Nature inspired node density estimation for molecular nanonetworks

Nano Communication Networks (

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Nano Communication Networks journal homepage: www.elsevier.com/locate/nanocomnet

Nature inspired node density estimation for molecular nanonetworks Taqwa Saeed a,∗ , Marios Lestas a , Andreas Pitsillides b a

Department of Electrical Engineering, Frederick University, Nicosia, Cyprus

b

Department Computer Science, University of Cyprus, Nicosia, Cyprus

article

info

Article history: Received 3 December 2016 Available online xxxx Keywords: Density estimation Molecular communication Nanonetworks

abstract The problem of estimating the node density in ad hoc networks is a significant one for protocol design. In molecular nanonetworks, the node density estimation problem poses additional challenges due to the limited processing and communication capabilities of the network nodes which necessitate the design of simple to implement distributed solutions, and the diffusion based communication channel which is different from traditional electromagnetic networks. In this work, inspired by the quorum sensing process, we propose and analyze a new node density estimation scheme based on synchronous transmission of all network nodes and measurement of the received molecular concentration. We show that when the synchronous transmission is performed in infinite space, a linear parametric model of the node density can be derived which can be used for estimation purposes. When, however, the transmission is performed over a finite space the model becomes time varying. To overcome the difficulties associated with the time varying nature we propose the use of periodic transmission which for large enough values of the period transforms the linear model into a static one. An online parameter identification technique is then introduced to estimate the node density using the derived linear static parametric models. The utilization of the node density estimates to adaptively regulate probabilistic flooding in network structures relevant to nanonetworks is then considered. The random geometric graph model and uniform grid structures are used to demonstrate how the node estimates can be used to dictate the desired rebroadcast probabilities, through analysis and simulations. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The node density is an important network parameter which greatly affects network properties such as congestion [1], network capacity [2], routing efficiency and delay [3] and power consumption [4]. It is thus an important problem to design node density estimation schemes. Density analysis is more challenging in infrastructure-less networks like MANETs, VANETs and molecular Nanonetworks [5] as the nodes are mobile in nature and the node density is thus a time varying parameter. A number of estimation schemes have been proposed in literature for MANETs, based on beacon message exchange, and the Received Signal Strength, and for VANETs based on the flow rate and speed information [6,5]. In [4] for example, the authors discuss a power control method for MANET clustering based on the network density, where nodes exchange packets to estimate the number of neighbors in a cluster and the distance to

∗

Correspondence to: 7, Y. Frederickou Str., Pallouriotissa, 1036 Nicosia, Cyprus. E-mail address: [email protected] (T. Saeed).

http://dx.doi.org/10.1016/j.nancom.2017.02.003 1878-7789/© 2017 Elsevier B.V. All rights reserved.

the farthest neighbor. According to the acquired information each node adjusts its transmission power to the minimum sufficient level. Another technique which was proposed for general wireless ad hoc networks and is based on the received power is [7]. The authors discuss a cooperative technique where each node shares its received power information with its own neighbors and hence a better estimation of the density is achieved. In [8] a density estimation scheme based on neighbor information is proposed for VANETs. The probe vehicle is assumed to find the number of its neighbors via exchanging ‘‘HELLO’’ packets, the distance between vehicles is assumed to have an exponential pattern and based on this assumption the equations to calculate the local density in the cases of One-Hop-Neighbor, Two-Hop-Neighbor and Cluster information are derived. In [6], the authors propose a node density estimation scheme for VANETs, by using distributed methods for system size estimation in P2P networks, a method which does not require high deployment and maintenance costs compared to infrastructure based solutions. This method relies on exchanging small messages between nodes which have position information to estimate the size, a result which is then divided by the area of the road to find the number of nodes per unit area. The work in [5] on

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the other hand proposes a solution to predict node density without the overhead and congestion which could be caused by exchanging beacon messages. The authors run simulations to provide nodes with two equations, one to relate the received signal strength to the number of nodes transmitting simultaneously and the second is to relate node density to the number of nodes transmitting simultaneously. The results of the simulation-based technique suggest that it is robust to different traffic flow conditions. In [9] the speed of the vehicle is used as an indication of the network density in order to develop an adaptive probabilistic broadcasting scheme. Low vehicle density implies high vehicle speed while high density is indicated by low vehicle speed. In the case of broadcasting, high densities cause the broadcast storm problem [10] which can be mitigated by broadcast suppression. There are several ways to do the latter. In [11] they have been categorized into probabilistic methods, area based methods and neighbor knowledge methods. Probabilistic broadcasting has been addressed extensively in the literature [12,9,13–15], in order to find a technique with which the optimum rebroadcast probability can be calculated. It has been pointed out in [9] that the critical probability after which high reachability is achieved is significantly affected by the density of the network. Thus, node density estimation is important when employing probabilistic broadcasting. Molecular nanonetworks is a recent network paradigm which comprises of a large number of highly mobile nodes in the scale of micrometers, known as nanonodes. Molecular Communication (MC) is a means of communication where information is carried in chemical signals using molecules [16]. Nanonetworks employing molecular communication pose a number of design challenges due to their high density and the diffusion based propagation model and many issues in protocol design remain largely unexplored [17]. Since nanonetworks are not expected to have large processing capabilities due to the small size of the network nodes a key requirement is the simplicity of implementation. In our recent work in [18] we have demonstrated that probabilistic flooding is a good candidate solution for information dissemination in nanonetworks and that for optimal performance node density estimations are required. However, MC represents a propagation model which is different than that of electromagnetic waves propagation and does not suffer from multipath fading. Therefore, the above mentioned techniques are not suitable for density estimation when MC is employed. In this work we propose a distributed and simple to implement method for node density estimation. The method is inspired from the quorum sensing process and relies on synchronous transmission of the network nodes. The method is analogous to using the signal strength for density estimation in traditional electromagnetic networks. However, in molecular communications the propagation model is different and the resulting analysis which is presented here is thus different. As mentioned above, the method relies on synchronous transmission of the network nodes which in case of infinite space leads to a linear static parametric model which allows for node estimation [18]. However, in this paper we build on our work in [18] to show that when the space is finite the linear model is time varying and not static which makes the estimation prone to errors since the initial transmission time is difficult to determine. We overcome this problem by employing periodic broadcasting and averaging over the period of rebroadcasting. This allows for the parametric model to be almost static with the constant of proportionality being affected by the frequency of rebroadcasting and the area in which rebroadcasting takes place. We show these relationships in both two dimensional and three dimensional structures. The linear static parametric models are then used as a baseline to design estimates of the node density using online parameter identification techniques. The estimation algorithms are shown analytically to converge to the true parameter value.

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In addition a discussion is offered regarding the implementation feasibility of such an algorithm using latest developments in the field of synthetic biology. Furthermore, we consider probabilistic broadcasting as an application where node density estimation is crucial. Information dissemination in nanonetworks must be performed using simple distributed techniques which suit the low computational and power capabilities of nanonodes [19]. Probabilistic broadcasting is a suitable candidate for high density networks like nanonetworks as it alleviates the broadcast problem [10]. In this paper, we provide design guidelines for probabilistic flooding in network structures which are relevant to the molecular nanonetwork paradigm. We consider the problem of adaptively regulating the rebroadcast probability using the obtained node estimates in random geometric graphs and in uniform grid networks. In random geometric graphs, which are meaningful due to the random movement of the molecular nodes and the MC paradigm, we show how recent analytical findings [20] can be used to determine the desired rebroadcast probability. In addition, in uniform grid networks which are meaningful in metamaterial nanonetworks [21] with molecular communication capability [22], we show using simulations that the high reachability requirement is critically affected by the node density and not the network size and desired rebroadcast probabilities are suggested for different cases in 2 and 3 dimensional structures. The paper is organized as follows, in Section 2 a review of the Quorum sensing process is provided, in Section 3 the linear static parametric models are derived which are used as a baseline to design the online estimation scheme in Section 4. In Section 5 design guidelines are provided as to how the node estimates can be used to adaptively regulate probabilistic flooding in molecular nanonetworks and finally in Section 6 we offer our conclusion and future research directions. 2. Quorum sensing Quorum sensing is a biological means of communication in bacteria. Based on the diffusion and reception of information molecules known as autoinducers bacteria can monitor, control and synchronize their group behavior. Studies have shown that the change in bacterial behavior is closely related to the population density of bacteria in the environment [23,24]. For instance, V. fischeri bacteria, which is responsible for the bioluminescence in squids, only emit light when its population exceeds a certain threshold. Many other physiological processes are controlled using QS, such as motility, antibiotics production and secretion of virulence. Quorum sensing is the system with which bacteria can communicate to synchronize and sense their density population in order to perform cooperative behaviors and hence have a multicellular-organism-like action [25]. When using quorum sensing, bacteria diffuse a type of information molecules, known as autoinducers, into the extracellular environment. The reception of autoinducers by other bacteria stimulates the production of more molecules of the same kind. Thus, high concentration of these molecules indicates high density population of bacteria. Bacteria keep monitoring the level of the signaling molecules concentration. When the concentration of autoinducers exceeds a certain limit, sensed by the community of bacteria within the environment, the group behavior of bacteria changes [24]. Also, the rate of production of autoinducers increases dramatically when the concentration of these particles exceeds a certain limit [26]. In some cases more than one type of autoinducers are used to enable inter species communication as in V. harveyi bacteria [25]. Quorum sensing is crucial when taking a group action that would fail if taken by an individual bacterium. For instance, when attacking

T. Saeed et al. / Nano Communication Networks (

an organism bacteria must attack as a group to be able to defeat the defense mechanism of the host, otherwise the host’s immune system will certainly eliminate individual bacterial activities [25]. Several computing techniques have been inspired by Quorum sensing [25,27]. In [26] for example Quorum sensing is represented as a synchronization mechanism among nanomachines. The authors define QS as a ‘‘collective communication’’ method where a broadcast message is encoded using the concentration of molecules and decoded by calculating the node density. In [28] QS is used to amplify the broadcast signal. The amplification phase starts after the synchronization between nodes is obtained using autoinducers, so that source node and its cluster will start broadcasting the same information signal. This group broadcasting is considered to amplify the information signal. A quorum sensing inspired method has been proposed in [29] to design a self-stopping computer worm by exchanging small messages between infected hosts which triggers the transmission of more similar messages when received, these messages are equivalent to autoinducers in biological systems. When the count of the received autoinducers exceeds a predefined limit all infected nodes halt their activities. In this work we aim to utilize a quorum sensing based scheme to estimate the density of nodes in nanonetworks. 3. Density estimation Inspired by the quorum sensing process, which nature utilizes to synchronously initiate density dependent procedures, we propose a technique for node density estimation which is based on measurements of the received number of molecules when transmitters simultaneously transmit molecules of the same species. Just like in Bacteria, nanonodes will start releasing information molecules (autoinducers) into the environment and monitor the concentration levels of these molecules in order to estimate the node density. This process may be done at the beginning of a communication session to calculate the critical probability when probabilistic broadcasting is used, or to calculate the optimum number of molecules to be transmitted. Also, it could be repeated periodically to update density information of the network. In the following, we present the mathematical analysis which allows one to derive the relationship between the concentration of the received molecules with the node density. The analysis considers uniformly deployed nodes in 2 and 3 dimensions. 3.1. Density estimation in infinite space (a linear parametric model) We consider a reference molecular nanonetwork model in the fashion of the one considered in [30]. In this model we assume molecular transmitters which are uniformly distributed in a specific area (2 dimensional case) or volume (3 dimensional case) such that their number per unit area or volume, denoted by n, is fixed. For illustration purposes we start with the 2-dimensional case and we then extend the analysis to the 3-dimensional case which is more common in realistic molecular networks. The molecular transmitters indexed by k and located at xk are assumed to have negligible size and emit molecules of identical types according to a signal sk (t ). As pointed out in [30], as soon as molecules are transmitted at transmitter k the concentration c (x, t ) of molecules at location x at time t is governed by Eq. (1):

∂ c (x, t ) = sk (t )δ(x − xk ). (1) ∂t The emitted signal sk (t ) of the transmitter k, propagates to any point x in space according to the diffusion equation (2) where ∇ 2 is the Laplacian operator and D is the diffusion coefficient.

∂ c (x, t ) = D∇ 2 c (x, t ). ∂t

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It must be noted that the diffusion equation relies on the assumption of independent Brownian motion of the sent molecules which are identical and indistinguishable. The concentration of molecules at location x at time t due to the emission of molecules by transmitter k according to the signal sk (t ) is governed by Eq. (3).

∂ c ( x, t ) = D∇ 2 c (x, t ) + sk (t )δ(x − xk ). ∂t

(3)

The solution of the above equation is given by: c (x, t ) = gd (rk , t ) ∗ sk (t ) ∞



gd (rk , τ )sk (τ − t )dτ

=

(4)

0

where rk is the Euclidean distance between the location of transmitter xk and the point x and gd (rk , t ) is given by: r2 k

gd (rk , t ) =

e− 4Dt

(4π Dt )

.

(5)

Our model also includes molecular receivers indexed by

v and located at xv . The receivers are assumed to be ideal

molecular concentration detectors. As pointed out in [30] this is an approximation of a real molecular receiver whose size is negligible compared to the propagation distances. The proposed estimation scheme utilizes the signal measured by a receiver v when all the transmitters transmit molecules according to signals sk (t ). In this case, the concentration c (xv , t ) measured by receiver v at time t is given by: c ( xv , t ) =

∞ 

gd (rk , t ) ∗ sk (t ).

(6)

k=0

If we assume identical sk (t ) signals which are Dirac-delta functions of intensity Q such that:



∞

sk (t )dt = Q

(7)

−∞

then the equation becomes:

 c ( xv , t ) = Q

r2



k ∞  Qe− 4Dt gd (rk , t ) = . (4π Dt ) k=0 k=0

∞ 

(8)

Let:

λ=

Q

(4π Dt )

,

l=

1 4Dt

.

(9)

In order to calculate the summation in Eq. (8) we consider a circular ring of radius r and thickness δ r as shown in Fig. 1. The receiver is assumed to be placed in the center of the ring. As the thickness δ r is infinitesimally small, and assuming that the number of transmitters residing in the ring is equal to the area times n, the concentration of molecules reaching the receiver as a result of 2

the transmissions within the ring is given by 2π nr λe−r l . To find the total concentration of molecules at the receiver, the molecule concentrations at the receiver due to all rings must be added as the thickness δ r tends to 0 resulting in the following integral: c (r , t ) =

∞ 

2π rnλe

−r 2 l

∞



2

2π rnλe−r l dr

=

(10)

0

k=0

which can be evaluated using integration by parts to yield: c (r , t ) = 2π nλ

∞



2

re−r l dr = π nλ 0

(2)

3

1 l

= Qn.

(11)

Since c (r , t ) can be measured at the receiver and Q is known, Eq. (11) constitutes a linear static parametric model of the

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molecules at time t at a distance r from the transmitting node is given by: c (r , t ) =

Q

−r 2

(4π Dt )

e 4Dt .

3 2

(15)

Let:

λ=

parameter n which is unknown and time varying. This model can c (r ,t ) be used to estimate n by the simple division of: Q . However, division is prone to errors and presents high computational complexity for nanodevices. These difficulties could be overcome by employing online parameter identification techniques from formal control theory as described later in this paper. 3.2. Density estimation in finite space In the above analysis all nodes are assumed to broadcast simultaneously. Here, we explore the possibility of only part of the nodes broadcast or that the nodes are deployed within a finite space as depicted in Fig. 1. This case is modeled by replacing the limits of the integrals in Eq. (11) by r1 to r2 instead of 0 to ∞. Thus, the integral becomes: c ( r , t ) = 2 π nλ

r2

−r 2 l

re

= nQ [e

−e 2

] = nQ w(t )

∞ 

(12)

2

nQ w(t − Tk).

(13)

k=0

And the result of the integration over one period becomes: c (r , t ) = nQ



t1 + T t1

∞ 

w(t − Tk).

l=

∞



1 4Dt

.

(16)

2

2π r 2 nλe−r l dr 0

= 2nπ λ

√ π

(17)

= Qn.

3

4l 2

(18)

Just like in the 2D case we consider integration with finite space limits r1 and r2 and the concentration at the receiver is then given by: c (r , t ) = 4nπ λ



r2

2

r 2 e−r l dr

r1



nQ

2

2

[r1 e−lr1 − r2 e−lr2 ] π Dt    √ √ π π + erf ( lr2 ) − erf ( lr1 )

= √

4l

where z (t )

where w(t ) = [e−r1 l − e−r2 l ]. We observe that the static parametric model is no longer static but is rather time varying due to the term w(t ). The time t is the time which has elapsed from the common time of broadcast of all the nodes. This is prone to errors in determining both the time of broadcast and the time which has elapsed (if at all possible to determine) due to difficulties in implementing timers in the molecular level. In order to avoid the inaccuracies and the implementation complexity associated with timers we pose the objective of developing a method with which the signal w(t ) becomes a constant. In this work we propose periodic transmission and integration over the period of transmission. We show that this approach renders the signal w(t ) slow varying and by suitable choice of the period constant. The approach does not depend on the initial time the integration is performed and this removes the necessity of timers. If periodic broadcast is performed with period T then Eq. (12) becomes: c (r , t ) =

c (r , t ) =

dr

−r22 l

,

4l

= nQz (t )

r1

−r12 l

(4π Dt )

3 2

Cylindrical rings of area 2π r 2 δ r are considered to determine the concentration of molecules at the receiver due to the simultaneous transmission of all nodes in the network. This induces the integral:

Fig. 1. Finite space model.



Q

(14)

k=0

3.2.1. 3-dimensional case We extend similar analysis for 3-dimensional nodes deployment, which yield similar results. Hence, the concentration of

= √ erf ( lr1 )]]. 4l

π

(19) √ 1 [[r1 e π Dt

−lr12

− r2 e

−lr22

] + [

π 4l

√

erf ( lr2 ) −

Considering periodic transmission with period T and integration over this period we get: c (r , t ) = nQ



t1 + T t1

∞ 

z (t − Tk).

(20)

k =0

3.3. Simulation results We assume node density of n = 15 625 e7 /m2 , intensity of Q = 100, and diffusion coefficient of D = 105 . First, we consider periodic transmission with 2 s between transmissions. However, a single transmission results in a slow decaying curve of the concentration of molecules. Therefore, short inter-transmission periods represent a logarithmic profile of the signal oscillations due to the build up residuals produced by the slow decaying signals as shown in Fig. 2 over 300 s of transmission. Hence, we then consider longer inter transmission period to overcome the logarithmic profile as more time is given to the signal to decay. We consider a period of 80 s between transmissions. In Fig. 3 which shows 4500 s of periodic transmission we observe that larger period implies a constant profile of the signal. We integrate taking into account only the last 800 s (10 transmissions) which results into a uniform change in the concentration of molecules as shown in Fig. 4. This constant value could then be used in Eqs. (14) and (20) to estimate the density using online parameter estimation as discussed in [18]. For further investigation, we plot the average concentration when values of the inter transmission period ranging from 1 to 80 s are used. The different average values are referred to as µ1 . As shown in Fig. 5, the period affects the concentration only when

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Fig. 4. Average concentration vs. time in 2-dimensional case. Fig. 2. Measured concentration vs. time over 300 s with 2 s inter transmission time.

Fig. 3. Measured concentration vs. time over 4500 s with 80 s inter transmission time.

its value is less than 10 s, after that the change in concentration is fairly constant. This shows that the concentration and in turn the density estimate is independent of the length of the time between transmissions. The period beyond which the signal is constant is approximately 10 s. This demonstrates that relatively high periods can be chosen which suggests that the communication overhead can be small. Fig. 6 demonstrates the effect of changing the considered finite space on the concentration. We change the value of r2 by increasing it up to ten folds and measure the corresponding concentration average value (referred to as µ2 ). The figure shows that the concentration increases as the considered area increases which is logical since a larger area would contain more nodes and thus more molecules. Periodic Transmission has been applied also in the 3-dimensional case. As shown in Fig. 7 the change in the concentration over time is nearly constant. In addition, the scheme exhibits the same behavior as in 2D case when the values of the frequency

Fig. 5. µ1 versus different values of inter transmission time in 2-dimensional case.

of transmission and r2 are changed as shown in Fig. 8 and Fig. 9 respectively. 4. Online parameter identification As pointed out in the previous section, Eqs. (11) and (17) comprise linear static parametric models of the unknown parameter n. This model can be used to estimate n by employing division so that c (r ,t ) n = Q . Nonetheless, simple division is prone to errors, especially when Q attains small values [31]. Another drawback of division is that it is associated with a relatively high degree of implementation complexity which might not be suitable for the limited computational capabilities of nanodevices. Therefore, we propose utilizing online parameter identification techniques from formal control theory which result in iterative estimation algorithms for the unknown parameter. In this section we demonstrate the derivation process. According to the aforementioned analysis the

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Fig. 8. µ1 versus different values of inter transmission time in 3-dimensional case.

Fig. 6. µ2 versus different values of r2 in 2-dimensional case.

Fig. 7. Average concentration vs. time in 3-dimensional case. Fig. 9. µ2 versus different values of r2 in 2-dimensional case.

linear parametric model is: y(t ) = Q (t )n(t )

(21)

where y(t ), Q (t ) can be measured and n(t ) is unknown and time varying. Let nˆ denote the estimate of n to be calculated online using the available measurements and define the estimate of the output yˆ as yˆ = Q nˆ .

(22)

Let ϵ denote the prediction error which is defined as

ϵ = y − yˆ = Qn − Q nˆ = Q (n − nˆ ).

(23)

Theorem 1. The adaptive law (25) guarantees that nˆ (t ) converges to n exponentially fast if and only if Q (t ) is persistently exciting i.e.  T 0 +t 2 Q (τ )dτ ≥ a0 T0 . t Proof. n˙˜ = n˙ˆ − n˙ = n˙ˆ and this together with (24) and (25) leads to

⇒ n˙˜ = γ Q ϵ = −γ n˜ Q 2 ,

ϵ = −˜nQ .

(24)

We then consider the cost function J (ϵ) = derive the following gradient algorithm: nˆ (0) = nˆ 0

n˜ (0) = nˆ 0 − n.

(26)

From the theory of linear time varying equations, the above differential equation can be solved to yield:

Defining n˜ = nˆ − n yields

n˙ˆ = γ ϵ Q ,

where γ is the adaptive gain which affects the convergence properties of the algorithm. The properties of the proposed algorithm are summarized in the theorem below.

(y−ˆnQ )2 2

=

ϵ2 2

to

(25)

n˜ (t ) = e−γ

t

2 0 Q (τ )dτ

n˜ (0).

(27)

The solution of the differential equation establishes exponential convergence of n˜ to 0 [31] which in turn establishes exponential convergence of nˆ to n.

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It is important to note that the relationship (21) represents a linear parametric model which can be employed as a baseline to develop several other iterative estimation techniques [31] which have better convergence properties at the expense, however, of increased complexity. Simplicity of implementation is a crucial requirement in nanonetworks which is the reason why the proposed technique has been chosen. In the next section we show that the proposed scheme is amenable for implementation at the biological level as its main components have been shown to be possible to realize using synthetic biology [32]. 4.1. Practical implementation feasibility In this section we discuss the feasibility of the physical implementation of the online estimation model at the molecular level. Fig. 10 [31] is a schematic representation of the estimation algorithm presented in (25) which shows that the required components are an adder, a multiplier and an integrator. This could be implemented either as an analog system or a digital system. So far, there has been considerable work in the area of synthetic analog and digital circuits, which indicates the nearfuture feasibility of the implementation of the system in Fig. 10. In [33,34] it has been pointed out that almost all logic gates have been implemented using synthetic biology by programming cellular behaviors using RNA. In addition, conversion of the analogue signals to digital for processing using the existing digital logic gates has been recently shown to be realizable in [35] where the authors suggest using bi-modal circuits which can utilize both analogue and digital computation, and they propose the design of Analogue to Digital Converters based on molecular concentration. Furthermore, the authors in [36] propose the use of orthogonal genetic logic gates to design programmable integrated circuits. Digital computation is less sensitive to noise [35] compared to analog, as digital computation is based on the comparison between the signal and a certain threshold. If the signal exceeds the threshold then it qualifies as a logical ‘‘1’’ whereas if it is below the threshold then it qualifies as a logical ‘‘0’’ [37]. A single logic gate can perform a simple binary operation, thus, a massive number of gates must be combined in order to perform complex computation. This is feasible in transistor gates where millions of gates can be integrated on a single chip. In biological systems on the other hand, the resources of space and energy are limited which complicates the composition of numerous biological logic gate circuits [34]. Moreover, communication between cells is usually based on graded information and not on binary-like information where only two values are considered. Analogue computation does not require high number of components and allows graded information [34] to be utilized. Significant efforts have been reported to realize analogue implementations of the components included in Fig. 10. The authors in [38] develop a logarithmic adder circuit which could mimic digital multiplication (AND gate). Also, they develop a design for a ratiometer circuit based on logarithmic difference calculations. Then, the authors combine the two molecular-based genetic circuits to perform law power computations. In [39] a multiplier is developed using a protein circuitry. Finally, an integral feedback control system which includes an integrator has been proposed in [40]. 5. Probabilistic flooding in nanonetworks In this section we consider probabilistic flooding as an information dissemination example where node density estimation can be used for effective protocol design. Probabilistic flooding is an attractive solution for information dissemination in nanonetworks due to its simplicity of implementation and effectiveness in alleviating the broadcast storm problem. Node density estimates can

Fig. 10. Implementation of the scalar adaptive law Eq. (25).

be used to adaptively regulate the desired rebroadcast probabilities in order to achieve high reachability with the smallest number of messages exchanged. In this section, we use analysis and simulations to present design guidelines as to how to choose the rebroadcast probability when estimates of the density are available, in network structures which are relevant to nanonetworks. In particular we investigate the probabilistic flooding problem in random geometric graphs and in uniform grid structures. 5.1. Probabilistic flooding in random networks Due to the random movement of molecular nodes in many applications and the diffusion based molecular communication paradigm, the random geometric model is often meaningful. The random geometric model assumes nodes randomly placed in a square box of area A, according to a Poisson point process of intensity λ. The graph is formed by assuming a set of edges which are formed between any two nodes whose toroidal distance is less than the transmission range r. The assumption of the toroidal distance avoids edge effects. If probabilistic flooding is applied in the considered graph with rebroadcast probability equal to ω, then it has been shown in [20] that the probability of all nodes receiving the message is given by: −α

Ψ (A, λ, r , ω) = γ e−e

+ ϵψ

(28)

where Ψ is the probability of all nodes receiving the message (i.e. success probability), A is the area where the nodes are deployed, λ

is the average node density, γ , 1 − e−λ π r , α = −λ∗ π r 2 − ln(Aλ + 1), λ∗ = λω + A1 and limλ→∞ ϵψ = 0. The latter condition (i.e. limλ→∞ ϵψ = 0) implies that as the node density increases the error goes to zero. This approximation is meaningful in nanonetworks where the node densities are typically high. If we assume the error to be zero then one can solve for ω to obtain the desired rebroadcast probability which would result in the success probability attaining a desired value Ψ . ∗

ω=

ln Ψ

λπ

r2

+

1 Aλ

−

ln ϵΨ

λπ r 2

.

2

(29)

The relationship above demonstrates the dependence of the desired rebroadcast probability on the density λ and how the estimates of the node density can be used to determine this rebroadcast probability. 5.2. Probabilistic flooding in grid networks In some applications the geometric random graph model assumed above may not be meaningful. It has been shown in [21]

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Fig. 11. Square transmission model.

Fig. 12. Circular transmission model.

that meta-material nanonetworks are characterized by fixed grid like structures. In such a network simple and fast information dissemination is a critical issue. Probabilistic flooding is a simple to implement solution which has been partially considered also in [41]. The question is how does one choose the rebroadcast probability in such networks. To this end we envision that the network nanonodes in the grid structure have capability for both molecular and electromagnetic communication in the fashion of [22]. Molecular communication may be employed to estimate the node density, as this is less susceptible to obstacle signal attenuation, and electromagnetic communication is employed for message dissemination. In this section we present simulations which can be used as guidelines as to how to choose the rebroadcast probability when the node density is known. At present, the simulation study is conducted on Matlab and involves calculation of the probability of all nodes receiving the broadcast message as the rebroadcast probability is changed for different values of the network size and the number of neighbors. We consider square grid structures of size n. Two transmission models have been considered to examine the effect of the transmission model. The circular model shown in Fig. 12 and the square model shown in Fig. 11. The parameter K refers to the number of tiers of neighbors reached in a single broadcast hop in one dimension. Figs. 11 and 12 show the K = 1 and K = 2 cases. When probabilistic broadcasting method is considered each nanonode is expected to: Upon receiving the message for the first time, the node decides to rebroadcast it with probability p and take no action with probability 1 − p. The way with which this is implemented is that a number is drawn from a uniform distribution in the range 0 to 1. If the drawn number is smaller than p then the broadcast takes place. The way with which the probability of success is computed is by measuring the recurrence of the event of all nodes receiving the message over numerous repetitions of the experiment. Five repetitions were chosen. 5.3. Two dimensional experiments A 2-dimensional network is considered at first. In Fig. 13 the probability of all nodes receiving the message is shown as a function of the rebroadcast probability when K = 1 (solid lines) and K = 2 (dashed lines) for multiple network sizes employing the square transmission model. There are areas where the success probability is high and areas where it is low. The transition from one area to the other becomes more and more abrupt as the

Fig. 13. Probability of success vs. rebroadcast probability for K = 1 (solid) and K = 2 (dashed) and for different sizes of the grid.

Fig. 14. Critical probability vs. size of the two dimensional network.

number of nodes increases. Also, a strictly increasing function of the rebroadcast probability is observed. Furthermore, networks of larger sizes seem to require higher rebroadcast probability values to achieve the same success probabilities. The smallest value of rebroadcast probability which can achieve a success probability equal to or higher than 80% is denoted as the Critical Probability. The critical probability thus, ensures high success probabilities with the least number of broadcasts, which mitigates the broadcast storm problem. In order to investigate the effect of changing the size of the network on the critical probability we measure the rebroadcast probabilities which fulfill the 80% success for each size from the results of Fig. 13. Fig. 14 demonstrates that networks of larger sizes require higher rebroadcast probabilities. Then, we double the value of K so that when square transmission is used a node would have 24 immediate neighbors and 12 when circular transmission is employed. The dashed graphs in Fig. 13 show the probability of all nodes receiving the message vs. the rebroadcast probability when K = 2. The increase in the probability of success is substantial when the compared to the K = 1 graphs (solid). For example, when the network of size 21 (i.e. 441 nodes network) is considered, we can observe that

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Fig. 15. Comparison between circular and square models for two different sizes.

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Fig. 17. Probability of success vs. rebroadcast probability for K = 1 & K = 2 in 3D.

Nevertheless, the value of K has the greater impact on the critical probability. The value of K depends directly on the transmission range and the node density. Therefore, when the transmission is fixed it is essential to estimate the node density to calculate the optimal rebroadcast probability. 6. Conclusion

Fig. 16. Circular transmission model in three dimensions.

when K = 1 a probability of 0.818 is required to achieve 80% success, while a probability of 0.390 only is sufficient to achieve the same success probability when K = 2 (dashed). The rationale of this observation is that more immediate neighbors represent more rebroadcast available options. This result indicates the criticality of node density on the critical probability required for efficient probabilistic flooding. The probability of success as a function of the rebroadcast probability is depicted in Fig. 15 when square and circular transmission models (dashed and solid lines respectively) are employed in two different sizes of the network. Lower rebroadcast probabilities are sufficient when the square model is used as it provides more neighbors. Nevertheless, the difference is not as dramatic as when the value of K is increased. 5.4. Three dimensional experiments We apply a similar simulation setup in the 3-dimensional network case. Therefore, one node has 26 immediate neighbors when the square transmission pattern is used and 6 when the circular pattern is used as indicated in Fig. 16. The probability of success vs. the rebroadcast probability is depicted in Fig. 17 for K = 1 (bold graphs) and K = 2 (dashed graphs). As expected, higher rebroadcast probabilities are required for larger networks. In addition, doubling the value of K results in the graphs shifting further to the left indicating the sufficiency of smaller rebroadcast probability values. Our findings indicate that the size of the grid and the parameter K both have an impact on the required critical probability.

In this paper we propose and analyze a new method for estimating the node density in molecular nanonetworks, inspired from the quorum sensing process which bacteria employ for communication purposes. The method is based on synchronous transmission of all network nodes and measurement of the received molecular concentration. We show that when the synchronous transmission is performed in infinite space, a linear static parametric model of the node density can be derived. When, however, the transmission is performed over a finite space the model becomes time varying. To overcome the difficulties associated with the time varying nature we propose the use of periodic transmission which for large enough values of the period, transforms the linear model into a static one. The derived parametric models are then used as a baseline to develop an estimation based on online parameter identification techniques. We finally consider the use of the node density estimates to adaptively regulate probabilistic flooding in molecular nanonetworks. Design guidelines as to how to choose the rebroadcast probability are presented in random geometric models and uniform grid structures. In the future, we aim at obtaining linear static parametric models when simplifying assumptions of the current formulation are relaxed as for example, the deterministic nature of the network nodes, and the circular geometry. In addition, the effectiveness of both the estimation method and probabilistic flooding will be investigated using more realistic simulation tools. References [1] F. Ye, R. Yim, J. Zhang, S. Roy, Congestion control to achieve optimal broadcast efficiency in vanets, in: 2010 IEEE International Conference on Communications, (ICC), IEEE, 2010, pp. 1–5. [2] R. Ramanathan, Making ad hoc networks density adaptive, in: Military Communications Conference, 2001. MILCOM 2001. Communications for Network-Centric Operations: Creating the Information Force. IEEE, Vol. 2, IEEE, 2001, pp. 957–961. [3] M. Artimy, Local density estimation and dynamic transmission-range assignment in vehicular ad hoc networks, IEEE Trans. Intell. Transp. Syst. 8 (2007) 400–412.

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Taqwa Saeed received the B.Sc. degree in Electronics Engineering from the University of Bahrain, Bahrain. She is currently perusing her Ph.D. in Electrical Engineering at the Frederick University, Cyprus. Her research interest include probabilistic broadcasting in VANETs and Nanonetworks.

Marios Lestas received the B.A. and M.Eng. degrees in Electrical and Information Engineering from the University of Cambridge UK and the Ph.D. degree in Electrical Engineering from the University of Southern California in 2000 and 2005 respectively. He is currently an Associate Professor at Frederick University. His research interests include application of non-linear control theory and optimization methods in Computer Networks, Nanonetworks, Transportation Networks, and Power Networks. He has participated in a number of projects funded by the Cyprus Research Promotion Foundation and the EU. Andreas Pitsillides received the B.Sc. (Hns) degree in Electrical Engineering from University of Manchester Institute of Science and Technology (UMIST) and Ph.D. in Electrical Engineering from Swinburne University of Technology, Melbourne, Australia, in 1980 and 1993 respectively. He is a Professor, Department of Computer Science, University of Cyprus, and heads the Networks Research Laboratory (NetRL). He has particular interest in adapting tools from various fields of applied mathematics such as adaptive non-linear control theory, computational intelligence, and recently nature inspired techniques, to address problems in communication networks.