In the previous chapters, we have studied the connectivity of large static networks and dynamic networks. Asymptotic analysis is a major tool used in the study of large networks. In the next several chapters, we continue to study the connectivity of small to medium sized networks whose number of nodes is not necessarily large enough for asymptotic results to apply.

In a very general setting, we consider a wireless network with nodes identically and independently distributed following a homogeneous Poisson distribution with known density ρ in a given d-dimensional area (d = 1, 2), denoted by S. A node at \(\boldsymbol{x}_{2} \in S\) is directly connected to a node at \(\boldsymbol{x}_{1} \in S\) with probability \(g\left (\boldsymbol{x}_{2} -\boldsymbol{ x}_{1}\right )\), i.e., following the random connection model. There are three related probabilities characterizing the connectivity properties of such a network: 1) \(\Pr \left (k\right )\) the probability that a randomly selected node is k-hops apart from another randomly selected node, i.e., the length of the shortest path from the first node to the second node measured by the number of hops is k; 2) \(\Pr \left (k\vert \boldsymbol{x}\right )\) the probability that a node \(\boldsymbol{x}\) apart from another node is connected to that node in exactly k hops; and 3) \(\Pr \left (\boldsymbol{x}\vert k\right )\) the spatial distribution of the nodes k-hops apart from another designated node. These three probabilities are related through Bayes’ formula. Given the probability density \(p(\boldsymbol{x})\) for the displacement \(\boldsymbol{x}\) between two randomly chosen node [144] and if one is computable, the other two will be computable using similar techniques. Therefore, we call these three probabilities collectively the probabilities of k -hop connection or hop count statistics.

As special cases of the above general setting, we study, respectively, the cases when d = 1, i.e., one-dimensional networks, or when d = 2, i.e., two-dimensional networks; when the connection model is a unit disk connection model or when the connection model becomes a log-normal connection model. When the probabilistic measures \(\Pr \left (k\vert \boldsymbol{x}\right )\) and \(\Pr \left (\boldsymbol{x}\vert k\right )\) depend only on the Euclidean distance x between nodes, we can write \(\Pr \left (k\vert x\right )\) for \(\Pr \left (k\vert \boldsymbol{x}\right )\) and \(\Pr \left (x\vert k\right )\) for \(\Pr \left (\boldsymbol{x}\vert k\right )\), instead of their displacement. Obviously, \(\sum _{k=1}^{\infty }\Pr \left (k\right )\) gives the probability that two randomly selected nodes are connected (via a multi-hop path). Assuming the unit disk connection model, \(\sum _{k=1}^{\infty }\Pr \left (k\vert x\right )\) gives the probability that a one-dimensional network with two nodes placed at both ends of a line segment of length x and other nodes in between is connected. We shall also show that the three measures and their variants can be used to derive other connectivity related properties of the network.

Solutions to the probability of k-hop connection problems can be used in a number of areas in wireless networks. The probability \(\Pr \left (k\right )\) is useful in estimating the overall energy consumption, lifetime, and capacity of a wireless sensor network [204, 205] because these measures are tightly related to the number of hops required to transmit a packet from its source to its destination. The probability \(\Pr \left (k\vert \boldsymbol{x}\right )\) can be used in the analysis of end-to-end delay, energy consumption, and reliability of packet transmission [54, 203] and the probability in vehicular networks that a vehicle can access the base station within a designated number of hops [196]. The probability \(\Pr \left (\boldsymbol{x}\vert k\right )\) is useful in estimating the distance between two nodes from their neighborhood information and obtaining variance of such estimate, which can then be used in localization [62].

A focus of our study on the connectivity of small to medium networks will be on the characterization of the three related probabilistic measures.

1 Probabilities of k-Hop Connection in One-Dimensional Ad Hoc Networks

In this section, we consider the special case of a one-dimensional network with nodes distributed on a semi-infinite line extending towards + x axis following a homogeneous Poisson distribution with density ρ. A pair of nodes are directly connected following the unit disk connection model with a transmission range r 0. Without loss of generality, place the origin on the leftmost node and designate the node as the source node. We study the conditional probability \(\Pr \left (k\vert x\right )\), i.e., the probability that a node located at x is k-hop apart from the source node.

Let ξ k , k = 1, …, be the furthest Euclidean distance between the source node and its k-hop neighbors, if exist. We first study the distribution of ξ k , k = 1, … and then use the distribution of ξ k , k = 1, … to derive \(\Pr \left (k\vert x\right )\).

We derive the distribution of ξ k , k = 1, …, recursively and start with ξ 1. Due to the Poisson distribution of nodes, it can be readily shown that for x ≤ r 0

$$\displaystyle{ \Pr \left (\xi _{1} \leq x\right ) = \left (1 - e^{-\rho x}\right )e^{-\rho \left (r_{0}-x\right )} }$$
(11.1.1)

where the term 1 − e −ρ x is the probability that there is at least one node in \(\left [0,x\right ]\) and the term \(e^{-\rho \left (r_{0}-x\right )}\) is the probability that there is no node in \(\left (x,r_{0}\right ]\). It follows that the probability density function of ξ 1, denoted by \(f_{\xi _{1}}\left (x\right )\), is given by

$$\displaystyle{ f_{\xi _{1}}\left (x\right ) = \frac{d\Pr \left (\xi _{1} \leq x\right )} {dx} =\rho e^{-\rho \left (r_{0}-x\right )} }$$
(11.1.2)

Due to the unit disk connection model being considered, it follows that \(f_{\xi _{1}}\left (x\right ) = 0\) for x > r 0 and \(\Pr \left (\xi _{1} \leq r_{0}\right ) = 1 - e^{-\rho r_{0}}\) is the probability that there exists a 1-hop neighbor of the source.

Now we move on to consider the distribution of ξ 2. Obviously ξ 2 has to satisfy the condition that r 0 < ξ 2 ≤ ξ 1 + r 0, it follows that conditioned on ξ 1, the cumulative distribution function of ξ 2 is given by

$$\displaystyle{ \Pr \left (\left.\xi _{2} \leq x\right \vert \xi _{1}\right ) = \left \{\begin{array}{lll} 0 &\text{for}&x \leq r_{0} \\ \left (1 - e^{-\rho \left (x-r_{0}\right )}\right )e^{-\rho \left (\xi _{1}+r_{0}-x\right )} & \text{for}&r_{0} <x \leq \xi _{1} + r_{0} \\ 1 - e^{-\rho \xi _{1}} & \text{for}&x>\xi _{1} + r_{0} \end{array} \right. }$$
(11.1.3)

The probability density function of ξ 2, conditioned on ξ 1, is given by

$$\displaystyle{ f_{\xi _{2}}\left (\left.x\right \vert \xi _{1}\right ) = \left \{\begin{array}{lll} 0 &\text{for}&x \leq r_{0} \\ \rho e^{-\rho \left (\xi _{1}+r_{0}-x\right )} & \text{for}&r_{0} <x \leq \xi _{1} + r_{0} \\ 0 &\text{for}&x>\xi _{1} + r_{0} \end{array} \right. }$$
(11.1.4)

Supposing that the (conditional) distributions of ξ 1, ξ 2, …, ξ k are known, we now consider the conditional distribution of ξ k+1. First note that under the unit disk connection model, the distribution of ξ k+1 only depends on the distributions of ξ k and ξ k−1, i.e., \(f_{\xi _{k+1}}\left (\left.x\right \vert \xi _{k},\ldots,\xi _{1}\right ) = f_{\xi _{k+1}}\left (\left.x\right \vert \xi _{k},\xi _{k-1}\right )\) because a node is a k + 1-hop neighbor of the source if and only if its location x satisfies ξ k + r 0 < x ≤ ξ k−1 + r 0. This is also illustrated in Fig. 11.1.

Fig. 11.1
figure 1

An illustration of possible region for k + 1-hop neighbors

Full size image

Similarly as before, it can be shown that

$$\displaystyle\begin{array}{rcl} & & \quad \Pr \left (\left.\xi _{k+1} \leq x\right \vert \xi _{k},\ldots,\xi _{1}\right ) \\ & & =\Pr \left (\left.\xi _{k+1} \leq x\right \vert \xi _{k},\xi _{k-1}\right ) \\ & & = \left \{\begin{array}{lll} 0 &\text{for}&x \leq \xi _{k-1} + r_{0} \\ \left (1 - e^{-\rho \left (x-\xi _{k-1}-r_{0}\right )}\right )e^{-\rho \left (\xi _{k}+r_{0}-x\right )} & \text{for}&\xi _{ k-1} + r_{0} <x \leq \xi _{k} + r_{0} \\ 1 - e^{\rho \left (\xi _{k}-\xi _{k-1}\right )} & \text{for}&x>\xi _{ k} + r_{0} \end{array} \right.{}\end{array}$$
(11.1.5)

and the probability density function of ξ k+1, conditioned on ξ k , …, ξ 1, is given by

$$\displaystyle{ f_{\xi _{k+1}}\left (\left.x\right \vert \xi _{k},\ldots,\xi _{1}\right ) = \left \{\begin{array}{lll} 0 &\text{for}&x \leq \xi _{k-1} + r_{0} \\ \rho e^{-\rho \left (\xi _{k}+r_{0}-x\right )} & \text{for}&\xi _{ k-1} + r_{0} <x \leq \xi _{k} + r_{0} \\ 0 &\text{for}&x>\xi _{k} + r_{0} \end{array} \right. }$$
(11.1.6)

Remark 170

It is evident from (11.1.5)that, under the unit disk connection model, the distribution of the furtherest distance of k + 1-hop nodes depends only on the distributions of the furtherest distances of the previous two hop nodes. The same conclusion may not be true for other connections models, e.g., the log-normal connection model and the random connection model.

It follows that the joint distribution of ξ k+1, …, ξ 1 can be obtained as

$$\displaystyle\begin{array}{rcl} f\left (\xi _{k+1},\ldots,\xi _{1}\right )& =& f\left (\left.\xi _{k+1}\right \vert \xi _{k},\ldots,\xi _{1}\right )f\left (\left.\xi _{k}\right \vert \xi _{k-1},\ldots,\xi _{1}\right )\cdots f\left (\xi _{1}\right ) \\ & =& f\left (\left.\xi _{k+1}\right \vert \xi _{k},\xi _{k-1}\right )f\left (\left.\xi _{k}\right \vert \xi _{k-1},\xi _{k-2}\right )\cdots f\left (\xi _{1}\right ){}\end{array}$$
(11.1.7)

and the probability density function (and the cumulative distribution function) of ξ k , k = 1, …, can be obtained from (11.1.7) using the total probability theorem.

Given the distribution of ξ k , k = 1, …, \(\Pr \left (k\vert x\right )\) can be readily obtained:

$$\displaystyle\begin{array}{rcl} \Pr \left (k\vert x\right )& =& \Pr \left (\xi _{k-1} <x \leq \xi _{k-1} + r_{0}\right ) \\ & =& \Pr \left (x - r_{0} \leq \xi _{k-1} <x\right ){}\end{array}$$
(11.1.8)

2 Connectivity of One-Dimensional Infrastructure Based Networks

In this section, we continue to investigate the connectivity of one-dimensional wireless networks with infrastructure support. Examples of such networks can be found in vehicular networks and wireless sensor networks, see [44, 76] for more examples. In vehicular networks, roadside infrastructure plays an important role in the reliable and timely distribution of critical information to vehicles on the road [14]. In wireless sensor networks, data sinks gather useful information collected by the sensors via multi-hop paths. Then, these data sinks may either store the data for later retrieval or aggregate and transfer the data immediately via a backbone network to the remote base station or Internet. An example is the sensor network deployed on Great Duck Island for habitat monitoring [136]. From the above examples, it can be summarized that an infrastructure-based wireless network has the following characteristics: (a) the communication between “ordinary” nodes (vehicles / sensors) and “powerful” nodes (roadside infrastructure / data sinks) is important for the core functions of the networks to be carried out properly; (b) the powerful nodes are inter-connected, either by wired or by wireless links and their location is usually deterministic; (c) the location of ordinary nodes is often random. In this section, we investigate the connectivity of this type of one-dimensional infrastructure based networks, which is also referred to as hybrid networks in the literature.

To characterize connectivity of one-dimensional infrastructure based networks, we propose a new concept of connectivity, which we term type-II connectivity, to distinguish it from connectivity of homogeneous networks studied earlier. We say that a network is type-II connected if every ordinary node in the network is connected, directly or via multi-hop paths, to at least one of a small subset of powerful nodes. Type-II connectivity problem is a broad topic. In addition to connectivity probability, we also analyze the average number of clusters. The average number of clusters can be considered as an alternative measure of network connectivity, which measures how fragmented a network is if it is not connected. It tells us how many additional powerful nodes are required for all ordinary nodes in a network to be connected to at least one powerful node with high probability. Assuming these additional powerful nodes are mobile, then there exist a number of ways [16] that they can connect the ordinary nodes together to achieve certain purposes such as maximizing the communication reliability between nodes [113] or balancing the traffic load among the nodes [128]. Such problems are important in network topology control and routing. Based on the above connectivity and clustering results, we obtain the optimum powerful node distribution that minimizes the average number of clusters and maximizes the asymptotic connectivity probability of the network.

More specifically, the network model studied in this section is defined as follows.

Definition 171

Denote by G(λ,n p ;L;r o ,r p ) a one-dimensional wireless network with two types of nodes: ordinary nodes and powerful nodes. Ordinary nodes are identically, independently, and Poissonly distributed with a known density λ in the interval [0,L]. There are n p ≥ 2 powerful nodes in the network, where two of them are placed at both ends of the interval and the rest are arbitrarily distributed in the interior of the same interval. A direct connection between two ordinary nodes (respectively, between an ordinary node and a powerful node) exists if their Euclidean distance is smaller than or equal to r o (respectively, r p ). Furthermore, all powerful nodes are assumed to be inter-connected to each other.

An example of our model is illustrated in Fig. 11.2.

Fig. 11.2
figure 2

An example of a wireless network with a mixture of ordinary nodes and powerful nodes. The distances between adjacent powerful nodes are denoted by w i , for 1 ≤ i ≤ n p − 1

Full size image

In the model, the powerful nodes divide the interval [0, L] into n p − 1 sub-intervals and each sub-interval i has length w i for 1 ≤ i ≤ n p − 1.

In general, we assume that r p  ≥ r o . This assumption is justified because it is often the case that a powerful node can not only transmit at a larger transmission power than an ordinary node,but it may also be equipped with more sophisticated antennas, which make it more sensitive to the transmitted signal from an ordinary node [172]. As will be shown later, in this section we mostly focus on the network with L = 1, i.e., on the unit interval. Using the scaling technique, the results for the network on the unit interval can be easily applied to a network on the interval [0, L] where L ≠ 1.

2.1 Characterization of Type-II Connectivity Probability

In this subsection we investigate the type-II connectivity probability of a network G(λ, n p ; 1; r o , r p ), i.e., on the unit interval. Under the unit disk connection model, the connectivity probability can be derived by first examining each sub-interval bounded by two consecutive powerful nodes.

Let \(\mathcal{A}_{i}(w_{i})\) be the event that sub-interval i with length w i is type-II connected under the assumption that r p  ≥ r o . It is trivial to show that \(\Pr \left (\mathcal{A}_{i}(w_{i})\right )\), the probability that \(\mathcal{A}_{i}(w_{i})\) occurs, is 1 when w i  ≤ 2r p . For w i  > 2r p , a realization of a type-II connected sub-interval i for r p  ≥ r o remains type-II connected if and only if after removing an interval of length r p − r o (and the ordinary nodes within that interval) from the left end and right end of sub-interval i respectively, the resulting sub-interval with r p  = r o is still type-II connected. Hence for w i  > 2r p ,

$$\displaystyle{ \Pr \left (\mathcal{A}_{i}(w_{i})\right ) =\Pr \left (\mathcal{A}_{i}^{eq}(w_{ i} - 2(r_{p} - r_{o}))\right ) }$$
(11.2.1)

where \(\mathcal{A}_{i}^{eq}(x)\) is the event that sub-interval i with length x is type-II connected under the situation that r p  = r o . In the next subsection we provide the derivation of \(\Pr \left (\mathcal{A}_{i}^{eq}(x)\right )\).

A network is type-II connected if and only if each sub-interval is type-II connected. Under the unit disk connection model, the event that one sub-interval is type-II connected is independent of the event that another sub-interval is type-II connected. Hence, the probability that a network with each sub-interval i having length w i is type-II connected (say, event \(\mathcal{B}(w_{1},\cdots \,,w_{n_{p}-1})\)) is

$$\displaystyle{ \Pr \left (\mathcal{B}(w_{1},\cdots \,,w_{n_{p}-1})\right ) =\prod _{ i=1}^{n_{p}-1}\Pr \left (\mathcal{A}_{ i}(w_{i})\right ) }$$
(11.2.2)

Based on (11.2.2), we can obtain the following theorem.

Theorem 172

Denote by \(\mathcal{B}\) the event that a random instance of G(λ,n p ;1;r o ,r p ) is type-II connected. Then, the probability that event \(\mathcal{B}\) occurs is

$$\displaystyle{ \Pr \left (\mathcal{B}\right ) =\int _{\mathbb{D}}\left (\prod _{i=1}^{n_{p}-1}\Pr \left (\mathcal{A}_{ i}(w_{i})\right )\right )f(\mathbf{w})\mathrm{d}\mathbf{w} }$$
(11.2.3)

where \(\mathbb{D} =\{ (w_{1},\cdots \,,w_{n_{p}-1}):\sum _{ i=1}^{n_{p}-1}w_{i} = 1\}\); \(f(\mathbf{w}) = f(w_{1},\cdots \,,w_{n_{p}-1})\) is the joint probability density function of the distances between adjacent powerful nodes; \(\Pr \left (\mathcal{A}_{i}(w_{i})\right )\) is given in ( 11.2.1 ) and \(\Pr \left (\mathcal{A}_{i}^{eq}(w_{i})\right )\) is given in ( 11.2.10).

Using (11.2.3), we can calculate the type-II connectivity probability of a network with any distribution of powerful nodes as long as f(w) of that distribution is known. For example, if the powerful nodes are uniformly distributed, then f(w) = (n p − 2)! [53]. If the powerful nodes are placed in an equidistant fashion, then (11.2.3) simplifies into

$$\displaystyle{ \Pr \left (\mathcal{B}\right ) = \left (\Pr \left (\mathcal{A}_{i}(w)\right )\right )^{n_{p}-1} }$$
(11.2.4)

with \(w = \frac{1} {n_{p}-1}\). In the following subsections, we derive \(\Pr \left (\mathcal{A}_{i}^{eq}(x)\right )\) and its asymptotic approximation where the superscript eq is used to highlight that the power nodes are placed in an equidistant fashion.

2.1.1 Exact Probability that a Sub-Interval Is Type-II Connectedfor r p  = r o

As mentioned earlier, the result for r p  = r o can be used to obtain the result for the general case where r p  ≥ r o . Let \(\mathcal{A}_{i}^{eq}(m_{i},w_{i})\) be the event that sub-interval i with length 0 < w i  ≤ 1 is type-II connected given that there are m i ordinary nodes in the sub-interval. Denote the common transmission range by r, i.e., r = r p  = r o . The derivation of \(\Pr \left (\mathcal{A}_{i}^{eq}(m_{i},w_{i})\right )\) relies on the following lemma from [79].

Lemma 173

Let [x,x + y] be a sub-interval of length y within [0,1]. Assume two of k given vertices have been placed at the borders of this sub-interval. Define two vertices to be neighbors if and only if they are at distance r or less apart, let \(\mathcal{Z}_{k,y,r}\) be the event that k − 2 vertices, corresponding to the remaining vertices and uniformly placed in [0,1], are inside [x,x + y] and “join” the borders, that is, the k vertices form a connected subgraph of length y; and let \(P(k,y,r) =\Pr (\mathcal{Z}_{k,y,r})\) . Then, for k ≥ 2,

$$\displaystyle{ P(k,y,r) =\sum _{ j=0}^{\min (k-1,\left \lfloor y/r\right \rfloor )}\binom{k - 1}{j}(-1)^{j}(y - jr)^{k-2} }$$
(11.2.5)

A sub-interval is type-II connected if all ordinary nodes within the sub-interval are connected to at least one of the two powerful nodes located at both ends of the sub-interval. Hence, event \(\mathcal{A}_{i}^{eq}(m_{i},w_{i})\) occurs with probability

$$\displaystyle{ \Pr \left (\mathcal{A}_{i}^{eq}(m_{ i},w_{i})\right ) = P(m_{i} + 2,1,\hat{r}) + m_{i}(m_{i} + 1)\int _{0}^{1-\hat{r}}P(m_{ i} + 1,\hat{x},\hat{r})\mathrm{d}\hat{x} }$$
(11.2.6)

where \(\hat{r} = \frac{r} {w_{i}}\) is the normalized transmission range and \(\hat{x} = \frac{x} {w_{i}}\) is the normalized distance of x. The two terms on the right-hand side of (11.2.6) represent the two possible cases of the event, as illustrated in Fig. 11.3. Figure 11.3a corresponds to the first term in (11.2.6), and Fig. 11.3b corresponds to the second term.

Fig. 11.3
figure 3

An illustration that sub-interval i is type-II connected. Subgraph (a ) shows that all ordinary nodes are connected to both powerful nodes. All spacings between any two adjacent nodes are not greater than the transmission range r. Subgraph (b ) shows an example where there is a big spacing with length s in the sub-interval, and s > r. All ordinary nodes located to the left (right) of the big spacing are connected to the left (right) powerful node

Full size image

Figure 11.3a shows a possible case where all m i ordinary nodes within sub-interval i are connected to both powerful nodes. That is, none of the m i + 1 spacings between the adjacent ordinary nodes and the two powerful nodes is larger than r. In this case, all ordinary nodes and the two powerful nodes in sub-interval i form a connected “subgraph” of length w i . From Lemma 173, the probability of this case is \(P(m_{i} + 2,1,\hat{r})\), where m i + 2 is the sum of the number of ordinary nodes and the two powerful nodes.

Figure 11.3b shows the other possible case where the m i ordinary nodes inside sub-interval i are connected to either one of the two powerful nodes but not both. Then, among the m i + 1 spacings between adjacent nodes, there is exactly one spacing with length s > r. Suppose temporarily that the big spacing of length s and the ordinary node attached to the left end of the big spacing are removed from sub-interval i, as illustrated in Fig. 11.4. Then, the m i − 1 remaining ordinary nodes and the two powerful nodes form a connected “subgraph” of length x = w i − s.

Fig. 11.4
figure 4

An illustration of the scenario that if the big spacing with length s and the ordinary node attached to the left end of the big spacing are removed (the “remove” operation) from the sub-interval i (except for the case when the big spacing is the left most spacing, and then the ordinary node attached to the right of the big spacing is removed), then the m i − 1 remaining ordinary nodes and the two powerful nodes form a connected “subgraph” of length x = w i − s

Full size image

A special case occurs when the big spacing is the leftmost spacing in the sub-interval. If this is the case, then we remove the ordinary node attached to the right end of the big spacing instead. The probability that the m i − 1 remaining ordinary nodes and the two powerful nodes, with the sub-interval having the new length of x, form a connected interval is given by \(P(m_{i} + 1,\hat{x},\hat{r})\) where \(\hat{x} = \frac{x} {w_{i}}\). Following the convention in [79] that nodes are treated as distinguishable, the event that a particular node i attached to the left end of the big spacing is removed together with the big spacing and the remaining nodes form a connected interval, and the event that a particular node j attached to the left end of the big spacing is removed together with the big spacing and the remaining nodes form a connected interval are treated as different events. Therefore, any of the m i ordinary nodes can be attached to the left end of the big spacing (or attached to the right end for the special case), and the big spacing can be any of the m i + 1 spacings in sub-interval i. As a result, the probability that events like Fig. 11.3b occur is then \(m_{i}(m_{i} + 1)P(m_{i} + 1,\hat{x},\hat{r})\), for \(\hat{x}\) ranging from zero to \(1 -\hat{ r}\). Therefore, we obtain the second term in (11.2.6).

After applying (11.2.5) into the second term of (11.2.6), we can get rid of the integral in the second term by moving the inner sum outside the integral. With further changes to the range of summation and integral, we obtain

$$\displaystyle\begin{array}{rcl} & & \quad m_{i}(m_{i} + 1)\int _{0}^{1-\hat{r}}P(m_{ i} + 1,\hat{x},\hat{r})\,\mathrm{d}\hat{x} \\ & & = (m_{i} + 1)\sum _{j=0}^{\min (m_{i},\left \lfloor 1/\hat{r}\right \rfloor -1)}\binom{m_{ i}}{j}(-1)^{j}\left (1 - (j + 1)\hat{r}\right )^{m_{i} }{}\end{array}$$
(11.2.7)

Using (11.2.7), and replacing the first term in (11.2.6) by (11.2.5), we can simplify (11.2.6):

$$\displaystyle\begin{array}{rcl} & & \quad \Pr \left (\mathcal{A}_{i}^{eq}(m_{ i},w_{i})\right ) \\ & & =\sum _{ j=0}^{\min (m_{i}+1,\left \lfloor w_{i}/r\right \rfloor )}(1 - j)\binom{m_{ i} + 1}{j}(-1)^{j}(1 - j \frac{r} {w_{i}})^{m_{i} }{}\end{array}$$
(11.2.8)

Since all sub-intervals bounded by powerful nodes are nonoverlapping segments with length w i and ordinary nodes are Poissonly distributed, m i is a Poisson random variable with mean w i λ, and m i and m j are mutually independent for i ≠ j. Let \(\mathcal{A}_{i}^{eq}(w_{i})\) be the event that sub-interval i with length 0 < w i  ≤ 1 is type-II connected. Then,

$$\displaystyle{ \Pr \left (\mathcal{A}_{i}^{eq}(w_{ i})\right ) =\sum _{ m_{i}=0}^{\infty }\Pr \left (\mathcal{A}_{ i}^{eq}(m_{ i},w_{i})\right )\frac{(w_{i}\lambda )^{m_{i}}} {m_{i}!} \exp (-w_{i}\lambda ) }$$
(11.2.9)
$$\displaystyle\begin{array}{rcl} & & =\sum _{ j=0}^{\left \lfloor w_{i}/r\right \rfloor }(1 - j)(-1)^{j} \frac{1} {j!}(j + w_{i}\lambda - jr\lambda ) \\ & & \ \quad \times (w_{i}\lambda - jr\lambda )^{j-1}\exp (-jr\lambda ) {}\end{array}$$
(11.2.10)

where from (11.2.9) we first exchange the order of the inner sum and the outer sum after we substitute \(\Pr \left (\mathcal{A}_{i}^{eq}(m_{i},w_{i})\right )\) by (11.2.8). Then, we substitute \(\binom{m_{i} + 1}{j} = \left. \frac{1} {j!} \frac{\mathrm{d}^{j}} {\mathrm{d}t^{j}}t^{m_{i}+1}\right \vert _{t=1}\), move the derivative outside the inner sum, and with some arithmetic steps we obtain (11.2.10). Note that (11.2.10) is still valid even when w i  ≤ 2r. That is, \(\Pr \left (\mathcal{A}_{i}^{eq}(w_{i})\right ) = 1\) when w i  ≤ 2r as expected.

2.1.2 Asymptotic Probability That a Sub-Interval Is Type-II Connected for r p  = r o

Eq. (11.2.10) is in a very complicated form which may prevent us from obtaining in-depth understanding on the relations among parameters that determine \(\Pr \left (\mathcal{A}_{i}^{eq}(w_{i})\right )\). In the following we derive a simplified asymptotic approximation for \(\Pr \left (\mathcal{A}_{i}^{eq}(w_{i})\right )\).

Let \(\hat{w} = w_{i}/r\) be the normalized length of sub-interval i by r, where r = r p  = r o as usual. Let μ = 2r λ be the average node degree ignoring the boundary effect. Let \(\phi (\hat{w}) =\Pr \left (\mathcal{A}_{i}^{eq}(\hat{w}r)\right )\). Then, (11.2.10) implies the following difference-differential equation.

$$\displaystyle{ \frac{\mathrm{d}^{2}} {\mathrm{d}\hat{w}^{2}}\phi (\hat{w}) + 2\beta \frac{\mathrm{d}} {\mathrm{d}\hat{w}}\phi (\hat{w} - 1) +\beta ^{2}\phi (\hat{w} - 2) = 0 }$$
(11.2.11)

where \(\beta = \frac{\mu } {2}\exp (-\frac{\mu }{2})\). Using (11.2.11), we can obtain the Laplace transform of \(\phi (\hat{w})\) as

$$\displaystyle{ \Phi (s) = \frac{1} {s +\beta \exp (-s)} + \frac{\beta \exp (-s)} {(s +\beta \exp (-s))^{2}} }$$
(11.2.12)

As s → 0, exp(−s) ≈ 1 − s. Substituting this approximation into (11.2.12), the inverse Laplace transform of the approximated equation is then

$$\displaystyle{ \phi (\hat{w}) \approx \left [ \frac{1 - 2\beta } {(1-\beta )^{2}} + \frac{\beta \hat{w}} {(1-\beta )^{3}}\right ]\exp (\frac{-\beta \hat{w}} {1-\beta } ) }$$
(11.2.13)

Figure 11.5 shows that (11.2.13) serves as a good approximation for the exact result in (11.2.10) provided μ ≥ 6, and virtually all values of \(\hat{w} \geq 2\), not just large values of \(\hat{w}\).

Fig. 11.5
figure 5

The type-II connectivity probability of a sub-interval given different normalized length of the sub-interval w i ∕r and the average node degree. The solid lines are the exact results, and the dashed lines are the asymptotic results

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Solving (11.2.13) for \(\hat{w}\) leads to

$$\displaystyle{ \hat{w} = \left \{-\mathrm{W}_{-1}[-(1-\beta )^{2}\exp (-(1 - 2\beta ))\phi (\hat{w})] - (1 - 2\beta )\right \}\frac{1-\beta } {\beta } }$$
(11.2.14)

where W−1[⋅ ] is the real-valued, non-principal branch of the LambertW function [47]. Given the required connectivity probability \(\phi (\hat{w})\), and the value of β, which is related to the ordinary node density λ and the transmission range r, we can use (11.2.14) to obtain \(\hat{w}\), the maximum distance between two adjacent powerful nodes so that the designated connectivity probability requirement is fulfilled.

2.2 Average Number of Clusters

Besides connectivity probability, another measure of interest is the number of clusters in the network. It is an indicator of how fragmented a network is. In our network model, all ordinary nodes that are connected to at least one of the powerful nodes belong to the same cluster. Denote by “main cluster” the cluster formed by the ordinary nodes which are connected to at least one powerful node. Other ordinary nodes which are not connected to any powerful node, if exist, form one or more “secondary clusters.” Therefore, there is always one main cluster and zero or more secondary clusters in a network. Note that a network is type-II connected if and only if there is no secondary cluster in the network. In this subsection we investigate the average number of clusters in a network G(λ, n p ; 1; r o , r p ) where the powerful nodes are placed in an equidistant fashion. The reason for focusing on equidistant powerful node distribution is that, as will be shown later, it gives the best performance in terms of minimizing the average number of clusters and maximizing the asymptotic type-II connectivity probability. Nevertheless, the analysis in the previous subsection has provided the conceptual basis of how the results can be generalized to having powerful nodes arbitrarily distributed.

Let C i (w) be the number of secondary clusters in sub-interval i with length w under the assumption that r p  ≥ r o . Note that \(w = \frac{1} {n_{p}-1}\) for equidistant powerful node distribution. We only consider w > 2r p to avoid triviality. Also note that ordinary nodes in sub-interval i, which are at most r p − r o Euclidean distance away from the powerful nodes, belong to the main cluster with probability 1 and any other ordinary nodes which are directly connected to these ordinary nodes are also directly connected to the powerful nodes. Consequently, the number of secondary clusters remains the same after we remove an interval of length r p − r o (and the ordinary nodes within that interval) from the left end and right end of sub-interval i, respectively, and then assume that r p  = r o . So, we have C i (w) = C i eq(w − 2(r p − r o )) where C i eq(x) is the number of secondary clusters in sub-interval i with length x under the special assumption that r p  = r o .

To obtain C i eq(x), let t i (x) be the number of spacings with length greater than r in sub-interval i of length x where r = r p  = r o as usual. Then, C i eq(x) and t i (x) have the following relationship:

$$\displaystyle{ C_{i}^{eq}(x) = \left \{\begin{array}{@{}l@{\quad }l@{}} t_{i}(x) - 1\quad &\mbox{ for }t_{i}(x) \geq 1 \\ 0 \quad &\mbox{ for }t_{i}(x) = 0 \end{array} \right. }$$
(11.2.15)

Assume that there are m i ordinary nodes in sub-interval i of length x and let 1 i, j (x) be an indicator function such that

$$\displaystyle{ 1_{i,j}(x) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\mbox{ if the }j\mbox{ -th spacing in sub-interval }i\mbox{ of length }x\mbox{ has length greater than }r\\ 0\quad &\mbox{ otherwise} \end{array}\right. }$$

where 1 ≤ j ≤ m i + 1. Then, the expected value of t i (x) given m i ordinary nodes in sub-interval i with length x is

$$\displaystyle\begin{array}{rcl} E[\left.t_{i}(x)\right \vert m_{i}]& =& E\left [\sum _{1\leq j\leq m_{i}+1}1_{i,j}(x)\vert m_{i}\right ] \\ & =& (m_{i} + 1)E\left [1_{i,j}(x)\vert m_{i}\right ]\;\;\mbox{ for any }j{}\end{array}$$
(11.2.16)
$$\displaystyle{ = (m_{i} + 1)(1 - \frac{r} {x})^{m_{i} } }$$
(11.2.17)

where \(E\left [1_{i,j}(x)\right \vert m_{i}]\) is equal to the probability that the j-th spacing in sub-interval i of length x has length greater than r. Since this probability is equal to the probability that the m i ordinary nodes fall into a smaller interval of length \(1 -\frac{r} {x}\) in sub-interval i, we obtain (11.2.17).

Since m i is a Poissonly distributed random variable with mean x λ, it follows immediately that

$$\displaystyle\begin{array}{rcl} E\left [t_{i}(x)\right ]& =& \ \sum _{m_{i}=0}^{\infty }E\left [\left.t_{ i}(x)\right \vert m_{i}\right ]\frac{(x\lambda )^{m_{i}}} {m_{i}!} \exp (-x\lambda ) {}\\ & =& \ (x\lambda - r\lambda + 1)\exp (-r\lambda ). {}\\ \end{array}$$

From (11.2.15) we have

$$\displaystyle{ E\left [C_{i}^{eq}(x)\right ] = E[t_{ i}(x)] - 1 +\Pr \left (t_{i}(x) = 0\right ) }$$
(11.2.18)
$$\displaystyle{ =\sum _{ j=2}^{\left \lfloor x/r\right \rfloor }(-1)^{j} \frac{1} {j!}(j + x\lambda - jr\lambda )(x\lambda - jr\lambda )^{j-1}\exp (-jr\lambda ) }$$
(11.2.19)

where from (11.2.18) to (11.2.19) we apply

$$\displaystyle{ \Pr \left (t_{i}(x) = 0\right ) =\sum _{ j=0}^{\left \lfloor x/r\right \rfloor }(-1)^{j} \frac{1} {j!}(j + x\lambda - jr\lambda )(x\lambda - jr\lambda )^{j-1}\exp (-jr\lambda ) }$$
(11.2.20)

which is obtained from the first term in (11.2.6) and simplified using the same procedure as that resulting in (11.2.10). Finally, let D(n p ) be the number of clusters in a network with n p powerful nodes equally spaced and assume r p  ≥ r o . Then,

$$\displaystyle{ D(n_{p}) =\sum _{ i=1}^{n_{p}-1}C_{ i}(w) + 1 }$$
(11.2.21)
$$\displaystyle{ =\sum _{ i=1}^{n_{p}-1}C_{ i}^{eq}(w - 2(r_{ p} - r_{o})) + 1 }$$
(11.2.22)

where \(w = \frac{1} {n_{p}-1}\). That is, we add up the number of secondary clusters in each sub-interval and one (and the only) main cluster in the whole network. Based on (11.2.22), we can obtain the following theorem.

Theorem 174

For G(λ,n p ;1;r o ,r p ) with powerful nodes placed in an equidistant fashion, the expected number of clusters in the network is then

$$\displaystyle{ E\left [D(n_{p})\right ] = (n_{p} - 1)E\left [C_{i}^{eq}(w - 2(r_{ p} - r_{o}))\right ] + 1 }$$
(11.2.23)

where \(E\left [C_{i}^{eq}(x)\right ]\) is given in ( 11.2.19 ) and \(w = \frac{1} {n_{p}-1}\).

2.3 The Optimal Distribution of Powerful Nodes

In this subsection, we shall show that the equidistant placement of powerful nodes will minimize the average number of clusters in a network and maximize the asymptotic type-II connectivity probability.

2.3.1 Minimizing the Average Number of Clusters

From (11.2.21), we have the average number of clusters in a network G(λ, n p ; 1; r o , r p ) given each sub-interval i has length w i , is

$$\displaystyle{ E\left [D(w_{1},w_{2},\cdots \,,w_{n_{p}-1})\right ] =\sum _{ i=1}^{n_{p}-1}E\left [C_{ i}(w_{i})\right ] + 1 }$$
(11.2.24)

where E[C i (w i )] is the average number of secondary clusters in sub-interval i. Finding the optimal powerful node placement to minimize the average number of clusters can be treated as a constrained optimization problem:

$$\displaystyle{ \begin{array}{ll} \mbox{ minimize}\;\;\;\; &E\left [D(w_{1},w_{2},\cdots \,,w_{n_{p}-1})\right ] \\ \mbox{ subject to}\;\;\;\;&\sum _{i=1}^{n_{p}-1}w_{i} = 1\end{array} }$$

In the following we prove that E[C i (w i )] is a convex function of w i .

Recall that for w i  > 2r p , we have E[C i (w i )] = E[C i eq(w i − 2(r p − r o ))]. From (11.2.18) we further have,

$$\displaystyle\begin{array}{rcl} E[C_{i}^{eq}(x)] = (x\lambda - r_{ o}\lambda + 1)e^{-r_{o}\lambda } - 1 +\Pr \left (t_{i}(x) = 0\right )& & {}\\ \end{array}$$

where \(\Pr \left (t_{i}(x) = 0\right )\) is given by (11.2.20). With some arithmetic steps we can derive the second derivative of E[C i eq(x)] and obtain

$$\displaystyle\begin{array}{rcl} \dfrac{\mathrm{d^{2}}} {\mathrm{d}x^{2}}E[C_{i}^{eq}(x)]& =& \frac{\mathrm{d^{2}}} {\mathrm{d}x^{2}}\Pr \left (t_{i}(x) = 0\right ) {}\\ & =& (\lambda e^{-r_{o}\lambda })^{2}\Pr \left (t_{ i}(x - 2r_{o}) = 0\right ) {}\\ & \geq & 0 {}\\ \end{array}$$

Hence, the second derivative of E[C i (w i )] is also greater or equal to zero for w i  > 2r p . It is trivial to show that the second derivative is zero for w i  ≤ 2r p as E[C i (w i )] = 0 in that range.

Since E[C i (w i )] is a convex function, using (11.2.24), \(E[D(w_{1},w_{2},\cdots \,,w_{n_{p}-1})]\) is also a convex function. Hence the optimization problem is a convex optimization problem. It is then straightforward to prove, e.g., using the method of Lagrange multipliers, that the minimum of the average number of clusters is achieved when \(w_{1} = \cdots = w_{n_{p}-1} = \frac{1} {n_{p}-1}\) and by convexity it is a global minimum.

2.3.2 Maximizing the Asymptotic Type-II Connectivity Probability

Using (11.2.13) we approximate the type-II connectivity probability

$$\displaystyle\begin{array}{rcl} \Pr \left (\mathcal{B}(w_{1},\cdots \,,w_{n_{p}-1})\right ) =\prod _{ i=1}^{n_{p}-1}\Pr \left (\mathcal{A}_{ i}(w_{i})\right ) \approx \prod _{i=1}^{n_{p}-1}\phi (x_{ i})& & {}\\ \end{array}$$

where x i  = w i − 2(r p − r o ), and

$$\displaystyle{ \phi (x_{i}) = \left \{\begin{array}{@{}l@{\quad }l@{}} \left [ \frac{1-2\beta } {(1-\beta )^{2}} + \frac{\beta x_{i}/r_{o}} {(1-\beta )^{3}} \right ]\exp (\frac{-\beta x_{i}/r_{o}} {1-\beta } )\quad &\mbox{ if }x_{i}> 2r_{o} \\ 1 \quad &\mbox{ otherwise} \end{array} \right. }$$

Since both expressions \(\frac{1-2\beta } {(1-\beta )^{2}} + \frac{\beta x_{i}/r_{o}} {(1-\beta )^{3}}\) and \(\exp (\frac{-\beta x_{i}/r_{o}} {1-\beta } )\) are log-concave on x i  ≥ 0, and the product of log-concave functions is a log-concave function [30], we have ϕ(x i ) is log-concave on x i and \(\Pr \left (\mathcal{B}(w_{1},\cdots \,,w_{n_{p}-1})\right )\) is also a log-concave function of the lengths of sub-intervals. Using this property, it can be readily shown that the maximum of the probability that the network G(λ, n p ; 1; r o , r p ) is type-II connected is also achieved when powerful nodes are distributed in an equidistant fashion.

2.4 Impact of Different Parameters on Connectivity

In this subsection, we investigate the impact of different parameters on the performance of a network G(λ, n p ; 1; r o , r p ). Note that all figures are plotted under the condition that powerful nodes are placed in an equidistant fashion.

First, Fig. 11.6 shows the probability that a network is connected given different values of λ, n p and r p  = r o  = 0. 05 under the unit disk connection model. The accuracy of the analytical results is verified by simulation results obtained from 40, 000 randomly generated network topologies. As the numbers of instances of random networks used in the simulations are very large, the confidence interval is too small to be distinguishable and hence ignored in this plot and the latter plots. The number of powerful nodes has been varied from 2 to 10. With r p  = 0. 05, the network will be fully covered by the powerful nodes for n p  > 10. It is shown that an increase in n p significantly improves network connectivity probability. The impact of λ on connectivity is rather interesting. When λ is small, the network connectivity probability initially drops as λ increases. That is because when the number of ordinary nodes is small, the probability that an ordinary node is connected to a powerful node via a multi-hop path is small and can be almost neglected. Therefore, an ordinary node has to be close to a powerful node in order to be connected. Thus, when the number of ordinary nodes is small, an increase in the number of ordinary nodes causes a drop in the probability that all ordinary nodes are connected to at least one nearby powerful node. As the number of ordinary nodes further increases, the probability that an ordinary node far away from a powerful node can establish a multi-hop path to the powerful node increases, which consequently causes an increase in the probability of having a type-II connected network. Note that this phenomenon, i.e., the increase of node density will first reduce the connectivity probability and then improve it, can also be verified by examining the first derivative of \(\Pr \left (\mathcal{B}\right )\) from (11.2.4) with regard to λ. In addition, the value of node density which minimizes the connectivity probability can be obtained numerically using the classical Newton’s method.

Fig. 11.6
figure 6

The type-II connectivity probability given different values of λ, n p , and r p  = r o  = 0. 05 under the unit disk connection model. The solid lines are plotted using (11.2.4), verified by simulation results obtained from 40, 000 randomly generated network topologies

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Note that the properties observed in Fig. 11.6, i.e., under the unit disk connection model, are also observed when the log-normal connection model is considered. In the log-normal connection model, two nodes separated by a Euclidean distance x are directly connected with probability

$$\displaystyle{ g(x) = Q\left (\frac{10\alpha } {\sigma } \log _{10}\frac{x} {r}\right ) }$$
(11.2.25)

where \(Q(y) = \frac{1} {\sqrt{2\pi }}\int _{y}^{\infty }\exp (-\frac{z^{2}} {2} )dz\) is the tail probability of the standard normal distribution, α is the path loss exponent, σ 2 is the shadowing variance, r is the transmission range ignoring shadowing effect. In order to further accurately model the direct connection between nodes, we consider channel correlation in our simulation. That is, we follow the approach in [78, 89, 194] and use an exponential model to model the fading correlations between wireless links. In the model, the received signals at two nearby nodes from the same transmitting node are correlated with correlation coefficient

$$\displaystyle{ \rho (x) =\exp \left (- \frac{x} {d_{corr}}\log _{e}2\right ) }$$
(11.2.26)

where x is the Euclidean distance between two receiving nodes, d corr is the de-correlation distance whose typical value is 20 m for the urban environment and 5 m for the indoor environment [194]. This is the well-known Gudmundson model. It works well for two-dimensional networks but it may not be able to accurately model some situations in one-dimensional networks. Particularly, consider a big obstacle located between a powerful node and two ordinary nodes in proximity where the wireless signals between the powerful node and the ordinary nodes cannot propagate through. One ordinary node cannot receive the signals from the powerful node implies that the other ordinary node also cannot receive the signals either. That is, two nodes “hiding” behind a big obstacle in a one-dimensional network are highly correlated compared to the two-dimensional case. Gudmundson model is less suitable in modeling such situation. Nevertheless, Gudmundson model is still used here due to its popularity in the literature.

Figure 11.7 is plotted using simulation results obtained from 40, 000 randomly generated network topologies, and following the correlated log-normal model. To have a fairer comparison between different shadowing effect assumptions, we adjust the density λ of ordinary nodes in each simulation so that the average node degree μ of an arbitrary ordinary node is preserved under different path loss exponent α and shadowing variance σ 2 settings. Ignoring the boundary effect, we have \(\mu = 2\lambda r_{o}\exp \left (\frac{1} {2}\left ( \frac{\sigma } {10\alpha }\log _{e}10\right )^{2}\right )\). The steps to derive the equation are omitted here. When σ = 0, the log-normal connection model reduces to the unit disk connection model and we have μ = 2λ r o . As a result, Fig. 11.6 can be directly compared with Fig. 11.7 as the former is also plotted with the average node degree ranging from 0 to 8. Figure 11.6 and 11.7 together show that the impact of the powerful nodes on the type-II connectivity probability under the correlated log-normal model has quantitatively little difference compared with the impact of the powerful nodes under the unit disk connection model. Furthermore, an increase in shadowing variance σ 2 will improve the connectivity probability even if the node density has been reduced to preserve the same average node degree. The better type-II connectivity probability observed under the log-normal model is consistent with the results in ad hoc networks without infrastructure support.

Fig. 11.7
figure 7

The type-II connectivity probability given different values of the average node degree μ, n p , r p  = r o  = 0. 05, and considering the correlated log-normal model. The dash lines are plotted using simulation results obtained from 40000 randomly generated samples

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Next we investigate the impact of λ and n p on the average number of clusters under the unit disk connection model. The analytical formula in (11.2.23) is verified by simulations obtained from 40, 000 randomly generated network topologies.

Figure 11.8 shows virtually an exact match of (11.2.23) with the simulation results. In addition, the curves in the figure also agree with the curves in Fig. 11.6 and show that the connectivity probability reaches its minimum when the average number of clusters is maximized; conversely, the connectivity probability approaches one when the average number of clusters approaches one.

Fig. 11.8
figure 8

The average number of clusters given different values of λ, n p , and r p  = r o  = 0. 05 under the unit disk model. The solid lines are plotted using (11.2.23), verified by simulation results obtained from 40, 000 randomly generated network topologies

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3 Notes and Further Readings

In this chapter, we studied connectivity of one-dimensional small to medium sized networks with and without infrastructure support respectively. The results will be useful for many real world applications modelable by one-dimensional networks, e.g., a vehicular network built along a highway or a sensor network deployed along the border of a defined region for intrusion detection.

Connectivity of one-dimensional wireless ad hoc networks has been extensively studied [55, 67, 77, 147]. Among the studies, Miorandi and Altman [147] assumed that there is a pre-determined node located at the origin. They investigated the probability of other nodes, which are either arbitrarily or uniformly distributed along a semi-infinite line, being connected (either directly or via multi-hop paths) to the node at the origin. Both the unit disk connection model and a Boolean model with random transmission range were considered. This scenario can be considered as a special case of type-II connectivity with only one powerful node placed at the origin.

Dousse et al. have conducted a study closely related to type-II connectivity in [61], considering one-dimensional networks under the unit disk connection model. The nodes are assumed to be Poissonly distributed on a line segment of length L with a known density. Two base stations are placed at both ends of the line segment. Based on the above model, they obtained analytically p(x), the probability that a node at distance x from the left base station is connected to at least one base station. Based on p(x), the authors concluded that the existence of base stations improves the probability that two arbitrary nodes are connected. The authors considered this line segment as a “reduced” version of a more general network with an infinite number of base stations placed every L units distance on an infinite line. Note that Dousse et al. analyzed the probability that a node at location x is connected to at least one base station, denoted by p(x), whereas in this chapter we analyzed the probability that all nodes are connected to at least one powerful node (or base station). It is not trivial to derive the probability that a network is type-II connected using p(x). The difficulty lies in the fact that the event that one node located at x is connected to a base station and the event that another node at y is connected to a base station are not independent, but correlated in a complicated way.

It is worth noting that despite numerous studies on network connectivity and advances in the field, analytical characterization of connectivity of one-dimensional networks assuming a more general connection model than the unit disk connection model (including the unit disk model where each node has a variable transmission range) remains an open problem. Figure 11.9 gives an illustration of the difficulty. Under a more general model, e.g., the log-normal connection model or the random connection model, a node may be connected to another node (via a multi-hop path) through a third node in the opposite direction of the other node. It becomes very difficult to enumerate and incorporate all such possibilities in the analysis of network connectivity.

Fig. 11.9
figure 9

An illustration of the difficulty in analyzing connectivity of one-dimensional networks assuming a more general connection model than the unit disk connection model. Solid lines represent direct connections between nodes. A node may be connected to another node via a multi-hop path through a third node in the opposite direction of the other node

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