Achieving Small World Properties using BioInspired Techniques in Wireless Networks
Abstract
It is highly desirable and challenging for a wireless ad hoc network to have selforganization properties in order to achieve network wide characteristics. Studies have shown that Small World properties, primarily low average path length and high clustering coefficient, are desired properties for networks in general. However, due to the spatial nature of the wireless networks, achieving small world properties remains highly challenging. Studies also show that, wireless ad hoc networks with small world properties show a degree distribution that lies between geometric and power law. In this paper, we show that in a wireless ad hoc network with nonuniform node density with only local information, we can significantly reduce the average path length and retain the clustering coefficient. To achieve our goal, our algorithm first identifies logical regions using Lateral Inhibition technique, then identifies the nodes that beamform and finally the beam properties using Flocking. We use Lateral Inhibition and Flocking because they enable us to use local state information as opposed to other techniques. We support our work with simulation results and analysis, which show that a reduction of up to can be achieved for a highdensity network. We also show the effect of used to create regions on average path length, clustering coefficient and connectivity.
1 Introduction
Decades of academic and industrial research in wireless networks [Akyildiz] has led to the tremendous growth of wireless networks requiring researchers to address manageability and scalability issues. Due to these issues, most of the research work has been oriented towards autonomous wireless networks. The autonomous behavior of the wireless nodes made decentralized computing and cost efficient topology deployment possible [Dressler]. It was also proved that selforganization of the network can lead to better performance.
An attractive model to achieve better network performance is the Small World network. Small world networks are characterized by reduced Average Path Length () and high Clustering Coefficient (). Here, the is the mean of between all pairs of nodes in the network. Consider a node, , with neighbors. In the subgraph of these nodes, the is defined as the fraction of links that exist to the maximum number of links that could have existed in the subgraph. Drawing inspiration from the experimental work of Stanley Milgram [Milgram], Watts et al [Watts] proposed a model that could achieve small world properties. In the model, Watts et al proposed, small world properties could be reached by randomly rewiring a few existing links within the network. Watts et al showed that the dynamics of these small world networks lie between that of a regular network and a random network [Watts, Wattsbook]. To prove the findings, however, Watts et al used a regular wired network and called the rewired links as shortcuts. Many complex real world networks such as internet, biological networks, food web and social networks also demonstrate small world properties [Barabasi, BarabasiAlbert, Newman]. In real world networks where there is a nonuniform distribution of nodes, these real world networks were shown to exhibit the properties of scalefree networks marked by power law degree distribution. Section 6.1 provides more details on small world networks.
In a wireless ad hoc network, achieving small world properties can help us in many ways. Having a low would increase the performance of the network in terms of communication [Brust, KleinbergSWP] (reduced traffic per unit area, reduced congestion and reduced signal interference), low latency and reduce the overall energy consumption in the network during the data communication. On the other hand, maintaining the would ensure connectivity to the neighborhood and would make the network resilient [Albert, GuidoniLoureiro]. However, Watts’ model cannot be applied directly to wireless ad hoc networks because of the spatial nature of such networks. In wireless ad hoc networks, addition of a shortcut between any two nodes should depend on the distance between two nodes. Helmy in [Helmy] first studied the effect of adding few distancelimited links in the network. He showed that, upon introduction of distancelimited links, wireless ad hoc networks show small world properties. He concluded that, when the shortcut lengths are th of the network diameter, there is a maximum reduction in the . Thus, proving that realization of small world properties in a wireless ad hoc network depends crucially on the length of shortcuts created among nodes. Another important factor in the realization of small world properties is the choice of nodes among which shortcuts are to be created. One method to obtain these nodes is that of preferential attachment [Simon, BarabasiAlbert], typically observed in real world networks, wherein links are created to nodes with high structural importance. It was shown that, analogous to real world networks, using preferential attachment for creation of distancelimited links in a spatial network resulted in reduced network diameter [BarthelemyMark, Manna]. This was accompanied by high clustering coefficient and a shift in the node degree distribution towards power law. These results motivate us to say that, creation of links to nodes having high structural importance in the network can result in the desired small world characteristics.
The creation of a wireless ad hoc network with the small world properties also depends on the manner in which distancelimited links are added. Such links can be added through different techniques like: 1) creating the directional beam using the same power as when the node was operating in the omnidirectional mode; 2) increasing the omnidirectional transmission range of the node; 3) introducing of few long wired links [Sharma]; 4) introducing special nodes with higher omnidirectional transmission range deterministically in the network [GuidoniLoureiro]; 5) using another antenna for beamforming in addition to the omnidirectional antenna.
Talking about the selforganization characteristics of the nodes, only techniques one and two mentioned above qualify. However, even though other techniques help in achieving desired network characteristics, they lack selforganization capabilities. In addition, the second technique suffers from the problem of early death of the node due to increased energy consumption. Thus leaving us only the first technique. Achieving reorganization or rewiring in a wireless ad hoc network through the first technique is hard due to the spatial nature of the wireless ad hoc network. Finding the beam direction, the beam length and determining the new neighborhood are primary issues associated with rewiring in a wireless ad hoc network. Our previous study, [Banerjee], proved that the use of distancelimited long links in wireless ad hoc network to achieve small world properties is beneficial, (Cf. Fig. 1).
Motivated by this, in this study, we investigate how we can increase connectivity, reduce the and almost maintain the in a nonuniformly distributed wireless ad hoc network. We thus propose an algorithm that achieves these goals by creating longrange directional beams between nodes that have low and high structural importance. The decentralized computing and selforganizing requirements of such an approach motivate us to draw inspirations from nature. We further propose that Lateral Inhibition [Lawrence, Nagpal, NagpalMamei, Afek] and Flocking [Reynolds], in conjunction with the centrality concept of graph theory, can provide valuable insights in building a solution to our problem.
We use Lateral Inhibition to create small logical regions within a network. The use of Lateral Inhibition not only reduces the message complexity but also enables us to apply the Flocking rule analogy successfully. We use analogy of Flocking rules to identify the nodes that beamform and the beam properties. According to the rules, explained later in section 6.4, it is important to identify stray nodes, align the nodes and move them towards the centroid of their neighborhood. Analogous to this, after region formation in a nonuniformly distributed wireless ad hoc network, we use Flocking rules to identify the beamforming nodes and direct the beams of these beamforming nodes towards the centroid of the region. The centroid node in the region has a high structural importance. Beamforming towards the centroid node of the region contributes towards reducing the because the centroid node of the region is the most connected node and has the highest Closeness Centrality measure. Thus, beamforming towards the centroid node is the preferential attachment behavior of the beamforming node, thereby making centroid finding a prerequisite to Flocking. In a distributed system where nodes only have local information and lack facilities, exact centroid node identification of the region is challenging. We can only make an estimate to the centroid node location in the region. We, therefore, use the selforganizing virtual coordinate scheme combined with the centrality concepts to identify the centroid nodes.
Thus, our algorithm design is such that it first identifies regions using Lateral Inhibition, then identifies the centroid nodes of the regions and then uses the analogy of flocking rules to identify the nodes that will beamform along with their beam properties. Section 2 gives a formal description of our proposed algorithm.
The organization of rest of this paper is as follows. Section 2 presents the assumptions used for the proposed algorithm along with the algorithm specifications. Section 3 presents the formal definitions. Section 4 and 5 discuss the simulation setup and the results respectively. For the readers who are unfamiliar with the concepts used in this paper, we provide a detailed description of the same in section 6. We finally conclude our work in the section 8 after providing insights to some future research directions in section 7.
2 Assumptions and Algorithm
2.1 Assumptions
To address issues mentioned in the Introduction, we focus ourselves towards the deployment of homogenous and autonomous wireless ad hoc nodes with no central entity controlling the nodes. This type of deployment enables us to easily apply selforganizing features, achieve global consensus with very limited local information, make any eligible node the group leader, make the system highly fault tolerant, ease the topological maintenance, lower the deployment cost and extend to incorporate the mobility of the nodes in the future. Further, the nodes are set to have an omnidirectional transmission range . We assume a nonuniform distribution of nodes generated using thinning process defined by Bettstetter et al [BettstetterGyarmati]. The nonuniform distribution of nodes allows us to realize scenarios that are more realistic. The algorithm proposed by Bettstetter et al proceeds by removing nodes which have less than neighbors within a transmission range (ref. section 6.6). Further, we assume the deployment of the nodes on a 2D plane of area .
As part of our network setup, our algorithm assumes each node to have an antenna consisting of isotropic elements. The use of single antenna element results into omnidirectional beam while use of more than one antenna element results into a longrange directional beam. A node, however, decides to use more than one antenna element using simple local rules mentioned later in this section. The nodes use beamforming only to transmit data but use omnidirectional beams for reception. We have used the Sector model
Further, we assume that the nodes lack facilities and global network knowledge. To achieve our goal, it is thus first essential to know what information can be used by the nodes. We limit a node to use local information along with that of its one hop neighborhood. Determining single hop neighborhood to build the local information is thus essential for the correct operation of the algorithm. Various studies have proposed many neighborhood discovery mechanisms, eg. [Vasudevan], and have carefully analyzed them. Therefore, for our approach, we assume that all the nodes have information about their neighborhood.
Notation  Meaning  Notation  Meaning 

simulation area  gradient  
network with set of vertices and set of  maximum gradient  
edges  Egocentric Betweenness of w.r.t. its  
region with set of  cluster  
vertices and set of edges  between node and  
number of regions formed  virtual coordinates of in the region  
node  updated virtual coordinates of in the  
node in region  region  
transmission radius  error margin  
Bettstetter transmission radius  max antenna elements available with  
average node density  number of antenna elements used by  
identification number of node  to beamform  
neighbor list of  set of centroid nodes reachable from  
neighbor list of in the region  with their that are within  
minimum number of neighbors used for  hops from when is not beamforming  
creating a nonuniform distribution  set of centroid nodes reachable from  
size of , i.e., degree of  with their when is beamforming  
set of all region heads  beam direction, i.e., the sector  
head node of the region  boresight direction  
set of all centroid nodes  beam length  
centroid node of the region  beam width  
set of all peripheral nodes  Average Path Length  
set of peripheral nodes in the region  Clustering Coefficient  
peripheral node  Uniform Linear Antenna Array  
peripheral neighbor of  Giant Strongly Connected Component  
Set of nodes neither in nor in  Giant In Component 
It is also essential to address the selforganizing paradigms, [Prehofer], to claim for the selforganizing behavior of the network. Prehofer et al’s [Prehofer] paradigms state: designing local rules to achieve global properties, implicit coordination, minimizing the use of historic information about the state of the network and designing an algorithm that changes with environment parameters. Our algorithm uses only locally available information to determine the beamforming nodes, beam properties and the regions. The nodes implicitly coordinate with their neighbors to determine the node with the highest from the centroid of the region. For a given region, the nodes also coordinate implicitly to determine the centroid node of that region. The current discussion focuses on a static network. In dynamic network scenarios, optimizing the extent of reconfiguration to deal with frequent changes in state information is likely to be a crucial factor. We leave this for future investigation but offer some insights in section 7.
We further describe the system model and the algorithm in the following sections.
2.2 System Model
Given a network, , where is the set of vertices and is the set of edges, we visualize as a network consisting of logical regions, , i.e., . Each region, , consists of the set of nodes, and , and set of edges, and . All vertices in are located within hops of a head node, . As a part of our algorithm, we use Lateral Inhibition to identify regions and regional heads.
We characterize the set of vertices, , into three sets. These are termed as the Peripheral node set, the Centroid node set and the Standard node set. We provide separate role to the nodes in these sets. The Peripheral nodes set () contains the nodes that beamform. The Centroid node set () contains the nodes towards which the nodes in the Peripheral node set beamform. We call the set of remaining nodes, , as the Standard node set. Further, we call nodes in these sets as the peripheral nodes, the centroid nodes and the standard nodes respectively.
Mathematically, Closeness centrality of a node, , in a graph is equal to , where is the between nodes and . The node having maximum Closeness Centrality is the centroid of the graph and has a high structural importance. For the vertex sets defined above, nodes in the set have lowest value of Closeness Centrality, i.e., . However, the nodes in the set have highest value of closeness centrality, i.e., }. A node in beamforms towards a node in in order to minimize the distance to other nodes and reduce .
The directional beam is modeled using Sector model, i.e., for a given directional beam length , the corresponding beam width, , is
(1) 
In realistic antenna model, as beam length of the directional antenna is dependent on the number of antenna elements used, , the corresponding value of used is .
Further, table 1 lists the notations used in this paper.
2.3 Algorithm
We divide our approach into two parts:

Use of Lateral Inhibition technique and selforganizing virtual coordinate scheme for the identification of regions and the centroid nodes of the regions, so that there are less message overheads and nodes can beamform towards the centroid node to achieve reduced . Section 2.3.1 provides more details.

Use of flocking rules to identify the nodes that beamform, to determine beam properties that realize small world properties and improve connectivity. Section 2.3.2 provides more details.
We describe these parts in detail in the next sub sections.
Region formation and Centroid finding
The Closeness Centrality [Freeman, Freemanlc] identifies the structural importance of the node in the network. The node with the highest Closeness Centrality value is the most central node in the network. Through this node, the spread of the information to other nodes is quick. To determine the Closeness Centrality of the node, the node requires the knowledge of other nodes in the region as suggested by the definition of Closeness Centrality, (ref. section 6.5.3). This makes the Closeness Centrality a global measure. Storing information about all the nodes in the network can consume a lot of node’s memory. When there is lack of global information, gathering such information can also be time consuming and the message complexity could be high. To overcome these problems, we create small logical regions. The creation of regions not only reduces the message complexity of the network but also reduces the effect on the due to the failure of a node, thereby making the network more manageable, efficient and tolerant to failures [BrustRibeiro]. Some algorithms designed in this direction were centralized. The Base Station chose the region heads based on the energy and the position of the nodes. Other techniques use either the transmission power or the degree or the mobility, eg., [BrustAndronache]. On the contrary to centralized approaches, some algorithms were either distributed, [Heinzelman], or probabilistic [Younis].
We thus divide this part into two, identification of regions using Lateral Inhibition and identification of centroid node in the region. As we only have local information, we use degree of the node in the Lateral Inhibition process.
For Lateral Inhibition, we consider that a node broadcasts and stores a message containing following information: the identity of the head node to which is associated (), its from and the degree of (), where . Initially, all the nodes, , consider themselves as heads, i.e. , and store their own information, i.e., , and . Each node, , then broadcasts this information to its neighbors, . Similarly, receives information from each of its neighbors and subsequently updates the information stored in it. Thus, a node replaces its stored values, if the stored degree, , is less than that of the received value and is less than , where is the gradient or the desired size of the regions. Further, if the stored and the received are same, the node decides to update the stored information based on lower value. If the is also same, then the node randomly decides to update the stored information to received information. The node then broadcasts the updated information after incrementing the by 1. Subsequently, removes itself from , i.e., , and inhibits itself from acting as the regional head. The process continues until all the nodes within hops from the maximum degree node reach a consensus about the head node. Due to , the algorithm assigns same to all the nodes within hops of the head node. We call the nodes having same to belong to one region, . The nodes lying at different from the virtually creates a gradient of different hops around , (Cf. Fig. e). In the end, the algorithm tags a node with no neighborhood as the head as it has remained uninhibited, (Cf. Fig. c). The regions created differ from other Lateral Inhibition algorithms, [Afek], in a way that our algorithm creates regions that are not limited to 1 hop, (Cf. Fig. c and Fig. d). However, the Lateral Inhibition technique does not guarantee that the head nodes identified above have a high Closeness Centrality value and are the most central nodes, (Cf. Fig. 2).
We thus now describe the steps for the centroid node identification in a given region, , created using Lateral Inhibition described earlier. Due to the global properties of the Closeness Centrality and unavailability of any facilities within the nodes, we take insights from existing algorithms on selforganizing virtual coordinate systems. In selforganizing virtual coordinate system, the nodes identify their own coordinates relative to their neighborhood in the network. We however, make use of selforganizing virtual coordinate system to calculate centroid of the region. Existing techniques on selforganizing virtual coordinate system include [Capkun, Caruso, Leong, Dabek, Rao, Priyantha, Watteyne, Awad]. These studies deploy various mechanisms to reach consensus. We use a method for achieving consensus on centroid location based on selforganizing virtual coordinate techniques that rely on averaging of local neighborhood values [Rao, Watteyne]. This allows us to limit the information required to a single hop, and thereby have minimum communication overheads.
Thus, in our algorithm, all nodes in assign themselves randomly selected virtual coordinates, . The identity of the nodes in the virtual coordinate system, however, remains the same. The nodes then communicate to their neighbors in these coordinates, i.e., . Using the coordinates of their local neighborhood, the nodes compute an average of the coordinates, , and broadcast the average coordinates to their neighbors. The neighbors in turn use these coordinates to compute a new average. This process continues until all nodes in the region reach consensus of having same average coordinates of the centroid.
The selforganizing virtual coordinate technique reveals the location of the centroid node in the selforganizing virtual coordinate system but not the identity of the node that is to be termed as centroid. In order to identify the centroid node of the region, nodes use their initially assigned virtual coordinates and the newly found average coordinates. Each node checks if , where is the error margin, and declares itself as the centroid. This process might result into multiple nodes declaring themselves as the centroid as two or more nodes can lie within the range of . To avoid this, a node also considers its Degree and Egocentric Betweenness
This broadcasting of the centroid node information enables the nodes to build for future use. is the set of centroid nodes within hops of the node , where . Algorithm 1 represents the algorithmic description of the region formation and the centroid identification process. The Fig. f shows the centroid nodes for the regions identified in the Fig. c.
Beamforming
In this part, we describe the steps involved in beamforming. According to the results of [Helmy], it requires only a small fraction of nodes with long link capabilities to achieve small world properties. In a selforganizing environment where all nodes possess beamforming capabilities, it is essential to identify nodes that create longrange beams along with the direction and the width of the beam. Flocking provides us with valuable insights in determining the answers to these questions. We use insights from the Alignment rule of Flocking to identify the set . Alignment in Flocking is the change in the direction of the node to match its neighbors, in other words the change in the orientation of the node. Further, Alignment rule is, the node has to decide to change the direction and has to find the new direction. We modify the Alignment rule and say that our Alignment rule is only limited to the decision of whether to create the beam or not. The Alignment rule we apply is, thus, to identify the set of peripheral nodes, in the region . Our algorithm uses the of the neighborhood nodes to decide whether or not the node is a peripheral node, , of the region . If all of the node have less than or equal to the node’s to the , then the node declares itself as a peripheral node. i.e., for a given region with centroid , . This implies that, a single unconnected node will become a peripheral node because it does not have any neighborhood. Further, we can also infer that two peripheral nodes can be neighbors of each other due to the equality in the condition.
The peripheral nodes randomly choose the number of antenna elements, , and use the above rules to beamform. Considering to be equal to in a Sector model, by keeping constant power as used for omnidirectional beam, we can easily compute from eq. (1) as . From this we infer that, to cover all the directions, minimum number of sectors that we need to consider is . The dependency of and on affects the connectivity of the network. The Fig. 5(a) shows the variation in and when . When is smaller, i.e., when we use less number of antenna elements, the probability of connecting to the neighbors is high as the beam is wider, (Cf. Fig. 5(b)). However, when is longer, i.e., when we use more antenna elements, the probability of connecting to a neighbor is low as the beam is narrower, (Cf. Fig. 5(c)).
As the number of sectors increase exponentially with an increase in the number of antenna elements, there is an increase in the time taken to decide the best sector. Checking all the sectors formed for all requires a test of sectors. The complexity of such a test is . This results into more energy consumption at the node. To reduce this energy consumption and the complexity to , our algorithm randomly selects the number of antenna elements, , and only tests the corresponding set of sectors.
Nonuniformity reduces the size of the giant component in the wireless ad hoc network. It is thus important for the nodes to find different network components and connect them using beamforming. Separation rule of Flocking provides us insight towards this problem. Separation rule states that the nodes should maintain certain distance with their neighbors. Our algorithm applies similar analogy to address the connectivity issue. We say, in order to increase connectivity, nodes create beam in different directions from their peripheral neighbors. Consider as a peripheral neighbor of then for all ’s, must hold. Here is the boresight direction. To make this decision, if of a decides to create the beam in certain direction, informs about the chosen direction before it actually creates the beam. then tries to create the beam in another direction. Further, gives preference to connect to the nodes in other region rather than that of its own. This increases the possibility of connecting to an isolated region. The Fig. 4 shows two node and which were initially neighbors of each other, create beams in different direction in order to increase connectivity.
Nevertheless, we still have to address the best direction of the beam and the knowledge of whether a has a node within its 1 hop. We address these problems next in this section.
To the abovementioned problem, we use analogy of Cohesion rule of Flocking to determine the best direction of the beam. In Flocking, Cohesion rule states that a node should move towards the centroid of the neighborhood to remain connected to all of its neighbors. We apply this definition of Cohesion in our algorithm because we want to bind a peripheral node with other nodes in minimum hops. From the previous section, we already know that the centroid node has the highest Closeness Centrality value in a given region. Directing the peripheral node’s beam towards the centroid node would help reduce the average distance of the peripheral node to other nodes of the region in which the centroid node lies.
Combining Separation and Cohesion rules as discussed above, we can say that, if the centroid node chosen by the peripheral node and the peripheral node itself were not connected initially, connecting them would help in increasing the connectivity, (Cf. Fig. 4). On the other hand, if the centroid node chosen by the peripheral node was within some hops from the peripheral node, it will lead to the reduction in the .
antenna elements are used.
antenna elements are used.
To account for choosing the correct centroid to connect, the peripheral node, , builds , a set of all centroid nodes reachable when it is beamforming. To determine , the peripheral nodes sweep through all the sectors () created with the chosen number of antenna elements except the sectors in which ’s have created the beam. If and , i.e., identified two or more potential centroid nodes, assuming the to these centroid nodes as the decision to connect to one of them is randomly made. However, if , i.e., no new centroid is found, the decides to connect to farthest centroid node in . As we know that is dependent on any reduction in this summation will lead to a reduced network path length. In order to have maximum reduction in the path length, the node should connect to the farthest centroid. If the farthest centroid node was the , then beamforms towards it. However, this decision also depends on the between and . Creating the beam toward the centroid that is less than two hops away will only reduce the initial neighborhood but not the . In this case drops the decision of being the peripheral node and remains omnidirectional. The Fig. 6(a) and the Fig. 6(b) depicts the same. In the Fig. 6(a), node is 5 hops away from while it is 4 hops away from and 2 hops away from the centroid of the region in which lies. Thus, in order to have a reduced path length, node decides to create beam towards . On the contrary, in the case when the node does not have the previously stored information about the centroid nodes and , the node considers to these centroid nodes as and randomly chooses one of them to connect to, (Cf. Fig. 6(b)).
Whenever a peripheral node creates a beam towards a centroid node that is more than 1 hop away, asymmetric link may arise. This is due to the fact that the of peripheral node is while of a centroid node is , in other words, . Due to this difference, peripheral nodes will not know if they got connected to the centroid of other region or not. We propose to solve this issue as, when a centroid node receives information about the node trying to connect to it, it just for one time instant, to acknowledge the reception, creates the beam back to the node. We do this after determining angle of incidence of the beam. This works well for both connected and unconnected components. Algorithm 2 represents a brief algorithmic description of beamforming using Flocking rule analogy. The Fig. 7 shows the new network created after running our algorithm on the network shown in the Fig. b.
3 Formal Definitions
Definition 1.
Assume a centroid of the region , and a node in which has the highest Closeness Centrality, then
(2) 
Definition 2.
The node with neighborhood of the region with centroid is a peripheral node .
Lemma 1.
The expected number of nodes remaining after applying the thinning processes, [BettstetterGyarmati], on a uniformly distributed network is
(3) 
where is the expected number of nodes remaining after the thinning process is applied, is the initial node density in a given area and is the incomplete gamma function.
Lemma 2.
The separation between any two head nodes is between where is the used to create the region, [NagpalMamei].
Proof.
Consider a head node with a gradient around itself. All the nodes within hops from the head node will be in its region. A node which is more than hops away will lie in another region. If in the neighboring region, a head node does not have any gradient around it, then the distance between the two head nodes in hops will be . On the other hand, if the neighboring region also has a gradient around it, then the distance between two head nodes in hops will be . ∎
Lemma 3.
The number of regions is equal to number of centroid nodes and each region has exactly one centroid node.
Proof.
Our algorithm computes the centroid of the region based on average of coordinates, Degree and Egocentric Betweenness of the node for each region. According to our algorithm, the nodes are termed as centroid if the node falls within range of the centroid coordinate estimation algorithm and have maximum sum of Degree and Egocentric Betweenness. If still there are multiple nodes that are termed as centroid nodes, the nodes randomly decide for being the centroid and thus only one node is chosen as centroid. The value of is thus an important factor in the estimation of the centroid node. Also, smaller will tend to provide better estimation of the centroid nodes. As there is only one centroid node per region, the number of centroid nodes is equal to the number of regions. ∎
Lemma 4.
If a node is not a centroid node, it is connected to a centroid node.
Proof.
Our algorithm identifies regions and their centroid nodes. An identified region is always connected, i.e., all the nodes in the identified region are connected to each other. Further, there is one and only one centroid node in a region, ref. lemma 3. Thus for a given region, all nodes that are not centroid are connected to the centroid node. ∎
Lemma 5.
An unconnected node is both the centroid node as well as the peripheral node.
Proof.
A single unconnected node does not have any neighborhood. It thus remains uninhibited at the end of the region formation phase and becomes the head. As it is lacking any neighborhood, the node does not have any gradient around itself and is the only node in the region. In this region, the average coordinates perfectly match the virtual coordinates of the node. Thus requiring no further computation to correctly identify the centroid node.
This node is also the peripheral node as the condition of Definition 2 holds true because of the unavailability of the neighborhood. ∎
Lemma 6.
For a node distribution and fully connected network with average node density and total number of nodes , then is bounded by and .
Proof.
From lemma 2, the hop distance between two heads is bounded by .
Case 1 (Lower Bound): When the heads are separated by hops, the number of regions formed are less. The number of nodes in one region is . Thus, the total number of nodes in all the regions is . As the total number of nodes are , . From lemma 3 ,
Case 2 (Upper Bound): When all the heads are separated by hops, the number of regions formed are more. A head in such a case is connected to only 6 other heads. This can be visualized as a hexagon with vertexvertex distance equal to and a node at the center of hexagon. Each of the vertex nodes are shared between 3 other hexagons. Thus, the total number of heads that are exclusive for the hexagon are +1 = 3. In other words, there are 3 heads in an area of . Thus, for the area=, =. ∎
Lemma 7.
Consider a network with components (), average density of the nodes as and number of nodes as for , is bounded by and , where .
Proof.
From lemma 2, the hop distance between two heads is bounded by .
Case 1 (Lower Bound): Consider component of the network. When the heads are separated by hops, the number of regions formed is less. The number of nodes in one region is . Thus, the total number of nodes in all the regions in the component is , where are the number of region in component. But as the total number of nodes were assumed to be , =. Thus for all the components, the number of regions formed is . From lemma 3, ,
Case 2 (Upper Bound): Upper bound to the number of regions arises when all nodes in the network are disconnected. Thus, all nodes in such a case will be uninhibited thereby becoming region heads. Thus . ∎
Lemma 8.
For a node distribution and fully connected network, and using lemma 6, the number of peripheral nodes in the network is bounded by and .
Proof.
Peripheral nodes are the nodes lying in the outer most gradient of the region. Thus, the number of nodes in the gradient of a region = =
Now using lemma 6, the number of peripheral nodes for all regions thus varies between and . ∎
Lemma 9.
4 Simulation setup
We use a simulation area of x to simulate our algorithm. and are set to and respectively to achieve the nonuniform distribution of node throughout the simulation area. The nonuniform node distribution enables us to visualize the real world scenarios. The range of average density, , of nodes per unit area is set to [x, x]. We make the choice of this range for after considering the percolation of the giant component for the nonuniform node deployment, (Cf. Fig. 8). Initially, each node operates in omnidirectional mode using antenna element with the omnidirectional radius as . We set the maximum number of antenna elements that the nodes are equipped with to . The separation between two antenna elements computed using frequency, . Through our simulations, we explore the effect on connectivity, and by varying the node densities and the gradient.
We use MATLAB to simulate our algorithm with a confidence interval of . We average All the results over topologies.
the effects of the gradient.
the effects of the gradient.
5 Results and Analysis
First, we prove the correctness of the centroid finding in the region. For this, we compute the relation between the nodes that have maximum SocioCentric Betweenness and the centroid nodes in the region. If the centroid node has the highest SocioCentric Betweenness in the region, then the algorithm found centroid node correctly, (Cf. Fig. 9). This depends on the value of the gradient. Larger gradients decrease the SocioCentric Betweenness rank of the centroid node in the region. As the gradient increases, more nodes are now associated to a region thereby increasing the possibility of occurrences of the bridge nodes (bridge nodes have high SocioCentric Betweenness value). Thus, we also calculate the distance in hops between the centroid node and the maximum SocioCentric Betweenness node. According to the results, (Cf. Fig. 9), for a , the percentage of centroid nodes that also have high SocioCentric Betweenness is more and all the centroid nodes in the network are within . The Fig. 9 however shows that for any more than of the time the centroid node is within hop distance to the maximum SocioCentric Betweenness value node and it is within hop of the time.
Further, we use to obtain the results when the Sector model is used in a nonuniformly distributed network, (Cf. Fig. 10). The Fig. a shows the effect of beamforming on the . The obtained in omnidirectional case is initially less than that obtained for the directional cases because the density of the nodes in the component is low. When the algorithm induces directional beams, due to the inclusion of the nodes in other network components, there is an increase in the . The for the directional case is less than that of the omnidirectional case when due to the fact that the nodes connect to the centroid node of other regions in the different component as well as in the same component. The gradient affects the . The lower the value of the gradient is, higher is the number of nodes that beamform, (Cf. Fig. c), leading to more shortcuts and in turn more reduction in the . For and , there is a reduction of almost 40% in the while there is a reduction of almost for , (Cf. Fig. f). However, for and when most nodes are unconnected, there is an increase of in due to the abovementioned facts.
The introduction of the longrange beams also causes the to change, (Cf. Fig. b). For very lowdensity networks, the for the directional case is less because beamforming leads to loss in the initial neighborhood. However, for higher density networks, the does not vary as much as the (Cf. Fig. f). For , there is a reduction of and for and respectively. However, for and any , the reduction in is almost . The for directional case for and is almost constant. This implies that the directional network shows modularity where is independent of and evolves towards hierarchical network [Ravaszv]. However, when the evolution towards hierarchical networks cannot be justified.
The number of components in the network can define connectivity. In a very lowdensity omnidirectional network, the number of disconnected components is higher, (Cf. Fig. e. The number of disconnected components increases to a certain maximum and then decreases as the density increases. This is because, for a high density, all nodes can find at least one neighborhood node within their reach. In addition, as the number of components decreases, the connectivity increases. For the directional case however, as nodes beamform towards different components with the objective of increasing connectivity, the number of disconnected components is less than that of the omnidirectional case.
The size of the giant component can also explain the connectivity of the network. For the directed graphs however, [Dorogovtsev] defined the giant component using the Giant Strongly Connected Component ()
The number of centroid nodes () depends on the value of the gradient, (Cf. Fig. d). For a lowdensity network, the value of the gradient does not matter while as the density increases the value of the gradient affects the number of regions formed. As the gradient increases, more nodes inhibit leading to less number of regions. The difference between the number of regions formed for and is of for while the difference for is very less.
The value of the gradient used also affects the number of peripheral nodes () identified, (Cf. Fig. c). For a low gradient value, as there are more regions, more nodes are included in because of the reduced neighborhood with respect to the region. However, when the value of the gradient is more, is less because there are more nodes in the region and the nodes have relatively more neighbors to check before making the decision of beamforming. greatly affects the number of unidirectional paths. However, it has an adverse effect on the . As the number of peripheral nodes increases, unidirectional paths between the nodes also increases leading to more loss in the . For and , the difference between the number of peripheral nodes is almost negligible. For , however, the number of peripheral nodes varies by more than as the regions formed for lower gradient are more.
Our algorithm affects the and the of the network when we use model, (Cf. Fig. 11). On the other hand, it does not affect and . No dependency of the model on and is rightly justified because these sets are built when the network was omnidirectional, (Cf. Fig. c, d). However, there is a reduction of almost and in the for higher gradient value and for low gradient value respectively. On the other hand, there is no considerable reduction in the . The reduction in the is only between to . Due to variation in for different in model (Cf. Fig. 16), the values obtained for the , the and connectivity are different from that of the Sector model. From the Fig. 14 we observe that, for higher density networks, the change in the for the model is more than that of the Sector model while the changes at a much lower rate.
Until now, we have shown that small world properties are achieved and connectivity be increased in a nonuniformly deployed network. However, it is also important to show the complexity of the algorithm. Due to the storage of three required data values in the region formation phase, neighborhood information and the knowledge about being the peripheral node for both itself and its neighbors is needed. Thus the required memory size is of the order O(3(d+r)+d+1) where is the size of the neighborhood and is the size of reachable centroid nodes. For highdensity network, reaching consensus in the region formation and the centroid finding phase is time consuming. However, for a lowdensity network, the algorithm reaches this consensus quickly.
6 Useful concepts and Related Work
In this section, we define useful concepts giving an overview of the related work. We first define small world concepts in the section 6.1 which form the basis of our research. The need of having long range links for achieving small world properties lead us to discuss beamforming in the section 6.2. We then define Lateral Inhibition in the section 6.3 and Flocking in the section 6.4. The definitions of centrality concepts are discussed in the section 6.5. Further, we discuss nonuniform deployment in the section 6.6.
6.1 Small World Network
Inspired by Stanley Milgram’s [Milgram] experiment of “six degrees of separation”, Watts et al [Watts] suggested a model for the creation of small world network. Watts et al in [Watts, Wattsbook] showed that rewiring edges of a regular network with a probability results into reduction in the of the network while there is very little change in the . Starting by choosing a random vertex and one of its edge to the vertex’s 1 hop neighbor with , Watts et al reconnected the edge to a random vertex in the remaining network. Watts et al then considered all other vertices for rewiring. The process of rewiring continued with the edges now connecting the two hop neighbors. This process continued until all the edges were considered. highly affected the rewiring process. Probability meant that no rewiring while meant complete rewiring of the graph. Using resulted into complete randomness in the network.
The small world model motivated many research studies, [Helmy, Barabasi, AlbertBarabasi, BarabasiAlbert], and many models were proposed. Newman,[NewmanReview, Newman], compiled a comprehensive list of the models on small world. Mostly, the researchers studied two kinds of network structures, one without network growth while another with the network growth. Researchers analyzed the scaling and performance issues for the growing networks [Barabasi, BarabasiAlbert]. Barabasi et al in [AlbertBarabasi, BarabasiAlbert] showed that small world properties also exists in a growing network and there is a preferential attachment of the nodes giving rise to “rich gets richer” property. Barabasi et al showed that the real world networks possess these properties. This led to the behavioral analysis of the networks. On the contrary, assuming spatial wireless ad hoc network without growth, Helmy [Helmy] performed the small world analysis and showed that rewiring of links does not change the structure of the network. Two other results shown in [Helmy] are significant in the context of this paper. First, the is reduced at a greater rate when shortcuts are 25% to 40% in length of the network diameter. Second, the rate of the reduction is more when there are only 0.2% to 2% shortcut links. The reduction rate stabilizes when there are more than 2% shortcut links.
6.2 Antenna Model and Beamforming
Authors of [Bettstetter, Balanis] provided an extensive study of antenna models and defined antenna gain using radiation intensity where angle is angle with the axis and with the plane as
(4) 
Considering antenna elements and isotropic radiators with same phase shift between them, researchers defined two basic antenna models Uniform Linear Array antenna model (), (Cf. Fig. 15), and Uniform Circular Array antenna model (). When , there is no superimposition of the radiation. This leads to a beam with omnidirectional characteristics. However, when , there is a constructive and destructive superimposition of the radiation due to the phase shift between the antenna elements. This leads to a beam with directional characteristics.
The gain pattern for the antenna model is only dependent on the number of antenna elements. It has no dependency on the boresight direction (, the direction of maximum radiation intensity, Cf. Fig. 16). On the other hand, for the antenna model, gain pattern is dependent on both the number of antenna elements and .
However, in wireless ad hoc networks, beamforming using model has been well studied. Classical beamforming techniques using model include Random Direction Beamforming () [Bettstetter, Vilzmann, VilzmannBettstetterHartmann] and beamforming based on the angle of incidence and packet flow. Bettstetter et al [Bettstetter] studied the use of with the path probability to improve the connectivity in the wireless networks. Vilzmann et al [VilzmannWidmer] derived low complexity techniques for beamforming and proposed Maximum Node Degree Beamforming (). In the nodes directed their beams towards the node that had maximum degree. The authors found that leads to less number of intercluster connections but had more intracluster connections. To overcome this drawback, the authors proposed Twohop Node Degree Beamforming (). In the nodes directed their beams towards the node that had maximum twohop neighborhood. The authors showed that outperforms both and . Other works on beamforming include [VilzmannWidmer, Kiese, Yu, Li]. However, most of these studies were concentrated on nodes that were uniformly distributed at random in the given area but very few among them talk about nonuniform distribution of the nodes. Considering all nodes use directional beams, [Bettstetter, Vilzmann, VilzmannWidmer, Kiese, Yu, Li] addressed connectivity very well but do not discuss the impact on the and the . Table 2 illustrates a comparison between these studies. On the other hand, studies related to the small world properties lack connectivity analysis for the nonuniformly distributed network. Table 3 illustrates comparisons between various studies performed in the direction of achieving small world properties in wireless ad hoc networks and our model.
Parameter\  Vilzmann  Widmer  Kiese  Yu  Li 
Reference  et al [Vilzmann]  et al [VilzmannWidmer]  et al [Kiese]  et al [Yu]  et al [Li] 
Transmission mode  Directional  Directional  Directional  Directional  Both 
Reception mode  Directional  Directional  Directional  Omnidirectional  Both 
Mobility  No  Yes  No  No  No 
Beam width  Depends on  Constant  Constant  Optional  Constant, 
beam direction  switched beam  
antenna  
Beam direction  Random  Optional  Optional  Optional  Random 
Antenna model  modeled  Sector  Keyhole  
as keyhole  
Node distribution  Uniform  Uniform and  NonUniform  Not specified  Uniform 
NonUniform 
Parameter\  Our Model  Banerjee  Guidoni  Helmy  Sharma  Verma 
Reference  et al [Banerjee]  et al [GuidoniLoureiro]  et al [Helmy]  et al [Sharma]  et al [Verma]  
Shortcut Creation  Rewiring  Rewiring  Addition  Addition  Addition  Addition 
Node distribution  NonUniform  Uniform  Uniform  Uniform  Uniform  Uniform 
External  No  No  High range    Wired  Two radios 
infrastructure  Sensor  for each node  
Global knowledge  No  No  Yes  Yes  Yes  Yes 
Density of nodes  Low  High  High  High    Low 
Shortcut Edge  Directed  Directed  Undirected  Undirected  Undirected  Directed 
Shortcut direction  Towards  Longest  Random,  Random  Random  Random 
centroid of  Traffic Flow  towards sink  
other region  path  
Shortcut length  Function of  Function of  Constant  Limited  Constant  Constant 
antenna  node density  
elements  
Shortcut width  Depends on  Depends on  Constant      Constant 
Shortcut Length  Shortcut Length  
Prob. of Shortcut  based  Based on  function of  
creation  on model  centrality  network size  
parameters  values  
Performance metric  Path length,  Path length,  Path length,  Path length,  Path length,  Path length, 
Clust. Coeff.  Connectivity  Clust. Coeff.  Clust. Coeff.  Energy  Clust. Coeff.,  
Connectivity  degree 
6.3 Lateral Inhibition
Lateral Inhibition is a process by which cells of animal tissues, based on the properties of neighbor cells, decide whether to perform a task or not. Lateral Inhibition ensures that the cells that perform the tasks are equidistant from each other. This helps in producing regular patterns throughout the surface. Lawrence [Lawrence] modeled Lateral Inhibition as, when a cell performs a task, it inhibits its neighbors within hops from performing that task thereby resulting into equally spaced uninhibited cells. Lateral Inhibition thus creates clusters where the cluster heads are uninhibited nodes distributed over an area. Nagpal et al [Nagpal, NagpalMamei] described a simple algorithm to achieve Lateral Inhibition. In the algorithm, the cells assign themselves a random number. Each cell starts to count backwards. If before reaching , a node receives an inhibition signal from the neighboring cell, the cell stops counting otherwise sends out an inhibition signal to all its neighbors. Nagpal et al [Nagpal, NagpalMamei] showed that the used to create the cluster greatly affects the number of clusters formed.
Recent studies revealed that Lateral Inhibition can be achieved in an optimal way [Afek]. Inspired by the tissue of the fruit fly, Afek et al [Afek] modeled distributed Lateral Inhibition using local information and requiring only two exchange mechanisms. These exchange mechanisms are, first, broadcasting a single control bit to the neighbors with certain probability and second, if the node receives no message from the neighbors, it sends out a control bit to inhibit its neighbors. As a variation to Nagpal et al’s algorithm, the algorithm used a probabilistic approach that varied over time in an increasing manner to perform Lateral Inhibition. The runtime complexity of the algorithm was of the order where was the number of nodes in the system. Due to single bit exchange messages over single hop, the algorithm had a low message complexity.
6.4 Flocking
Flocking, [Reynolds], was first modeled by Reynolds in order to simulate the birds’ behavior. In nature, flocking is observed in many other social living organisms like cattle, fishes and humans. Reynolds, while modeling Flocking, termed each social entity as a boid and formulated three very simple rules, (a) Alignment (b) Separation and (c) Cohesion. Reynolds defined Alignment rule as the direction matching of a boid with its neighbors. He defined Separation rule as the collision avoidance with neighborhood boids and Cohesion rule as the tendency of a boid to remain as close to its neighbors as possible and not stray. The Fig. 17(a), shows that the boid orients itself in the direction in which its neighbors were moving. The Fig. 17(b), shows that the boid has to move away from the neighbors in order to avoid collision while the Fig. 17(c) shows that the boid moves towards the centroid of the neighbors in order to remain close to its neighborhood. Couzin in [Couzin] formulated mathematical explanation of these rules. Due to the motion of a boid, velocity and displacement were associated with the boid. Alignment rule was modeled using the direction of a boid while Separation and Cohesion were modeled using both velocity and the displacement.
Recent studies have revealed the use of Flocking in solving various problems in wireless ad hoc networks. Antoniou et al [Antoniou] used Flocking to provide efficient congestion control mechanism by computing the congestion at the neighbor nodes while [Kadrovach] used the Separation rule for the efficient placement of nodes to maximize the coverage area.
6.5 Centrality
Decades of research on network and graph theory has led researchers to derive many fundamental concepts related to the importance of a node in the network. The concept of centrality was one such concept that was developed and used to address the topological characteristics of the network nodes. Proposed definitions of centrality measures include those that use global parameters as well as those that only use local information. Some examples of global centrality measures are SocioCentric Betweenness [Freeman, Freemanlc] and Closeness Centrality [Freeman] while Degree Centrality [Freeman] and Egocentric Betweenness Centrality [Everett, Daly] are examples of the local centrality measure.
SocioCentric Betweenness Centrality
The SocioCentric Betweenness Centrality, [Freeman, Freemanlc], is the measure of the number of shortest paths passing through the node thereby expressing the most important node in the network and through which most of the communication takes place. The SocioCentric Betweenness is a frequency measure and requires the global network knowledge. Usually nodes with high degree and those that are acting as the bridge nodes tend to have relatively high SocioCentric Betweenness. Mathematically the SocioCentric Betweenness of a node is
(5) 
where is the number of shortest paths between any two nodes that pass through while is the total number of shortest paths in the network.
Egocentric Betweenness Centrality
Aiming to compute the Betweenness centrality using local properties, [Everett, Daly] proposed the Egocentric Betweenness Centrality measure. Everett in [Everett] computed the Egocentric Betweenness using upper diagonal adjacency matrix . is created considering 1 hop neighborhood of the node . Consider to be the identity matrix, then the sum of the inverse of all nonzero elements in along is the Egocentric Betweenness of the node.
Marsden in [Marsden] performed an empirical study to find the relation between the two types of Betweenness, the SocioCentric and the Egocentric Betweenness, and found that the Egocentric Betweenness is strongly correlated to the SocioCentric Betweenness and it can be used when global network information is lacking.
Closeness Centrality
The Closeness Centrality [Freeman] on the other hand is the measure of how fast a node can transfer data to all the nodes. The Closeness Centrality is the fraction of shortest distance between a node to all other nodes in the network. Assuming be the shortest distance between node and , the Closeness Centrality of is
(6) 
A node with the highest Closeness Centrality value is the centroid of the network.
As all the centrality measures convey different information, it is not necessary that a node having high value for one centrality measure also have high values for the others. Many other types of centralities, such as, Bridging Centrality, Eigen Vector Centrality and Spectral Centrality also exist. We refrain ourselves from describing them in detail. However, Katsaros, [Katsaros], provided a brief survey on these centrality measures.
6.6 NonUniform distribution of nodes
Many nonuniform deployment strategies have been proposed, [Weijen, LeBoudec, Hu, Aitsaadi, Riihijarvi, BettstetterGyarmati]. We take insights from Bettstetter et al, [BettstetterGyarmati], node deployment strategy. Bettstetter et al proposed the use of thinning process to generate a nonuniform node deployment. The authors started with uniform distribution of nodes in a given region, then pruned the nodes based on two factors, transmission radius, , and the number of neighbor nodes, . If the node had at least neighbors within , the node was not removed else it was removed. Schilcher et al, [SchilcherGyarmati], formulated and measured the degree of nonuniformity of this pruned network. Schilcher et al divided the region into smaller subregions and estimated the number of nodes in the subregion. The estimated value was then used to calculate the nonuniformity index, . The Fig. b shows the deployment achieved when the thinning process is applied to the deployment shown by the Fig. a. The Fig. c shows the density distribution of nodes using kernel method.
7 Future Work
A Number of extensions to our algorithm can be visualized. Identifying the optimal gradient size to choose for the determination of minimal peripheral set of nodes is one way of extending our work. We are currently working on how we can apply game theory to successfully find the minimal peripheral set. We believe that by applying game theory nodes can determine what the suitable gradient size is and can reduce asymmetric links further.
We would also like to extend our algorithm to support dynamic environment and asynchronous operation. Dynamic environments are likely to result in frequent changes to the state of the node. Any change in the state of the node would require reconfiguring in the network using the proposed algorithm. Information available at the neighborhood nodes would be helpful in learning about the previous configuration. This learning could be docitive [Giupponi], meaning, partial learning from the neighborhood states could make nodes infer about the previous good configuration so that reconfiguration can be done easily and quickly. This will also help us to address the unaddressed paradigms of [Prehofer]. Further, we would like to address network lifetime of the network when implementing our algorithm.
8 Conclusion
In this paper, we have presented an algorithm for achieving small world properties using beamforming and bioinspired techniques in a wireless ad hoc network. Our algorithm works using locally available information and does not require the knowledge of the network. We have also removed the possibility of requirement of any external infrastructure for achieving our goal. Through our algorithm, we have shown how isolated communities can collaborate and connect with each other to achieve better and faster communication. BioInspired techniques like Lateral Inhibition helped us to form communities within the network for the reduced message complexity while the Flocking analogy helped us to determine beam properties. Our results show that for both theoretical and realistic antenna models and relatively highdensity networks, there is a reduction in the by almost to for . On the other hand, reduction in the is between to . Our results also show improvement in the connectivity. The increase in the size of the for the nonuniformly distributed directional network is around for high density network while it is around for relatively low density networks.
References
Footnotes
 Sector model approximates realistic antenna models
 Egocentric Betweenness approximates the SocioCentric Betweenness very well in the absence of global knowledge [Marsden]
 in a directed graph is the length of the largest cycle in the graph component.
 is the set of nodes in the component which can connect to .