i_{n}
,
j_{n}
). We prove lemma 1 based on induction.

Basis:n= 1, W(i1,j1) satisfying W(i1,j1) = min{ W(i1,j1), W(i2,j2), …, W(in,jn)}, andd(i1,j1) ≤rmax. Therefore, nodei1andj1is the neighbor of each other satisfyingi1⇔j1.

Induction: Assume that lemma 1 holds for all pairs (ig,jg) (g= 0,1,…,n1). We need toprove lemma 1 also holds for the pair (in,jn). Nodejnsatisfyingjn∈NV(in) whend(in,jn) ≤rmax, and there exists a pathl= (w0=i,w1,w2,…,wn1,wn=j) from nodeito nodejsatisfying (wg,wg+1) ∈E(G0)(g= 0,1,…,n1) . Applying the induction hypothesis to each node pair (wg,wg+1)(g= 0,1,…,n1), we havewg⇔wg+1. Therefore,in⇔jn.
Theorem 1
.
G
_{0}
is connected if
G
is connected.
Proof
: Suppose
G
is connected. For any two nodes
i
,
j
∈
V
(
G
), there exists a path
l
= (
w
_{0}
=
i
,
w
_{1}
,
w
_{2}
,…,
w
_{n1}
,
w_{n}
=
j
) from node
i
to node
j
, where (
w_{g}
,
w
_{g+1}
) ∈
E
(
G
)(
g
= 0,1,…,
n
1) and
d
(
w_{g}
,
w
_{g+1}
) ≤
r_{max}
. Since
w_{g}
and
w
_{g+1}
satisfy
w_{g}
⇔
w
_{g+1}
by lemma 1, we have
i
⇔
j
.
 5.2 Average Connectivity of Nodes in Each Cluster
Node connectivity refers to the number of connecting edges of each node. In the network, the connecting edges of each node can be divided into edges for receiving data packets and edges for transmitting data packets. There is only one edge for transmitting data packets, and multiple edges can be used to receive data packets. Therefore, node connectivity refers to the number of edges for receiving data packets.
Fig. 9
is the illustration of node connectivity. Based on the definition of node connectivity, node connectivity of node 1 is 2, node 4 is 1, node 2 and node 3 are 0. The average connectivity of nodes refers to the ratio of the total number of edges for receiving data packets and the total number of nodes in the network.
Illustration of node connectivity
Assume that the number of nodes in each cluster is
n
_{0}
. Since there is only one leader node in each cluster, the number of sensor nodes is
n
_{0}
1. Based on the multihop transmission principle, nodes transmit data packets to the next hop neighbor. Therefore, there is only one edge for transmitting data packets of each sensor node, which means that the total edges for transmitting data packets in each cluster are
n
_{0}
1. Based on the principle that connecting edge is a bidirectional edge, edges for transmitting data packets of a node are also the edges for receiving data packets of another node. Therefore, the total edges for receiving data packets are
n
_{0}
1, and the average connectivity of nodes is (
n
_{0}
 1)/
n
_{0}
.
 5.3 The Time Complexity of MCHTC
Since the number of leader nodes is small compared with the total number of nodes in the network, the time complexity of MCHTC is mainly determined by the time complexity of topology setup phase in clusters. The average number of nodes in each cluster can be expressed as
N
/
m
, and
m
is the number of leader nodes in section 4.1.2.
In the phase of topology setup in clusters, if new node added in chain needs to traverse all nodes which are waiting to be added in chain, the time complexity is the maximal time complexity. The number of nodes which the first node added in chain needs to traverse is
N
/
m
 1, the number of nodes which the second node added in chain needs to traverse is
N
/
m
 2 …… Similarlily, the number of nodes which the penultimate node needs to traverse is 1, and the last node will de added in chain directly. Therefore, the maximal time complexity
of topology setup in clusters is as follow:
Therefore, the maximal time complexity
of topology setup in clusters is
O
(
N
^{2}
) .
If new node added in chain only needs to traverse one neighbor, the time complexity is the minimum time complexity
, which is as follow:
Therefore, the minimum time complexity
of topology setup in clusters is
O
(
N
) .
In the phase of topology setup among leader nodes, it needs to execute
m
1 outer loops to obtain the node pairs with the minimum weight, and
m
inner loops need to be executed to update the leader nodes which have been added in tree. Therefore, the time complexity of topology setup among leader nodes is
m
(
m
 1) .
In summary, the maximal time complexity of MCHTC is
O
(
N
^{2}
) , and the minimum time complexity is
O
(
N
) .
 5.4 The Number of Leader Nodes and Network Energy Consumption
Based on the schematic diagram of data transmission among leader nodes in section 4.2.2, leaf leader nodes only need to transmit data packets to their parent nodes, inner leader nodes not only need to receive and aggregate data packets transmitted from child nodes, but also need to transmit the fused data packets to the parent nodes, and root leader node only needs to transmit the fused data packets to the base station.
In the topology setup phase among leader nodes, there is only one root leader node. Assume that the number of leader nodes that are in the end of chain is
m
_{1}
, the number of leader nodes that are in chain is
m
_{2}
, which satisfies
m
_{1}
+
m
_{2}
=
m
. The number of leaf nodes in
m
_{1}
is
n
_{1}
, and the number of leaf nodes in
m
_{2}
is
n
_{2}
. Based on the radio energy dissipation model in section 3.3 and the energy consumption model of chain topology in section 3.4, we analyze the relationship between the number of leader nodes and the network energy consumption as follows, and meanings of the parameters are the same parameters as in section 4.1.2.

The energy consumption of root leader node in chain is:

The energy consumption of leaf leader nodes which are in the end of chain is:

The energy consumption of inner leader nodes which are in the end of chain is:

The energy consumption of leaf leader nodes in chain is:

The energy consumption of inner leader nodes in chain is:

The energy consumption of sensor nodes which are in the end of chain is:
The energy consumption of sensor nodes in chain is:
where
a
_{1}
,
a
_{2}
,
a
_{3}
are the number of child nodes of corresponding leader nodes. Therefore, the network energy consumption is as in Eq. (14).
where
m
_{1}

n
_{1}
=
β
_{1}
m
,
m
_{2}

n
_{2}
=
β
_{2}
m
,
β
_{1}
is the proportion of inner leader nodes which are in the end of chain,
β
_{2}
is the proportion of inner leader nodes which are in chain, and satisfies 0 <
β
_{1}
< 1 , 0 <
β
_{2}
< 1 .
m
_{1}
=
αm
, where
α
is the proportion of leader nodes which are in chain satisfying 0 <
α
< 1. Therefore, Eq. (14) can be expressed as Eq. (15).
The derivation of Eq. (15) is:
We can obtain that the number of leader nodes and the network energy consumption are unrelated. The same conclusion can be obtained when root leader node is in the end of chain.
 5.5 Energy Consumption of “Long Distance” in Each Cluster
Assume that the distribution area is a square, and the side length is
L
. The subareas are approximately small squares. The area of distribution area is
L
^{2}
, and the area of each cluster is approximately
L
^{2}
/
m
. The average number of nodes in each cluster is
N
/
m
. The diagonal length is
l
_{0}
, distance between nodes in each cluster satisfies
d
≤
l
_{0}
. Based on the relationship between diagonal length and area of square, we can obtain
. The diagonal length
l
_{0}
decreases with the increasing of the number of subareas.
We analyze the energy consumption of “long distance” based on the maximal energy consumption of a node. Assume that leader node and a sensor node are located in the two vertices of the diagonal respectively, and the sensor node is not in the end of chain. The distance between the sensor node and the leader node is
, distance between other nodes is
. The energy consumption of the sensor node is as in Eq. (16).
We analyze the average energy consumption of nodes in each cluster as follows. Assume that leader node is in the end of chain, but is not the root leader node, and it is the leaf leader node of the minimum cost tree. The energy consumption of the leader node is
E_{ld}
, the energy consumption of sensor nodes in the end of chain is
, the energy consumption of sensor nodes in chain is
. Therefore, the average energy consumption of a node in each cluster is as in Eq. (17).
Based on Eq. (16) and Eq. (17), the difference between
and
E_{ave}
is as in Eq. (18).
The derivation of Eq. (18) is:
If
, we can obtain
, which means that thereexists an optimal number of leader nodes to ensure a minimum value of
E_{diff}
(
m
).
Based on the Eq. (10) in section 4.1.2, the residual energy threshold of round
i
is
, and the threshold of round
i
+1 is
. Therefore, the energy consumption threshold
of each round is as in Eq. (19).
If
E_{diff}
≤
, the impact of “long distance” on the energy consumption of nodes in clusters is within an acceptable range, while if
E_{diff}
>
, the “long distance” has a negative impact on the energy consumption of the network.
6. Simulations and Results
To evaluate performance of MCHTC discussed in the previous section, simulations are presented by MATLAB and its performance is compared with LEACH, PEGASIS, and IEEPB. We describe performance metrics, simulation setup and simulation results in this section.
 6.1 Performance Metrics
The following metrics are used to capture performance of the proposed algorithm and to compare with other algorithms.
Network lifetime
: Wireless sensor networks is a data collection network, failure of any node may lead to the incompleted data and unconnected network. Therefore, in this paper, the metric of network lifetime is the round that appears the first death node.
Energy consumption
: The total energy consumed by nodes receiving and sending the data packets.
Network energy balance
: In this paper, we use the same definition of network energy balance as in
[26]
. Based on the definition of network lifetime, to prolong network lifetime, we need to prolong the round that appears the first death node. Assume that round appears the first death node is
r_{i}
, and the energy consumption of any node
j
is
E_{j}
. To prolong network lifetime, we can obtain Eq. (20).
Eq. (20) presents that when the first death node appears in the network, the energy consumption of all nodes in the network is equal to the total energy consumption of the network, which means that the network energy is not wasted.
Assume that
is the residual energy of node
i
when the first death node appears in the network, the energy load factor is as in Eq. (21).
The average energy load factor is as in Eq. (22).
Based on Eq. (21) and Eq. (22), the
Network Energy Balance
(EB) is as in Eq. (23).
We can obtain that the smaller the difference between
load_{i}
and
load_{ave}
, the more balanced the energy consumption of nodes.
 6.2 Simulation Setup
100 nodes are randomly distributed in the monitoring area with the size of 100
m
×100
m
in the simulation. The coordinates of the base station is (50,150) . The basic parameters for the simulation are as shown in
Table 2
.
Simulation parameters
 6.3 Simulation Results
 6.3.1 The Topology of Network
Fig. 10
is the network topology that does not execute topology control. If topology control is not executed, nodes will communicate with neighbors with the maximal transmission power, a lot of redundant links will exist in the network. The energy consumption of a node is large, and leads to a short network lifetime.
Topology that does not execute topology control
Fig. 11c
and
Fig. 11d
are the topologies derived by MCHTC when
p
= 0.05. If
p
= 0.05, the
Region Partition Parameter
satisfies
t
= 3, which means that node distribution area is divided into 9 subareas. Therefore, 9 short chains are formed in the network. The soild blue dots in figures are the leader nodes elected in each cluster. The solid red lines are the links between leader nodes, which presents that a tree topology is formed by leader nodes. Compared with
Fig. 10
, the topology derived by MCHTC reduces a lot of redundant links between nodes, the transmission power of nodes only needs to satisfies that nodes can reach the farthest neighbor when broadcasting messages. Compared with the topology derived by PEGASIS, the multichain topology derived by static clustering not only reduces the transmission delay of data packets, but also preserves the superiority that multihop topology can reduce the communication distance between nodes and energy consumption of nodes.
In MCHTC, according to the actual requirements of the distribution area, the
Region Partition Factor p
can be adjusted to change the number of clusters and the length of chains formed in clusters.
Fig. 11a
,
11b
are the topologies derived by MCHTC when
p
equals to 0.03, 0.07 respectively and nodes have the same distribution.
p
is independent of the distribution of the nodes. No matter whether nodes have the same distribution or not, executing MCHTC will form a hierarchical topology as mentioned above, and changing
p
will change the number of chains and the tree topology formed by leader nodes.
Fig. 11c
,
11d
are the topologies derived by MCHTC when
p
equals to 0.05 and nodes have different distribution.
Topology derived by p = 0.03
Topology derived by p = 0.07
Topology derived by p = 0.05
Topology derived by p = 0.05
 6.3.2 Network Lifetime
Fig. 12a
shows the trend of total number of nodes that remain alive over simulation runs.
Fig. 12b
is the network lifetime corresponding to different number of leader nodes.
Table 3
shows the number of rounds when 1%, 10%, 50%, and 90% of nodes die.
Table 4
shows the percentage of prolonged network lifetime with different
Region Partition Factor
.
Number of active nodes per round
Comparison of the network lifetime
Number of rounds when different proportions of nodes die
Number of rounds when different proportions of nodes die
Percentage of prolonged network lifetime with differentRegion Partition Factor
Percentage of prolonged network lifetime with different Region Partition Factor
We can obtain that the network lifetime is the shortest under the topology derived by LEACH. Singlehop communication between cluster heads and the base station results in a rapid energy consumption of cluster heads. In PEGASIS and IEEPB, the death rate of nodes is relatively slow at first, but accelerates when the network enters into the decline phase. Due to that there exists longchain problem between some nodes in PEGASIS, and long communication distance leads to premature death of some nodes, the netwok lifetime is shorter than IEEPB and MCHTC. In IEEPB, although the method of constructing chain topology could avoid the longchain problem in PEGASIS, it has a negative effect on the network energy balance, because some nodes may have more child nodes, while others may have fewer or no child nodes. Nodes with more child nodes have a larger energy consumption, which will shorten the lifetime of the network. The network lifetime is the longest under the topology derived by MCHTC, and the death rate of nodes is relatively slower when network enters into the decline phase. When all nodes die in other three kinds of algorithms, there are certain number of nodes alive in MCHTC. Therefore, MCHTC not only prolongs the network lifetime, but also balances the energy consumption of nodes. It is because the communication cost , the residual energy of nodes and distance are considered when nodes look for the next hop neighbor. The simulation result in
Fig. 12a
is obtained when
p
= 0.05.
Because there is only one leader node in PEGASIS and IEEPB, the network lifetime of PEGASIS and IEEPB are reference values which are presented by the dotted lines in
Fig. 12b
. Compared with LEACH, MCHTC significantly prolongs the network lifetime when the number of leader nodes is the same in the network. Therefore, multihop topology can reduce the energy consumption of nodes compared with singlehop topology. It is very important for the energyconstrained wireless sensor networks.
We can obtain from
Table 3
that the network lifetime is the longest when
p
= 0.05. When
p
= 0.01, the topology derived by MCHTC is the same as PEGASIS. In MCHTC, nodes select next hop neighbor based on
Energy Distance
, the network lifetime increases compared with PEGASIS. However, the problem of long distance between nodes leads to a slightly lower network lifetime compared with IEEPB. But we can not ignore that the negative impact of long distance on the network lifetime is reduced when nodes look for the next hop neighbor based on
Energy Distance
.
Table 4
shows the percentage of prolonged network lifetime with different
Region Partition Factor
. When
p
=0.01, MCHTC extends network lifetime 34.24% and 5.42% respectively compared with LEACH and PEGASIS. However, the network lifetime is shorter than IEEPB, because when
p
=0.01, the network is not divided into subareas, a chain is formed in the network, and there still exists longchain problem between some nodes. We can obtain from the
Table 4
that when
p
=0.05, MCHTC has the best performance of network lifetime, it extends network lifetime 51.37%, 18.87% and 9.93% respectively compared with LEACH, PEGASIS and IEEPB. When
p
=0.03, 0.05, 0.07 and 0.09 respectively, several short chains are formed in the network, and a tree topology is formed by leader nodes. Compared with IEEPB, MCHTC extends network lifetime 4.81%, 9.93%, 8.25% and 7.42% respectively, it shows that multichain topology and multihop transmission can prolong the network lifetime as well as avoid the condition that some nodes have more child nodes while others have fewer or no child nodes.
 6.3.3 Energy Consumption
Fig. 13
demonstrates the residual energy of the network during the simulation rounds. LEACH consumes more energy because the cluster heads collect data packets from sensor nodes and transmit the fused data packets directly to the base station. Compared with PEGASIS, IEEPB and MCHTC only slightly reduce the energy consumption, and the performance of IEEPB and MCHTC is nearly the same. It is because the topology derived by PEGASIS has significantly reduces the distance between nodes, and has a better performance in terms of energy consumption and network energy balance. Based on the theoretical analysis in section 5.4, in MCHTC, the number of leader nodes is unrelated with the energy consumption, therefore, the number of leader nodes does not have an impact on the network energy consumption. At the same time point, the network residual energy in PEGASIS, IEEPB and MCHTC is higher than LEACH, which means that the energy consumption of each round in multihop topology is smaller, and the energy efficiency is higher.
Energy consumption of the network
 6.3.4 Network Energy Balance
Fig. 14a
is the network energy balance with different
Region Partition Factor p
. The parameter
p
is the theoretical percentage of cluster nodes in LEACH. In MCHTC, parameter
p
is the
Region Partition Factor
which determines the number of clusters. The energy balance of PEGASIS and IEEPB are the reference values which are presented by dotted lines.
Comparision of network energy balance
Based on the definition of EB in section 6.1, the smaller the EB, the more balanced the energy consumption of nodes in the network. LEACH has a poor performance in terms of EB because sensor nodes communicate with cluster heads directly, and the energy consumption of nodes which are far from cluster heads is large. On the contrary, the energy consumption of nodes which are near the cluster heads is small. Cluster heads transmit data packets to the base station directly, which leads to a large energy consumption of cluster heads.
The performance of PEGASIS is better because there is only one leader node elected to transmit data packets directly to the base station. The way PEGASIS electing leader node can ensure that all nodes in the network are elected as leader node after
N
rounds and nodes only communicate with the nearest neighbor.
The chain formed in IEEPB can avoid long distance between nodes, however, increase the number of comparisons and some nodes have multiple child nodes, which has a negative impact on the network balance. Therefore, the performance of IEEPB is poorer than PEGASIS. In MCHTC, when 0.01≤
p
≤0.02, the topology derived by MCHTC is the same as PEGASIS, and has a better performance in terms of EB compared with PEGASIS and IEEPB. However, with the increasing of
p
, the number of leader nodes increases, which means that the number of nodes with large energy consumption increases. Therefore, it has a negative impact on EB. When 0.03≤
p
≤0.06, the performance of MCHTC is between PEGASIS and IEEPB, which means that the multichain topology not only can avoid long distance between nodes, but also will prevent nodes from having multiple child nodes. when 0.07≤
p
≤0.09, MCHTC has a poorer performance compared with PEGASIS and IEEPB, but better than LEACH.
Fig. 14b
is the energy balance with different number of leader nodes. As mentioned above, the energy balance of PEGASIS and IEEPB are reference values which are presented by dotted lines. The performance of MCHTC is significantly better than LEACH when the number of leader nodes is the same in the network.
Comparision of network energy balance with the same number of leader nodes
 6.3.5 Energy Consumption Analysis of “Long Distance” in Clusters
Based on the theoretical analysis in section 5.5,
Fig. 15
is the comparison of
and
E_{ave}
. To ensure the uniform distribution of the abscissa, we use
Region Partition Parameter t
to present the abscissa. The difference between
and
E_{ave}
decreases with the increasing number of clusters. We analyze whether
E_{diff}
is within an acceptable range as follows.
Comparison of energy consumption
In this paper,
r_{max}
= 1200,
E
_{0}
= 0.5J. Based on Eq. (19), the energy consumption threshold of each round is approximately 4.17×10
^{4}
. We can obtain that
E_{diff}
is not within the acceptable range when the distribution area is not divided. Therefore, the energy consumption of “long diatance” has a negative impact on the energy consumption of the network.
E_{diff}
decreases with the increasing number of clusters, and satisfies
E_{diff}
≤
. Therefore, the multichain topology derived by MCHTC can to some extent avoid the negative effect of “long distance” in clusters.
7. Conclusion
In this paper, we have proposed a Multichain Based Hierarchial Topology Control Algorithm (MCHTC) for wireless sensor networks. The main goal of MCHTC is to minimize energy consumption and prolong network lifetime based on certain network energy balance. MCHTC can meet the requirements for energysaving applications. MCHTC divides the network area into several subareas based on Region Partition Factor by static clustering. The communication cost, the residual energy and distance are considered when nodes select the next hop neighbor to reduce the energy consumption of nodes. MCHTC organizes nodes in each cluster into a chain, and leader nodes form a tree topology to avoid long distance between them. when electing leader nodes, candidate leader nodes are elected based on the residual energy threshold. Leader nodes are elected based on the residual energy of candidate leader nodes and the distance between themselves and the base staion. The number of clusters can be adjusted by changing the Region Partition Factor based on the actual requirements of distribution area. Simulation results show that compared with LEACH, PEGASIS and IEEPB, the proposed algorithm can reduce the energy consumption and prolong the network lifetime based on certain network energy balance.
BIO
Tang Hong obtained his phD degree in Computer Software & Theory from the University of Chongqing in 2003. He received his Master degree on Photoelectric Signal Process from Sichuan University. He is currently a Professor at the Chongqing University of Posts and Telecommunications, China. His current research interests include computer networks and telecommunication technologies.
Wang Huizhu received her B.E. degree on Telecommunication Engineer from Chongqing University of Posts and Telecommunications, China in 2013. She is a postgraduate in Telecommunications & Information College, Chongqing University of Posts and Telecommunications. Her research interests include wireless sensor networks and complex network models.
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