Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations

Continuum Line-of-Sight Percolation on Poisson-Voronoi Tessellations

Quentin Le Gall Orange Labs Networks, 44 avenue de la République 92320 Châtillon quentin1.legall@orange.com Bartłomiej Błaszczyszyn Inria - ENS, 2 rue Simone Iff CS42112 75589 Paris Cedex 12 bartek.blaszczyszyn@ens.fr Élie Cali Orange Labs Networks, 44 avenue de la République 92320 Châtillon elie.cali@orange.com  and  Taoufik En-Najjary Orange Labs Networks, 44 avenue de la République 92320 Châtillon taoufik.ennajjary@orange.com
Abstract.

In this work, we introduce a new model for continuum line-of-sight percolation in a random environment given by a Poisson Voronoi tessellation. The edges of this tessellation are the support of a Cox process, while the vertices are the support of a Bernoulli process. Taking the superposition of these two processes, two points of are linked by an edge if and only if they are sufficiently close and located on the same edge of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a subcritical phase as well as a supercritical phase exist under general assumptions. We also give numerical estimates of the critical parameters of the model. Our model can be seen as a good candidate for modelling telecommunications networks in a random environment with obstructive conditions for signal propagation.

1. Introduction

1.1. Background and motivation

Bernoulli bond percolation, introduced in the late fifties by Boradbent and Hammersley [13], is one of the simplest mathematical models featuring phase transition. Since then, this model has been generalized in many different ways, making percolation theory a broader and still very active research topic today.
Since the seminal work [19] of Gilbert, random graphs have been a key mathematical tool for the modelling of telecommunications networks. Good connectivity of the network is then interpreted by percolation of its associated connectivity graph. Over the years, Gilbert’s original model has been refined and lots of mathematical models for telecommunications networks are now available in the literature.
In our previous work [24], we introduced and numerically studied a new mathematical model for modelling telecommunications networks in obstructive urban environments: we think of the urban network as being supported by a haphazard system of streets, modelled as a Poisson Voronoi tessellation (PVT) . The edges of are the support of a Cox process modelling users of the network, and the vertices of are the support of a Bernoulli point process modelling relays. Considering the superposition of these two point processes, say , we define a connectivity graph whose nodes are the points of and where two points are joined by an edge if and only if they are sufficiently close and located on the same edge of . This is how we model obstructive conditions for signal propagation. Good connectivity of the network is then interpreted as percolation of the aforementioned connectivity graph.

In this paper, we study this new model on a more theoretical approach. Our main findings are the following ones:

  • Minimal parameter for the Bernoulli process: There exists a minimal parameter of the Bernoulli process under which percolation of the connectivity graph cannot happen with positive probability, regardless of all other parameters.

  • Above the former threshold, two connectivity regimes exist:

    • Non-trivial subcritical phase : If the connectivity range is not too large, then there exists a subcritical phase for percolation of the connectivity graph.

    • Non-trivial supercritical phase : For sufficiently large and finite positive linear intensity of the Cox process, percolation happens with some positive probability.

The rest of this paper is organised as follows: We begin with recalling some related works in Subsection 1.2. Then, we present our network model and introduce convenient notations in Section 2. In Section 3, we state our theoretical results in more detail and proceed with their proofs. In Section 4, we present results of numerical simulations to illustrate our main mathematical results. Finally, we conclude and give perspectives for future work in Section 5.

1.2. Related works

In [19], Gilbert introduced percolation in a continuum setting by considering a planar homogeneous Poisson point process where two points are joined by an edge if and only if they are separated by a distance gap less than a given threshold. This model has at the time been considered to be a good candidate for representing a telecommunications network, with the range of the stations being taken into account as a parameter. The Poisson case has now extensively been studied [2, 26] and Gilbert’s model has recently been extended to other types of point processes, among which sub-Poisson [7, 8, 9] , Ginibre [18] and Gibbsian [22, 28].
The study of percolation processes living in random environments has only been considered recently and outlined that many standard techniques from Bernoulli or continuum percolation cannot be applied in a random environment setting. As a matter of fact, new tools and techniques had to be introduced. In this regard, the paper from Bollobás and Riordan [11] on the threshold of Voronoi percolation in the plane is pioneering. Later on, [1, 29] brought additional results concerning this model. Other percolation models [31], tessellations [12] and other random graphs [3, 4, 6] have also been considered. A more general study of Bernoulli and first-passage percolation on random tessellations has been conducted in [32, 33].
A natural extension of Gilbert’s model in a random environment setting is obtained by considering a Cox point process, i.e. a Poisson point process with a random intensity measure. Percolation of Gilbert’s model in such a setting has theoretically been studied for the first time in [21].
In Gilbert’s original model, connectivity between two network nodes only depends on their mutual Euclidean distance. This assumption has proven to be quite simplistic for the modelling of real telecommunications networks, where physical phenomena such as interference, fading or shadowing are at stake, making connectivity between two nodes to occur depend on other factors. As a matter of fact, other extensions of Gilbert’s work have been considered for a more accurate modelling of telecommunications networks. In particular, percolation of the signal-to-interference-plus-noise ratio (SINR) model, where a connection between a pair of points does not only depend on their relative distance anymore but also on the positions of all other nodes of the network, has theoretically been studied in [16]. SINR percolation for Cox point processes has only been explored very recently [30]. On a more applied perspective, random tessellations have turned out to yield good fits of real street systems, as has been proven in [20]. Percolation thresholds of the Gilbert graph of Cox processes supported by random tessellations have numerically been investigated in [14], yielding other interesting applications for telecommunication networks.
Recently, mathematical models of so-called line-of-sight (LOS) networks have been introduced, modelling telecommunication networks in environments with regular obstructions, such as large urban environments or indoor environments. Nodes of the network are then connected when they are sufficiently close and when they have line-of-sight access to one another, in other words if no physical obstacle stands between them. In [17], asymptotically tight results on -connectivity of the connectivity graphs arising from such models are studied. Bollobàs, Janson and Riordan [10] extend these results by introducing a line-of-sight site percolation model on the discrete square lattice and the two-dimensional -torus and asymptotical results for the critical probability were derived as well. Interesting connections to Gilbert’s continuum percolation model were also investigated. However, the study of line-of-sight percolation in a continuum setting with a random environment has not, as far as we know, been studied yet.
It is in light of these recent developments that we introduced a new percolation model for Cox processes supported by Poisson-Voronoi tessellations (PVT) in our previous work [24].

2. Model definition and main results

2.1. Network model

Consider a probability space and state space , where is the usual Borel -algebra of .

Let and be a homogeneous planar Poisson point process (PPP) in the state space with intensity . Consider the Poisson-Voronoi tessellation (PVT) associated with . In particular, is stationary and isotropic. By analogy with a telecommunications network, will be called street system from now onwards. Moreover, note that is scale-invariant.
Denote by the edge-set of and by the vertex-set of . Furthering the aforementioned analogy, the elements of (respectively ) will be called street segments (respectively crossroads).
Let be the stationary random measure defined such that:

  • , where denotes the 1-dimensional Hausdorff measure of . In other words, for a Borel set , is the total edge length of contained in and so can be seen as a Lebesgue-like measure on the edges of , rescaled in such a way that the total -measure of a 1-area window is 1.

The users, equipped with mobile devices, are modelled by a Cox process driven by the random intensity measure . In other words, conditioned on a given realization of the street system , is a PPP with mean measure . In particular, the number of users on a given street segment is a Poisson random variable with mean and the numbers of users on two disjoint subsets of are independent random variables.
The relays (either representing physical antennas or additional users not modelled through ) are modelled by a doubly stochastic Bernoulli point process on the set of crossroads with parameter , so that one can write:

where and where denotes the Dirac measure at . In other words, conditioned on (or, equivalently, ) each crossroad is retained (resp. erased) independently from all others with probability (resp. ).
Moreover, we also assume that the processes of users and of relays are conditionnally independent given their random support, i.e. . We denote the superposition of users and relays.

The key network parameters are:

  • The user density .

  • The relay proportion .

  • The connectivity range .

The network is then modelled by the connectivity graph defined in the following way:

  • is undirected.

  • The vertex set of is given by the points of .

  • The edge is drawn if and only if and are located on the same street segment and of mutual Euclidean distance less than . In other words:

    (1)

Figure 1 illustrates an example of a realization of our network model and of its connectivity graph in a bounded observation window. The blue lines represent the random environment (PVT) supporting the Cox process of users illustrated by red points. The Bernoulli process of relays is illustrated by the green points. Finally, possible connections are highlighted in orange and illustrate the connectivity mechanism given by (1).

Figure 1. Illustration of a realization of the connectivity graph of our model for LOS networks supported by a random environment in a continuum setting.

In this paper, our main concern is the percolation regime of the connectivity graph . In other words, the main question we address is: Are there critical values of the parameters for which percolation of occurs with positive probability?

2.2. Definitions and Notations

We begin with introducing a few notations and definitions which will be useful for the purposes of our developments.

For and , we denote as customary the Euclidean distance between and by:


For , , we denote by the square of side centered at . We note that this is exactly the definition of the closed ball with center and radius for the infinite norm of :

For simplicity, whenever , we will write to mean .

We will use the following convenient notation for the length of a street segment or a subset of street segment: let and . Then we denote the length of by .

We will also need the concepts of stabilization and essential asymptotic connectedness given in [21] for investigating spatial dependencies of random measures.
To that end, we denote by the space of Borel measures on , equipped with the evalutation -algebra [23, Section 13.1], i.e. the smallest -algebra making the mappings measurable for all Borel sets . For a (possibly random) Borel measure on and , we denote the restriction of to by . We also adapt the definition of the support of a measure as follows: let be a (possibly random) Borel measure on . The support of is the following set:

The concept of stabilization for investigating spatial dependencies of random measures has been introduced in many different ways in the literature. We use the same definition as the one introduced in [21]:

Definition 1.

[21, Definition 2.3] A random measure on is called stabilizing if there exists a random field of stabilization radii defined on the same probability space as such that:

  1. are jointly stationary

  2. for all , the random variables

    are independent for all bounded measurable functions

    and finite such that .

Remark 1.

Throughout the rest of this paper, we assume the random variables to be -measurable, as has been done by the authors of [21].

We slightly modify the definition of asymptotical essential connectedness given in [21] for the sake of simplicity and use the following definition:

Definition 2.

Let be a random measure. Then is essentially asymptotically connected if there exists a random field such that is stabilizing for and if for all , whenever , the following assertions are satisfied:

  1. is contained in a connected component of

As a matter of fact, we introduce the following notation for convenience: for and we denote:

whenever is a stabilizing random measure for the stabilization field .

The following result is stated in [21, Example 3.1] for a slightly modified version of Definition 2. It is easy to check that it adapts in our case as follows:

Proposition 1.

Let , where is the PVT generated by an homogeneous stationary Poisson point process. Then is stabilizing and essentially asymptotically connected for the following stabilization field:

where is the PPP having generated .

Finally, as is customary in any percolation problem, we need to define concepts of openness and closedness for the study of connected components. In our model, these definitions are adapted to crossroads and parts of street segments (possibly the whole street segments themselves) as follows:

Definition 3 (Open/Closed crossroad).

Say a crossroad is open if it is an atom of the point process , i.e. , or, equivalently:

Say is closed if it is not open, i.e. , or, equivalently:

Definition 4 (Open/Closed street segment).

Let be a street segment and let be a non-empty subset of .
Say is open if either of the two following set of conditions are satisfied:

  1. OR

Say is closed if is not open, i.e.:

3. Results

Our results concern the phase transition of the connectivity graph corresponding to the model presented in Section 2.1 as well as a minimality condition on to make percolation to occur possible. Namely, we have the following:

Theorem 1 (Minimality condition on ).

Let be the Voronoi site percolation threshold (a theoretical estimate is given in [27] and a numerical one is given in [5]). Let . Then, for all and , the connectivity graph does not percolate with probability 1.

Concerning the sub-critical phase, we have the following:

Theorem 2 (Existence of a non-trivial sub-critical phase).

Assume . For given , denote by the critical user density required for percolation of the connectivity graph :

Then, denoting , we have that .
In other words, if the connection radius is not too large, there exists a subcritical phase for percolation of the connectivity graph when all crossroads are equipped with a relay.

A straightforward consequence of Theorem 2 is the following:

Corollary 1.

Let be defined as in Theorem 2. Then, whenever , for all , we have:

Finally, we also were able to get a matching super-criticality result:

Theorem 3 (Existence of a non-trivial super-critical phase).

Let . Then, for sufficiently large and finite and sufficiently large , percolates. In other words, for any , there exists a non-trivial super-critical phase.

While Theorem 1 is quite straightforward, Theorems 2 and  3 require the use of renormalization techniques similar to the ones exposed in [21] and the domination by product measures theorem [25, Theorem 0.0].
We will carry the proofs of the former theorems in the rest of this section.

3.1. Proof of Theorem 1


Let denote the usual Poisson-Voronoi site percolation threshold, as defined in  [5, 27]. It is known that . Note that by stationarity, is independent of the intensity of the PPP generating the considered PVT. Hence, consider site percolation on the PVT with parameter and denote by the associated graph.
Now, note that for given , all and , the edge-set of the connectivity graph of our model is a subset of the edge-set of .
Hence: . Now, by definition of , does not percolate. This concludes the proof of Theorem 1. ∎

3.2. Proof of Theorem 2


Proving Theorem 2 is equivalent to proving that there exists such that whenever , does not percolate if is sufficiently small but positive.
As customary in any continuum percolation problem and as has been done in [21], we will use a renormalization argument and introduce a discrete percolation model constructed in such a way that if the discrete model does not percolate, then neither does . Proving the absence of percolation of the discrete model will then be done via appealing to [25, Theorem 0.0].

To this end, for , say a site is n-good if the following conditions are satisfied:

Say a site is -bad if it is not -good.

Our first claim is the following:

Lemma 1.

Percolation of implies percolation of the process of -bad sites.

Proof. Assume percolates and denote by a giant component of . Since is infinite, we have: .
Denote . Note that is composed of open street segments and open crossroads. Therefore, since is crossed by for all , has to be crossed by some open non-empty , for some . Hence is -bad, and is a infinite component of -bad sites.
Now, obviously is connected in . Without loss of generality, we can assume that the ’s have been indexed in the order they are encountered when exploring . In other words, when going along the path of given by , if, at some point, we encounter , then the next site we encounter following our exploration along is .
We claim that is almost surely a connected path in . Indeed, assume for a contradiction that this is not the case. Then, with positive probability, it is possible to find a realization of satisfying the following property: there exists such that the two neighbouring squares and crossed by only share a vertex, as in Figure 2 (these two squares obviously neighbouring by definition of the indexation convention we have chosen). Denote by the intersection of two such consecutive squares with the infinite connected component . We thus have and that . Then, necessarily, since , belongs to the boundary of and we thus have . Denote by the polar coordinates of . Denoting the Palm probability of the stationary segment process of the edges of , its intensity, and the Euclidean ball of radius centered at the origin, we have, by the inversion formula of Palm calculus:

(2)
(3)

where we have used the isotropy of in (2). But it is known that when is a PVT, .

Figure 2. Two consecutive -bad sites have to share a common side

Hence we get a contradiction, and so any two consecutive squares and must share a common side. As a matter of fact, is a connected path in and we have therefore found an infinite path of -bad sites. Hence, the process of -bad sites percolates.

By Lemma 1, it suffices to prove that the process of -bad sites does not percolate if and are sufficiently small. This will be done via appealing to [25, Theorem 0.0].
The conditions of [25, Theorem 0.0] are valid for so-called -dependent random fields:

Definition 5.

Let be a discrete random field. Let . Then is said to be -dependent if for all and all finite with the property that , the random variables are independent.

As we shall see thereafter, the process of -bad sites previously defined is -dependent:

Lemma 2.

For , set . Then is a -dependent random field.

Proof. As a starting point, note that . It is therefore equivalent to prove that the process of -good sites is -dependent.
For , set . Let be such that . We want to show that the random variables are independent. Since we are dealing with indicator functions, this is equivalent to showing that:

Now, we have:

(4)
(5)

where we have used -measurability of the random variables in (5).

For , set . According to Definition 4, for a given , the event only depends on the configuration of the random measure and of the Cox process inside the square . Therefore, given , the events only depend on , . Since we have , then the squares are disjoint. Moreover, given , has the distribution of a Poisson Point Process. Thus, by Poisson independence property, the events are conditionally independent given . Hence (5) yields:

(6)

Set . Then is a deterministic, bounded and measurable function of . Moreover, the set is finite and satisfies:

Since the infinite norm is always upper bounded by the Euclidean norm, we have , and so satisfies:

Hence, by condition (3) in the definition of stabilization (Definition 1), the random variables appearing in the right-hand side of (6) are independent. This yields:

(7)
(8)

where we have used the aforementioned independence property in (7) and -measurability of the ’s in (8). This concludes the proof of the lemma.

Now that we have proven that the process of -bad sites is -dependent, in order to apply [25, Theorem 0.0], it remains to prove that the probability for a site to be -bad is sufficiently small when and are chosen sufficiently small. Equivalently, as has been done in [21], we need to prove the following:

By stationarity, it suffices to prove that:

Now, note that we have:

On the one hand, we note that the quantity does not depend on and and by Definition 1, we have , thus yielding:

(9)

Therefore, it remains to deal with the quantity . We have:

For , denote:

We have: . Thus:

by -measurability of the events ,  .

Given , has the distribution of a Poisson point process with mean measure and only depends on once is fixed. Therefore, given , the events depend on the number of Cox points on distinct edges and so, by the Poisson independence property, these events are conditionally independent given . Thus:

(10)

For all such that , we have that , and moreover the event is increasing in with . So, by monotone convergence, we have that for all such that , . As a matter of fact:

Noting that , we can apply dominated convergence in (10), we get:

(11)

Note that for any fixed , the event is decreasing in . Therefore, again by monotone convergence, we have for all :

Hence . Combining this with (11) yields:

(12)

Hence, we obtain:

(13)

To conclude, we have:

Using (9) and (13), we finally get:

Hence, by [25, Theorem 0.0], the process of -bad sites is stochastically dominated from above by a sub-critical Bernoulli process when and are sufficiently small. In particular, the process of -bad sites cannot percolate when and are sufficiently small. In other words, there exists such that . This concludes the proof of Theorem 2. ∎

3.3. Proof of Corollary 1


Let be defined as in Theorem 2. Let and . For fixed , there are obviously fewer possible connections in the connectivity graph than in . In other words, the vertex-set (resp. edge-set) of is a subset of the vertex-set (resp. edge-set) of . Thus we have:

Therefore:

By Theorem 2, we have that whenever . This concludes the proof of Corollary 1. ∎

3.4. Proof of Theorem 3


As in the proof of Theorem 2, we will use a renormalization argument to prove that percolates if and are chosen sufficiently large.
To this end, let us first define a discrete model in such a way that percolation of the discrete model will imply percolation of the continuum one. For , say a site is -good if the following conditions are satisfied:

  1. , i.e. the square contains a full street segment (not just a subset of a street segment)

  2. There exists such that is open, in the sense of Definition 4. In other words, there exists an open edge which is fully included in the square .

  3. Every two open edges are connected by a path in , i.e. and are connected by a path of the connectivity graph staying inside of the larger square if all crossroads are open.

  4. , i.e. all crossroads in are open, in the sense of Definition 3.

As in the proof of Theorem 2, for , say a site is -bad if it is not -good.

The -good sites have been defined so as to satisfy the following:

Lemma 3.

Percolation of the process of -good sites implies percolation of the connectivity graph .

Proof. Let be an infinite connected component of -good sites. Consider a finite path of -good sites . Then, by condition in the definition of -goodness, we can find an open edge , for each . Let . Then for some and . Since is connected in , we have . By symmetry, we can assume . Thus:

Therefore, we have and so implies . Since we also have and that and are both open, by condition (4) in the definition of,-goodness, and are connected by a path in . But since is an -good site, by condition (5) in the definition of -goodness, all crossroads inside of are open. Therefore, the path also connects and in . Iterating this process gives rise to an infinite connected component in . This concludes the proof of Lemma 3.

As a matter of fact, proving Theorem 3 amounts to proving that the process of -good sites percolates for sufficiently large and .
As has been done in the proof of Theorem 2, we will stochastically dominate the process of -good sites by a Bernoulli process via the means of [25, Theorem 0.0]. To check that the conditions of this theorem are satisfied, we will need to use a slightly modified version of -goodness more adapted to the use of the aforementioned theorem, as follows:

For , say a site is weakly--good if the following conditions are satisfied:

  1. , i.e. the square contains a full street segment (not just a subset of a street segment)

  2. There exists such that is open, in the sense of Definition 4. In other words, there exists an open edge which is fully included in the square .

  3. Every two open edges are connected by a path in , i.e. and are connected by a path of the connectivity graph staying inside of the larger square if all crossroads are open.

  4. , i.e. all crossroads in are open, in the sense of Definition 3.

Note that the difference between -goodness and weakly--goodness only lies in the first condition being weakened into . All other conditions remain unchanged.

Then the following is clear:

Lemma 4.

, is -good is weakly--good. Therefore, we have the following:

Moreover, since the condition (1) in the definition of -goodness has not been used in the proof of Lemma 3, the following is also clear:

Lemma 5.

Percolation of the process of weakly--good sites implies percolation of the connectivity graph .

The reason why considering almost--good sites instead of -good sites will turn out to be easier for our purposes is the following:

Lemma 6.

For , set . Then is an -dependent random field.

Proof. In the same way as in the proof of Lemma 1, it suffices to prove that for all finite