{'fraction_non_alphanumeric': 0.03867626830649729, 'fraction_numerical': 0.012177360627882515, 'mean_word_length': 4.289749625508239, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fundamentals of Clustered Molecular Nanonetworks

We present a comprehensive approach to the modeling, performance analysis, and design of clustered molecular nanonetworks in which nano-machines of different clusters release an appropriate number of molecules to transmit their sensed information to their respective fusion centers. The fusion centers decode this information by counting the number of molecules received in the given time slot. Owing to the propagation properties of the biological media, this setup suffers from both inter- and intra-cluster interference that needs to be carefully modeled. To facilitate rigorous analysis, we first develop a novel spatial model for this setup by modeling nanomachines as a Poisson cluster process with the fusion centers forming its parent point process. For this setup, we first derive a new set of distance distributions in the three-dimensional space, resulting in a remarkably simple result for the special case of the Thomas cluster process. Using this, total interference from previous symbols and different clusters is characterized and its expected value and Laplace transform are obtained. The error probability of a simple detector suitable for biological applications is analyzed, and approximate and upper-bound results are provided. The impact of different parameters on the performance is also investigated.


I. INTRODUCTION
Molecular communications (MC) has attracted significant research interest due to its numerous applications in biological and communication engineering such as nanoscale sensing, drug delivery, and body area networks [1]. In MC, biological nano-machines (NMs) release molecules into the environment to realize nano-scale information transfer. Information can be encoded in the concentration or type of the molecules or at the time instants at which they are released. A detector decodes the sent information of an NM based on the number of captured molecules. Some examples of modulation and intersymbol interference (ISI) mitigation techniques can be found in [2]- [5]. These works focus on a single pair of transmitting and receiving NMs. On the other hand, nano-networking and Internet of Bio-Nano Things (IoBNT) paradigms require information collection from multiple arbitrarily-located NMs sensing potentially different phenomena [6]. Hence, interaction and interference of released molecules of multiple NMs over time play a crucial role in the performance characterization and design of bio-nanonetworks.
Several prior works consider single source interference [7] and fixed-location configurations with a limited number of interfering NMs [8], [9]. Due to the randomness and irregularity in the locations of NMs, stochastic geometry approaches based on the Poisson point process (PPP) assumption have also been considered for the modeling and analysis of nanonetworks [10]- [15]. However, these early stochastic geometry-based works lack in two important aspects that inspired this paper. First, these works did not explore the dependence of detectors on interference even though interference causes significant increase in the molecule count. Second, despite the usefulness of PPP for modeling the uniform deployments of NMs, it cannot model deployments where fusion centers (FCs) are located at places with high NM density to sense and process their information. The latter is particularly important for capturing the fact that many biological phenomena, such as abnormalities, are localized and multiple cooperative NMs, e.g., each monitoring a different feature of a target, might be required for fast, accurate, and diverse sensing [16]. In such deployments, it is important to take into account non-uniformity as well as the correlation that exists between the locations of the NMs and FCs. Models based on Poisson cluster processes (PCPs) [17,Sec. 3.4] have recently been well studied for wireless networks over two-dimensional (2D) regions [18]- [23]. In these works, the network usually follows a Thomas cluster process (TCP) [17,Definition 3.5] or Matérn cluster process (MCP) [17,Definition 3.6]. Further, the formalism for establishing distance distributions in PCPs is well-known [24] and has been applied extensively in the recent years to derive key distance distributions for both Thomas and Matérn cluster processes [18], [25]- [27].
In this paper, we consider a bio-nanonetwork consisting of multiple NMs forming clusters around their respective FCs in a three-dimensional (3D) biological media 1 . In each time slot, one NM is assumed to transmit its sensed information to its respective FC. This not only results in inter-cluster interference from the simultaneous transmissions in the current time slot but also both inter-and intra-cluster interference from the transmissions corresponding to the previous time slots. A rigorous analysis of this setup required a spatial model that is rich enough to capture its salient features while being tractable enough to enable mathematical analysis of the aforementioned interference and its effect on the system performance. Inspired by this, we develop a novel approach based on PCPs to facilitate tractable modeling and analysis of 3D clustered bionanonetworks. We first characterize relevant distance distributions of 3D PCPs that are inspired by this setup and provide closed-form results for 3D TCPs. In particular, we identify a specific structure for 3D TCPs that provides a remarkably simple distribution. Both ISI and interference from interfering NMs in different clusters are characterized, and the expected value and Laplace transform of intra-and inter-cluster interferences are derived. Then accordingly, a low-complexity detector for FCs is designed that can efficiently adjust to the interference. Finally, the performance of the detector is analyzed and tractable approximations are provided. We also investigate the impact of different parameters of the system model on the performance in terms of the error probability and the expected value of interference. As expected, our analysis reveals that a higher intensity of cluster centers or distance of a reference NM to the center of its cluster has a degrading effect on the performance. Also, increasing the time slot duration or the difference of released molecules for different symbols decreases the error probability.

II. SYSTEM MODEL
In this section, we provide a mathematical model of the bionanonetwork, including the spatial distribution of the NMs and FCs, the molecular propagation model, and the transmission scheme.

A. Spatial Model
We consider a 3D clustered bio-nanonetwork as shown in Fig. 1, where the locations of NMs are modeled as a PCP in R 3 . A 3D PCP Φ can be formally defined as a union of offspring points in R 3 that are located around parent points (i.e., cluster centers). The parent point process is a 3D PPP Φ p with intensity λ p , and the offspring point processes (one per parent) are conditionally independent. The set of offspring points of x ∈ Φ p is denoted by N x , such that Φ = ∪ x∈Φp N x , and the probability density function (PDF) of each element being at a location y + x ∈ R 3 is f Y (y). After characterizing theoretical results in terms of general distribution f Y (y), we specialize the results to the 3D TCP where the points are distributed around cluster centers according to an independent Gaussian distribution In the center of each cluster, there is an FC that detects, gathers and processes the transmitted information of the NMs of the same cluster. In our analysis, we will model each FC as a ball of non-zero radius r 0 that will place some restrictions on the placement of FCs and NMs. For instance, NMs cannot lie inside the FCs. While the exact analysis of this modified setting is complicated and will lead to significant loss in tractability, we will include the non-zero radius r 0 in many components of our analysis, thus capturing these additional restrictions while maintaining tractability.
It is worth noting that the proposed PCP-based setup reduces to a single cluster setup if the parent point process intensity goes to zero. In fact, the analysis of this special case has not yet appeared in the literature, which further reinforces the generality of this work. For brevity, we will not write separate corollaries for the single-cluster case with the understanding that they can be easily deduced from the main results by substituting λ p = 0 (since our analysis will assume the typical FC at the origin, which will be retained even under this limit).

B. Propagation Model
We consider a time-slotted transmission with time slots of duration T . This implicitly assumes perfect synchronization of all NMs, which is a common assumption in the MC literature. While we will not delve into synchronization aspects in this paper, interested readers are advised to refer to [28], [29] for more information on how synchronization is achieved in MC networks.
An NM at a location z can release X z ∈ {x 1 , ..., x M } molecules at the beginning of a time slot to transform log 2 (M ) information bits, each with equal probability 1 M [30]. All NMs are assumed to release the same type of molecule. The molecules released by the NMs propagate through the biological medium and are observed at the FCs. We assume that a molecule is absorbed at an FC and then counted when it hits the surface of the FC. NMs to FC propagation in a reference cluster is illustrated in Fig. 2. Let p iL (d) denote the i-th hitting probability in an L-th time slot, i.e., the probability that the molecule with distance d to the FC and released at the i-th time slot hits the FC during the L-th time slot. The hitting probability depends on the medium and type of detector [2]. For instance, for a general 3D environment with a point-source NM and a spherical absorbing FC with radius r 0 2 , the hitting probabilities are given by [31], [32] and where where D is the diffusion coefficient, which depends on the temperature, viscosity and size of the transmitted molecule. The parameter µ denotes the reaction rate constant of molecular degradation, and erfc(x) = 2 √ π ∞ x e −t 2 dt. When there is no molecule degradation in the environment, i.e., µ = 0, the function g(t, d) simplifies to [33] In general, sequence-based channel estimators can be trained to estimate the hitting probability [32]. Before we conclude this discussion, please note that the proposed spatial model is not limited to the choice of this propagation model (which is a popular one) and can be easily applied to other biological communications by choosing appropriate propagation models for those settings.

C. Transmission Scheme
In order to mitigate interference in each cluster of our clustered MC setup, only one of the NMs of the cluster is scheduled to release molecules according to its information availability status 3 . Let the NM at y i x in the cluster with x ∈ Φ p is selected to release X x+y i x molecules in the i-th time slot. According to the propagation model and due to the fact that the input-output relationships between the NMs and the FCs can be modeled as independent Poisson channels with additive Poisson noise [2], [3], at the end of L-th time slot, we can model the number of received molecules at the FC at where λ 0 T is the mean of the number of noise molecules during the time slot. In (6), the effect of ISI due to the transmissions in the previous i ∈ {1, ..., L − 1}-th time slots is also included.

III. DISTANCE DISTRIBUTIONS
In this section, we present the following theorem on the conditional PDF of the distance of any (arbitrary) element in the set N x of the cluster centered at x ∈ Φ p to the origin o and the subsequent corollary as a special case of the theorem for N o . These results will be used later in derivations of the expected value of interference and its Laplace transform in Section IV. As noted before already, even though the formalism for establishing distance distributions in PCPs is well-known and the distance distributions for 2D PCPs have been extensively studied in the recent years (inspired by wireless applications), similar investigations for 3D PCPs have not been performed yet. For instance, in the below result, we demonstrate that the aforementioned distribution admits a closed form solution for the 3D TCP, which was primarily enabled by the 3D assumption.
Theorem 1: Conditioned on x , i.e., the distance of the parent point x from the origin, the PDF of the distances d = y + x , ∀y ∈ N x , is which is specialized for TCP as Proof: See Appendix A. Corollary 1: The PDF of the distances d = y , ∀y ∈ N o , is which is specialized for TCP as Proof: By putting x = 0 in Theorem 1, we get the results. However, for the TCP result, we achieve 0 0 and need to use L'Hopital's rule as This completes the proof.
IV. FUSION CENTER DETECTOR DESIGN Our analysis will focus on the performance of the typical cluster/FC, which we term as the reference cluster/FC in this paper. The conditioning of this FC at a specific location can be made rigorous by using the idea of Palm distribution. For the PPPs, it suffices to invoke Slivnyak's theorem one of whose consequences is that conditioning on the presence of a point of a homogeneous PPP at some location is equivalent to adding a point to this PPP at that location (which we assume origin without loss of generality, due to stationarity) [17]. Therefore, thanks to Slivnyak's theorem, we simply add the reference FC to the origin along with its cluster N o . Also, we consider the L-th time slot, where the reference FC desires to detect the information of a reference NM at location y 0 in the reference cluster. Then, from (6), the number of received molecules at the reference FC can be rewritten as where is the total interference. The intracluster interference I intra L from NMs inside the reference cluster and inter-cluster interference I inter L from NMs of other clusters are given by and Due to the unknown transmitted symbols and locations of NMs, the value of I L is unknown to the FCs. However, using the statistical characteristics of the interference I L , we propose an approximate maximum likelihood (ML) decision rule aŝ where the interference I L is approximated by its expected value E {I L }. It can be obtained as where for E I intra L , we have where the term ∞ r0 f d (y|0)dy = P( y > r 0 ) is for the condition that NMs are outside the reference FC, i.e., the ball with radius r 0 centered at the origin. From (2), the term (17) is equal to Then, for E I inter L , we have where (a) follows from the fact that the distance of each NM in the cluster x ∈ Φ p to the origin is i.i.d. with distribution f d (.| x ) [18], [22] and the term ∞ r0 f d (y| x )dy = P( y + x > r 0 ) is for the condition that NMs are outside the ball with radius r 0 centered at the origin. Then, (b) follows from the Campbell's theorem for PPPs [17] and the fact that FCs have at least 2r 0 distance from each other. From (2)-(3), the term For relatively large time slot duration when ISI can be ignored, the expected value of interference is given as in the following lemma. Lemma 1: For the TCP model and T → ∞, the expected value of I L is simplified to where and Proof: The result is obtained from the following facts: where g(t, y) is the channel function in (4) and ∞ r0 where (c) comes from the variable changes t = . Also, we have the fact From (21), it can be seen that the expected interference linearly increases with λ p and 1 M M j=1 x j . Assuming the order x 1 < x 2 < ... < x M , it can be shown that the rule in (15) can be further simplified tô where the thresholds are as This simple detector has low complexity to be implemented on an NM with no memory requirements 4 .

V. PERFORMANCE ANALYSIS
In this section, we derive the error probability of the detector proposed in Section IV for the considered PCP setup. In order to put the novelty of this analysis into context, please note that the error probability analysis in the literature has been limited to the PPP-based models (assuming M = 2) [12]- [14]. In fact, even the single-cluster setup (which is a special case of our setup) has not been studied yet. From the technical standpoint, we obtain a new form of the Laplace tansform of interference, propose new approaches to evaluate derivatives that appear in this analysis, as well as propose new special cases and a closed-form upper bound.
The error probability of the proposed detector can be written as where, defining th 0 = 0, th M = ∞, and the Laplace transform (LT) of I L as L I L (s) = E e −sI L , we have which follows from E e −s(p LL ( y0 )xj +I L +λ0T ) = e −s(p LL ( y0 )xj +λ0T ) L I L (s) , ∀s, and the derivative property of the LT as E X m e −sX = (−1) m d m L X (s) ds m , ∀m, for a random variable X. In (30), a denotes the closest upper integer to a. Also, L I L is given in Theorem 2.
Proof: From the Leibniz product rule [34], the term in (30) can be obtained as can be calculated from (38) and the Faà di Bruno's formula as where the summation i is over all i-tuples of nonegative integers (i 1 , · · · , i i ) satisfying the constraint 1 . i 1 + 2 . i 2 + · · · + i . i i = i. Replacing (41) in (40) and (40) in (30) proves the final result. As the form given in (30) is rather unwieldy, we next provide a closed-form upper-bound for (30).
Proof: See Appendix C. It is notable that this upper-bound can be also used for other setups when we replace related LT of interference. In the following lemma, the design and analysis results are simplifed for the ON/OFF keying modulation [3], [12], [14], [30].
Lemma 3: In the special case M = 2 with x 0 = 0 and where W is the Lambert function [36], the error probability is simplified to where and P(E|x 1 ) = e −(p LL ( y0 )ξ+λ0T ) L I L (1).
Proof: See Appendix D.

VI. NUMERICAL RESULTS
In this section, we provide numerical results for specific scenarios of clustered bio-nanonetworks with the parameter values given in Table 1, unless otherwise stated. We also provide Monte Carlo simulations to verify the accuracy of the results. For the simulations, we include exclusion zones for all the FCs to exactly model the clustered setup described in Section II.A. Recall that the exclusion zone for the reference FC was incorporated in the analysis too. Therefore, this true simulation and our analysis differ in terms of exclusion zones for the interfering FCs, which were considered in the simulation but ignored in the analysis. However, numerical comparisons of these two cases will shortly reveal that ignoring exclusion zones of the interfering FCs in the analysis was without any loss of accuracy. Also, while we presented the analytical results for a general M , here we focus on M = 2 as it is the usual present implementation case for molecular communications [3], [11]- [14], [30] and the effect of different parameters can be described more cleanly.
In Fig. 3, the analytical results in (29)  probability increases as the number of time slots increases. It is because the interference from ISI increases. Other observation is related to the increase of the diffusion coefficient D that can significantly decrease the error probability. It is due to the direct effect of increase in D on the increase in the hitting probability p LL in (3) and accordingly the improvment of the link from the reference NM. The error probability as a function of time slot duration T for y 0 = 2.5r 0 and 3r 0 is studied in Fig. 4. It is observed that as T increases, the error probability decreases. It is due to the fact that the hitting probabilities p iL in (2) decrease, which decreases the ISI. Also, when the reference NM is located closer to the reference FC, the performance improves since p LL increases. In Fig. 5, the expected number of interference molecules in (16) and the threshold given in (28) are shown as a function of T for y 0 = 2.5r 0 . We can observe that as T increases, the expected number of interference molecules increases and the threshold follows the increasing trend by unit step function increments. Also, the duration of constant thresholds increases.
In Fig. 6, the error probability as a function of the parent intensity λ p is plotted for σ = 3r 0 and 5r 0 . The error probability increases with the increase in λ p . It is because the number of interfering clusters increases. Also, increasing σ improves the performance. This is intuitive because the probability of the event that the reference FC is located farther from the NMs of the reference cluster increases with σ and this leads to decrease in p iL in (2). In Fig. 7, the expected number of interference and the threshold are shown as a function of λ p for σ = 5r 0 . We observe that the number of interference molecules increases linearly with λ p and the threshold follows this increase.
Figs. 8 and 9 present the effect of the number of released molecules for the second symbol, i.e., x 2 , on the error probability for µ = 0.1 and 0.8 s −1 , and on the expected number of interference and the threshold for µ = 0.1, respectively. It is observed that by increasing x 2 (increasing x 2 − x 1 ), the error probability decreases. This is because from (28), the decision area for each symbol expands. However, the slope of this improvement in performance decreases, which is due to the increase in interference. Also, increasing µ decreases the error probability due to a decrease in interference. It is also observed that the threshold can adjust itself with the increase in x 2 .

VII. CONCLUSIONS
In this paper, we used tools from stochastic geometry to develop the first comprehensive framework for the modeling, analysis, and design of MC in bio-nanonetworks whose NMs form clusters around their respective FCs. In order to capture the coupling in the locations of NMs and their respective FCs, we modeled NMs as a PCP with the FCs forming the parent point process. This departs significantly from the known  approaches that rely on PPP-based models and often ignore the complex dependence of detectors on interference. For the proposed model, we first characterized the distributions of the distances from a reference FC to various intra-and inter-cluster NMs. We also identified a specific structure for 3D TCPs that provided a remarkably simple expression for    the distance distribution in the TPC setting. Then, based on the expected value of intra-and inter-cluster interferences, we proposed a simple detector for FCs that is suitable for biological applications. Using the developed tools, we also provided analytical expressions and easy-to-use approximations for the error probability of the proposed detector. Our analysis revealed that decreasing the intensity of cluster centers or the distance of the NM from the center of its cluster and also increasing the time slot duration or number of released molecules improve the performance in terms of the error probability.
APPENDIX A PROOF OF THEOREM 1 where (a) is due to the fact that f Y (y) is rotationally invariant. It is notable that the result of (48) is dependent to the norm of x. Then, by taking a derivative of the CDF F d (y| x ), the conditional PDF f d (y| x ) is obtained with the help of the Leibniz integral rule [34] and simplifications. For the special case of TCP, by substituting the following f Y from (1) as into (7), we obtain Then, we obtain which leads to the final result.

APPENDIX B PROOF OF THEOREM 2
The LT of the interference is where for L I intra L , we have where the condition that y is more than r 0 is applied. For L I inter L , we have L I inter L (s) = E e −sI inter where (a) follows from the fact that the distance of each NM in the cluster with x ∈ Φ p to the origin is i.i.d. with distribution f d (.| x ) [18], [22] and (b) follows from the PGFL of PPPs [17] and the minimum distance between FCs, i.e., 2r 0 .

APPENDIX C PROOF OF THEOREM 3
From (30), the conditional probability of error can be written as e −(p LL ( y0 )xj +I L +λ0T ) (p LL ( y 0 )x j + I L + λ 0 T ) k k! (a) = 1 − P(H j > p LL ( y 0 )x j + I L + λ 0 T ) where (a) comes from chi-squared distributions H j−1 and H j with 2 th j−1 and 2 th j degrees of freedoms, respectively, and (b) is from the Alzer's lemma [37] on chi-squared distributions H j−1 and H j with parameters η j−1 = th j−1 ! Under this case, we have th 1 = 1, which leads to the final error probability result from (29)- (30). In the following, we prove that th 1 = 1 iff (44) holds. From (28), th < 1 if which leads to which can be simplified to ξ < ξ 0 , where ξ 0 is the solution of Then, (58) can be rewritten as From the Lambert finction definition [36], (59) leads to and then which completes the proof.