Spatial Motifs for Device-to-Device Network Analysis (DNA) in Cellular Networks

12/23/2017 ∙ by Tengchan Zeng, et al. ∙ University of Florida Georgia Southern University Virginia Polytechnic Institute and State University 0

Device-to-device (D2D) communication is a promising approach to efficiently disseminate critical or viral information. Reaping the benefits of D2D-enabled networks is contingent upon choosing the optimal content dissemination policy subject to resource and user distribution constraints. In this paper, a novel D2D network analysis (DNA) framework is proposed to study the impacts of frequently occurring subgraphs, known as motifs, on D2D network performance and to determine an effective content dissemination strategy. In the proposed framework, the distribution of devices in the D2D network is modeled as a Thomas cluster process (TCP), and two graph structures, the star and chain motifs, are studied in the communication graph. Based on the stochastic properties of the TCP, closed-form analytical expressions for the statistical significance, the outage probability, as well as the average throughput per device, are derived. Simulation results corroborate the analytical derivations and show the influence of different system topologies on the occurrence of motifs and the D2D system throughput. More importantly, the results highlight that, as the statistical significance of motifs increases, the system throughput will initially increase, and, then, decrease. Hence, network operators can obtain statistical significance regions for chain and star motifs that map to the optimal content dissemination performance. Furthermore, using the obtained regions and the analytical expressions for statistical significance, network operators can effectively identify which clusters of devices can be leveraged for D2D communications while determining the number of serving devices in each identified cluster.

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I Introduction

The number of mobile devices, such as smart-phones and tablets, has significantly increased, and almost half a billion mobile devices were added in 2016 [2]. This growth in mobile devices has been accompanied by a rise in the need for pervasive wireless connectivity in which users require to access their content of interest anywhere, anytime, and on any device. One effective approach for such large-scale content dissemination is to use device-to-device (D2D) communication links, which can enable mobile devices to communicate directly without infrastructure [3, 4, 5]. Compared with broadcasting contents by base stations (BSs), using D2D communications for content dissemination can help better serve users at cell edges and users that are not in the coverage of BSs [3, 4, 5, 6, 7, 8, 9]. Moreover, as each user can have different interests on contents, the information broadcasted by BSs may not be useful for every user [10, 11, 12]. In addition, D2D-based content dissemination can offload traffic from BSs and, thus, help alleviate the network congestion [13, 14, 15, 16, 17]. However, when deploying D2D communications for content dissemination, one must address two key challenges [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]: a) designing content dissemination strategies for choosing the number of serving devices that can serve as seeds for disseminating contents, and b) developing metrics to assess the content dissemination performance under different network topologies.

First, in order to choose the optimal set of serving devices, one can leverage a communication graph composed of D2D links and make use of graph-theoretic properties to identify the most influential devices that can store popular content and to choose the number of serving devices that maximizes the overall system throughput [7, 8, 9, 10, 11, 12]. For example, in [7] and [8], the authors use degree centrality, defined as the number of connected D2D links for every node, and identify the set of nodes with high degree centrality as serving devices. Meanwhile, the works in [9] and [10] focus on the betweenness centrality, which measures the fraction of shortest paths passing through a focal node, to identify the set of serving devices for D2D content dissemination purposes. Moreover, the authors in [11] use the closeness centrality, computed by the sum of distance between a node and all other nodes in the graph, to establish the strategy for choosing the best set of serving devices. Furthermore, in [12]

, the authors first build the adjacency matrix for D2D communication network, and then choose the set of serving devices by using the eigenvector corresponding to the largest eigenvalue in the adjacency matrix.

Although the works in [7, 8, 9, 10, 11, 12] exploit the graph-based properties, these works solely focus on the low-order network connectivity, which can only capture the features at the level of individual nodes and edges. In other words, these properties just measure the accumulation of influence from one device on other devices or D2D links. However, for content dissemination, understanding how information propagates among multiple nodes is critical, as such propagation can directly capture the influence of one device on a group of devices. Moreover, the works in [7, 8, 9, 10, 11, 12] do not conduct performance analysis for content dissemination under different network topologies.

To analyze the performance of content dissemination using D2D and derive tractable performance metrics for coverage and data rate, it has become customary to use stochastic geometry techniques [13, 14, 15, 16, 17]. For example, in [13], the authors derive the expression of end-to-end signal-to-interference-plus-noise ratio (SINR) for a relay assisted D2D network where devices are distributed according to a Poisson point process (PPP). Also, the authors in [14]

derive the outage probability for an arbitrary D2D link in a model where D2D receivers are uniformly distributed in a circular region around the transmitters. Moreover, in

[15], to capture the notion of device clustering, the authors model the locations of the devices as a Poisson cluster process (PCP), and develop expressions for coverage probability and area spectral efficiency. Furthermore, performance metrics, such as outage probability and average data rate, are also analyzed for systems where the distributions of BSs and devices follow PPP as done in [16] and [17].

Nevertheless, these works in [13, 14, 15, 16, 17] do not consider multicast and multi-hop communications, which can extend the range of content dissemination and achieve a better performance than single-hop D2D links [18]. In addition, these works are restricted to the modeling and analysis of D2D communication, and do not consider any content dissemination policy or the choice of the optimal set of serving devices. Although multicast and multihop communications have been considered, respectively, in [19] and [20], these works do not study more practical scenarios in which both types of communications exist. Moreover, the authors in [19] and [20] are restricted to D2D systems with uniformly and independently distributed BSs and devices. However, due to the similarity of users’ content interests, devices can group together within given areas, e.g., libraries, and form clusters instead of being independently and uniformly distributed [15].

The main contribution of this paper is to propose a graph-based D2D network analysis (DNA) framework to determine the content dissemination strategy in clustered D2D networks where both multicast and multi-hop communications exist. In particular, we first develop a framework, based on stochastic geometry, that models the distribution of device groups within the D2D network as a cluster process. Then, we build a D2D communication graph based on the distance distribution among devices and explore frequently occurring network subgraphs, also known as graph motifs, to capture content propagation patterns in a group of devices. Next, we theoretically analyze the occurrences for two types of graph motifs, i.e., chain (multi-hop) and star (multicast) motifs, and conduct rigorous performance analysis for D2D and cellular devices. Finally, based on the theoretical analysis, we determine guidelines for designing effective content dissemination strategies in D2D-enabled cellular networks. To our best knowledge, this is the first work that exploits network motifs to analyze the performance of content dissemination in D2D-enabled cellular networks. The novelty of this work lies in the following key contributions:

  • We explore occurring network structures, known as graph motifs, to capture content propagation patterns among a group of devices and theoretically analyze the occurrences of different motifs in D2D networks. In particular, we model the distribution of devices as a Thomas cluster process (TCP), where the devices are normally scattered around the central points. Then, based on the distance distribution between two arbitrary devices in the network, we formulate a distance-based graph. Also, we derive tractable expressions for the statistical significance to capture the occurrences of two types of motifs, chain and star motifs that respectively capture multi-hop and multicast communications.

  • We conduct a comprehensive performance analysis for both D2D chain and star motifs. In particular, we derive closed-form and tractable expressions for the average throughput for D2D and cellular devices, in presence of motifs. Moreover, we also derive the outage probabilities for both chain and star motifs and provide analytical bounds for the derived expressions.

  • Extensive simulation results are used to corroborate the analytical results. In particular, we can observe the influence of different system topologies on the occurrence and the outage probability of motifs and the system throughput. Moreover, the results highlight that as the statistical significance of motifs increases, the system throughput will initially increase, and, then, decrease.

  • The proposed framework provides important guidelines for designing effective content dissemination strategies in D2D-enabled cellular networks. In particular, using the derived expressions and the simulation results, network operators can determine the statistical significance regions for different motifs that map to the optimal system throughput. Moreover, by comparing the motifs’ statistical significance in each cluster with the optimal regions, the operators can identify which clusters can be optimally leveraged for D2D communications. In addition, based on the analytical derivations on the statistical motif occurrence, one can also determine the number of serving devices in each identified cluster.

The rest of the paper is organized as follows. Section II presents the system model and key assumptions. In Section III and Section IV, we conduct graph motifs based analysis for D2D networks and develop the performance-related metrics. Simulation results are provided in Section V and conclusions are drawn in Section VI.

Ii System Model

Consider a cellular network composed of a single BS and a number of devices that can communicate with the BS over cellular links as well as directly exchange information with one another via D2D communications, as shown in Fig. 3LABEL:sub@fig:f2. Here, devices with injected contents from the BS are referred to as seeding nodes, and those that obtain the content from nearby devices are called non-seeding nodes. Moreover, due to the similarity of users’ interests, devices can concentrate at the same area, such as a library or a stadium, and the grouped devices will form clusters [8, 9, 10, 11, 12]. Such a cluster-based model has been recently introduced in [11] as an effective and practical approach to model D2D communication networks and study content dissemination.

(a) Spatial motifs in the representative cluster.
(b) Different clusters existing within a geographical area.
Figure 3: System model that follows a TCP.

In addition, as done in [21], to avoid the interference between cellular and D2D links, we consider a D2D overlay system in which the available bandwidth is partitioned into for D2D links and for cellular links, where .

To capture the non-uniform, cluster-based distribution of devices in the network, we consider a TCP composed of a parent point process and a daughter point process [22]. In particular, as shown in Fig. 3LABEL:sub@fig:f1, we model the distribution of the parent points (center points of the clusters) as a PPP with density

. Also, the daughter points (devices) in each cluster are normally scattered with variance

around the corresponding parent point [22]. Note that the locations of daughter points, with respect to the corresponding parent point, are independent and identically distributed (i.i.d.) variables [22].

Ii-a D2D Communication Graph

As shown in Fig. 3LABEL:sub@fig:f2, we arbitrarily select a cluster with parent point , considered as a representative cluster . In cluster , seeding nodes can disseminate popular contents to other devices via D2D links. Nonetheless, due to limited transmit power and different interests on contents, devices belonging to cluster will not communicate with other clusters. Hence, in cluster , we can model the D2D network as a directed acyclic graph , whose vertices are devices in the set and whose edges, within the set , are D2D links among devices in the cluster. In particular, each D2D link can be defined as a edge, , , where device and device are the transmitter and the receiver, respectively. Also, as done in [23] and [24], we assume that no edge exists between a pair of vertices in graph if their distance exceeds the maximum value . Moreover, if there exists a graph with and , the graph is a subgraph of . For analytical tractability, we assume that the number of devices in each cluster is . As will be clear from the following discussion, although the total number of devices in each cluster is the same, the number of D2D links varies from one cluster to another, which maintains generality of the model.

By focusing on cluster , we can observe that groups of D2D links can form various subgraphs with different structures, which can show how the content propagates among a group of devices. For example, as shown in Fig. 3LABEL:sub@fig:f2, two D2D links can share the same transmitter, and the receiver of one link can act as the transmitter of another D2D link. To characterize D2D subgraphs appearing in the network, we exploit the notion of graph motifs, defined as frequently occurring subgraphs. The frequency of occurrence of motifs is typically measured with respect a baseline system such as a random graph built by randomly assigning D2D links to pairs of arbitrary devices [25]. In addition, to capture the occurrence of different motifs, one can measure the statistical significance for each motif using the notion of a Z-score [26]:

(1)

where is the number of occurrences of a given motif in the network, and and

denote, respectively, the mean and the standard deviation of motif’s occurrence in the baseline system. For the baseline system, we use a random graph model that shares the same total number of D2D links existing in graph

, but randomly rewires each link between an arbitrary pair of devices [27]. For example, if there are motifs where one seeding node sends data to two non-seeding nodes, the total number of edges in the baseline system will be . As a result, the edge between any pair of devices in the baseline system occurs with the same probability

. Using the properties of the binomial distribution, we can derive

and , where is the floor function of and denotes the number of -combinations in a set with elements. Furthermore, we can observe that , when , indicating that the -score is an increasing function of the motif occurrence parameter .

In addition, we assume that each seeding device can disseminate content to multiple non-seeding devices, and non-seeding devices are capable of forwarding the content to others. Therefore, as illustrated in Fig. 3LABEL:sub@fig:f2, for any group of three devices, we can observe two types of communication motifs – the star motif and the chain motif. The star motif can be viewed as a multicast link in which the seeding device disseminates its content to two non-seeding devices, and is captured by two edges sharing the same first element in the communication graph. Meanwhile, the chain motif can be considered as a two-hop D2D link. In particular, the seeding device will first send its content to a non-seeding device in one time slot, which functions as a relay propagating the content to another non-seeding device in the next time slot. Similarly, for the chain motif in the graph, there exists two edges connecting three devices, and the second element of one edge is the first element of the next edge. Here, we focus on three-device motifs, since larger motifs can always be constructed using smaller motifs, and, hence, our framework can extend to any other graph structures.

Ii-B Channel Model and Interference Analysis

We model the channel for the D2D link , , in cluster centred at as a Rayleigh fading channel. Hence, the received power is given by , where is the transmit power of each device,

is the channel gain that follows an exponential distribution with mean equal to

, is the corresponding distance between any two devices and , and is the path loss exponent. Similar to the bandwidth allocation in [28], in a star motif, we assume that the two D2D links will split the frequency resources evenly and each link uses half of the bandwidth, . Meanwhile, in a chain motif, the first and second D2D links will reuse the same resource, , at two consecutive time slots. Therefore, due to the co-channel deployment of D2D links, the receiving nodes will encounter interference from other D2D links, irrespective of their motif types. We designate the interference on device generated from devices in cluster as the intra-cluster interference , and the interference from devices within other clusters as the inter-cluster interference . The two types of interference are given by:

(2)

where the indicator variable if device is transmitting data to other devices via D2D link; otherwise, . For a large D2D network, the thermal noise power at the receiver will be substantially smaller than the received interference power [29]. Thus, the expression of the SINR of the D2D link can be approximated as

(3)

Since we consider a single-cell scenario and the BS will transmit contents to seeding devices in different time slots, the seeding devices will experience no interference and their signal-to-noise ratio (SNR) will be:

(4)

where is the transmission power of the BS, is the channel gain that follows an exponential distribution with mean equal to , represents the distance between the BS and device , and captures the power of the background noise. Moreover, we can find the achievable data rate for D2D links and cellular links, according to , where is the assigned bandwidth.

Ii-C Relationship between the Occurrence of Motifs and the D2D System Throughput

For each cluster, the BS can collect the occurrence of different communication motifs and the data rate of each device. Due to the co-channel deployment of D2D links, interference will increase as the frequency of occurrence of any given motif increases. As a result, as shown in Section II-A, the -score is an increasing function of the occurrence of motifs, and, hence, the interference will increase as the -score increases. Thus, for clusters having higher -scores, content dissemination via D2D communications will experience a high interference, thereby degrading the dissemination throughput. On the other hand, for clusters with a smaller -score, only a limited number of D2D links can be formed, resulting in a low spectral reuse. Therefore, we need to find the -score region, which can yield an optimal system performance. This -score region can allow identifying clusters that can be used for D2D communications by comparing any given -score of the clusters with the optimal region. Moreover, the number of seeding devices in each identified cluster can be determined based on the relationship between the -score and the occurrence of motifs in the network.

However, due to the lack of closed-form expressions that link the -score and the wireless throughput, directly finding the -score region that yields the optimal system performance is challenging. Nevertheless, we can observe that the value of the -score depends on number of devices , their locations, and the maximum communication distance . In addition, the system throughput is also a function of these three parameters. Therefore, instead of finding a direct relationship between the -score and the system throughput, one can alternatively determine the analytical expressions of the -score and the system throughput as functions of their common parameters. Using such analytical expressions, the BS can collect data on the -score and the system throughput corresponding to the same parameter settings, and further observe how the system throughput changes as the -score of motifs varies. The changes of the system throughput as function of the -score can enable network operators to disclose the relationship between the occurrence of motifs and the system performance, and further determine the -score region which maps to the optimal system throughput. Next, we will first leverage the distance distribution among devices and the stochastic properties inherent in the TCP to derive tractable expressions for the -scores of chain and star motifs and the system throughput. Also, we will further identify the hidden relationship between motifs and the system performance based on simulation results and determine the content dissemination strategy for clustered networks.

Iii Statistical Significances of Star and Chain Motifs

In this section, for an arbitrarily chosen group of three devices in cluster , we use the distance distribution between any two randomly and uniformly selected points to derive the probability of the content dissemination pattern among three devices being either a star or chain motif. Based upon the probability of occurrence, we can derive closed-form expressions for the -scores.

Iii-a Distance Distribution

We arbitrarily select a group of three devices in cluster , and due to the stationarity of the TCP, we treat an arbitrary device out of these three devices as a typical point located at the origin. In addition, we assume that the location of the parent point is at , and the other two devices are located at and relative to the parent point. Therefore, according to the definition of the TCP in Section II, and

are distributed according to a symmetric normal distribution with the variance

around the parent point , and the parent point also follows a zero mean symmetric normal distribution with the variance relative to the origin point. Additionally, , , and are i.i.d. variables, .

Furthermore, we denote as the distance set from the typical point to another uniformly selected point in the same cluster. In particular, the realization of is . Conditioned on the distance from the typical point to the parent point

, the probability density function (PDF) of

follows a Rician distribution [29]:

(5)

where is the modified Bessel function of first kind with zero order. Conditioned on , the elements in are i.i.d. variables as shown in [29]. Moreover, the authors in [29] also conclude that, when the distances from the typical point to the parent points of other clusters are a fixed value, the distances between the typical device at the origin and the devices in other clusters are also i.i.d. variables and follow Rician distributions. As it will be clear from the following discussion, we use this property to calculate inter-cluster interference. In addition, according to the definition of the TCP, the distance follows a Rayleigh distribution with scale parameter and with the following PDF:

(6)

Iii-B Probability of Occurrence of Motifs

Based on the distance distribution obtained in Section III-A, we can derive the probability with which a group of three arbitrary devices will form either a chain or star motif. First, we can observe that, for both chain and star motifs, there is a node capable of directly communicating with the other two devices. Hence, if we consider a device with two direct D2D links as a typical point located at the origin, we can express the distance from the typical point to the other two uniformly and randomly selected devices as the norm of , and , where , , and . In a TCP, and

are zero mean complex Gaussian random variables relative to

. Hence, , , , and . In addition, , and . Thus, and follow the Rayleigh distribution. Due to the common element, , in and , the distances and are also correlated. Based on the distribution and correlation between and , next, we can obtain the joint probability of both and being smaller than .

Theorem 1.

Given that and follow Rayleigh distributions and are correlated, the joint probability of both and being smaller than is:

(7)

where refers to the lower incomplete gamma function, defined as .

Proof:.

To calculate the probability of both and being smaller than the distance threshold , we first calculate the correlation existing between and as

(8)

where and are the standard deviations for and , respectively. Then, due to the fact that both and follow Rayleigh distribution, we are able to have the joint cumulative density function (CDF) as [30]

(9)

where represents to the complete gamma function, defined as . After replacing and with and simplifications, we can obtain the joint CDF shown in (7). ∎

In (7), represents the probability that a typical device can build D2D links with two other devices uniformly and randomly chosen from the cluster, irrespective of the link direction. Since a receiving device cannot access the content from multiple transmitters at the same time, and a transmitter can spread content to at most two devices simultaneously, is the probability with which any three devices can form either the star motif or the chain motif. For ease of exposition, we use the term “three-node motifs” to refer to the union of the star and chain motifs, and replace with hereinafter. Moreover, we assume that the probability of one three-node motif being a star motif is , , and the probability of forming a chain motif will be .

Iii-C -scores for chain and star motifs

To calculate the -scores for the chain and star motifs, we must derive the occurrence of these two motifs in cluster and compare it with the baseline system. In particular, the maximum number of groups of three devices is , where any arbitrary two groups do not share common devices and devices in each group are randomly and uniformly selected. Therefore, we can express the probability of groups of three-node motifs existing in a cluster as . Using the properties of the binomial distribution, the expected number of three-node motifs in the network with distance will be . Accordingly, the expected numbers of occurrence for the star and chain motifs are give by

(10)

For the baseline system, the total number of D2D links is equal to the one in the considered system. In particular, since the number of D2D links in each three-node motif is two, the expected number of D2D links in the baseline system is . Therefore, we can derive the probability of one pair of arbitrary devices being connected by a D2D link in the baseline system as follows:

(11)

In this case, the probability with which three devices form a three-node motif is . Furthermore, the probabilities of being a star motif or a chain motif in the baseline system are , and , respectively. Similarly, by using the binomial distribution, we can express the mean and the standard deviation for the star motif and the counterparts and for the chain motif in the baseline system as

(12)
(13)
(14)
(15)

After obtaining the occurrence information for both motifs in cluster and the baseline system, we can derive the -scores for both motifs.

Remark 1.

By using the analytical results in (1), (10) and (12)-(15), the closed-form expressions of -scores for both motifs in the D2D system can be determined. In fact, (10) captures the expected number of occurrences for chain and star motifs in cluster . Also, (12)-(15) represent the mean and standard variance of the occurrences for both motifs in the baseline system.

As observed from Theorem 1, the joint probability is a function of the total number of devices , the distribution variance , and the maximum allowable distance . Therefore, we can conclude that the value of the -score is also dependent on , , and .

Iv Expected Average Throughput and Outage Probability

To analyze the performance of the content dissemination strategy under different clustered network topology parameters, we use the distance distribution properties inherent in the TCP to calculate the expected throughput for D2D and cellular devices. Moreover, the outage probabilities for the chain and star motifs are also derived.

Figure 4: Illustration of the inter- and intra-cluster interference to the representative star motif.

Iv-a Expected Throughput for Non-Seeding Devices in the Star Motif

As shown in Fig. 4, we choose an arbitrary star motif as a representative star motif in cluster . In motif , the two receiving devices, and , will access data via D2D links from the seeding device, , located at the origin. Moreover, we assume that the set of seeding devices in star motifs as in the cluster , and the number of elements in cannot be greater than .

Next, we take device as an example and conduct Laplace transforms for the intra- and inter-cluster interference encountered by device . Based on the derived Laplace transforms, we can study the performance-related metrics, such as the outage probability and the expected throughput. According to the stationarity of the TCP and the symmetry between the two receiving devices in the star motif, the Laplace transforms of the intra- and the inter-cluster interference for device can apply to another device and receiving devices in other star motifs.

Lemma 1.

For device in the star motif, the Laplace transform of the intra-cluster interference can be expressed as

(16)

where , .

Proof:.

See Appendix -A. ∎

By using the Bernoulli’s inequality [31], we can obtain a lower bound on the intra-cluster interference over device , as given by the following corollary.

Corollary 1.

The lower bound of the intra-cluster interference will be given by:

(17)
Proof:.

See Appendix -B. ∎

The Laplace transform of the inter-cluster interference and its lower bound are derived next.

Lemma 2.

For device in the star motif, the Laplace transform of the inter-cluster interference is given by

(18)

where , .

Proof:.

See Appendix -C

Corollary 2.

The lower bound for the inter-cluster interference can be expressed as

(19)
Proof:.

See Appendix -D. ∎

Considering that content dissemination within a star motif will not occur an outage if and only if the SINR for both D2D links in the star motif exceed the minimum requirement , then the outage probability for the star motif can be derived as follows.

Theorem 2.

The outage probability for the star motif is given by:

(20)

where , ,

Proof:.

We first calculate the SINR distribution for a D2D link to obtain the probability of a D2D link meeting the minimum SINR requirement:

(21)

where follows the fact that the channel gain for a Rayleigh fading channel. In , we substitute with and use polar coordinates. Note that the probability in (2) can also apply to the link between devices to , due to the symmetry between devices and . As outage will not occur when the SINR of both links exceeds the threshold , we use the probability in (2) for both D2D links in the star motif to derive the outage probability in (20). ∎

Since , will yield . Thus, after replacing with in (2), we can have the CDF of the throughput for receiving devices in the star motif.

Corollary 3.

Based on the distribution of the SINR, the CDF of the throughput for receivers in the star motif can be derived as follows:

(22)

By using the relationship between the CDF and the expected value, we are able to obtain the expected throughput of receiving devices in the star motif in the following corollary.

Corollary 4.

The expected throughput for receivers in the star motif is given by:

(23)

Iv-B Expected Throughput for Non-Seeding Devices in the Chain Motif

As illustrated in Fig. 5, we also choose an arbitrary representative chain motif in cluster . In motif , the seeding device first transmits its data to , which is at the origin and will subsequently propagate the data to device . Similar to the receiving devices in the star motif, device in a chain motif also accesses its content from the transmitter at the origin. Therefore, the Laplace transforms of the intra- and inter-cluster interference given by (16) and (18), and the probability in (2) can be directly applied to the D2D link between device and . In contrast, due to the different assigned bandwidth, the CDF and the expected throughput must be re-calculated. Moreover, unlike device , device located at the origin has different distance distribution to the serving device and the devices that generate interference from receiving nodes in the star motifs. Hence, to obtain the expression of the throughput, we need to derive the Laplace transforms of the intra- and inter-cluster interference for receiving device in the chain motif.

Figure 5: Illustration of the inter- and intra-cluster interference to the representative chain motif.
Lemma 3.

For the typical device in the chain motif, the Laplace transform of the intra-cluster interference, conditioned on the distance from the typical point to the parent point , is given by

(24)

where , .

Proof:.

See Appendix -E. ∎

Corollary 5.

After using the proof in Appendix -B, the lower bound of the intra-cluster interference can be expressed as:

(25)
Lemma 4.

For the device in the chain motif, the Laplace transform of the inter-cluster interference is given by

(26)

where , .

Proof:.

See Appendix -F. ∎

Corollary 6.

After using the proof in Appendix -D, the lower bound of the inter-cluster interference experienced by the link from to is given by

(27)

As the D2D link between devices and and the link between and share the same bandwidth and the locations of and follow the same point process, the interference encountered by these two devices and can be correlated [22]. In the following theorem, we consider such interference correlation and calculate the outage probability for the chain motif.

Theorem 3.

Taking into the interference correlation between two devices and , the outage probability of the chain motif can be given by:

(28)

where , and represents the Laplace transform of the fading distribution.

Proof:.

See Appendix -G. ∎

Even though Theorem 3 characterizes the interference in presence of correlation, one can see from (3) that this expression can be computationally complex to derive. As such, in order to obtain a more tractable result, hereinafter, we consider a special case where there is no correlation between the interference encountered by devices and . Such a case has been considered in recent works such as [13] and [32]. Next, the simplified outage probability for the chain motifs is calculated.

Corollary 7.

When there is no correlation between the interference experienced by two receivers in the chain motif, the outage probability of chain motifs is

(29)

where , , and , .

Proof:.

For the D2D link from device to , the probability of the D2D link from device to meeting the minimum SINR threshold is

(30)

In , we first make a change of variables by setting , and, then, we perform one de-conditioning in terms of and another de-conditioning for , and finally convert the coordinates from Cartesian to polar. Based on the distance distribution, we have and . Then, since the outage will not occur if and only if the SINR of both links exceed the threshold level, we can obtain (7). ∎

Similar to Corollary 3, we can replace with in (2) and (7) to obtain the throughput CDF for the two D2D links in the chain motif. To calculate the throughput, we can observe that the chain motif is equivalent to a two-hop network. According to the work in [33], if a non-seeding node is connected to the seeding node through a multi-hop link with length , the achievable throughput of the non-seeding device is limited by times the minimum of all D2D link rates over the multi-hop communication network. Therefore, we can derive the expected throughput of non-seeding devices and in the following corollaries.

Corollary 8.

Since device is directly connected to seeding node , the expected throughput is