Consensus of the Distributed Varying Scale Wireless Sensor Networks

Consensus problems are investigated for a type of the distributed varying scale wireless sensor network (VSWSN), where the scale of the network is increasing or decreasing due to the new nodes joining in or the invalid nodes quitting from WSN, respectively. In order to demonstrate the communicating behaviors more realistic, we offer the node attached component sequence for each valid node under the general sleep algorithm. Based on the component sequences, several concepts, such as the local limited intersection connection (LLI connection) and the global limited intersection connection (GLI connection), are provided to display the connectivity of the varying topology. Under certain conditions, the designed protocol ensures that all sensors arrive at the component consensus or the global consensus if the communicating graph is LLI connected or GLI connected, respectively. By the basic theoretical analysis, several consensus criterions are obtained. In the meanwhile, the consensus regions are investigated. The numerical example shows the reliability of the proposed results.


Introduction
Consensus is of great importance in many applications of wireless sensor networks (WSNs) [1], such as the clock synchronization [2], data fusion [3,4], and formation control [5].To achieve the consensus, it requires that all nodes of the networks achieve the special value, while they may begin with the different initial values.
In the past few years, consensus problems have been widely investigated, and many exciting results have been obtained.The early work of consensus problems can be found in [6], where the consensus problems are investigated for the fixed scale networks.The most further researches focus on optimizing the algorithms and the communicating topology, such as in [7][8][9][10], and the optimal algorithms are taken to solve the consensus.Other researchers devote to optimize the topology, such as Lin and Jia solve the average consensus problems of multiagents with time delay in the jointly connected topology [11]; Atrianfar and Haeri deal with the average consensus problems in the weak connected topology [12]; consensus problems are studied under the sleeping-awaking method [13,14].
In recent years, some consensus results of the scale-free networks, where the scale of the networks may be increasing, are obtained in the exciting literatures [15,16].
And so far, consensus regions of the networks have attracted much attention by some researchers, such as Li et al. who propose an observer-type protocol to solve the consensus problem, and meanwhile an unbounded consensus region was provided [17].Based on the connected network, consensus regions are investigated in the multiagent dynamical systems [18,19].
It should point out that in the traditional consensus literatures, the scale of the network is fixed or increasing.Namely, the newly joined nodes and the invalid removed nodes have not been considered.In fact, the varying scale wireless sensor network (VSWSN) exists in many applications.For example, after the limited power exhausted, the nodes become invalid, or when the new nodes join the network, the scale of WSN is changed, and the related consensus regions are indefinite.However, up to now, consensus problems of VSWSN with the consensus regions have not caused much attention.
The main purpose of this paper is to investigate the consensus problems of VSWSN and its consensus regions.In the traditional literatures, the connected topology, the weak connected topology, or the jointly connected topology is the basic condition for the network communication.Different from the traditional literatures, in VSWSN, while some new nodes join in the network and some nodes quit from the network, VSWSN may neither connected, nor jointly connected, nor weak connected, and the communicating behaviors among those nodes are indefinite and random.The communicating graph of VSWSN which we proposed in this paper is displayed by the node attached component sequence.It turns out that if every two nodes have chance to connect, namely, the topology is the local limited intersection connected (LLI connected) or the global limited intersection connected (GLI connected), and the intersection topology has the enough dwell time, then under the designed protocol, VSWSN achieves the component consensus or the global consensus.
Moreover, the consensus regions are different from the traditional literatures.In the traditional literatures, the consensus value is determined by the state of the fixed node set.In VSWSN, the consensus value is varying according to the varying node set.So the considered systems in this paper are more general than that of network with the fixed node set, which the topology is the special case of the global limited intersection connection.
The outline of this paper is listed as follows.In Section 2, some basic concepts and notations are introduced.In Section 3, we provided the main results.For each certain topology, the consensus criterions and the consensus regions are investigated in detail.For the varying topology in which some nodes may be added while some other nodes may be removed, the consensus criterions and the consensus regions are also provided.In Section 5, a numeral example shows the reliability of the proposed results.In Section 6, several conclusions are obtained.
Under LLI connected and GLI connected, we are going to discuss the consensus problems of VSWSN in the following sections.

Problem Statement.
In many applications, the communication of VSWSN is based on the multiple components.In this paper, the consensus problems which we considered are the component-based communication.
For   ∈ (), let   (, ) be the state of sensor or node   , where   (, ) ∈   ,  ∈  + .Suppose the state of   is given by where  is a constant,    () is the consensus protocol and is given by where ã () is a positive constant.For all   () ⊂ (), under the dynamic (3) and protocol (4), the state of component   () which attaches on node   can be displayed as follows: where n = (     ) n ×n  , and      is defined as () is the vector state of the valid nodes that come from the former topology  − 1, where 0 <  < 1, d  refers to the maximal degree node of   (), then VSWSN ( 5) is said to be the component potential consensus.In addition, if   () = () and ( 9) holds, then VSWSN ( 5) is said to be the global potential consensus.
Proof.Note that ( Similarly, we can get the result when   () = ().
Remark 8.If VSWSN ( 5) is the component potential consensus or the global potential consensus, then the errors among the adjacent nodes are decreased, but it does not mean that system achieves consensus, in any case, the consensus is constrained by the dwell time   .
Assumption 1.We assume that for each intersection topology , their exists  > 0, such that   > .
Remark 9.If a sensor leaves from its neighbors and becomes an isolated node, and its state may not keep in coordination with other nodes.Note that if the communicating graph is LLI connected or GLI connected, the node has chance to keep in coordinating with other nodes.
The primary purpose of this paper is to provide the basic consensus analysis of (5) for the aforementioned scenarios.Because the scale of the network is a variable, different from the traditional literatures which investigate the consensus problems are based on the global structure, to study consensus problems, we depend on the component sequences.In the meanwhile, the consensus regions are gotten in the following sections.

Main Results
In this section, we are going to discuss the component potential consensus and the global potential consensus firstly.Then the relations between the component potential consensus and the component consensus, the global potential consensus and the global consensus are studied.

Consensus Investigation for WSN in a Certain Topology.
In this subsection, we study the potential consensus problems of the certain topology.For convenience, we focus on the component   (), its topology index is dropped and denoted by   , and its state is denoted by  n().The error of  n() is denoted by () =  n() − 1 n×1 ⊗  0  ().To express the state of (), we depend on the eigenvalues.Let eigen( n −  n) be the eigenvalue set of the matrix ( n −  n), let   be the th eigenvalue of  n −  n, and  = max{|  | |  ∈   ()}.
To achieve the potential consensus, it requires that the error is decreasing.To keep the decreasing error, it requires that  takes in some values, and the following discussion focuses on how to take the value of parameter   to get the needed .
The proof is similar to that of Theorem 10, and it is omitted.
Remark 12.If  = 1 and () is the fixed node set, then the related consensus problems discussed here are the same as in literatures [6,20].However, when || > 1, the protocol (15) and the protocol listed in the literatures [6,20] cannot insure subsystem (5) to achieve the consensus, and the following section will solve the problems.
In (15), if   is equal to a certain value and () is the fixed node set, then system converges to one equilibrium point.In order to get the wide consensus regions, the value of   takes in (0, (1 + )/2  ), and the similarly reasons are taken by the Theorems 13 and 14 as follows.
Based on Theorems 7 and 10, we get the following theorem.
Proof.According to Geřsgorin disc theorem, we obtain Similarly to the proof of Theorem 10, based on Theorem 7, it is known that subsystem (20) is the component potential consensus.This completes the proof.
Similarly to Theorem 13, we obtain theorem as follows.

Consensus Investigation for WSN in the Varying Topology.
Based on the discussion of the potential consensus, in this subsection, we investigate the consensus problems of the node attached component in VSWSN.
(ii) Similarly, we can get the result when −1 <  ≤ 0; we also can show that VSWSN is the global consensus when WSN is GLI connected.This completes the proof.(5) is LLI connected, then the following statements hold.(i) If  ≥ 1,    ,  satisfy (21), and if (34) holds, then under (19), subsystem (20) is the component consensus.
Proof.Similar to the proof of Theorem 16, it is omitted.

The Consensus Regions
In fact, the former sections discuss consensus problems that require the state of each node keeps in coordination with the average value  0 (, ), and the value of  0 (, ) is decided by the parameters    and .The regions of    and  mean that the changeable value of  0 (, ) has a widely range.Hence, in this section, we investigate the consensus regions.Definition 18.Under protocol (4) or (19), the region of    and , such that VSWSN (5) is the component consensus or the global consensus, is called the consensus region.

Numerical Example
Example 21.Consider two node attached components   () and   () of 1  , 3  (as shown in Figure 1), where 0, 1, and 2 are the part topologies in sequence,   () takes value in 0 or 1, and   (1) =   (1) is the intersection topology.For   () and   (), the related node set sequences are shown as follows: Suppose that the state of   is shown as in (3), and consider the following three questions: (i) when  = 0.6 or  = −0.6, in topology 0, under protocol (15), determine whether system (5) achieves the component potential consensus or not; (ii) when  = 5 or  = −5, in topology 0, under protocol (19), determine whether system (20) achieves the component potential consensus or not; (iii) suppose that WSN is LLI connected, the partial node attached components of 1, 3 are in topology sequence 0, 1, and 2 (as shown in Figure 1), and  = 0.6, under protocol (15), determine whether system (5) achieves the component consensus or not.When WSN is GLI connected, two node attached components of 1, 3 are in topology sequence 0, 1 (as shown in Figure 1), under protocol (15), determine whether system (5) achieves the global consensus or not.
For the above questions, under the related protocol, we can get (34) holds firstly.Then we propose the following analysis.

Conclusions
This paper has investigated the consensus problems of VSWSN.According to the local limited intersection connected and the global limited intersection connected, it has provided several criterions of the component consensus and the global consensus.All gotten results have been based on the structures of the node attached component sequences.Based on the structure, the results cannot be influenced by the increasing-decreasing node set.In the meanwhile, it has given the consensus regions.

Figure 1 :Figure 2 :
Figure 1: For nodes 1 and 3, there are two related node attached components.

Figure 3 :Figure 4 :
Figure 3: In topology 0, node attached components of 1 and 3 keep the component potential consensus, respectively.

Figure 5 :Figure 6 :
Figure 5: In topology 0, node attached components of 1 and 3 keep the component potential consensus, respectively.

Figure 7 :
Figure 7: In topology 2, node attached components of 1 and 3 keep the component potential consensus, respectively.