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Keywords:

  • random geometric graph;
  • connectivity;
  • percolation;
  • diameter;
  • spanning ratio

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well-known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these. The randomness comes from the underlying distribution of vertices in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 45–66, 2014

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

It is sometimes necessary to sparsify a network: given a connected graph, one wants to extract a sparser yet connected subgraph. In general, the protocol should be distributed, in that it should not involve any global optimization or coordination for obvious scaling reasons. The problem arises for instance in the formation of Bluetooth ad-hoc or sensor networks [24], but also in settings related to information dissemination (broadcast or rumour spreading) [4, 8].

In this paper, we consider the following simple and distributed algorithm for graph sparsification. Let inline image be a finite undirected graph on inline image vertices and edge set E. A random irrigation subgraph inline image of Gn is obtained as follows: Let inline image be a positive integer. For every vertex inline image, we pick randomly and independently, without replacement, cn edges from E, each adjacent to v. These edges form the set of edges inline image of the graph Sn (if the degree of v in Gn is less than cn, all edges adjacent to v belong to inline image). The main question is how large cn needs to be so that the graph Sn is connected, with high probability. Naturally, the answer depends on what the underlying graph Gn is.

When inline image is the complete graph then for constant inline image, Fenner and Frieze [11] proved that Sn is c-connected (for both vertex- and edge-connectedness) with high probability. This model is also known as the random c-out graph. In a subsequent paper, Fenner and Frieze [12] considered the probability of existence of a Hamiltonian cycle. They showed that there exists inline image such that a Hamiltonian cycle exists with probability tending to 1 as n tends to infinity. In a recent article Bohman and Frieze [1] proved that inline image suffices.

Apart from the complete graph, the most extensively studied case, and arguably the most important for applications, is when inline image is a random geometric graph defined as follows: Let inline image be independent, uniformly distributed random points in the unit cube inline image. The set of vertices of the graph inline image is inline image while two vertices i and j are connected by an edge if and only if the Euclidean distance between Xi and Xj does not exceed a positive parameter rn, i.e., inline image where inline image denotes the Euclidean norm. Many properties of inline image are well understood. We refer to the monograph of Penrose [20] for a survey. The graph inline image was introduced in the context of the Bluetooth network [24], and is sometimes called the Bluetooth or scatternet graph with parameters inline image, and cn. The model was introduced and studied in [13, 19, 10, 6, 22].

We are interested in the behavior of the graph inline image for large values of n. When we say that a property of the graph holds with high probability (whp), we mean that the probability that the property does not hold is bounded by a function of n that goes to zero as inline image. Equivalently we say that a sequence of random events En occurs with high probability if inline image. There are two independent sources of randomness in the definition of the random graph inline image. One comes from the random underlying geometric graph inline image and the other from the choice of the cn neighbors of each vertex.

Since we are interested in connectivity of inline image, a minimal requirement is that inline image should be connected. It is well known that the connectivity threshold of inline image is inline image where inline image, where inline image is the Lebesgue measure of the unit ball in inline image. See [21, 15] or Theorem 13.2 in [20]. This means that Gn is connected with high probability if rn is at least inline image where inline image while Gn is disconnected with high probability if rn is less than inline image where now inline image. We always consider values of rn above this level.

When inline image is constant, the geometry has very little influence: For instance, Dubhashi, Johansson, Häggström, Panconesi, and Sozio [9] showed that when inline image is independent of n, inline image is connected with high probability. The case when rn is small is a more delicate issue, since the geometry now plays a crucial role. Crescenzi, Nocentini, Pietracaprina, and Pucci [6] proved that in dimension d = 2 there exist constants inline image such that if inline image and inline image, then inline image is connected with high probability.

Arguably the most interesting values for rn are those just above the connectivity threshold for the underlying graph inline image, that is, when rn is proportional to inline image. The results of Crescenzi et al. [6] show that for such values of rn, connectivity of inline image is guaranteed, with high probability, when cn is a sufficiently large constant multiple of inline image. In this paper we show that this bound can be improved substantially. For the given choice of rn, there is a critical cn for connectivity. It is quite easy to show that no connectivity can take place (whp) for constant cn, and that for inline image for a sufficiently large λ, the graph is connected whp (because the maximal cardinality of any ball of radius r is whp inline image). The objective of this paper is to nail down the precise threshold. Our main result is the following theorem.

Theorem 1. There exists a finite constant inline image, depending on d only, such that for all inline image, and with

  • display math

we have, for all inline image,

  • display math

The proof shows that connectivity occurs at the same threshold for the presence of inline image-cliques. It might be a bit surprising that the threshold is virtually independent of γ. The threshold (in cn) is also independent of the dimension d. This is probably less surprising since cn counts a number of neighbors and the number of visible vertices in a ball is of order inline image, independently of d, for the range of rn we consider.

The structure of the paper is the following: In Section 2 we prove a lower bound on the critical value of cn needed to obtain a connected graph whp given a value of rn in the range where connectivity could be achieved. In Section 3 we show that inline image is connected whp where rn is proportional to inline image and cn is just above the corresponding value obtained in Section 2 nailing down the precise threshold in that case. Finally in Section 4 we obtain an upper bound on the diameter of inline image for the same values of rn as in Section 3 but with a slightly larger value of cn. In particular, we show that if cn is a sufficiently large constant times inline image then the diameter of inline image is inline image which is the same order of magnitude as for the underlying random geometric graph.

A final notational remark: To ease the reading for the rest of the paper we omit the subscript n in the parameters r and c as well as in most of the events and sets we define that depend on n.

2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

The aim of this section is to prove a lower bound on the value of c needed to obtain connectivity whp for a given value of r. First we need a lemma on the regularity of uniformly distributed points. Let inline image be the number of vertices in a set inline image. We consider inline image to provide a sufficient margin of play. It is an interesting problem to consider smaller values of γ. We expect that the results also hold for that case. However, the methods that we use don't allow us to go closer to the critical radius for connectivity.

Lemma 1. (Ball density Lemma). Let inline image, where inline image is the volume of the unit ball in inline image. Then for each inline image, there exist constants inline image such that the following event, which we denote by inline image, occurs whp:

  • display math

for every inline image.

Proof. We use the binomial Chernoff bound: If inline image and inline image then

  • display math

where we write inline image, for reference see [5, 17].

The expected cardinality of the set of vertices in a ball inline image is inline image, where inline image takes care of the border effect and inline image is the volume of the unit ball in dimension d. Therefore the number of vertices inline image is stochastically between inline image and inline image. Thus, we have for any inline image

  • display math
  • display math

We choose inline image so that inline image and inline image so that inline image. Define the event inline image. We can apply a union bound to obtain

  • display math

if inline image. Repeating the argument for balls of radius 2 r and r/2 we need inline image where inline image.

The next theorem shows that for any value of r above the connectivity threshold of the random geometric graph one cannot hope that Sn is connected unless c is at least of the order of inline image. In particular, when r is just above the threshold (i.e., it is proportional to inline image) then c must be at least of the order of inline image. We say that the vertices at distance less than r from Xi are the visible neighbors of i (i.e., the neighbors of i in Gn) and that inline image is the visibility ball of i. Note that the following result implies the lower bound of Theorem 1.

Theorem 2. Let inline image and inline image be such that

  • display math

Then inline image is not connected whp. (In the case of inline image, we define inline image.)

Note that in the range of r considered, we do have inline image.

Proof. Note that we can assume inline image otherwise c = 0 so every vertex is isolated, the graph is disconnected and there is nothing to prove. We will use this fact at the end of the proof. We show that there exists an isolated inline image-clique whp. The proof is an application of the second moment method. Let inline image be the random family of subsets of inline image given by

  • display math

Denote by I(Q) the indicator of the event that the vertices in Q form an isolated clique in Sn. Then inline image is the number of isolated inline image-cliques. First we condition on all the vertices inline image. The only randomness we consider are the choices of each vertex among their visible neighbors. Let inline image be the event described in Lemma 1 which holds whp. In the following we work conditionally on inline image assuming inline image holds. Throughout, we use several auxiliary functions inline image with the property that inline image for all inline image.

Define inline image as the indicator for the event that no vertex inline image chooses to link to a vertex inline image, and inline image as the indicator for the event that every inline image avoids choosing the vertices in Q as an endpoint of any of its c links. Clearly inline image. Furthermore, conditionally on inline image, the variables inline image and inline image are independent (because they involve the choices of disjoint sets of indices). On inline image,

  • display math
  • display math
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We can write the first equality in the display above because inline image for all n sufficiently large. Let inline image then we also have

  • display math
  • display math
  • display math
  • display math
  • display math
  • display math

where we use the bound inline image (if inline image then there exists inline image such that inline image). Also, we used the facts that inline image and inline image.

Moreover, on inline image, we can lower bound the size of inline image by choosing i0 and counting the sets inline image such that all the points inline image are inside inline image since this implies that all the distances between them are less than r. Note that this counts each set c + 1 times. So, we have

  • display math
  • display math

Thus, the expected number of isolated inline image-cliques may be lower bounded as

  • display math
  • display math
  • display math
  • display math
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when inline image holds. Therefore, when inline image,

  • display math
  • display math
  • display math
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since inline image. When inline image, the proof is analogous, if we substitute inline image by 1 and inline image by 0 in the previous equation.

To finish the proof we need to upper bound the variance N to ensure that inline image with high probability. Note that if inline image and inline image, then inline image because Q and inline image cannot be isolated cliques at the same time. Now, in the case inline image the random variables inline image and inline image are independent and we obtain, for any inline image such that inline image holds,

  • display math
  • display math
  • display math
  • display math
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For the variance we have

  • display math
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If the vertices in Q and inline image are far enough apart (i.e., when there is no vertex inline image such that both inline image for some inline image and inline image) then the choices involved in I(Q) and inline image are independent. Thus, we need only sum over pairs in

  • display math

since all other terms vanish. Therefore,

  • display math
  • display math
  • display math
  • display math
  • display math
  • display math

where we upper bound the size of inline image by choosing i0 and j0 and counting the sets inline image and inline image such that all the points Xi for inline image are inside inline image (since all of them have to be at distance r from inline image) and Xj for inline image are inside inline image. So, on inline image we have

  • display math
  • display math
  • display math

The last inequality holds because inline image by the remark at the beginning of the proof. Finally, on inline image, applying Chebyshev's inequality we get

  • display math

as inline image. This completes the proof since inline image holds whp by Lemma 1.

3 CONNECTIVITY NEAR THE CRITICAL RADIUS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

In this section we prove the remaining part of Theorem 1. We consider inline image with inline image. We only need to prove that Sn is connected whp when c is above the threshold since Theorem 2 implies that Sn is disconnected whp when c is below it.

Theorem 3. Let inline image and suppose that

  • display math

Then inline image is connected whp.

We first give a high-level proof using a combinatorial argument which reduces the problem of connectivity to the occurrence of four properties that will be shown to hold in a second part.

We tile the unit cube inline image into cells of side length inline image. A cell is interconnected and colored black if all the vertices in it are connected to each other without ever using an edge that leaves the cell. The other cells are initially colored white. Two cells are connected if they are adjacent (they share a inline image-dimensional face) and there is an edge of Sn that links a vertex in one cell to a vertex in the other cell. Two cells are *-connected if they share at least a corner.

Consider the following events:

  1. All cells in the grid are occupied and connected to all their neighbors. (2 d for cells in the inside, less than 2 d for cells on the boundary.)
  2. The largest *-connected component of white cells has cardinality at most q.
  3. The smallest connected component of Sn is of size at least s.
  4. Each grid cell contains at most inline image vertices.

Proposition 4. Suppose that (i)–(iv) above hold. Assuming further that q, s and λ are positive functions of n such that

  • display math

then, for all sufficiently large n, the graph Sn is connected.

Proof. The proof uses a percolation-style argument on the grid of cells. We define a black connector as a connected component of black cells that links one side of the cube inline image to the opposite side.

  1. There exists a black connector in the cell grid graph: Note that by a generalization of the celebrated argument of Kesten [18], either there is a black connector, or there is a white *-connected component of cells that prevents this connection from happening (one of the two events must occur). In dimension 2, this blocking *-connected component of white cells is a path that separates the two opposite faces of interest; in dimension d, the blockage must be a inline image -dimensional sheet (see also [14, 3]). In any case, the *-connected component of white cells, if it exists, must be of size at least inline imagein order to block any black connector. Since the largest *-connected component of white cells has size at most q, and inline image for n large enough, a black connector must exists. The black components of size less than 1/ r are now recolored gray. Note that this leaves at least the black connector component, of size at least 1/ r.
  2. Next we show that all remaining black cells are connected. Note that this implies that the corresponding vertices of Sn belong to the same connected component. This collection of vertices of Sn is called the black monster. Assume for a contradiction that there exists two connected components of black cells that are not connected together, say K and inline image. Then they must be separated by a *-connected component of white cells, and in particular there must exist some white cells. Now consider K, one of these two components of black cells. Let inline image be the *-connected components of white cells of the (vertex-) boundary of K in the grid. Each one of these boundaries separates K from one of the components of the complement of K in the grid, see Lemma 2.1 from [7]. Clearly, one and only one of inline image, without loss of generality inline image, suffices to separate K from inline image, see Fig. 1. By definition, removing inline image from the grid creates some connected components of cells, one of them containing K and other containing inline image. Let inline image be the one containing inline image. Without loss of generality, we may assume that the size of inline image is at most inline image (otherwise we may replace inline image by K). Note also that inline image contains at least 1/ r cells, for inline image itself contains that many cells. By the isoperimetric theorem on the finite grid inline image due to Bollobás and Leader [2], the (vertex-) boundary of inline image, inline image (inside the finite grid) consists of at least inline image white cells. In particular, since inline image is *-connected, there exists a *-connected component of white cells containing at least a constant times inline image cells. By assumption, inline image, and thus, no such separating white *-connected chain can exist for a sufficiently large n.
  3. Each vertex connects to at least one vertex of the black monster: To prove this, consider any vertex j, outside of the black monster, and write C for the component of Sn it belongs to. If any vertex of C lies in the black monster, then j is connected to the black monster and we are done. So we now assume that all vertices of C belong to white or gray grid cells. Adjacent vertices in C lie in the same cell, or two *-adjacent cells. Let K be the *-connected component of all grid cells visited by vertices of C. Enlarge K by adding all grid cells that reach K via a white *-connected chain of cells. The resulting *-connected component of white and gray cells is called inline image, see Figure 2. By assumption, it contains at least inline image cells, since it covers the connected component C of Sn (by properties (iii) and (iv)). So we have exhibited a fairly large *-connected component of cells that are not black; the only issue is that it might not be fully white, and we wish to isolate a large white *-connected component is order to invoke property (ii) for a contradiction. Call a cell of inline image a border cell if one of its 2 d neighbors in the grid is black. Clearly, border cells must be white, because no gray cell can have a black neighbor. Now, inline image is surrounded either by border cells, or by pieces of the boundary of the cube. The argument in (b) shows that there is a component of inline image containing inline image white cells. By property (ii), this is impossible. This finishes the proof.
image

Figure 1. The two components K and inline image separated by the *-connected piece of the boundary inline image. The boundary of K is colored in gray and inline image in dark gray. The boundary of inline image is inline image.

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image

Figure 2. The component C with its corresponding *-connected component of occupied cells K. We enlarge K by adding all the connected white and gray cells to get inline image. The border cells of inline image are colored in dark gray.

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To show properties (i) through (iv) we further subdivide each cell into inline image cubes of side length inline image which we call “minicells”. We need two auxiliary results, one similar to Lemma 1 for the number of vertices in each minicell, and another about the connectivity of adjacent pairs of minicells.

Lemma 2. (Cube density Lemma). Grid the cube inline image using cubes of side length inline image. Let inline image. Then for each inline image, there exist constants inline image such that the following event, which we denote by inline image, occurs whp:

  • display math

Proof. Given a fixed cube C, the number of vertices N(C) is distributed as inline image. Thus, writing inline image, we have

  • display math
  • display math

Choose inline image and inline image to be the solutions of inline image smaller and greater than 1 respectively. Define the event inline image. We can apply a union bound over all the cells to obtain

  • display math

because inline image and inline image so that inline image.

Lemma 3. (Cube connectivity). With high probability, all minicells are occupied and connected to their 2 d adjacent neighbors.

Proof. From Lemma 2, when the event inline image holds all cardinalities of the minicells are at least inline image (and at most inline image) whp. We condition on any point set with this distributional property, leaving only the choices of the c neighbors as a random event. Consider two neighboring minicells C and inline image in any direction. By the choice of inline image we have inline image for any inline image and inline image.

When inline image holds each ball inline image has cardinality at most inline image. By independence, the probability that all vertices in inline image miss those in C with their c choices is not more than

  • display math

Since there is a total of inline image minicells, the union bound shows that the probability that two neighboring minicells do not connect tends to zero.

We now show (i) through (iv) in four lemmas, leaving the hardest one, (iii), for last. We show all these properties with λ a sufficiently large constant depending upon γ, inline image, and inline image, leaving wide margins. Properties (i) and (iv) will follow easily from their minicell related statements above.

Lemma 4. (Part (iv)). Each grid cell contains at most inline image vertices with high probability, where inline image.

Proof. By Lemma 2 we have that every minicell of side length inline image has less than inline image vertices whp. This implies immediately that every cell contains at most inline image vertices.

Lemma 5. (Part (i)). With high probability, all cells in the grid are occupied and connected to their 2 d adjacent neighbors.

Proof. It suffices to consider two adjacent minicells in the boundary of the cells.

Lemma 6. (Part (ii)). The largest *-connected component of white cells has cardinality at most inline image whp.

Proof. We start by bounding the number of *-connected components of cells of a fixed size k. Fix an integer inline image, and let inline image be the infinite Δ-ary rooted tree (every vertex has Δ children). Let inline image be the number of subtrees of inline image containing the root and having exacly k vertices. It is well-known (see [23], Theorem 5.3.10) that inline image.

The number of cells *-adjacent to any fixed cell is at most inline image, thus the number of *-connected components of size k containing a specified cell is at most inline image. To see this it suffices to consider a spanning tree of the component and to note that for any graph G with maximum degree Δ, the number of subtrees with k vertices containing a fixed vertex v is not larger than the corresponding number in inline image. Overall, the number of *-connected components of size k is at most inline image since there are at most inline image starting cells.

Assume that we can then show that the probability that a cell is white is at most p. In that case, the probability that there is a *-connected component of size k or larger is not more than

  • display math(1)

by the union bound and because the colors of the cells are independent, given the location of the vertices. If we can show that

  • display math

then inline image suffices to make the probability bound (1) tend to zero.

We now prove that for n large enough, the probability that a specified cell is white is at most inline image. By the preceding arguments, this will complete the proof of the lemma. Recall that a cell is colored white if the graph induced by the vertices lying inside the cell is not connected.

We subdivide the cell into minicells of side length inline image. We know from Lemma 3 that all adjacent minicells are connected whp. Then if every minicell was connected inside, the whole cell would be black. Therefore, if we can bound the probability of the subgraph inside a minicell being disconnected by inline image the probability that the cell is white is inline image by a union bound.

Consider now a fixed minicell C and take any vertex v inside. Let inline image be the subset of the c neighbors of v that fall in C. Consider then all c choices of the vertices in inline image that fall in C as well, and that are not in inline image. Call that second collection inline image. We show that with high probability, all the remaining vertices select at least one vertex from inline image. Each of the remaining vertices selects in any of its c choices a vertex in inline image with probability at least

  • display math

when inline image holds. The probability that some vertex does not select any neighbor from inline image is at most

  • display math
  • display math

If all vertices select a neighbor inside inline image, then clearly, all vertices are connected (and within distance six of each other, pairwise: two vertices of inline image are within distance four, and any two neighbors of these are within distance six), and the cell is black. As a consequence, the probability of a having a white cell given the event inline image is thus bounded from above by

  • display math

where inline image is a constant to be selected later. Note that, for any inline image, the second term in the upper bound is smaller than inline image for all n large enough.

Finally, then, we consider inline image and inline image and condition on the event D. This implies that inline image. Now, for inline image to be small, one of the following events must occur: either inline image is small, or inline image is not small but inline image is small. Note that by definition inline image is stochastically larger than a

  • display math

Let inline image then for n large enough the above distribution is stochastically larger than a random variable Z distributed as inline image. We repeat a similar argument and note that inline image is stochastically larger than a Z-fold sum of independent binomial random variables, each of parameters c and inline image. Thus, assuming D and for n large enough, inline image is stochastically larger than a inline image.

So gathering the preceding observations, we obtain

  • display math
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This shows that for n large enough, inline image, as required.

Lemma 7. The smallest connected component of Sn is of size at least inline image whp.

Proof. It is in this critical lemma that we will use the full power of the threshold. The proof is in two steps. For that reason, we grow Sn in stages. Having fixed inline image in the definition of

  • display math

we find an integer constant L (depending upon inline image – see further on), and let all vertices select their c neighbors in rounds. In round one, each vertex selects

  • display math

neighbors uniformly at random without replacement. Then, in each of the remaining inline image rounds, each vertex chooses L further neighbors within its range r, but this time independently and with replacement, with a possibility of duplication and selection of previously selected neighbors. This makes the graph less connected (by a trivial coupling argument), and permits us to shorten the proof. Note that

  • display math

After the first (main) round, we will show that the smallest component is whp at least inline image in size, for a specific inline image. We then show that whp, in each of the remaining rounds, each component joins another component, and thus the minimal component size doubles in each round. After the last round, the minimal component is therefore of size at least

  • display math

which in turn is larger than inline image for all n large enough.

So, on to round one. Let Nh count the number of connected components of Sn of size exactly h obtained after round one. By definition, inline image for inline image. We show by the first moment method that whp the smallest component after round one is of size at least inline image for some inline image.

Let inline image be the event described in Lemma 1. If inline image holds, the number of sets of h vertices that can be connected is bounded from above by inline image since we count subgraphs of the visibility graph with maximum degree inline image.

Given a fixed set inline image of indices, one can only form a connected component if all the h vertices choose their neighbours among the remaining vertices in the set. Assuming inline image, the probability of this is at most

  • display math

Therefore,

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We can rewrite the upper bound as

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Note that f(h) is decreasing for inline image where inline image because

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for n sufficiently large since inline image. For such ρ, and n large enough, the upper bound is thus maximal at inline image. We have shown that

  • display math

for n large enough, since we have

  • display math
  • display math
  • display math
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This means we can take inline image. Define the event inline image of having a component of size h. Finally, the probability that a component of size at most inline image exists after round one is bounded from above by

  • display math
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For the final act, we tile the unit cube into minicells of side length inline image. Consider a connected component having size t after round one, where inline image. (Note that, for n large enough, any component of size at least inline image already satisfies the lower bound of inline image we want to prove.) Let the vertices of this component populate the cells. The i-th cell receives ni vertices from this component, and receives mi vertices from all other components taken together. The cell is colored red if inline image and blue otherwise. First note that not all cells can be red, since that would mean that inline image. In one round, each vertex chooses L eligible vertices in its neighborhood independently and with replacement. Consider two neighboring cells i and j (in any direction or diagonally) of opposite color (i is red and j is blue). Conditional on inline image, the probability that these cells do not establish a link between the size t component and any of the other components is at most

  • display math
  • display math
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Consider finally the situation that all cells are blue. Then the probability (still conditional on D) that no connection is established with the other components is not more than

  • display math
  • display math
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Since there are not more than n components to start with, the probability that any component of size between inline image and inline image fails to connect with another one is bounded from above by

  • display math

where inline image. The probability that we fail in any of the inline image rounds is at most equal to the probability that D fails plus

  • display math

by choosing L large enough that inline image. Thus, whp, after we are done with all rounds, the minimal component size in Sn is at least

  • display math

This concludes the proof of Lemma 7.

4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

In the previous sections, we have identified the threshold for connectivity near the critical radius. The connectivity is of course an important property, but the order of magnitude of distances in the sparsified Sn graph should also be as small as possible. Here we show that in the same range of values of r as in Theorem 1, as soon as c is of the order of inline image the diameter of the graph Sn is inline image which is clearly best possible as even the diameter of inline image cannot be smaller than inline image. This improves a result of Pettarin, Pietracaprina, and Pucci [22].

Given a connected graph embedded in the unit cube inline image and two vertices u and v (points in space), let inline image denote the Euclidean distance between u and v when one is only allowed to travel in space along the straight lines between connected vertices in the graph (this is the intrinsic metric associated to the embedded graph). Of course inline image, and one defines the spanning ratio as

  • display math(2)

One would ideally want the spanning ratio to be as close to one as possible. In the present case, this definition is not very relevant, since there is a chance that points that are very close in the plane are not connected by an edge. In particular one can show that, with probability bounded away from zero, there is a pair of points at distance inline image for which the smallest path along the edges is of length inline image, so that for some inline image, whp,

  • display math

(To see this, consider the event that for a point Xi, one other point falls within distance inline image and there are no other points within distance inline image.) This justifies introducing the constraint that the points in the supremum in (2) be at least at distance r. Hence the following modified definition of spanning ratio:

  • display math

The next theorem shows that the spanning ratio is within a constant factor of the optimal.

Theorem 5. There exist a constant inline image such that for any inline image, if

  • display math

there exists a constant K independent of n such that inline image whp. This implies the fact that the diameter of Sn is at most inline image.

The idea of the proof of Theorem 5 is the following. Partition the unit square into a grid of cells of side length inline image. We show that, with high probability, any two vertices i and j, such that Xi and Xj fall in the same cell, are connected by a path of length at most five. On the other hand, by Lemma 3, with high probability, any two neighboring cells contain two vertices, one in each cell, that are connected by an edge of Sn. These two facts imply the statement of the theorem. We prove the former in Lemma 9 below. The bound for the diameter follows immediately from the fact that, with high probability, starting from any vertex, a point in a neighboring cell can be reached by a path of length 6 and any cell can be reached by visiting at most inline image cells.

Just like in the arguments for the lower and upper bounds for connectivity, all we need about the underlying random geometric graph Gn is that the points inline image are sufficiently regularly distributed. This is formulated as follows: A moon is the intersection of two circles, one of radius r and the other of radius r/2 such that their centers are within distance 5 r/4 (see Fig. 3). Denote by inline image the moon with centers x and y.

Lemma 8. (Moon density Lemma). Let inline image, where c1 is the infimum of the volume of a moon. Then for each inline image, there exist constants inline image such that the following event, which we denote by inline image, occurs whp:

  • display math

for every inline image and every center of a cell y within distance 5 r/4 of Xi.

Proof. Since any moon has volume at least in inline image the number of vertices inline image is stochastically between inline image and inline image. Thus, we have for any inline image

  • display math
  • display math

Let inline image so that inline image and inline image so that inline image. Define the events inline image where inline image. Note that there exists a constant Cd that only depends on d such that inline image. So, we can apply a union bound to obtain

  • display math

if inline image where inline image.

image

Figure 3. For any point Xi in a box of edge length inline image, if inline image is a neighbor selected by i, then h may select neighbors within distance r/2 of the center of the square from the shaded regions–the so-called “moons”. The volume of any moon is at least a constant times rd. The figure shows two possible positions of Xi in a box with a corresponding neighbor and moon.

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The key lemma is the following.

Lemma 9. Fix inline image such that inline image occurs with inline image. Let i, j be such that Xi and Xj fall in the same cell of the grid. If inline image then

  • display math

where inline image denotes the distance of i and j in the graph Sn.

Proof. Let inline image denote the set of all vertices k such that inline image and Xk is within Euclidean distance r/2 of the center of the grid cell that contains Xi. The outline of the proof is the following: It suffices to show that Mi contains a large constant times inline image vertices. Since the same is true for Mj and any two vertices in inline image are within Euclidean distance r, with high probability there exists an edge between Mi and Mj, establishing a path of length 5 between i and j. Let Ni denote the set of c neighbors picked by i. Then each inline image chooses its c neighbors. Those that fall in the moon defined by Xh and the center of the cell belong to Mi, see Fig. 3.

Next we establish the required lower bound for the cardinality of Mi. Clearly, inline image is at least as large as the number of neighbors selected by the vertices in Ni that fall in R, the ball of radius r/2, centered at the mid-point of the cell into which Xi falls. Denote by inline image the c vertices belonging to Ni. Then

  • display math

h1 picks its c neighbors among all vertices within distance r. The number of those neighbors falling in R has a hypergeometric distribution. Since we are on D, inline image stochastically dominates H1, a hypergeometric random variable with parameters inline image. To lower bound the second term on the right-hand side, and to gain independence, remove all c neighbors picked by h1. Then inline image stochastically dominates H2, a hypergeometric random variable with parameters inline image (independent of H1). Continuing this fashion, we obtain that inline image is stochastically greater than inline image where the Hi are independent and Hi is hypergeometric with parameters inline image. Since inline image, this may be bounded further as inline image is also stochastically greater than inline image where the inline image are i.i.d. hypergeometric random variables with parameters inline image.

Clearly, inline image. We may bound the lower tail probabilities of inline image by recalling an observation of Hoeffding [16] according to which the expected value of any convex function of a hypergeometric random variable is dominated by that of the corresponding binomial random variable. Therefore, any tail bound obtained by Chernoff bounding for the binomial distribution also applies for the hypergeometric distribution. In particular,

  • display math

Thus, by the union bound, we obtain that

  • display math

Thus, we have proved that with high probability, for every vertex i, the number of second generation neighbors (i.e., the neighbors selected by the neighbors selected by i) that end up within distance r/2 of the center of the grid cell containing i is proportional to inline image. In particular, if i and j are two vertices in the same cell, then both Mi and Mj contain at least inline image vertices. If two of these vertices coincide, there is a path of length 4 between i and j. Otherwise, with very high probability, at least one vertex in Mi selects a neighbor in Mj, creating a path of length five. Indeed, the probability that all neighbors selected by the vertices in Mi miss all vertices in Mj, given that inline image and inline image are both greater than inline image and inline image is at most

  • display math

which goes to zero faster than any polynomial function of n. (Here we used that fact that inline image for a sufficiently large n.) Finally, we may use the union bound over all pairs of at most inline image pairs of vertices i and j to complete the proof of the lemma.

ACKNOWLEDGMENTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References

The authors would like to thank the referees for their valuable comments that contributed significantly to improve the quality and clarity of this manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
  5. 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
  6. 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
  7. ACKNOWLEDGMENTS
  8. References
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