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For understanding and controlling spreading in complex networks, identifying the most influential nodes, which can be applied to disease control, viral marketing, air traffic control, and many other fields, is of great importance. By taking the effect of the spreading rate on information entropy into account, we proposed an improved information entropy (IIE) method. Compared to the benchmark methods in the six different empirical networks, the IIE method has been found with a better performance on Kendall’s Tau and imprecision function under the Susceptible Infected Recovered (SIR) model. Especially in the Facebook network, Kendall’s Tau can grow by 120% as compared with the original IE method. And, there is also an equally good performance in the comparative analysis of imprecise functions. The imprecise functions’ value of the IIE method is smaller than the benchmark methods in six networks.

The phenomena of spreading can be seen everywhere in nature [

The identification of influential nodes is of great significance in fields of epidemic and rumor control, targeted advertising, and air traffic planning [

Most of the previous methods assume that the node’s influence depends on its own importance. But there is another key factor that cannot be neglected, namely, the neighbors’ importance. On the basis of this idea, Guo et al. [

(Color online) The sample network in the figure has 16 nodes as well as 25 edges. The SIR model can be used to simulate the nodes’ real spreading influence, that is, the amount of the infected nodes can be considered as spreading influence. We set

The original IE method assumes that the influence of the node should be obtained through the information entropy of its neighbors. In the IIE method, we argue that the spreading rate and the number of neighbors could adjust the initial information entropy. We can fulfill the identification of the influential nodes by using the final information entropy, namely, the IIE method. The details of the IIE method can be interpreted below.

In general, an undirected network

Equation (

To describe the IIE method in more detail, we set

(Color online) The figure shows how to calculate the IIE. There are four neighbors for node 1.

There are six empirical networks used to evaluate the performance of the IIE method. The US air network [

The e-mail network [

There are six fundamental statistical attributes in the six networks, such as

Network | ||||||
---|---|---|---|---|---|---|

US air | 332 | 2126 | 12.8 | 0.024 | −0.208 | 0.625 |

Polblogs | 643 | 16097 | 7.09 | 0.052 | −0.217 | 0.232 |

1133 | 5451 | 9.62 | 0.055 | 0.078 | 0.22 | |

Soc-hamsterster | 2426 | 16630 | 13.71 | 0.023 | 0.047 | 0.537 |

4039 | 88234 | 43.69 | 0.009 | 0.063 | 0.605 | |

LastFM | 7624 | 27806 | 7 | 0.037 | 0.017 | 0.219 |

Site^{1}.

For this paper, the node spreading influence is simulated with the SIR model [

Kendall’s Tau [

The imprecision function

For this paper, we selected six real networks to test the IIE method. According to different networks, we set

At first, we test the influence of different values of

(Color online) The figure shows the effect of parameter

From Figure

To check the efficiency of the IIE method, the

(Color online) The figure shows a comparison between the ranking list produced by the SIR model, and the ranking lists produced by the

Figure

(Color online) The figure shows the improvement of ratio

As can be seen from Figure

(Color online) This experiment compares different methods by using the imprecision functions

For controlling the spreading process, one of the basic tasks is to estimate the spreading influence and identify the influential nodes. By considering the information entropy and spreading rate of the target nodes, we proposed an improved information entropy (IIE) method. The IIE method takes the spreading rate and the number of the target node’s neighbors into account. And, those information dominate the new information entropy. According to the simulation results, the IIE method achieves a better performance than the IE method, and the IIE method (

Compared to the benchmark methods of the six networks, accuracy of the IIE method can be more satisfactory on identifying the influential nodes, while it poses some inevitable challenges. One of the challenges is that the IIE method merely takes the influence of the spreading rate for the target node into consideration and neglects the impact from target node’s neighbors. The distance

The datasets used in the present study are available from the first author upon reasonable request (

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (no. U1733203), Safety Foundation of CAAC (no. AQ20200019), and Foundation of CAFUC (no. J2020-084).