Research Article | Open Access
Asymptotic Distribution of Isolated Nodes in Secure Wireless Sensor Networks under Transmission Constraints
The Eschenauer-Gligor (EG) key predistribution is regarded as a typical approach to secure communication in wireless sensor networks (WSNs). In this paper, we establish asymptotic results about the distribution of isolated nodes and the vanishing small impact of the boundary effect on the number of isolated nodes in WSNs with the EG scheme under transmission constraints. In such networks, nodes are distributed either Poissonly or uniformly over a unit square. The results reported here strengthen recent work by Yi et al.
A wireless sensor network is composed of a collection of wireless sensors distributed over a geographic region. A wireless sensor network can be an integral part of military command, control, communication, computing, intelligence, surveillance, reconnaissance, and target system. It has been the subject of intense research in recent decades. Asymptotic analysis, valid when the number of nodes in the network is large enough, has been useful for understanding the characteristic of the network.
In many applications, a large number of wireless sensors are independently and uniformly deployed in the sensor field. They can be deployed by dropping from a plane or delivered in a missile. To model such a randomly deployed wireless sensor network, it is natural to represent the sensor nodes by a finite random point process over a network square area . In addition, due to the short transmission range of communication links, two wireless sensors can build a link if and only if they are within each other’s transmission range. Assume all sensors have the same transmission range of radius , then the induced network topology is a -disk graph in which two nodes are joined by an edge if and only if their distance is at most . This model is proposed by Gilbert  and referred to as a random geometric graph, denoted by .
However, in many applications, a wireless sensor network is composed of low cost sensors. Due to the limited capacity, traditional security schemes and key management algorithms are too complicated and not feasible for such a system. The Eschenauer-Gligor (EG)  key predistribution scheme is a widely recognized way to secure communication. In this scheme, in a WSN with sensors and sensor set , the EG scheme independently assigns a set of distinct cryptographic keys, which are selected uniformly at random from a pool of keys, to each sensor node; the set of keys of each sensor is called the key ring and is denoted by for sensor . The EG scheme is denoted by a random key graph in [3–6], in which an edge exists between two nodes and if and only if they possess at least one common key; that is, .
In a WSN using the EG scheme under transmission constraint, two sensors and establish a direct link between them if and only if they share at least one key and their distance is no greater than . We denote the event establishing this direct link by . If we let the graph model such a WSN, it is obvious to see is the intersection of random key graph and random geometric graph with nodes uniformly distributed over a square ; namely, where represents the parameters , and together.
Let be the location of point . A direct link exists in if both of the following two conditions are satisfied: where represents the Euclidean norm.
We let be the probability of key sharing between two sensors and note that is also the edge probability in random key graph . It holds that
Let be the probability that a secure link exists between two sensors in the WSN with EG scheme under practical constraint; obviously, is the edge probability in . It holds that . It is simple matter to show and , if , . Therefore if and , we get in view of (6).
The secure wireless sensor network with nodes uniformly distributed is modeled by graph , in which two nodes have an edge if their Euclidean distance is at most and have a secure link if their key rings have at least one common key. Compared to the work done by Yi et al. , in this paper, by using a different method, we establish that the number of isolated nodes in the secure wireless sensor network has an asymptotic Poisson distribution whether the nodes are induced by a uniform point process or Poisson point process.
To model transmission constraints, we use the popular disk model , and under the disk model, two nodes are directly connected if and only if their Euclidean distance is smaller or equal to a given threshold , where parameter is termed as the transmission range.
The following notations are used throughout the paper:(i) if ;(ii) if ;(iii) if ;(iv) if there exist sufficiently and such that, for any , ;(v) if there exist sufficiently and such that, for any , ;(vi)the notation “” stands for the natural logarithm function.
The rest of the paper is organized as follows. Section 2 reviews related work. Section 3 comparatively studies the distribution of isolated nodes in WSNs employing the EG scheme with nodes Poissonly or uniformly distributed over a unit square . Through the study, it shows that under certain conditions the impact of boundary effect on the number of isolated nodes is negligible. Finally, Section 4 summarizes conclusions and discusses prospects of establishing the distribution of isolated nodes and tighter connectivity thresholds for secure wireless sensor networks under improved conditions.
2. Related Work
With regard to the sensor distribution, we consider that nodes are independently and uniformly deployed in unit square area . The disk model induces a random geometric graph [4, 12–15] which is denoted by ; an edge exists between two sensors if and only if their Euclidean distance is no more than . Extensive research has been done on random geometric graphs. The connectivity of random geometric graphs has been studied by Dette and Henze , Penrose , and others [12, 18–20]. For a uniform point process over a unit area square , Dette and Henze  showed that, for any constant , if , then graph has no isolated nodes with probability asymptotically. Eight years later, Penrose  established that if a random geometric graph induced by a uniform point process or Poisson point process has no isolated nodes, then it is almost surely connected. Besides the overall connectivity, some applications are concerned with whether there exists a giant connected component. Continuum percolation is a useful theorem in analyzing threshold phenomena. Ammari and Das  focused on percolation in coverage and connectivity in three-dimensional space and found out whether the network provides long distance multihop communication.
For random key graph , Godehardt and Jaworshi  focused on the distribution of the number of isolated vertices in . Some partial results concerning the connectivity of random key graphs were given in [6, 7, 23]. In , Rybarczyk gave asymptotic tight bounds for the thresholds of the connectivity, phase transition, and diameter of the largest connected component in random key graphs for all ranges of . Other related works regarding model have been reported. For example, Bloznelis et al.  treated the evolution of the order of the largest component. Connectivity and communication security aspects of in various important settings are also studied in [7, 25, 26]. Although some properties of secure WSNs with the EG scheme have been extensively studied in [4–7, 13, 27], most research [5–7, 27] unrealistically assumes unconstrained sensor-to-sensor communications; that is to say, any two sensors can communicate regardless of the distance between them.
Recently, there is interest in random graphs in which an edge is determined by more than one random property, that is, intersection of different random graphs. The intersection of Erdős-Rényi random graph  and random geometric graph has been of interest for quite some time now. Recent work on such random graphs is by [20, 29] where connectivity properties and the distribution of isolated nodes are analyzed. And the intersection of random graphs and random key graph is considered in . Such a graph is constructed as follows: a random key graph is first formed based on the key distribution and each edge in this graph is deleted with a specified probability.
The intersection of random key graph and random geometric graph (i.e., ) is first studied in . Di Pietro et al.  have shown that, under the scaling , the one law that this class of random graphs is connected follows if and . Another notable work is due to Krzywdziński and Rybarczyk , where the authors have improved this result and established that, in , if with , , and without any constraint on , then is almost surely connected. And Krishnan et al.  demonstrated that if with and , then is almost surely connected. Recently, Tang and Li  and Zhao et al.  presented the first zero-one laws for connectivity in ; these laws improve the results [4, 13] significantly and help specify the critical transmission ranges for connectivity. Also the distribution of isolated nodes in is considered by Yi et al.  and Pishro-Nik et al. , where the network with nodes distributed uniformly over a unit disk or a unit square .
In addition to random key graphs and random geometric graphs, the Erdös-Rényi graph  has also been extensively studied. An Erdös-Rényi graph is defined on a set of nodes such that any two nodes establish an edge independently with probability . As already shown in the literature [5–7, 27], random key graph and Erdös-Rényi graph have similar connectivity properties when they are matched through edge probability; that is, . Hence, it would be tempting to exploit this analogue and conclude that the distribution of isolated nodes in is similar to that of in , whether the nodes are distributed Poissonly or uniformly on a unit square .
3. Main Result
In this section, we study the expected number of isolated nodes in WSNs with the EG scheme under transmission constraints with nodes either Poissonly or uniformly on a unit square . The number of isolated nodes is a key parameter in the analysis of network connectivity. A necessary condition for a network to be connected is that the network has no isolated nodes, and this may be possibly true for the intersection of random key graph and random geometric graph.
In order to obtain the distribution of isolated nodes in , we prove the same result for its Poissonized version, graph , where the only difference between and is that the node distribution of the former is a homogeneous Poisson point process with intensity on a unit square while that of the latter is a uniform point process.
3.1. Expected Number of Isolated Nodes in
In graph , let denote the event that node is isolated, and let denote the intersection of and the disk centered at position with radius . When node is at position , the number of nodes within area follows a Poisson distribution with mean , and to have an edge with in graph , a node not only has to be within but also has to share at least a key with node . Then the number of nodes neighboring to at follows a Poisson distribution with mean , and the probability that such number is 0 is equal to . Integrating over , then the probability that the node is isolated is given by
Theorem 1. Suppose that , , and nodes are Poissonly distributed on a unit square with the maximum transmission radius for some constant . Then the expected number of isolated nodes in converges asymptotically to as .
Proof. Let denote the number of isolated nodes in graph . By (7), we know holds. To compute based on , we divide in a way similar to that by Li et al.  and Wan and Yi . Specially, is divided into , and , respectively, as illustrated in Figure 1 (note that ). consists of all points each with a distance greater than to its nearest edge to , whereas is a square of size at the four corners of . We further divide into and as follows. In , contains points whose distance to the nearest edge of is no greater than , while the remaining area is . Then the expected number of isolated nodes is given by The four summands in (8) represent, respectively, the expected number of isolated nodes in the central area , in the boundary area along the four sides of , and in the four corners of . In the following analysis, we will show that the first term approaches as , and the remaining terms approach 0 an .
Consider the first summand in (8). It is clear that, for any position , we have , and note that , to getFor the second term in (8), we introduce some notation as follows. For any position , we let the distance from to the nearest edge of square be , where . For , clearly is determined by , and we denote it by . It is easy to obtainSince consists of four retangles, each of which has length and width , it follows that For simplicity, we write as . ThenIn view of (14), we further haveFor , it holds from (11) and (12) thatBy (15) and (16), it follows thatApplying (17) into (14),From (10) and (11), we obtain and . ConsiderUsing and (6) in (19), we deriveUsing (20) in (13), we getFrom , and with denoting (note that ),We obtainNow we consider . For , when the distance from to the nearest edge of square is , where , the function expression of still is . is increasing with for by . Thus, when the distance from to the nearest edge of equals , the area reaches its minimum value: With denoting , then . And with , we derive From , , and (22), it followsFor the last term in (8), we have for any and to getwhere (22), , and are used in reaching (27).
As a result of (8),(9), (23), (26), and (27), we prove The parameter in Theorem 1 is a constant, or it can depend on , in which case . The following corollary can be established.
Corollary 2. In graph under conditions , , and with or a constant , then for any constant such that holds.
Now we examine the impact of boundary effect on the number of isolated nodes in since the network area in our analysis is a square. The square accounts for the real-world boundary effect whereby some transmission region of a sensor close to the network boundary may fall outside the network field. In contrast, the torus eliminates the boundary effect. The analysis of impact of the boundary effect is done by comparing the number of isolated nodes in and the number in a network with nodes Poissonly distributed on a unit torus with a pair of nodes separated by a toroidal distance . Denote such network on a unit torus by . The following Theorem can be established.
Theorem 3. Suppose that , , and nodes are Poissonly distributed on a unit torus with the maximum transmission radius for some constant . Then the expected number of isolated nodes in converges to as .
Proof. Let denote the number of isolated nodes in graph , and let denote the intersection of and the disk centered at position with radius . The probability that node is isolated is . Since is a unit torus, it holds that for any . ThenUsing the condition , we obtain
Theorem 4. Suppose that , , and nodes are Poissonly distributed on a unit square with the maximum transmission radius for some constant . Then the number of isolated nodes in due to the boundary effect converges asymptotically to 0 as .
Proof. Comparing Theorems 1 and 3, it is noted that the expected numbers of isolated nodes on a unit torus and on a unit square , respectively, asymptotically converge to the same constant as . Now we use the coupling technique  to construct the connection between and . Consider a graph , and the number of isolated nodes in that graph is . Remove each connection of the above graph with probability , independently of the event that another connection is removed. Due to , then . We further note that only connections between nodes near the boundary will be affected. Denote the number of newly appearing isolated nodes by ; namely, is the number of isolated nodes due to the boundary effect; it is straightforward to show that is a nonnegative random integer. Further, such a connection removal process results in a random network with nodes Poissonly distributed with density . That is, a random network on a unit square with boundary effect is included. The following equation result holds: By Theorems 1 and 3 and the above equation, it can be shown that Due to the nonnegativity of ,
3.2. Distribution of the Number of Isolated Nodes in
In this subsection, we analyze the distribution of the number of isolated nodes in . For this purpose, we give some definitions. Let Poi() be a Poisson random variable with parameter .
Let be a finite set of indices and let be a family of random indicator variables. We say are positively related (see ), if, for each , there exist random indicator variables with the distributions such that for every .
A useful result obtained by Stein-Chen method is the following.
For , let if node is isolated in and . Therefore, is the number of isolated nodes in as defined in Theorem 3. We will demonstrate the asymptotic Poisson distribution of by employing the Stein-Chen method . The result is given in Theorem 6.
Theorem 6. Suppose , , , and nodes are Poissonly distributed on a unit torus with the maximum transmission radius for some constant . Then the distribution of the number of isolated nodes in converges to a Poisson distribution with mean as .
Proof. The triangular inequality for the total variation distance implies By a coupling argument  and Theorem 3, we have Combining this with (36), we now only need to proveFirst, we claim that are positively related. To see this, define where is a graph with or compared to , where represents the key ring which is adjacent to in . Conditional on the isolation of node in , any node is not adjacent to or in . Hence, we have For every , if then . Consequently, we get .
By Lemma 5, the binary nature, and exchangeability of the random variables involved, we find thatThe cross term in (41) is given by , and we have With denoting , combining (29), (41), and (42) readily giveswhere , , , and (22) are used in reaching (43), along with (36), (37), and (43), which concludes the proof.
We now consider the asymptotic distribution of the number of isolated nodes in . From Theorem 4, holds, and using Slutsky’s Theorem , the following result on the asymptotic distribution of can be readily obtained.
Theorem 7. Suppose , , , and nodes are Poissonly distributed on a unit square with the maximum transmission radius for some constant . Then the distribution of the number of isolated nodes in converges to a Poisson distribution with mean as .
3.3. Distribution of the Number of Isolated Nodes in
We derive the distribution of isolated nodes in by using standard Poissonization technique [11, 14]. The idea is that the result about the distribution of isolated nodes for follows once we establish the result with Poissonization, that is to say, once we obtain the distribution of isolated nodes for . See the following lemma for rigorous argument.
Lemma 8. Suppose that , , and nodes have the same maximum transmission radius for some constant . Then with denoting , where is an arbitrary constant with , then node is isolated in if and only if is an isolated node in .
Proof. We will use the standard de-Poissonization technique [11, 14, 18] to prove Lemma 8. Let denote the number of nodes in graph ; clearly follows a Poisson distribution with mean , for any positive , and from Chebyshev’s inequality, we haveWithout loss of generality, we take as an integer. With , substituting into (46), we getHence, holds almost surely.
When , we construct a coupling between graph and graph ; let graph be the result of adding nodes uniformly distributed on to graph . Let be the node set of ; clearly, is a subset of , where is the node set of . In addition, it is straightforward to see that the edge set of is also a subset of that of . Then under coupling , graph is a subgraph of .
Let denote the set of isolated nodes in and , respectively. To prove Lemma 8, we show It is straightforward to see By (47), we will prove if we can derive To prove (48) and (49), with , we consider the coupling under which is a subgraph of .
First, we consider (48) with ; event happens if and only if there exists at least one node such that and ; that is to say, is isolated in but is not isolated in . Then there exists at least one node in such that and are neighbors in . Due to , noting that is the edge probability in , then with denoting the number of isolated nodes in , as an easy consequence of the union bound,In order to apply Corollary 2 to , for sufficiently large, the following condition holds:We demonstrate (51) in view of where And sufficiently large are used in the final step. Then with (51), for any constant , we use Corollary 2 to getNote that , and . Then from (50), with satisfying , it follows thatSecond, in order to prove (49) with , we consider event that occurs if and only if there exists at least one node such that and . With , then is isolated in , which along with leads to and . Let denote the probability that a node is isolated in ; it follows via a union bound thatSince , according to the proof of Theorem 1, it is easy to show . With and , thenIn conclusion, since , the above discussions lead to establish Lemma 8.
Theorem 9. Suppose , , , and nodes are uniformly distributed on a unit square with the maximum transmission radius for some constant . Then the distribution of the number of isolated nodes in converges to a Poisson distribution with mean as .
Noting that the number of isolated nodes in a network is a nonnegative integer, the following result can be obtained as a consequence of Theorems 7 and 9. Notice that, in formulating this result, we drop the assumption that originally is a constant and allow it instead to be -dependent.
Corollary 10. Suppose , , and nodes are Poissonly (uniformly) distributed on a unit square with the maximum transmission radius for some . A necessary condition for graph to be asymptotically connected is .
4. Conclusion and Future Work
Yi et al.  considered that a wireless ad hoc network consists of nodes distributed independently and uniformly in a unit disk or a unit square . They used Brun’s sieve to show that, for graph or , if and , the number of isolated nodes asymptotically follows a Poisson distribution with mean . Pishro-Nik et al.  also obtained such result on asymptotic Poisson distribution with condition generalized to .
In this paper, we discuss the distribution of isolated nodes in WSNs employing the widely Eschenauer-Gligor key predistribution scheme under transmission constraint with nodes distributed either Poissonly or uniformly on a unit square or a unit torus . Using the coupling technique, it is shown that the impact of the boundary effect on the number of isolated nodes vanishes small as . We obtain that the numbers of isolated nodes in graph and graph asymptotically follow a Poisson distribution with mean with and .
In practice WSNs, is expected to be several orders of magnitude smaller than , so it often holds that . We believe that the following conjecture is true.
Conjecture 11. Suppose that , , and nodes are uniformly (Poissonly) distributed on a unit square with the maximum transmission radius for some constant . Then the distribution of the number of isolated nodes in graph