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Research Article

# Connectivity threshold of Bluetooth graphs

Article first published online: 4 SEP 2012

DOI: 10.1002/rsa.20459

Copyright © 2012 Wiley Periodicals, Inc.

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#### How to Cite

Broutin, N., Devroye, L., Fraiman, N. and Lugosi, G. (2014), Connectivity threshold of Bluetooth graphs. Random Struct. Alg., 44: 45–66. doi: 10.1002/rsa.20459

#### Publication History

- Issue published online: 25 NOV 2013
- Article first published online: 4 SEP 2012
- Manuscript Accepted: 1 JUN 2012
- Manuscript Revised: 30 MAY 2012
- Manuscript Received: 2 MAR 2011

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- Cited By

### Keywords:

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

### Abstract

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

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
- 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
- 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
- ACKNOWLEDGMENTS
- 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 be a finite undirected graph on vertices and edge set *E*. A *random irrigation subgraph* of *G*_{n} is obtained as follows: Let be a positive integer. For every vertex , we pick randomly and independently, without replacement, *c*_{n} edges from *E*, each adjacent to *v*. These edges form the set of edges of the graph *S*_{n} (if the degree of *v* in *G*_{n} is less than *c*_{n}, all edges adjacent to *v* belong to ). The main question is how large *c*_{n} needs to be so that the graph *S*_{n} is connected, with high probability. Naturally, the answer depends on what the underlying graph *G*_{n} is.

When is the complete graph then for constant , Fenner and Frieze [11] proved that *S*_{n} 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 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 suffices.

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

We are interested in the behavior of the graph 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 . Equivalently we say that a sequence of random events *E*_{n} occurs with high probability if . There are two independent sources of randomness in the definition of the random graph . One comes from the random underlying geometric graph and the other from the choice of the *c*_{n} neighbors of each vertex.

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

When is constant, the geometry has very little influence: For instance, Dubhashi, Johansson, Häggström, Panconesi, and Sozio [9] showed that when is independent of *n*, is connected with high probability. The case when *r*_{n} 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 such that if and , then is connected with high probability.

Arguably the most interesting values for *r*_{n} are those just above the connectivity threshold for the underlying graph , that is, when *r*_{n} is proportional to . The results of Crescenzi et al. [6] show that for such values of *r*_{n}, connectivity of is guaranteed, with high probability, when *c*_{n} is a sufficiently large constant multiple of . In this paper we show that this bound can be improved substantially. For the given choice of *r*_{n}, there is a critical *c*_{n} for connectivity. It is quite easy to show that no connectivity can take place (whp) for constant *c*_{n}, and that for for a sufficiently large λ, the graph is connected whp (because the maximal cardinality of any ball of radius *r* is whp ). 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 , depending on d only, such that for all , and with

*we have, for all* ,

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

The structure of the paper is the following: In Section 2 we prove a lower bound on the critical value of *c*_{n} needed to obtain a connected graph whp given a value of *r*_{n} in the range where connectivity could be achieved. In Section 3 we show that is connected whp where *r*_{n} is proportional to and *c*_{n} 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 for the same values of *r*_{n} as in Section 3 but with a slightly larger value of *c*_{n}. In particular, we show that if *c*_{n} is a sufficiently large constant times then the diameter of is 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

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
- 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
- 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
- ACKNOWLEDGMENTS
- 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 be the number of vertices in a set . We consider 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 , where is the volume of the unit ball in . Then for each , there exist constants such that the following event, which we denote by , occurs whp:

*for every* .

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

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

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

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

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

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

Theorem 2. Let and be such that

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

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

Proof. Note that we can assume 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 -clique whp. The proof is an application of the second moment method. Let be the random family of subsets of given by

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

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

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

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

Moreover, on , we can lower bound the size of by choosing *i*_{0} and counting the sets such that all the points are inside 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

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

when holds. Therefore, when ,

since . When , the proof is analogous, if we substitute by 1 and by 0 in the previous equation.

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

For the variance we have

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

since all other terms vanish. Therefore,

where we upper bound the size of by choosing *i*_{0} and *j*_{0} and counting the sets and such that all the points *X*_{i} for are inside (since all of them have to be at distance *r* from ) and *X*_{j} for are inside . So, on we have

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

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

### 3 CONNECTIVITY NEAR THE CRITICAL RADIUS

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

In this section we prove the remaining part of Theorem 1. We consider with . We only need to prove that *S*_{n} is connected whp when *c* is above the threshold since Theorem 2 implies that *S*_{n} is disconnected whp when *c* is below it.

Theorem 3. Let and suppose that

Then 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 into cells of side length . 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 -dimensional face) and there is an edge of *S*_{n} 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:

- 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.) - The largest *-connected component of white cells has cardinality at most
*q*. - The smallest connected component of
*S*_{n}is of size at least*s*. - Each grid cell contains at most vertices.

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

then, for all sufficiently large n, the graph S_{n} 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 to the opposite side.

- 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 -dimensional sheet (see also [14, 3]). In any case, the *-connected component of white cells, if it exists, must be of size at least in order to block any black connector. Since the largest *-connected component of white cells has size at most*q*, and 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*. - Next we show that all remaining black cells are connected. Note that this implies that the corresponding vertices of
*S*_{n}belong to the same connected component. This collection of vertices of*S*_{n}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 . 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 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 , without loss of generality , suffices to separate*K*from , see Fig. 1. By definition, removing from the grid creates some connected components of cells, one of them containing*K*and other containing . Let be the one containing . Without loss of generality, we may assume that the size of is at most (otherwise we may replace by*K*). Note also that contains at least 1/*r*cells, for itself contains that many cells. By the isoperimetric theorem on the*finite grid*due to Bollobás and Leader [2], the (vertex-) boundary of , (inside the finite grid) consists of at least white cells. In particular, since is *-connected, there exists a *-connected component of white cells containing at least a constant times cells. By assumption, , and thus, no such separating white *-connected chain can exist for a sufficiently large*n*. - 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*S*_{n}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 , see Figure 2. By assumption, it contains at least cells, since it covers the connected component*C*of*S*_{n}(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 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, 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 containing white cells. By property (ii), this is impossible. This finishes the proof.

To show properties (i) through (iv) we further subdivide each cell into cubes of side length 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 using cubes of side length . Let . Then for each , there exist constants such that the following event, which we denote by , occurs whp:

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

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

because and so that .

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 holds all cardinalities of the minicells are at least (and at most ) 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 in any direction. By the choice of we have for any and .

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

Since there is a total of 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, , and , 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 vertices with high probability, where .

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

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

The number of cells *-adjacent to any fixed cell is at most , thus the number of *-connected components of size *k* containing a specified cell is at most . 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 . Overall, the number of *-connected components of size *k* is at most since there are at most 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

- (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

then 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 . 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 . 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 the probability that the cell is white is by a union bound.

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

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

If all vertices select a neighbor inside , then clearly, all vertices are connected (and within distance six of each other, pairwise: two vertices of 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 is thus bounded from above by

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

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

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

So gathering the preceding observations, we obtain

This shows that for *n* large enough, , as required.

Lemma 7. The smallest connected component of S_{n} is of size at least 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 *S*_{n} in stages. Having fixed in the definition of

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

neighbors uniformly at random without replacement. Then, in each of the remaining 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

After the first (main) round, we will show that the smallest component is whp at least in size, for a specific . 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

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

So, on to round one. Let *N*_{h} count the number of connected components of *S*_{n} of size exactly *h* obtained after round one. By definition, for . We show by the first moment method that whp the smallest component after round one is of size at least for some .

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

Given a fixed set 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 , the probability of this is at most

Therefore,

We can rewrite the upper bound as

Note that *f*(*h*) is decreasing for where because

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

for *n* large enough, since we have

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

For the final act, we tile the unit cube into minicells of side length . Consider a connected component having size *t* after round one, where . (Note that, for *n* large enough, any component of size at least already satisfies the lower bound of we want to prove.) Let the vertices of this component populate the cells. The *i*-th cell receives *n*_{i} vertices from this component, and receives *m*_{i} vertices from all other components taken together. The cell is colored red if and blue otherwise. First note that not all cells can be red, since that would mean that . 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 , the probability that these cells do not establish a link between the size *t* component and any of the other components is at most

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

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

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

by choosing *L* large enough that . Thus, whp, after we are done with all rounds, the minimal component size in *S*_{n} is at least

This concludes the proof of Lemma 7.

### 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. A LOWER BOUND FOR CONNECTIVITY ON THE WHOLE RANGE
- 3 CONNECTIVITY NEAR THE CRITICAL RADIUS
- 4 UPPER BOUND FOR THE SPANNING RATIO AND DIAMETER
- ACKNOWLEDGMENTS
- 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 *S*_{n} 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 the diameter of the graph *S*_{n} is which is clearly best possible as even the diameter of cannot be smaller than . This improves a result of Pettarin, Pietracaprina, and Pucci [22].

Given a connected graph embedded in the unit cube and two vertices *u* and *v* (points in space), let 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 , and one defines the spanning ratio as

- (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 for which the smallest path along the edges is of length , so that for some , whp,

(To see this, consider the event that for a point *X*_{i}, one other point falls within distance and there are no other points within distance .) 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*:

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

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

there exists a constant K independent of n such that whp. This implies the fact that the diameter of S_{n} is at most .

The idea of the proof of Theorem 5 is the following. Partition the unit square into a grid of cells of side length . We show that, with high probability, any two vertices *i* and *j*, such that *X*_{i} and *X*_{j} 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 *S*_{n}. 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 cells.

Just like in the arguments for the lower and upper bounds for connectivity, all we need about the underlying random geometric graph *G*_{n} is that the points 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 the moon with centers *x* and *y*.

Lemma 8. (Moon density Lemma). Let , where c_{1} is the infimum of the volume of a moon. Then for each , there exist constants such that the following event, which we denote by , occurs whp:

for every and every center of a cell y within distance 5 r/4 of X_{i}.

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

Let so that and so that . Define the events where . Note that there exists a constant *C*_{d} that only depends on *d* such that . So, we can apply a union bound to obtain

if where .

The key lemma is the following.

Lemma 9. Fix such that occurs with . Let i, j be such that X_{i} and X_{j} fall in the same cell of the grid. If then

where denotes the distance of i and j in the graph S_{n}.

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

Next we establish the required lower bound for the cardinality of *M*_{i}. Clearly, is at least as large as the number of neighbors selected by the vertices in *N*_{i} that fall in *R*, the ball of radius *r*/2, centered at the mid-point of the cell into which *X*_{i} falls. Denote by the *c* vertices belonging to *N*_{i}. Then

*h*_{1} 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*, stochastically dominates *H*_{1}, a hypergeometric random variable with parameters . To lower bound the second term on the right-hand side, and to gain independence, remove all *c* neighbors picked by *h*_{1}. Then stochastically dominates *H*_{2}, a hypergeometric random variable with parameters (independent of *H*_{1}). Continuing this fashion, we obtain that is stochastically greater than where the *H*_{i} are independent and *H*_{i} is hypergeometric with parameters . Since , this may be bounded further as is also stochastically greater than where the are i.i.d. hypergeometric random variables with parameters .

Clearly, . We may bound the lower tail probabilities of 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,

Thus, by the union bound, we obtain that

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 . In particular, if *i* and *j* are two vertices in the same cell, then both *M*_{i} and *M*_{j} contain at least 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 *M*_{i} selects a neighbor in *M*_{j}, creating a path of length five. Indeed, the probability that all neighbors selected by the vertices in *M*_{i} miss all vertices in *M*_{j}, given that and are both greater than and is at most

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

### ACKNOWLEDGMENTS

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

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

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