In physics, many physical phenomena are related to the field model which describes the interaction relationship between different objects. The interaction between two different entities is manifested by the operation of the energy. The physical properties of field can be used to depict the physical phenomenon in field space. The relationships between different variables reveal the ways of energy conversion in the field [15]. Therefore, the features of the field can be used to solve similar practical problems.
In the preliminary work, we put forward an influence calculation model of users in online social network based on field theory by using the data of Sina microblog. A comparative analysis was carried out on the influence calculation model through experiments. The experimental results showed that this model had a better performance in user classification [16]. This paper proposes a node importance evaluation method in wireless sensor network based on the energy field model, in which the influence transfer refers to the data transmission between different nodes and the transmission process refers to the process of influence transferring along the network topology. The process is regarded as the accumulation of node influence. The relationship between nodes and the node influence is analyzed with the characteristics of nodes in the process of data transmission. The node influence in network generates or attenuates when the data transmit among nodes. In other words, when a node sends data, a certain part of energy of the node is consumed. Finally, the importance of the node is evaluated by the influence of the nodes which generated in the process of data transmission in wireless sensor network.
All the nodes in the target network which are set in indirect structure have the same functional characteristics. When N
_{
i
} sends a data message, its related N
_{
j
} will forward the message. If N
_{
j
} is not the destination node of the message, N
_{
j
} will forward the message. It means that N
_{
i
} has exerted an influence on N
_{
j
} during data transmission. In the forwarding process, the influence of the two nodes N
_{
i
} and N
_{
j
} will increase. Therefore, the influence of N
_{
i
} on N
_{
j
} is larger than that of N
_{
j
} on N
_{
i
}. N
_{
j
} receives message from node N
_{
i
}, which shows the influence of node N
_{
i
} indirectly. In other words, N
_{
j
} will get influence from node N
_{
i
} while forwarding the message received from N
_{
i
}. Meanwhile, N
_{
j
} will feed back some influence to node N
_{
i
}. When the message forwards through nodes one after another, the influence will transfer and generate feedback constantly between relevant nodes, as shown in Fig. 1.
In wireless senor network, the operation of the energy should be obtained when data is transferred between different nodes. Then, the influence of the corresponding nodes is calculated by calculating the parameters in the data transmission process in the network layer. For any two nodes, N
_{
i
} sends the message and N
_{
j
} receives the message. The energy consumed for transmitting k bit data is expressed through the universal wireless communication energy consumption model as follows [17]:
$$ {E}_{\mathrm{send}}\left(k,d\right){E}_{\mathrm{elec}}k+{\varepsilon}_{\mathrm{amp}}k{d_{\mathrm{ij}}}^2 $$
(1)
The energy consumed for receiving k bit data is:
$$ {E}_{\mathrm{receive}}(k)={E}_{\mathrm{elec}}k $$
(2)
where E
_{elec} is the transmission coefficient of radio frequency, ε
_{amp} is the amplification coefficient of sending device, d
_{ij} is the data transmission radius of the node. The congestion of node can be written as follows:
$$ J=\frac{C_{\mathrm{dsend}}}{C_{\mathrm{receive}}+C{}_{\mathrm{send}}} $$
(3)
where C
_{dsend} is the number of unsent messages of the node, C
_{receive} is the number of received messages, and C
_{send} is the number of sent messages.
The distance between N
_{
i
} and N
_{
j
} is 1; thus, the influence of N
_{
i
} can be expressed as I
_{
r = 1}. When the distance between N
_{
i
} and N
_{
j
} is larger than 1, the influence of N
_{
i
} is expressed as I
_{
r > 1}. The influence in the expressions generates in one data acquisition period. The influence generated during message transmission is:
$$ \left\{\begin{array}{l}{I}_{r=1}={f}_1\left({N}_i,{C}_r\right)\hfill \\ {}{I}_{r>1}={f}_2\left(I\left({N}_{fa}\right),r\right)\hfill \end{array}\right. $$
(4)
where the value of I
_{
r=1
} is related to the node properties and the number of messages C
_{
r
} that N
_{
j
} forward from N
_{
i
}. The value of I
_{
r>1} depends on the distance r between N
_{
j
} and N
_{
i
} as well as the influence of father N
_{
fa
} (the node near sink is father node).
When r = 1, the formula of influence derived from field theory is:
$$ {I}_{r=1}=G\frac{C_r}{C_p}\left({E}_{\mathrm{send}}+{E}_{\mathrm{receive}}\right) $$
(5)
where C
_{
r
} is the number of messages N
_{
j
} forward from N
_{
i
} in one data reception period and C
_{
p
} is the number of messages received from N
_{
j
} in one data reception period. The ratio of C
_{
r
} to C
_{
p
} refers to the correlation coefficient of N
_{
j
} to N
_{
i
}. G refers to the density of nodes around N
_{
i
} in one data reception period. E
_{send} refers to the energy of N
_{
i
} consumed for sending k bit data. E
_{receive} refers to the energy of N
_{
j
} consumed for receiving k bit data. G can be expressed as follows:
$$ G=\frac{K_i}{K_{\max }} $$
(6)
where K
_{
i
} refers to the actual sides connected with N
_{
i
} and K
_{max} refers to the possible maximum sides connected with N
_{
i
}. k
_{
i
} is the degree of N
_{
i
}. Then, K
_{
max
} can be expressed as:
$$ {K}_{\max }=\frac{k_i\left({k}_i1\right)}{2} $$
(7)
When r > 1:
$$ {I}_{r>1}=I\left({N}_{fa}\right)P\left({J}_{fa},r\right) $$
(8)
where I(N
_{
fa
}
) is the influence of the father nodes and P is the forward probability. The larger the surplus energy, the higher the probability of becoming a forwarding node. J
_{
fa
} is the congestion rate of the father node. Considering the message may impair in forwarding, the forward probability is:
$$ P={\left({P}_E{J}_{fa}\right)}^r $$
(9)
where P
_{
E
} refers to the surplus energy ratio of the node:
$$ r=\left{e}^{\mu {d}_{ij}}\right $$
(10)
where μ is the weighted value of the range attenuation. The influence of Ni is:
$$ I={\displaystyle \sum_n{I}_{r=1}}+{\displaystyle \sum_m{I}_{r>1}} $$
(11)
where n refers to the number of nodes which are away from node N
_{
i
} with the distance r equaling 1 and m refers to the number of nodes causing the influence with the distance r greater than 1. The influence of node N
_{
i
} could be denoted by the influence of these nodes in wireless sensor network, as shown in Fig. 2.
There are four steps to evaluate node importance based on energy field model:

(1)
Calculate the energy consumption in the process of data transmission in the network

(2)
Choose the parameters which are related to the node in the process of data transmission

(3)
Under the given experimental environment conditions, calculate the I
_{
r=1
} and I
_{
r>1
}

(4)
Calculate the influence of all nodes in the whole network by expression (11) and find out the influential nodes according to the results