Borel Cayley Graph-Based Topology Control for Consensus Protocol in Wireless Sensor Networks

Borel Cayley graphs have been shown to be an efficient candidate topology in interconnection networks due to their small diameter, short path length, and lowdegree. In this paper, we propose topology control algorithms based onBorel Cayley graphs. In particular, we propose two methods to assign node IDs of Borel Cayley graphs as logical topologies in wireless sensor networks. The first one aims at minimizing communication distance between nodes, while the entire graph is imposed as a logical topology; while the second one aims at maximizing the number of edges of the graph to be used, while the network nodes are constrained with a finite radio transmission range. In the latter case, due to the finite transmission range, the resultant topology is an “incomplete” version of the original BCG. In both cases, we apply our algorithms in consensus protocol and compare its performance with that of the random node ID assignment and other existing topology control algorithms. Our simulation indicates that the proposed ID assignments have better performance when consensus protocols are used as a benchmark application.


Introduction
An adhoc wireless sensor network (WSN) is a self-organized and distributed network consisting of a large number of small and light sensor nodes [1,2].A sensor node includes a processor, a wireless radio, and various sensors to monitor and sense environmental parameters such as temperature, moisture, and pressure.In a WSN, sensor nodes interchange information and collaborate with each other to achieve a common mission.The flexibility, fault tolerance, high sensing fidelity, low cost, and rapid deployment characteristics of sensor networks create many new and exciting application areas for remote sensing [3].Examples of ad-hoc wireless sensor networks applications include building monitoring [4], environmental sensing [5][6][7], traffic monitoring [8], and surveillance [9].Some WSN applications require very dense networks.Hundreds to several thousands of nodes can be deployed throughout a sensor field.For example, some machine diagnosis applications use up to 3000 nodes in a 100 m by 100 m area [10] or sensors can be deployed within tens of feet of each other for object tracking [11].The topology of a large and dense sensor network is important to its performance.For example, topology control algorithms are essential in reducing energy consumption and radio interference [12], thus expanding the network's lifetime.
According to [13], topology control algorithms can be classified as location-based, direction-based, or neighborbased approaches.In all these approaches, a network topology is formed in an adhoc manner based on sensor nodes' location, direction, or some sort of neighbor ordering.However, since these approaches do not generate a predefined graph, there is no guarantee on the topology's overall properties such as a bounded diameter and average path length.
We approach topology control with predefined graphs.Structured graphs have been studied for a long time and are good candidates for interconnection networks [14,15].Various graph-based interconnection networks have been applied to wavelength division multiplexed optical networks [15], satellite constellations [16], and chip design [17].In peer-to-peer overlay network schemes, various structured graphs are investigated and compared to unstructured P2P overlay network [18].Examples of P2P overlay networks include k ring lattices with Chords [19], de Bruijn graphs with Koorde, and distance halving [20,21].There is theoretic analysis to apply de Bruin and Cayley graphs to P2P [22,23].Graph-based wireless sensor networks have also been explored by other researchers [24][25][26][27][28].In general, a predefined graph topology with its deterministic connection rule facilitates performance analysis.In addition, some offer symmetry, hierarchy, and hamiltonicity, and can have a constant low degree and existing distributed routing protocol, all preferable properties for WSNs.
We focus on Borel Cayley graphs, a member of Cayley graph family [29].In [30], it was shown that Borel Cayley graphs have potential to be efficient logical topologies for dense WSNs because of their small degree and short diameter.Also, BCGs are symmetric graphs, a property that enables distributed routing [31].One of BCG applications is the consensus protocol, a distributed node-to-node message exchange rule to drive nodes to an agreement for a quantity of interest [32].BCGs showed good performance when compared to mesh, torus, and small world networks [33].
In this paper, we propose two ID assignment algorithms for applying BCGs as logical topologies in WSNs.The first one, the chordal ring based node ID assignment algorithm, uses a specific representation (the chordal ring representation) of BCG to assign node IDs of the entire graph.Its goal is to minimize long distance communications.In the second algorithm, the distributed node ID swapping assignment, we consider the realistic constraint of network nodes' finite transmission constraint, and thus the imposition of the entire BCG is not always possible.
In the former case, when the goal is to minimize communication distance, the proposed chordal ring-based node ID assignment algorithm requires a smaller radio range (57% that of the random ID assignment).In the latter case, when the nodes are subject to a finite transmission range and the imposition of the entire BCGs is not feasible, the proposed distributed node ID swapping assignment imposes more edges of the original graph (43% more edges in comparison to random ID assignment).
It must be noted that a shorter paper describing an initial version of Section 5 was published in [34].The main differences between two papers are the addition in this paper of the experiment to evaluate a consensus protocol comparing with known topology control algorithms.
This paper is organized as follows.Section 2 reviews the basic concept of BCGs and consensus protocol.Section 3 presents the goals of this paper.Section 4 describes our proposed algorithm to reduce the average communication distance.Section 5 presents our proposed algorithm to maximize connections of BCGs under transmission range constraints.Section 6 shows our topology control's performance in a consensus protocol.Conclusions are presented in Section 7.

Preliminaries
In the following, we provide a definition of Cayley graphs, Borel subgroup, and Borel Cayley graphs.

Definition 1 (Cayley graph [29]). A graph
for some  ∈ , where (, * ) is a finite group and  ⊂  \ {}. is called the generator set of the graph and  is the identity element of the finite group (, * ).
The definition of a Cayley graph requires vertices to be elements of a group but does not specify a particular group.
Definition 2 (Borel subgroup).If  is a Borel subgroup of general linear 2 × 2 matrices set, then where a fixed parameter  ∈   \ {0, 1},  is prime, and  is the order of .That is,  is the smallest positive integer such that   = 1 (mod ).
Definition 3 (Borel Cayley graph (BCG) [29]).Let  be a Borel subgroup, and let  be a generator set such that  ⊆ \{}; then  = B(, ) is a Borel Cayley graph with vertices for the 2×2 matrix elements of  and with directed edge from V to  if  = V * , where  ̸ = V ∈ ,  ∈ , and * is a modulo- multiplication chosen as a group operation.Definition 4 (GCR [29]).A graph  is a generalized chordal ring (GCR) if nodes of  can be labeled with integers modulo the number of nodes , and there is a divisor  of  such that node  is connected to node  if and only if node  +  (mod ) is connected to node  +  (mod ).
The connection rules of elements are defined by connection constants.Connection constants of  and  +  are the same according to the definition.There are four connection constants when the graph is four regular.For example, Figure 1(a) shows a degree 4 GCR with 21 nodes and  = 3 classes.The connection rules can be defined as follows: let  = 0, 1, 2, . . ., 20.For any  ∈ , if  mod 3 =: "0" :  is connected to  + 3,  − 3,  + 4,  − 10 (mod 21) , "1" :  is connected to  + 6,  − 6,  + 7,  − 4 (mod 21) , "2" :  is connected to  + 9,  − 9,  − 7,  + 10 (mod 21) . ( The connection constants for class 0 are +3, −3, +4, and −10. A chordal ring (CR) is a special case of GCR, in which every node has +1 and −1 modulo  connections.In other words, all nodes on the circumference of the ring are connected to form a Hamiltonian cycle.A Hamiltonian cycle is a graph cycle through a graph that visits each node exactly once.

Proposition 5. For any finite Cayley graph with vertex set
and any  ∈  such that   = , there exists a GCR representation of C with divisor  = /, where  is the identity element.

Proposition 6. All degree-4 Borel Cayley graphs have CR representations.
The proofs of these propositions are given in [29] and not repeated here. is referred as the transform element.  are class representing elements.
For the simplicity of GCR representation, we chose  and   for simple representation as follows [29]: Any vertex V ∈  is represented with  and   as follows [29]: BCGs are defined over a group of matrices.The systematic representation of BCGs from the group domain to the integer domain is useful for routing because nodes are defined in the integer domain and there is a systematic description of connections.The node ID representation in GCR (ID  (V)) is denoted as follows [29]: where  is .Symmetry or vertex-transitivity is a preferable attribute for an efficient interconnection network topology.Informally, a symmetric or vertex-transitive graph looks the same from any node.This property allows to use an identical routing table at every node.Mathematically, this implies that for any two nodes  and  in the graph, there exists an automorphism of the graph that maps  to .This property is very useful for practical implementation of interconnection networks.Most of the well-known interconnection graphs, such as the toroidal mesh, hypercube, and cube-connected cycle, exhibit such property.
In Table 1, we list the parameter values for BCGs used in this paper.Parameters  and  determine  and BCG parameter .Parameters  1 and  1 were used to construct the first generator.Parameters  2 and  2 were used to construct the second generator.Using two different generators and their inverse, we construct undirected BCGs.We arbitrarily chose parameters   and   for generators.

Problem Statement
A WSN can be represented as a graph, where each node of the graph corresponds to a sensor node and each edge represents a radio connection between nodes.Obviously, for dense WSNs, the number of physical neighbors (nodes within radio range) is large.For ease of description, we call the topology representing the physical neighbors a host graph.Besides representing the number of neighbors within communication reach of each other, an edge of a host graph is also weighted by the communication distance or cost between nodes.We are interested in the problem of imposing a BCG with a small degree on a dense WSN.We call the BCG topology the target graph, and the host graph after the imposition, the resultant graph.We consider the two following cases: (i) the radio range of each node covers the whole sensor deployment area.In this case, any ID assignment always produces a resultant graph with a fully connected BCG topology.But depending on the ID assignment, the communication distances between neighbors vary.So, in this case, the goal of this ID assignment is to reduce the communication distance between connected nodes; (ii) the radio range does not cover the whole sensor deployment area.A careful node ID assignment is required to produce a resultant graph as similar as possible to the target graph (a BCG topology).In this case, the goal is to find the ID assignment that establishes the most communication edges following BCG connection rules.

Minimize the Average Communication Distance When
the Radio Range Covers the Deployment Area.Our node ID assignment problem can now be described in terms of finding the assignment that yields the minimum sum of weights of the resultant graph after imposing a target graph onto the host graph.Figure 2 illustrates these terminologies.Figure 2(a) is the host graph with each node's physical neighbors and weights on the edges that represent the connections communication cost.Let a host graph  = (, ) in the Euclidean space represent the underlying network before applying our proposed methods, with  being the set of sensor nodes and  representing the set of communication distances between nodes.We assume that the radio range of each node in the host graph is large enough to cover the whole deployment area.So the host graph is a fully connected graph.The weight of (, V), denoted by  V , represents the Euclidean distance between nodes  and V.The weight between two nodes  and V with coordinates (  ,   ) and (V  , V  ) is computed as ISRN Sensor Networks

Completing Communication Edges with Finite Radio
Range.This node ID assignment problem can be described as finding the maximum number of edges of the resultant graph after imposing a target graph onto the host graph.Figure 3 illustrates these terminologies.Figure 3(a) is the host graph that shows the physical neighbors of each node.The edge (, V) is determined by the radio range and the Euclidean distance between nodes  and V.The distance between two nodes  and V with coordinates (  ,   ) and In the remaining sections, we will propose algorithms to solve these two problems: (a) reducing the average of communication ranges of BCG-based networks and (b) completing as many as possible communication edges that follow BCG connection rules.Each problem is dealt with a distributed method.Regardless of the application type, a distributed method is preferred in wireless sensor networks.

CR Representation and Node ID Conversion between GCR and CR.
Recall that nodes of ID  in a chordal ring (CR) have +1 and −1 modulo  connections.In other words, all nodes on the circumference of the ring are connected to form a Hamiltonian cycle.In the CR representation of BCG, the transform element and class representing elements are function of BCG parameters.Let T be a transform element in CR, and let a  be the class representing elements in CR.Then, any vertex V ∈  is represented by T and a  as follows: Since there is no systematic representation for CR and no conversion method between GCR ID and CR ID, we propose (a) a CR representation in the integer domain and (b) a conversion method between GCR ID and CR ID.The number of classes, , can be different in the CR and GCR domains [29].Therefore, we use   for the number of classes in the GCR domain and   in the CR domain.The node ID representation in the CR domain (ID  (V)) is given as follows: where   is  or  and ID  (V) is the node ID of V in the CR domain.By combining ( 5) and ( 8), the conversion formulation from CR ID to GCR ID is where ID  (V) is the node ID representation in the GCR domain.
Similarly, the conversion formulation from GCR ID to CR ID is Based on the set of class representing elements a given in [29], the integers   and   can be calculated from Algorithm 1.Note that Algorithm 1 does not have any infinite loop since the matrix V is T   multiplied by a   .We already discussed the relationship between GCR ID and the CR ID.The GCR and CR representations of BCG have different advantages.The GCR representation of BCG has an optimal routing algorithm to identify the shortest paths between any sources and destination pairs [31].On the other hand, the CR representation supports a Hamiltonian cycle and a simple suboptimal routing algorithm.Therefore, we assign IDs in the CR domain to minimize the communication distance and then map them to the GCR domain for optimal routing performance.

Algorithm.
The chordal ring-based method (CR assignment) consists of three main steps: (1) making a Hamiltonian cycle in the CR domain; (2) converting node IDs from the CR domain to the GCR domain; (3) establishing edges.
Figure 4 illustrates the CR assignment algorithm.There, the algorithm starts by choosing the lowest weighted edge of sensor node ID 0 and then assigns the ID 1 plus node ID 0 to the corresponding node (Figure 4(b)).The next node selects the edge with the lowest weight among edges that are not connected to already assigned nodes.The algorithm repeats until the last node ID is assigned (Figure 4(c)).Then, the node IDs are mapped from the CR domain to the GCR domain (Figure 4(d)).Finally, the actual connections are established using GCR connection constants following the BCG connection rule (Figure 4(e)).Algorithm 2 summarizes the CR assignment.
The CR assignment guarantees that at least the lowest weighted edges of consecutive node IDs are selected as communication links, except for the edge between the starting node and the last assigned node.As a result, we optimize two out of four edges per node excluding first and last ones.We expect the total weights of edges from the CR assignment to be smaller than those of a random ID assignment.Moreover, Require: T is nonsingular.
the CR assignment is a fully distributed algorithm and does not require preassigned node IDs.

Average Communication Distance.
We define the average communication distance as the average weights of the target graph communication distance between pairs of nodes.Our goal is to minimize the average communication distance.To evaluate our assignment algorithm, we calculated the expected communication distance analytically.From (6), the expected communication distance between randomly chosen positions is as follows: where  and  represent the horizontal and vertical dimensions, respectively.Figure 5 shows the average communication distance for each node ID assignment algorithm where ,  = 100 m.The random assignment algorithm uniformly and randomly assigns a unique BCG node ID to all the sensors.We call the resultant network topology with Random assignment BCG-0 and the resultant network topology with CR assignment BCG-1.The average weight of the random assignment and the expected distance are both approximately equal to 52.
We also calculated the standard deviation for the average communication distance from 100 CR assignment samples.The results show that selection of the initial node does not affect the average communication distance.For graphs of sizes  = 272, 506, 1081, 1474, and 2265, the standard deviations for the CR assignment were 0.58, 0.53, 0.62, 0.72, and 0.60, respectively.

Distributed Node ID Swapping Assignment
In the previous section, we showed the CR assignment algorithm for reducing the average communication distance without radio range constraint.In this section, we propose the distributed node ID swapping assignment for completing communication edges with radio range constraint.The distributed node ID swapping assignment (Dist-swap assignment) consists of four main steps executed by the current node: (1) broadcast its node ID to its physical neighbors; (2) collect IDs of physical neighbors that can be swapped; (3) select the best-fit node ID to be swapped with; (4) swap node IDs and update each logical neighbor table of its physical neighbors.

Terminologies.
We denote the logical node ID of node  by  id and define the following.
(i) ( id ): set of logical neighbors of node ID  id in the target graph; (ii) (): set of logical IDs of node 's physical neighbors; (iii) ( id , ) = |( id ) ∩ ()|: the number of logical neighbors of node ID  id that are also physical neighbors of node .
We define four different packet types.
(1) Token packet: forwarded by nodes and used to initiate the node swapping algorithm by the node recipient.Also contains counter reptCnt.
(2) Info packet: used by the current node holding the token to broadcast  id to its physical neighbors so as to identify candidate nodes to be swapped.
( (4) Swap packet: used by the current node  to announce to its physical neighbors swapping with node .Contains the pair ( id ,  id ).Nodes having those node IDs in ( ) need to update the swapped IDs.

Assumption.
We assume that (a) a host graph is a connected graph (there is a path between all node pairs), (b) nodes have preassigned unique IDs ranging from 0 to  − 1, (c) () (a set of logical IDs of its physical neighbors) at each node  ∈  is obtained before the swapping process, and (d) the order of nodes to perform operation is based on token method.

Algorithm.
The Dist-swap assignment is performed by one node at a time using a token method (i.e., there is one token in a network, and only the node holding the token executes the swapping algorithm).To prevent an infinite loop, we rely on a token counter (repCnt) that tracks the number of times the token is being passed around.At the beginning of the Dist-swap assignment, the node that starts the operation sets repCnt to repTotal that was heuristically selected based on previous simulation studies.As the token travels around the network, each node decreases repCnt by one before forwarding the token to the next node, and the Dist-swap assignment ends once repCnt reaches zero.We describe in detail the Dist-swap assignment as follows: (i) the following describes the Dist-swap assignment operation after node  receives a Token packet.
(1) A node  receives a Token packet.(2) Node  collects the Request packets.
(a) If there is no Request packet received, forward the Token packet randomly to one of its physical neighbors. (

ISRN Sensor Networks
(1) A node  receives an Info packet.
(2) Node  calculates the number of its logical neighbors among its physical neighbors if it was assigned node ID  id .
(a) If that number is larger than or equal to the number of logical neighbors of the current ID, send the Request packet.(b) Otherwise, ignore the Info packet.
To determine whether or not to reply with a Request packet, node  checks the number of its logical neighbors that would be assigned to its physical neighbors if it were assigned node ID  id .If it is larger than that with the current node ID, node  sends a Request packet.
(iii) The following shows the operation when node  receives a Swap packet.
(1) A node  receives a Swap packet that contains node IDs ( id ,  id ).
(a) If  id is equal to  id , change  id to  id and send a Swap packet with node IDs ( id ,  id ) to its physical neighbors.(b) If  id is equal to  id , ignore the Swap packet.(c) Otherwise, change  id to  id in ().
Swap packet contains node IDs ( id ,  id ).If  id is the same as  id , then the Request packet sent by  is accepted.So, node  changes its node ID to  id and sends a Swap packet with ( id ,  id ) to its physical neighbors so that they update their ( )s.When  id is the same as  id , node  ignores that packet.Otherwise, node  updates () to reflect that node ID  id is changed to  id .
First, node  receiving a Token packet decides whether or not to perform the swapping process based on the current number of logical neighbors among its physical neighbors ((5, )).The number of logical neighbors among the physical neighbors is not the same as the maximum number of logical neighbors 4 since () = {4, 6, 8, 13, 15}, (5) = {1, 3, 7, 9} and (5, ) = 0. So, node  sends an Info packet to its physical neighbor nodes.Swapping node IDs is beneficial to node  because (8, ) is zero and (5, ) is two, which means node  gains two more logical neighbors after swapping IDs with node ID 5. Thus, node  sends the Request packet to node .Node  also sends a Request packet to node  through the same process.Finally, node  determines to  swap IDs with node  because (8, ) + (5, ) = 4 is greater than (15, ) + (5, ) = 3.

Completeness of Resultant Graph.
Recall that an efficient ID assignment method should incorporate as many as possible target graph edges in the resultant graph from a given host graph, where sensor nodes are limited to a specific radio range.Therefore, to measure the performance of our assignment methods, we define   , the ratio of the number of edges of the resultant graph over that of the target graph.That is, For a given radio transmission range, the assignment method with the largest   is the most efficient since most edges of the target graphs are incorporated in the resultant graph.We define the resultant network topology with Distswap assignment BCG-2.Figure 7 shows   for BCGs target graphs with  = 1081.The Dist-swap assignment (BCG-2) shows larger   than that of the random assignment (BCG-0).Especially, with 50 m radio range in 100 m by 100 m area, it showed 43% more edges.
When applying BCGs to wireless sensor networks, we also need to consider network connectivity.Because the Distswap assignment maximizes   without considering network connectivity, we apply a random node selection algorithm on processed nodes that do not have the right number of logical neighbors.We define the resultant graph from this method BCG-3.Table 2 summarizes resultant graphs from our proposed algorithms.
Figure 8 shows the percentage of connected network function of assignments for 100 network samples.BCG-0 with 105 m radio range results in a 100% connected network.BCG-2 and BCG-3 require 65 m and 10 m radio   range, respectively, to produce a connected network.All connections of BCG-2 are satisfied by the connection rules of BCGs.However, BCG-3 has non-BCG connections since some are logical neighbors randomly selected for improved connectivity.

Simulation
6.1.Consensus Protocol.We use the consensus protocol as a benchmark to evaluate our topology control algorithms.Consensus protocol is a distributed node-to-node message exchange rule to reach a network-wide agreement over a quantity of interest (e.g., average of sensory data).Consensus protocol has a long history in distributed computing and has been used in a variety of applications.Moreover, it has been widely accepted as a reliable measure of data fusion performance of network topologies [35,36].In WSN, consensus protocol research focused on time synchronization and gossip algorithms [37][38][39].Readers interested in more detail on consensus protocol and its application are referred to [40].
The consensus agreement value can be the average, the maximum, the minimum, or any function and only depends on the initial states of the nodes in the network.Furthermore, the speed of consensus is a good measure of the efficiency of a network topology to distribute information.In this study, we consider the distributed average consensus protocol proposed in [40] which we summarize next.
Let us consider a network system of which logical network topology is represented by an unweighted, undirected graph.Each node V ∈  in the system communicates its state value  V to its immediate neighbors (V) :=  :  V, ∈ (), where  is an edge set.At each iteration , nodes exchange their current state values  V () with their immediate neighbors.Given the state values   () received from their neighbors  ∈ (V), each node  updates its state according to where  is typically 0 < 1/ < 1/2 max () and  max () is the largest nodal degree.We set  to 2 ×  max () + 1 in this paper.Consensus convergence can be controlled by  value.However, finding an optimal  value is out of scope for this paper.
Following the method in [41], we use the average consensus protocol (14) to measure the information fusion performance of the network generated by the proposed topology control protocol.BCGs have already been shown to exhibit better consensus protocol performance than mesh, torus, and small world networks [33].However, that research compared only topology level not considering nodes' physical geometric information and radio range.In this paper, we compare a BCG with other network topologies when applied to wireless sensor network.

Simulation
Setup.We evaluated our proposed topology control algorithms in terms of consensus speed and power consumption.The simulations were executed on 100 host graphs, each with sensors uniformly and randomly distributed over a 100 m × 100 m area.
Consensus protocol was initialized with nodes state value set to integers randomly chosen between −5 and 5 inclusive.We declared a network topology to have reached an agreement once all node values equal the average of all initial state values within a precision of 0.001.The performance of the consensus protocol was measured as the number of steps needed to reach a network-wide consensus.We also computed the energy needed for all the nodes in the network to reach a consensus using the average consensus protocol.To do this, we utilized the radio model described in [42].For the sake of simplicity, we only considered the energy consumptions from data transmission (  ) and reception (  ) defined as follows: where  is the path loss exponent which, in typical, ranges from 2 to 6.The constants  1 ,  2 , and  are the energy dissipated by the transmitter module, transmit amplifier, and the receiver module, respectively.We denote the estimated distance between nodes V and  by (V, ) and the length of the message by .We set  = 2, equivalent to the free-space pass loss model, and assume that a sensor node V, regardless of the topology control protocols considered, can adjust its transmission power to reach its neighbors .The parameters used in the simulation are summarized in Table 3.
Using the average consensus protocol, a node transmits and receives to and from each of its neighbors at every iteration.Thus, the node's energy consumed by the network at the th iteration is given by where deg(V) is the number of logical neighbors, (V, ) is a logical connection, and  denotes the edge set.The average nodal energy consumption is given by  0 = ∑ V∈  0 (V)/||.We call this power consumption model Power model 0.
In wireless sensor networks, a network can support multicasting routing to transmit at once to multiple nodes inside the radio range.Then power consumption is calculated not by each communication edge but by the maximum distance communication edge as follows: We call this power consumption model Power model 1.

Comparison between BCG-0 and BCG-1.
Figure 9 shows the histograms of the communication distance for  = 1081 generated by the proposed node ID assignment algorithms.The histogram of the CR assignment (BCG-1) exhibits a right skewed distribution with a high frequency of short edges.The Dist-swap assignment (BCG-2) shows that all connection distance are under 80 m.We obtained similar histograms for the network sizes listed in Table 1.We compared power consumption between the random assignment and the CR assignment because they require a radio range that covers the whole node deployment area.Figure 10 summarizes the resulting consensus protocol energy consumption.We found that BCG-1 consumed 8% less energy than BCG-0 with power model 0 and 2% less with power model 1.

Comparison between Our Proposed Topology Control
and Existing Topology Controls.We compared our topology control algorithms with existing topology controls.The evaluation was done in terms of diameter, average path length, consensus steps, and nodal energy consumption.We used the following topologies for comparison against our approach.
(i) Max: network where all nodes within the maximum radio range are logical neighbors.Without any topology control, a network is called Max topology.
(ii) E-MST [43]: the Euclidean minimum spanning tree is a minimum spanning tree whose edge weight is the distance between nodes.This algorithm minimizes the summed weight of edges and generates a connected network.
(iii) K-Neigh [44]: network where the number of neighbors of a node equals to or is slightly below a given value .
(iv) Gabriel Graph [45]: a planar graph (no edges cross one another) supports geographical routings and is defined such that an edge (V, ) exists if no other node is inside the circle with the diameter V.
Figure 11 provides illustrations of the topologies constructed with (a) the random assignment (BCG-0), (b) the Dist-swapping assignment (BCG-2), (c) Max, (d) E-MST, (e) K-Neigh, and (f) Gabriel graph.For ease of description, we show sample graphs with  = 272.However, we simulated with  = 1081 in this section.
Figures 12 and 13 show the diameter and average path length function of the transmission range, respectively.We only compared the cases where the resultant graph is a connected graph.From these figures, we confirm again that the BCGs have the smallest diameter and the shortest average path length for a given radio range except Max topology.This is because as a radio range increases, BCGs have a constant number of logical neighbors, while Max topology has much more logical neighbors than others.
From Figure 14, at radio range 30 m, consensus protocol convergence for BCG-3 was 35.5, 2932, and 15.9 times faster than that of Gabriel, E-MST, and K-Neigh topologies.At radio range 50 m, convergence of BCG-3 was 81.2, 6756.9, and 39.9 times faster than that of Gabriel, E-MST, and K-Neigh topologies.Even though Max topology has the best consensus steps, due to its large number of logical neighbors, it consumed 23 times more nodal power than BCG in Figure 15.
When considering multicast routing (Power model 1), the Max topology has an obvious advantage as only the maximum radio range is used in the power consumption computation.However, even with this advantage, our simulations showed that BCG continues to outperform the Max topology in Figure 16.This better performance can be explained by the   fact that the Max topology has a larger number of logical neighbors and hence larger power is consumed at the receiver modules.

Conclusions
Our goal was to use Borel Cayley graphs, a family of predefined graph topology, to overlay on dense wireless sensor  networks for topology control.In this paper, we proposed node ID assignment methods to reduce the communication distance between nodes or to increase the number of logical connections following predefined graph connection rules in a distributed manner.In particular, we proposed the chordal ring-based node ID assignment method and the distributed node ID swapping assignment method.

Figure 2 (
b) is the target graph.Figures2(c) and 2(d) show two example resultant graphs, where each graph has a different sum of weights, depending on how node IDs were assigned.

2 Figure 2 :
Figure 2: Graph representation with radio range enough to cover the whole deployment area.
Figure 3(b)   is the target graph.Figures3(c) and3(d)  show two different ways to impose the target graph on the host graph, where each results in a different number of edges, depending on how node IDs were assigned.
Figure 1(b) shows the CR representation of a Borel Cayley graph in the integer domain.The graphs shown in Figure 1 represent the same BCG represented in both the CR and GCR domains.

Figure 3 :
Figure 3: Graph representation with finite radio range.

Figure 4 :
Figure 4: Illustration of CR assignment.Note that each alphabet represents a physical sensor node.
(a) If |( id )| = ( id , ), forward the Token packet randomly to one of its physical neighbors.(b) Otherwise, send an Info packet to all its physical neighbors.

Figure 6 :
Figure 6: Host graph used as example of Dist-swap assignment.

Figure 7 :
Figure 7: Edge construction percentage with distributed node ID assignment.

Figure 8 :
Figure 8: Connected network percentage with node ID assignments.

Figure 9 :
Figure 9: Histograms of communication distance by the proposed algorithms with  = 1081.

Figure 13 :
Figure 13: Comparison of average path lengths.
First, node  determines whether or not it needs to execute the swapping process based on its current number of logical neighbor IDs in the physical neighbors and its target number of logical neighbors in the target graph.If those are not the same, node  sends an Info packet to its physical neighbors.After receiving Request packets from its physical neighbors, node  determines which node ID swapping is the most beneficial.Because the Request packet from node  contains  id , ( id , ), and ( id , ), node  can calculate the change in the number of logical neighbors from the physical neighbors when node IDs  id and ) Node  creates a list of candidate nodes.A node  is a candidate node if ( id , ) + ( id , ) − ( id , ) − ( id , ) > 0.(4) Node  sends Swap packet with ( id ,  id ).(5) Node  sends Token packet to node .idare swapped.If a candidate node for swapping exists, node  sends a Swap packet with its own node ID and the selected node ID to its neighbors.(ii)The following illustrates the operation after node  receives an Info packet.

Table 2 :
Summary of our proposed algorithms.

Table 3 :
Radio model parameters.