductively, it suffices to consider the case . But for , the claim is immediate. Indeed, as observed above, we have . ∎

Even for large values of the intensity , the probability that the site is -good for all decays exponentially fast in . Therefore, we have to deal with the occasional occurrence of defects. In the following, we say that a set of sites is -connected if it forms a connected set in the graph whose vertices are given by and where are connected by an edge if . We need a crude upper bound for the number of steps needed to traverse a set of cubes associated with a -connected set of -good sites.

Lemma 15.

Let and be a finite -connected set of -good sites. Furthermore, let be such that , for some . Then and are -connectable for .

Proof.

If , then the definition of -goodness implies that and are -connectable for . Next, if are such that , then, again by the definition of -goodness, there exist with , , and . Hence, the proof of Lemma 15 is completed by an elementary induction argument on the length of the path in connecting and . ∎

The next step is to combine Lemmas 14 and 15 into an upper bound that is useful in situations where the -connected -bad components associated with the sites , only cover a small proportion of these sites. More precisely, let be the union of the -connected -bad components associated with the sites , . If is -good, then we define its -connected -bad component to be empty. Note that is almost surely finite provided that is sufficiently large.

Let denote the unbounded connected component of . Then, consists of -connected components . Let denote the outer boundary of , i.e., consists of all such that for some . Note that is -connected, since the outer boundary of any -connected set is again -connected, see [aofpp, Lemma 2.23] (related results can be found in [deu96, bCon]).

Next, we identify subsets of that form linear arrangements of -good sites. To be more precise, we construct two finite increasing subsequences and of inductively as follows. If , then we put . Otherwise, choose as the first site that is not contained in . Furthermore, let be the last site after that is not contained . If , then put and terminate the construction. Otherwise, by definition, there is some such that . Define . If , then put and terminate the construction. Otherwise, define and continue inductively. See Figure 2 for an illustration of this construction.

Figure 2. Construction of the sequences and

We make two crucial observations. First, the sites are -good for all and . Second, if then the sites are contained in the -connected set . This allows us to make use of Lemma 15.

To summarize, we have derived bounds on the number of hops for traversing linear arrangements of -good cubes and for making detours around defects. These bounds are sufficient for our purposes provided that neither nor are contained in . In that situation, we need the following auxiliary result, where we write for Minkowski addition.

Lemma 16.

Let and be such that . Then, there exists such that and are -connectable for , where is a constant depending only on the dimension .

Proof.

Loosely speaking, we proceed as follows. Since is contained in , it is -connectable to the boundary of for some . Then, we make use of the observation in [yao, Lemma 3.4] that the minimum such cannot be too large in comparison to . To be more precise, let be some path in consisting of hops of distance at most such that is contained in . We note that there is a constant with the following property. There exists a finite subset of consisting of at most elements and such that for every there exists with . If there exist with for every and such that for every there exists at most one with , then the claim follows from the observation that . Hence, it remains to transform into a path with that property. This can be achieved by using Lawler’s method of loop erasure [saw].

To be more precise, let be the largest index such that . In particular, and . Now the construction proceeds inductively by defining as the path obtained by pasting the paths and , where is the loop erasure of the path . ∎

Let be the unique integer contained in the interval . Combining Lemmas 1416, we see that and can be connected using at most

(1)

hops. In order to translate this observation into an upper bound for , it is important to have some control on the size of the random variables and . In the following, we write for the probability that a fixed site is -bad. In particular,

(2)
Lemma 17.

If , then .

Proof.

Since any site in is -adjacent to an -bad -connected component intersecting , we have that

Furthermore, as shown in [deu96, Lemma 2.3], is stochastically dominated by , where is a family of iid random variables such that has the distribution of the size of the open -connected component at the origin when considering Bernoulli site percolation with parameter . The number of -connected subsets of sites containing the origin and consisting of exactly sites is bounded above by , see [penrose, Lemma 9.3]. Therefore,