1.1 Problem statement and objectives

Data aggregation is one of the main functionalities of sensor nodes. The sensors normally sense the data from the environment and transmit them to the base station or sink node. The regularity of sending the data and the number of sensors that send the data depend on the particular application. Data aggregation is outlined as the method of aggregating the data from multiple sensors in order to abolish redundant transmission and process the data about the sensed environment, after which combined information to the base station is provided. Data aggregation needs a technique for combining the sensed data into high-quality transferable information. It involves aggregation algorithms, which collect the sensed data from multiple sensors nodes and transmit the data to the base station for further processing. The various design issues of the data aggregation algorithm are discussed here. By resolving these issues, an aggregation algorithm can be implemented in all applications.

Applying the lightweight cryptography algorithm to secure the network communication is preferable when the sensor network is deployed through event-driven applications.

1.2 Contributions made in this article

This article proposes a hybrid communication for energy-efficient data (HCED) aggregation in WSNs. The proposed algorithm provides energy-efficient data aggregation among the sensor nodes. This algorithm has three parts: i) an energy-efficient aggregation tree construction, ii) a collision-free data aggregation algorithm and iii) an asymmetric key cryptography algorithm for secure data aggregation.

The first phase proposes an energy-efficient tree construction algorithm based on the binary search tree model. It follows a two-hop routing model for data aggregation. To resolve the energy-related issues, most researchers have proposed the distributed algorithm for data aggregation. Therefore, to resolve the energy and delay issues, the research have been determined to design distributed tree-based data aggregation. Although many distributed solutions that have been proposed for cluster-based data aggregation, collision issues have been reduced on cluster-head selection. Since each node in a binary tree model has a maximum of two nodes, which reduce collision issues, using the binary tree model is highly recommended.

In second phase, a hybrid delay-efficient data aggregation algorithm, which reduces the energy consumption by performing fast data aggregation is proposed. This hybrid data aggregation algorithm is a combination of distributed and centralized aggregation mechanisms to perform simultaneous data aggregation from sensor nodes.

Finally, secured data aggregation has been achieved through an asymmetric cryptography technique proposed in HCED. The proposed encryption algorithm reduces the computation overhead of the network more than the existing models.

The rest of this paper is organized as follows. Section 2 briefly describes some of the existing schemes that resolve data aggregation issues for WSNs. Section 3 provides the details about the network models and assumptions made regarding this proposed model. A brief discussion on the proposed HCED approach is presented in Section 4. Section 5 discusses how the HCED scheme will resolve the maximum data aggregation issues in an energy-efficient way. The simulation results and perfor-mance analysis of the proposed algorithm is discussed in Section 6. Finally, Section 7 provides the conclusion along with the contributions made in this paper.

2.1 Centralized approach-based schemes

In centralized approach-based schemes, the sink or base station collects the global information of the network topology and computes the optimal shortest path of the sensors before deploying them for data aggregation.

^{Mpitziopoulos, Gavalas, Konstantopoulos and Pantziou (2007)} have proposed a near-optimal scheduling algorithm (NOSA) for identifying an appropriate number of mobile sensor nodes and their near optimal paths using the Esau-Williams heuristic. In NOSA, the parallel deployment of multiple agents is suggested, where each agent visits the subset of nodes. NOSA outperforms the single agent-based approaches (e.g., LCF, GCF and GA) in terms of data fusion cost and overall response time, but it experiences high computational complexity in determining the agents’ itineraries.

In order to reduce high computational complexity, ^{Chen Gonzalez, Zhang, and Leung (2009)} has proposed a multi-agent scheduling (MAS) algorithm. To reduce the latency, the authors have proposed the MAS scheme because it helps in the collection of concurrent sensor data. These algorithms differ in cluster-group formation methods. In MAS, the authors used an angle gap for clustering all the sensor nodes in a particular direction as a single group. This approach does not describe how to determine the optimal angle gap threshold. ^{Cai, Chen, Hara, and Shu (2010)} and ^{Chen, Cai, Gonzalez, and Leung (2010)}, has been proposed a genetic algorithm base mobile agents itinerary planning algorithm to form the clusters. ^{Wang, Chen, Kwon, and Chao (2011)} has proposed a genetic algorithm-based approach which has the limitation of its higher computational overhead. These algorithms assume that the set of sensor nodes to be aggregated by the sink are predetermined, which limits the application scope of the network.

^{Konstantopoulos, Mpitziopoulos, Gavalas, and Pantziou (2010)} have proposed a greedy tree-based scheduling algorithm (TBSAs) to identify near-optimal paths for multiple agents. This algorithm is a centralized algorithm where the sink statically determines the number of aggregators and their schedules. The main theme of the TBSA algorithm is to divide the area around the sink into concentric zones, so that it can construct the near-optimal path tree from inner zones to outer zones. The main limitation of a centralized approach-based data aggregation scheme is that it uses static aggregation scheduling algorithms because it is based on an old view of the network topology. This approach lacks dynamic recovery from node or communication link failures.

2.2 Distributed approach-based schemes

The distributed approach-based data aggregation protocol helps the aggregators to change the route dynamically according to the current state of the network.

^{Xu and Qi (2006}, ²⁰⁰⁸⁾ proposed a dynamic data aggregation algorithm for target-tracking applications. Energy consumption, information gain and the remaining energy of a node represent the cost function used in this method for selecting the next node. The cluster-head aggregates the data from its cluster, and then returns to the sink node. This algorithm is more time-consuming and may face difficulties in returning to the sink without additional forwarding information. In MADD, the authors ^{Karaboga et. al (2012)} and ^{Chen, Kwon, Yuan, Choi, and Leung (2007)} have proposed mobile agent-based directed diffusion, where an aggregator node visits a subset of nodes. In this model, the sink uses the first phase of the directed diffusion algorithm proposed by ^{Intanagonwiwat, Govindan, and Estrin (2000)} to determine the clusters. However, the actual data aggregation is carried out by dispatching an aggregator that sequentially visits the subset of nodes.

In software agent-based directed diffusion, node visits are determined at the sink node, although the authors ^{Shakshuki, Malik and Denko (2008)} cannot access the procedure. This method uses the routing cost and the remaining energy of node in order to select the next node to be visited by an aggregator. The main limitation of the scheme is that, it depends on a directed diffusion scheme, which incurs extra communicational overheads for data aggregation and is only appropriate for request-based data aggregation applications. ^{Gupta, Misra, and Garg (2012)} has proposed a multiple mobile aggregator with dynamic scheduling-based data dissemination (MMADSDD) protocol, where nodes are organized in fixed regions and each aggregator is responsible for collecting aggregated data from each region. The route of an aggregator is dynamically located at each node using the cost function. MMADSDD adapts to unexpected node failures during data aggregation from the aggregator to the sink, although it consumes slightly more energy than TBID proposed by ^{Konstantopoulos et al. (2010)}.

^{Boudia, Senouci, and Feham (2015)} proposed an additive homomorphism encryption and an aggregate message authentication code (MAC) to provide end-to-end confidentiality and end-to-end integrity, respectively. SASPKC adopts state full public key encryption (SPKE) for efficiency in terms of computation and communication costs. SASPKC aggregates not only cipher texts but also signatures, while end-to-end data confidentiality and integrity security services are provided by using symmetric homomorphism encryption and an aggregate MAC, respectively. While considering new attacks, such as selective forwarding, SASPKC does not support node mobility. The main contribution of ^{Sun, Luo, and Das (2012)} is the proposed combination of a trust mechanism, data aggregation and fault tolerance to enhance data trustworthiness in wireless multimedia sensor networks, which consider both discrete and continuous data streams. ^{Ho, Wright, and Das (2012)} proposed a framework to detect compromised nodes in WSNs, as well as applied software attestation for the detected nodes. They reported that the revocation of detected compromised nodes cannot be performed due to the high risk of false positives in the proposed scheme.

^{Rezvani, Ignjatovic, Bertino, and Jha (2015)} proposed a novel collusion attack scenario against the number of existing IF algorithms. The authors have proposed an improvement to the IF algorithms in terms of providing an initial approximation of trustworthiness of sensor nodes. When compromised aggregators are involved in data aggregation, this model fails in protecting the data. Moreover, this method is only suitable for the new deployment of data aggregation. In multi-channel scheduling algorithms, the authors ^{Ghosh, Incel, Kumar, and Krishnamachari (2009)} dedicated their efforts to the aggregation scheduling problem in WSNs when multiple frequency channels are available. The authors then demonstrated that finding the minimum number of channels required in the network to alleviate all interference is NP-hard. The NP hardness in minimizing the scheduling latency in an arbitrary network, with respect to multiple channels, has been proved. The work proposed by ^{Joe, Choy, and Sheriff (2010)} formulates the scheduling problem of minimizing the overall data transmission delay. The characteristics of optimal scheduling were studied, followed by the evidence from the lower bound of optimal performance. Two scheduling policies were proposed in the work. The decision in one policy was made based on the current system state, while the predication of future system conditions was also taken into consideration in the other policy.

^{Li, Guo, and Prasad (2010)} offer a solution for the distributed aggregation scheduling problem in WSNs with respect to minimum latency. Compared with centralized solutions, a distributed scheduling plan has its own advantages. In this paper, an algorithm based on vertex coloring was proposed with a proven delay of 4R+2Δ -2.The minimum latency aggregation scheduling in WSNs with multiple sinks proposed by ^{Yu and Li (2011)} has been investigated. In contrast to what is found in the prior literature, this model proposed a dynamic selection of a sink node based on the shortest path by sensor nodes for the purpose of minimizing transmission latency. Two approximation algorithms with bounded latency were proposed. ^{Zeydan, Kivanc, Comaniciu, and Tureli (2012)} have proposed an adaptive and distributed routing algorithm for correlated data gathering, as well as exploiting the data correlation between nodes using a game theoretic framework. Routes are chosen in order to minimize the total energy expended by the network using the best response dynamics in relation to local data. The cost function used for the proposed routing algorithm takes into account the energy, interference and in-network data aggregation. This paper specifically addresses the problem of effective energy minimization, although the quantitative analysis of delay minimization has not been resolved.

^{Yao, Cao, and Vasilakos (2015)} proposed an energy-efficient delay-aware lifetime-balancing protocol for data collection in WSNs. The authors proved that the problem formulated by EDAL is NP-hard, as well as proposed a centralized and distributed heuristic to reduce its computational complexity. Although EDAL achieves significant energy consumption and delay, this model only proposed to resolve energy issues in heterogeneous networks. ^{Choa and Hsiao (2014)} proposed a structure-free and energy-balanced (SFEB) data aggregation protocol, which consists of two phases. In first phase, the proposed model designates some nodes as aggregators in order to gather as many packets as possible. Then, these aggregators send the collected packets back to the sink in phase two. Sensor nodes that fail to send data to aggregators will also transmit their packets to the sink in phase two. This model requires location information of sensor nodes to avoid structure-based data aggregation. Location information can be obtained by applying a localization protocol. ^{Alshahrany, Abbod, Alshahrani, and Alshahrani (2016)} has proposed a web based data collection framework which integrates data from homo-geneous and heterogeneous networks. But integrating WSN and RFID is a complex design process when supporting real-time applications. Sensor semantic network ontology has been used to form the cluster for data aggregation. Hence the cluster head should have good energy and computational capability. When implementing this model in remote and autonomous network appli-cations, it leads to more energy wastage and complex process on cluster formation.

^{Karaboga et al. (2012)} proposed a novel energy-efficient clustering mechanism based on an artificial bee colony algorithm in order to prolong the network lifetime. The artificial bee colony algorithm, which simulates the intelligent foraging behavior of honey bee swarms, has been successfully used in clustering techniques. The proposed ICWAQ protocol uses efficient and fast searching features of the ABC algorithm in order to optimize clustering of the nodes in the cluster-head selection process of defining routing gateways. Since this algorithm uses random cluster-head selection, it leads to collision problems. This algorithm is not suitable for routing mobile networks due to MAC layer issues.

2.3 Opportunistic routing

Opportunistic routing is widely known to have substantially better performance than unicast routing in wireless networks with lossy links. The opportunistic routing proposed by ^{Mao, Tang, Xu, Li, and Ma (2011)} allows any node that overhears the transmission to participate in forwarding the packet. The routing path is selected on the fly and completely opportunistic based on the current link quality situations. As stated by ^{Chakchouk (2015)}, the existing opportunistic routing proposals can be categorized into three main classes: geographic, link state aware and probabilistic routing.

The geographic opportunistic routing proposals proposed by ^{Wang, Chen, and Li (2012)}, ^{Zeng, Yang, and Lou (2009)} are location-centric. Hence, they are practical for scenarios where the knowledge of the nodes location is necessary, such as fire detection, gas leakage monitoring, and rescue operations. However, these protocols may not be efficient in terms of delay and reliability.

Link-state-aware opportunistic routing protocols has been proposed later by ^{Rozner, Seshadri, Mehta, and Qiu (2009)} and ^{Han, Bhartia, Qiu, and Rozner (2011)} to take other parameters, like link quality and bandwidth, into account. These protocols provide higher throughput for Wireless Mesh Networks and Internet applications. ^{Sudarsono, Huda, Fahmi, Al-Rasyid, and Kristalina, (2016)} has proposed an environmental health monitoring application through WSN with secure data communication access. For the communication module, they have used IEEE802.15.4-based communication which is legacy model for state of the art IoT applications. Moreover their proposed model is more specific to small scale applications where energy and security need not to be considered. When implement this model in large remote environment applications, the key sharing and renewal mechanisms which is proposed has leads to more energy consumption.

^{Conan, Leguay, and Friedman (2008)} and ^{Liu and Wu (2012)} have proposed a Probabilistic opportunistic routing which is suitable for dynamic wireless networks, since it copes with the high mobility property of these networks by using online inference and prediction schemes to estimate the links’ qualities and availabilities. But in more stable/stationary wireless networks, it has been observed that opportunistic routing design could be further refined and optimized using optimization-based formulations for candidate relays selection and prioritization.

However, with the proliferation of wireless heterogeneous networks and the security risks, a lot of research work still needs to be developed in order to adapt opportunistic routing to these new challenging environments. This is particularly important, since some opportunistic routing applications, like disaster manage-ment, are prone to be deployed in similar environments.

To overcome all the drawbacks of the existing models, the authors ^{Gopikrishnan and Priakanth (2016)} proposed a unique solution namely, hybrid secure data aggregation (HSDA), which can resolve security and energy issues because it provides highly secure data aggregation in an energy-efficient way. The HSDA implements an end-to-end symmetric key cryptography for secure authentication by using a shared public key, as well as hop-by-hop asymmetric key cryptography with the private keys of each node for data integrity and confidentiality. Although the authors have successfully resolved the maximum issues in HSDA, multiple algorithms implementation renders HSDA algorithm as inefficient in terms of energy utilization.

2.4 Motivation

With regard to WSNs, data aggregation routing protocols have always been a major research topic, although the unique solutions that resolve maximum issues, such as collision, delay, energy and security factors, are not addressed by existing protocols. The literature survey conducted in relation to this article analyzed all the data aggregation issues, with the best solutions presented for the existing models. EECD ^{Zeydan et al. (2012)} presents an effective solution, which resolves energy and collision problems. When the network density is greater, EECD fails in reducing the aggregation delay and produces less significant throughput. A centralized and distributed algorithm, which resolves energy and delay issues, has been proposed in EDAL ^{Yao et al. (2015)}, although the authors specifically presented their protocol only for heterogeneous networks. To avoid structure-based data aggregation, the authors have presented a modified aggregator-based data aggregation known as SFEB ^{Chao and Hsiao (2014)}. Although energy-efficient data aggregation has been achieved in SFEB, it fails with regard to fast data aggregation due to the mediatory implementation of the localization algorithm. In ABC, the authors ^{Karaboga et al. (2012)} have presented a cluster-based data aggregation algorithm, which mainly concentrates on reducing aggregation delay. The randomized cluster-head selection in ABC results in insecure data aggregation. To overcome all the drawbacks of existing models, as well as to resolve the maximum issues in order to obtain a unique solution, ^{Gopikrishnan and Priakanth (2016)} have proposed HSDA. Although the HSDA resolves all data aggregation issues, the energy utilization of this algorithm motivates to further reduce the energy utilization by reducing the communication and computational overheads of sensor nodes.

3.1 The network model

We represent a WSN by a connected graph G (V, E), where the vertex set V corresponds to the nodes in the network, while the edge set E corresponds to the wireless links between nodes (we use “edge” and “link” interchangeably hereafter). Let n and l be the cardinalities of V and E, respectively. There is a special node s ∈ V, which represents the sink that collects data from the whole network. To simplify the network model, a few reasonable assumptions has adopted as follows:

3.2 Energy model

The energy model used in this paper is similar to the energy model used in ^{Chen et al. (2009)}. The energy that is consumed in order to transmit k-bit data from S_i to S_j over a distance d is given by,

where Eelec is the energy dissipated in communicating 1- bit data, while ∈fs, ∈mp are energies necessary for the amplifier in the free space and the multipath model, respectively. The energy required for the reception of k-bit data by a sensor node S_j is given by,

4.1 HCED phase-1: aggregation tree construction

In the HCED approach, the tree construction follows a binary search tree construction method. In this phase, each node begins the tree construction process when a node receives the construction request message Con_Req (P_N, P_LN, P_RN) from the parent node. After receiving a construction request from other nodes, each node broadcasts a Node_Msg with the following two values: one is the node id, and the other is the residual energy of this node within radio range r. At the same time, each node receives the Node_Msg messages from its neighbor nodes. Based on the Node_Msg broadcasting, the neighbor nodes will be discovered along with their residual energy by each node N_i. The response time of each neighbor node in the sensing limit of each node will be registered in the routing table of node N_i. If node ʋ has the sensing limit of d meters, ʋ will identify all the neighbors Nʋ=V∈G existing within d meters. Similarly, all the nodes present in the network will identify their neighbors and their distances during the broadcast.

After the neighbor discovering process, the data aggregation tree construction begins at each node. As stated earlier, this HCED approach implements a binary search tree-based algorithm for constructing an energy-efficient data aggregation tree. Initially, the construction request begins from sink node S. The sink node is the root node that is responsible for aggregating data from other nodes with high energy resources. The root node first identifies the left child, which has the shortest response time among all the neighbor nodes and in turn identifies its right child, which has the second shortest response time in the routing table. If the node with the least response time is already rooted with other parent nodes, the second least responsive node will be assigned as the left child, while the third least responsive node will be assigned as the right child.

In a binary search tree, a node can have a minimum of zero and a maximum of two child nodes. Now that the root node has attained its maximum child limits, the construction control can be moved to the left child of the root node through a Con_Req (P_N, P_LN, P_RN) message. The left child node then follows the same binary search tree construction algorithm until it finalizes its left and right child nodes. In turn, the construction control is moved to the right child of the root node and performs the same procedure until it finalizes its right and left child nodes. This process will continue until all the nodes finalize their child nodes. The construction control subsequently moves to the next node or the next level only when a node finalizes its left and right child node or when a node has only one left child. When a node finds its child nodes and their parent nodes, the construction control is moved to the left child. If all the neighbors of a node are associated with another parent node, that node is assigned as a leaf node. When all the nodes identify their child and parent nodes, the tree construction process can be stopped.

Algorithm 1 has been developed as per the description of the tree construction phase. This algorithm must be implemented for all sensor nodes that are deployed within the network, after which the algorithm execution begins at the root node with a Node_Msg broadcast to the network. In Algorithm 1, the following assumptions are made:

4.2 HCED Phase-2: collision free data aggregation algorithm

After the successful construction of the energy-efficient data aggregation tree, the root node is responsible for initiating the data aggregation process among the network. The initiation of data aggregation process will vary based on the type of application. If the application is event-driven, the node that identifies the event within its sensing area will begin the data-sending process towards the root node through their root nodes. In a time-driven application, the root node initiates the data aggregation process based on the time interval.

Case1. If the application is event-driven; the data aggregation process will only follow the routing table when sending the data towards the root node. In this case, a node N_i, which has sensed the data, follows a three-step procedure:

Here the proposed energy-efficient data aggregation tree is enough to aggregate the data in an efficient way.

Case2. If the application performs the data aggregation based on the time interval, the data aggregation process will be initiated by the root node. In this case, before the data aggregation process initiated at the root node, it should ensure that:

When all the nodes have the required information, the root node will initiate the data aggregation process by broadcasting its public key Q_k among the network. When a node receives the public key Q_k, the following applies:

Similarly, all the nodes perform the same procedure in order to collect and send the data to the root node.

In this proposed system, the collision-free data aggregation algorithm will be executed by all the nodes in the network. Initially, this algorithm requires the routing table (generated by Algorithm 1), encrypted data and the list of nodes U_A={N_n1, N_n2...N_ni} to be aggregated in the network. The time slot T_S is the time required by the sink node to aggregate data from all the nodes; T_S will be initiated to zero. First, the algorithm verifies whether the node is a leaf node or not. If the node Ni is a leaf node, as well as a left node for another parent node, it will send the data to its parent node during the first time slot. Then, the node Ni will be removed from the set UA. In turn, the node, which is a leaf node, as well as a right node for another parent node, will send the data to its parent node during the second time slot. This node will also be removed from the UA. This aggregation process will be continued until all the nodes are removed from the UA. Once all the data are aggregated by the sink node, the value of T_S is the time required by the algorithm to aggregate the data from all the nodes.

4.3 HCED Phase-3: encryption and decryption algorithm

This third phase of the proposed HCED algorithm describes the encryption and decryption algorithm used to provide secure data aggregation. After completion of the data aggregation tree construction process by Algorithm 1, the public key generation algorithm (Algorithm 3) will be executed by the root node as an initial process of data aggregation. In Algorithm 3, the initial hour (T_h) and minutes (T_m) of the data aggregation process are the key factors. By performing the mathematical calculation on T_h and T_m, Algorithm 3 finds the public key Q_k for a single-time data aggregation process. Whenever the data aggregation process is initiated by the root node, it generates a new public key Q_k and broadcasts it among the network through a data aggregation request.

The encryption algorithm (Algorithm 4) used in this proposed model will be executed by all the nodes in the network. If the node is a leaf node, it will first execute the encryption algorithm in order to encrypt the plain data PT_Ni and the MAC M_Ni. The MAC is the authorized code to ensure the integrity of the message for each node. The MAC is appended with the actual data before it is sent to other nodes. To reduce the communicational overhead among the network, this encryption algorithm only requires the public key Q_k of the root node. In order to provide a high level of confidentiality with the data, each node generates its own private key P_k based on the node ID and the public key Q_k. The proposed method does not have any separate algorithm for generating the private key to reduce the computational overhead on the nodes. Instead, it generates the private key, which then encrypts the plain data and the MAC by using Q_k and P_k in a single encryption algorithm.

After aggregating the data from all the nodes, the root node will execute the decryption algorithm (Algorithm 5) to find the actual data. Before decrypting all the data, the root node must have the encrypted cipher data (CT_Ni) and encrypted MAC (EM_Ni) for each node N_i. Since the public key is generated by the root node, it has the information about Q_k, but it does not have the information about P_k. The key idea in this decryption algorithm is to find the private key P_k using the node ID by following the same procedure. As such, Algorithm 5 first identifies the private key P_k for each node N_i, then it decrypts the encrypted data CT_Ni and encrypted MAC EM_Ni.

5.1 Maintaining the aggregation tree

The data aggregation tree scheduling can be re-constructed periodically by all the nodes. This paper proposed a detailed algorithm that can be employed on every node. So that, each node can easily re-construct the energy efficient data aggregation tree. When energy level of nodes is below data transmission threshold, an active sensor periodically broadcasts its energy level to its neighbors. Based on the energy level each node can be re-constructing the tree for better performance.

In case of node failure due to energy or any other issues, the node cannot do broadcasting. Whenever a parent node is not receiving any broadcasting reply or data from its left or right child, the parent node will broadcast a request message to its entire neighbors for re-constructing the aggregation tree based on the available active neighbor nodes. And the information about the failure node will be appended to the request message. So that, node failure can be managed by the root or sink node.

6.1 Theoretical analysis

As discussed in relation to the proposed algorithm in Section 5, the proposed HCED approach is constructing a binary tree-based energy-efficient data aggregation tree. Hence, it is required to prove that the proposed data aggregation has constructed a binary tree. Let the proposed data aggregation algorithm construct a G tree.

Theorem 1. If G is a tree, then every two nodes are joined in a unique path:

Proof: Consider P₁ and P₂ are the two paths between the nodes N₁ and N₂.Let x and y be the first and second points on both paths when tracing the two points from node N1 to node N2. The paths between x and y on P₁ and P₂then make a cycle. But this cannot happen if G is acyclic.

Now it is mandatory to prove G is acyclic. A G tree is acyclic only when the tree is connected.

Theorem 2. G is connected:

Statement: If every two nodes of G are joined by a unique path, then G is connected.

Induction: Let n=e+1.

Proof: Assume it is true for less than n points. Removing any edge from G breaks G into two components, since the paths are unique. (Theorem 1) when the sizes are n₁ and n₂, such that n₁+n₂=n. By using the induction hypothesis, n₁=e₁+1 and n₂=e₂+1. However:

Hence, G is connected.

Theorem 3. G is acyclic:

Statement: If G is connected, then G is acyclic.

Proof by contradiction: Suppose G has a cycle of length k. Then, there are k points and k edges on this cycle. Since G is connected for each node v that is not on the cycle, this results in the shortest path from v to a node on the cycle. Each such path contains an edge e_v that is not found on any other. Thus, the number of edges is at least e>= (n-k)+k=n, which contradicts the assumption n=e+1.

Corollary 1.

A graph G with n vertices is a tree T if any of the following conditions is satisfied:

Theorem 4.There is at least one grandparent node between any two parent nodes in the HCED aggregation tree:

Proof: Consider a node P_N, which becomes a parent during the role assignment process. Without any loss of generality, suppose that P is included in the neighbor list of grandparent U = {N_n1,N_n2...N_nn} ∈ G (V,E). According to Algorithm 1, P changes its role to a child if P is the least or the second least node in the sorted neighbor list U_s={N₁,N₂...N_n}. However, the instructions in Algorithm2 can only remove children and grandchildren, as opposed to the parent node (like P_N), from U_s. As such, P_N is rolled as a left child or right child in the neighbor list of grandparents, meaning that there is at least one grandparent node between any two parent nodes in the proposed aggregation tree.

Theorem 5.Under the two-hop interference model, the sink can receive all the aggregated data in most O(R+∆) time slots:

Note that under the two-hop interference model, any two x and y senders are unable to be a communication neighbor (otherwise, x will cause interference in the receiver of y).As such, given ∆ neighbors of a node, need at least ∆ / five time slots to simply let each of these ∆ neighbors transmit once. Thus, the following theorem applies.

Theorem 6.Under the two-hop interference model, for any data aggregation method, it will take at least max(R; ∆/5) time slots for the sink to receive the aggregated data:

For the k-hop interference model, where k ≥ 3, then any two x and y nodes that are neighbors of a node u clearly cannot transmit simultaneously. Thus, ∆ is a lower bound on the delay in data aggregation.

6.2 Path cost analysis

The maximum path cost from any leaf node to a parent node is D_i.

The maximum path cost from any leaf node to sink is D_sink.

The initial staggered time out is calculated as follows: if T_iis the stagger timeout for the aggregator node i, then

where T_ci is the cascading timeout, which depends on the level of the parent node. This gives the initial timeout, as in the cascade timeout, for each node, which is the same for all the nodes in the same level. The T_Siis the aggregation timeout of the node, which depends on the number of children it has.

Here, h denotes the hop distance of the node A_Nj, ∆ is the depth of the tree, T is the data generation period or the dead line and T_TD is the one-hop delay between the levels. It depends on the queuing delay, MAC delay, processing delay for aggregation function and the transmission delay.

It is assumed to be 0.1 seconds as used in existing model ^{Yu and Li (2011)}. After introducing this initial delay in the aggregation timer, the aggregation timer is fired for every T seconds, thus enabling the collection of packets generated in that round from all the nodes in the collection tree.

6.3 Packet loss analysis

The data loss analysis of the proposed system has been analyzed in the part dealing with the results analysis. The throughput of the sensor network using the HCED approach is evaluated in terms of the number of successful packet deliveries. It is directly related to packet loss. The packet loss ratio of the network can be calculated as follows:

Finally, the traffic overhead of the proposed protocol is also analyzed in terms of communication and computational overhead. To evaluate the traffic overhead of the distributed approach in a WSN, the average amount of traffic transmitted within the network is tested.

6.4 Complexity analysis

If the aggregation tree has an n-node and one root, the left sub tree will have an i-node, while the right sub tree has an (n-i-1) node as per the proposed tree construction. Hence, the complexity of the proposed data aggregation will be defined as:

The root contributes 1 to the path length of each of the order n-1 nodes. Therefore, the average overall complexity for n-nodes will be:

where Tn is Onlog⁡n. Therefore, the average complexity of data aggregation over the network which has n-nodes is:

7.1 Communicational overhead

The communicational overhead of a network describes the collision problem in the network. The HCED approach is a unique solution that resolves maximum issues including collision. Here, the simulation has been carried out in order to evaluate the collision in terms of energy. The simulation model initially consists of 100 nodes, while the simulation is continued until up to 100 packets are received. Figure 3(a) represents the additional average energy required by the sensor nodes when aggregating the 100 packets. The HCED approach saves a minimum of 3.5% more energy than other models regarding the communicational overhead when the number of nodes is 100. It also saves a minimum of 7.3% more energy when the number of nodes is 600. Figure 3(b) represents the additional average energy required by the sensor nodes when aggregating the 600 packets. The two-hop tree-based data aggregation resolves the collision problem, meaning that the proposed model reduces the communicational overhead.

Fig. 3 Communicational overhead (a) Communicational overhead of the network when number of packets = 100 (b) Communicational overhead of the network when number of packets = 200 

7.2 Computational overhead

The additional energy required by a sensor node to execute the algorithms in a data aggregation model can be represented as computational energy. This computational overhead will be analyzed in relation to the proposed model in two cases. Case 1 will be evaluated until up to 100 packets are received from 100 to 600 nodes. The computational overhead comparison for the Case 1 will be described in Figure 4(a). Case 2 will be evaluated until up to 200 packets are received from 100 to 600 nodes. Figure 4(b) describes the computational overhead of the proposed model. When 100 packets are aggregating from 100 nodes, the proposed model reduces the computational overhead between 11% and 90% more than other models. When receiving the 200 packets, the HCED approach will save between 14% and 29% more energy than other models. When compared to all the other models, the proposed model will have less computational energy due to the simplified algorithms used for data aggregation and tree construction.

Fig. 4 Computational overhead (a) Computational overhead of the network when number of packets = 100 (b) Computational overhead of the network when number of packets = 200 

7.3 Performance on energy consumption

Two simulation scenarios have been considered in order to evaluate HCED performance regarding energy. In the first scenario, the number of nodes varies between 100 and 600 in increments of 100 nodes. The simulation was continued until 100 packets of data were received, after which the average energy consumption was evaluated to aggregate 100 packets. Based on the energy consumption, the results were compared with other models and are described in Figure 5(a). From the simulation results, it can be seen that the collision-free binary tree-based data aggregation model in the HCED approach reduces a minimum of 2.4% less energy consumption than the HSDA and a maximum of 36% less energy consumption than the ABC model when the number of nodes is 100. When increasing the number of nodes up to 600, the resultant energy consumption of the HCED approach is 11% less than the HSDA and 104% less than the ABC model.

Fig. 5 Overall Energy consumption (a) Overall Energy consumption of the network when number of packets = 100 (b) Overall Energy consumption of the network when number of packets = 200 

In the second scenario, the simulation study was conducted until 200 packets were received, after which the average energy consumption was evaluated. The simulation results in this scenario have been studied and are described in Figure 5(b). When the number of packets is increased, the proposed model saves 6% more energy than the HSDA model and 38% more than the ABC model when there are 100 nodes involved. When the number of nodes is 600 and the number of packets is 200, the HCED approach saves 6.9% more energy than the HSDA model and 38% more than the ABC model when there are 100 nodes involved. When the number of nodes is 600 and the number of packets is 200, the HCED approach saves 6.9% more energy than the HSDA model and 91% more than the ABC model. This comparison study on energy consumption with other models proves that the proposed model improves energy utilization because of its collision-free binary tree-based tree construction and simplified data aggregation algorithm.

7.4 Performance on delay

Delay can be represented as the number of seconds required to aggregate the data from all the nodes in a network. The delay performance of the proposed model was evaluated and analyzed by comparing it with other models in Figure6. By varying the number of nodes from 100 to 600 and aggregating the 100 packets of data, the average delay was calculated and analyzed with other models in Figure 6(a). When aggregating 100 packets from 100 nodes, the HCED approach reduces the delay by a further 1.7% to 6.3% compared with other models. When there are 600 nodes, the HCED approach reduces the overall delay by 3.4% to 20% compared with other models. Figure 6(b) presents a detailed comparison regarding delay when aggregating 200 packets.

Fig. 6 Overall Delay in sec (a) Overall Delay of the network when number of packets = 100 (b) Overall Delay of the network when number of packets = 200. 

7.5 Performance on throughput

The throughput of a network can be described as the average of successful data aggregation over a wireless communication channel. The throughput is usually measured in bits per second or data packets per time slot. The performance evaluation of the proposed model can be analyzed in terms of the number of packets successfully received by the sink node.

Figure 7 shows a detailed analysis of the HCED approach’s throughput compared with other models.

Fig. 7 Throughput (a) Throughput of the static network when number of packets = 300 (b) Throughput of the dynamic network when number of packets = 300. 

The throughput of an HCED approach has been evaluated in two cases. In the first case, the throughput has been calculated by varying the number of nodes from 100 to 600, in increments of 100 nodes, on the static network where the node positions are fixed. When it is assumed that the sink node needs to aggregate 300 packets from the network, the simulation results in this case show that the static network with 100 nodes achieves a 99.3% throughput, which is 1% to 5% higher than existing models. When the number of nodes is increased to 600, the network achieves a 91.6% throughput, which is 3% to 9% higher than all existing models. Figure 7(a) shows the throughput performance of the HCDA regarding a static WSN. In the second case, the throughput performance of the proposed model has been evaluated in a dynamic network, in which all the nodes have mobility. When the number of nodes is 100, the proposed model achieves a 97.6% throughput, whereas it achieves 86.66% when the number of nodes is 600. In conclusion of this case, the HCDA has been shown to work well in dynamic environments, showing 3% to 8% improvement in throughput compared with existing models. Figure 7(b) describes the throughput analysis of a dynamic network with existing models.

7.6 Performance on security

The security performance of the proposed model has been analyzed in terms of integrity and confidentiality of the aggregated data. Let consider a malicious node CN_i, which is also deployed within the network without registering at the sink node. Since node CN_i may be the neighbor of other nodes, it will also be a part of a tree construction process. When the aggregating process begins, the node CN_i will also obtain the public key, after which it encrypts the data in its own way and sends it to the sink node. Even though it uses a duplicate node ID, which is already registered with the sink, the sink node will easily identify the malicious node CN_i when decrypting the MAC for each node. This ensures the data integrity of the proposed model. Let AN_i be the attacker node that exists in the network with the intention to modify the actual data by hacking and decrypting the actual data sent among the network. Assume that node AN_i knows the public key and also have a duplicate node ID. The data aggregated by the sink node ensures its originality by the MAC as well as the node ID of each node. The decryption algorithm used by the sink node is a secured algorithm, which cannot be hacked and implemented by any AN_i. Therefore, any node AN_i cannot decrypt any of the original data. Therefore, the proposed HCED approach ensures the data confidentiality by providing a highly secured encryption algorithm.

The simulation results and mathematical analysis of the proposed HCED with compared to other model proves that the use of binary tree based aggregation tree provides better performance on collision avoidance than other tree structures. The implementation of hybrid aggregation algorithm which is a combination of distributed and centralized algorithm in HCED reduces the delay latency. Moreover the combination of symmetric and asymmetric cryptography technique which is used in proposed model proves the improvement in secure data aggregation than other models.