Topology potential theory is a new community detection theory on complex network, which divides a network into communities by spreading outward from each local maximum potential node. At present, almost all topology-potential-based community detection methods ignore node difference and assume that all nodes have the same mass. This hypothesis leads to inaccuracy of topology potential calculation and then decreases the precision of community detection. Inspired by the idea of PageRank algorithm, this paper puts forward a novel mass calculation method for complex network nodes. A node’s mass obtained by our method can effectively reflect its importance and influence in complex network. The more important the node is, the bigger its mass is. Simulation experiment results showed that, after taking node mass into consideration, the topology potential of node is more accurate, the distribution of topology potential is more reasonable, and the results of community detection are more precise.

Most complex networks show community structure; that is, groups of vertices that have a higher density of edges within them and a lower density of edges between groups [

Gan et al. [

Han et al. [

Zhang et al. [

Topology potential calculation is the foundation and key step for the above topology-potential-based community detection methods. In a given network

In formula (

On one hand, a node’s mass reflects its inherent properties, such as importance and influence. Different nodes have different inherent properties. For example, in social network, the importance of different people is significantly different, and public figures obviously have more influence than general people.

On the other hand, (

In order to solve the above problems, this paper puts forward a mass calculation method for complex network nodes, which is inspired from the idea of PageRank [

This paper is organized as follows: Section

Apparently, matter particle has its inherent mass. But how to weigh the mass of network nodes? A node’s mass should reflect its importance and influence in the complex network. The more important a node is, the bigger its mass should be. Inspired by the idea of PageRank algorithm, this paper puts forward a mass calculation method for complex network nodes.

The PageRank algorithm has been successfully used by Google to evaluate the importance of web pages. Each web page is assigned a PR value to reflect its importance. The algorithm claims that the PR value of a web page can be measured by the number and importance of web pages linking to this page. Generally speaking, the more web pages link to this page, the more important it is. The contributions of these web pages are different: the more important these pages themselves are, the more contribution they make to this page.

Similarly, the importance of a network node can be measured by the number and importance of its neighbor nodes. The more neighbor nodes the node has, the more important it is. The more important its neighbors themselves are, the more important the node is.

In a given network

Definition

This paper selected a representative social network—Zachary network to analyze the relationship between damping factor

The relationship between damping factor

Damping factor |
Number 1 node | Number 2 node | Number 3 node | Number 4 node | Number 5 node | Number 6 node | Number 7 node |
---|---|---|---|---|---|---|---|

0.00 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |

0.10 | 1.395255 | 1.126644 | 1.112363 | 1.021542 | 0.966275 | 1.013809 | 1.013809 |

0.20 | 1.747079 | 1.237061 | 1.219573 | 1.041998 | 0.935492 | 1.025750 | 1.02575 |

0.30 | 2.061667 | 1.335101 | 1.323713 | 1.062184 | 0.907004 | 1.035298 | 1.035298 |

0.40 | 2.343918 | 1.424051 | 1.426751 | 1.082954 | 0.880128 | 1.041800 | 1.041800 |

0.50 | 2.597738 | 1.506913 | 1.530693 | 1.105297 | 0.854066 | 1.044337 | 1.044337 |

0.60 | 2.826218 | 1.586712 | 1.637788 | 1.130467 | 0.827759 | 1.041493 | 1.041493 |

0.70 | 3.031637 | 1.666869 | 1.750830 | 1.160193 | 0.799622 | 1.030909 | 1.030909 |

0.80 | 3.215021 | 1.751764 | 1.873724 | 1.197029 | 0.766916 | 1.008239 | 1.008239 |

0.90 | 3.374151 | 1.847568 | 2.01265 | 1.244893 | 0.724111 | 0.964480 | 0.964480 |

1.00 | 3.495605 | 1.963056 | 2.178759 | 1.309709 | 0.658029 | 0.878151 | 0.878151 |

As can be seen from Table

Figure

The relationship between damping factor

Node mass calculated by our method can effectively reflect the importance and influence of nodes in complex network. The more important a node is, the bigger its mass is. After taking node mass into consideration, the topology potential of node will be more accurate, and the distribution of topology potential will be more reasonable. Now that mass

This section will empirically analyze the influence of node mass on three typical topology-potential-based community detection methods. These three methods come from literature [

Simulation program was implemented using scientific computing software MATLAB in the Windows environments. The experiment data include two complex networks: one is a real world network—Dolphin social network, which comes from

For each network, there are two schemes: one is “without mass” scheme, which ignores node difference and sets

The artificial complex network is generated by the LFR-Benchmark generator. The node number is 100, the edge number is 230, the average degree is 4.6, and the implanted community number is 2. The structure of the artificial complex network is shown in Figure

Artificial complex network.

Table

The topology potential of artificial network nodes with two schemes.

Node | Without mass | With mass | Node | Without mass | With mass |
---|---|---|---|---|---|

Node 1 | 2.4140 | 1.8383 | Node 11 | 2.5175 | 2.0233 |

Node 2 | 2.1035 | 1.4705 | Node 12 | 2.2415 | 1.6444 |

Node 3 | 2.5520 | 1.7553 | Node 13 | 2.5520 | 1.3730 |

Node 4 | 2.5175 | 1.8146 | Node 14 | 2.5865 | 1.4425 |

Node 5 | 2.5175 | 1.8885 | Node 15 | 2.4140 | 1.9363 |

Node 6 | 1.9140 | 1.2719 | Node 16 | 2.2415 | 1.9283 |

Node 7 | 2.6900 | 1.9954 | Node 17 | 2.4140 | 1.5655 |

Node 8 | 2.6210 | 1.7898 | Node 18 | 3.3280 | 1.5591 |

Node 9 | 2.2760 | 1.5757 | Node 19 | 3.0175 | 1.6472 |

Node 10 | 2.0000 | 1.4835 | Node 20 | 2.8105 | 1.4862 |

Table

The top 20 nodes of the artificial complex network.

Serial number | Without mass | With mass | Serial number | Without mass | With mass |
---|---|---|---|---|---|

1 | Node 99 | Node 99 | 11 | Node 91 | Node 91 |

2 | Node 97 | Node 97 | 12 | Node 90 | Node 88 |

3 | Node 100 | Node 100 | 13 | Node 85 | Node 85 |

4 | Node 98 | Node 94 | 14 | Node 87 | Node 87 |

5 | Node 95 | Node 98 | 15 | Node 88 | Node 86 |

6 | Node 94 | Node 96 | 16 | Node 86 | Node 90 |

7 | Node 96 | Node 95 | 17 | Node 81 | Node 84 |

8 | Node 93 | Node 93 | 18 | Node 82 | Node 76 |

9 | Node 92 | Node 89 | 19 | Node 84 | Node 81 |

10 | Node 89 | Node 92 | 20 | Node 78 | Node 78 |

The artificial complex network contains two communities: the community

The Dolphin social network describes the frequent associations between 62 dolphins in a community living off Doubtful Sound, New Zealand. The structure of the Dolphin social network is shown in Figure

Dolphin social network.

Table

The topology potential of Dolphin nodes with two schemes.

Node | Without mass | With mass | Node | Without mass | With mass |
---|---|---|---|---|---|

Node 1 | 6.2895 | 4.5944 | Node 11 | 5.3610 | 3.9025 |

Node 2 | 7.0212 | 5.4907 | Node 12 | 2.3973 | 1.7321 |

Node 3 | 4.1513 | 2.9848 | Node 13 | 2.3973 | 1.6296 |

Node 4 | 4.1605 | 2.8547 | Node 14 | 6.4585 | 5.5909 |

Node 5 | 2.3973 | 1.7321 | Node 15 | 9.6098 | 7.6395 |

Node 6 | 3.8699 | 3.2161 | Node 16 | 6.4678 | 4.7977 |

Node 7 | 5.4456 | 4.6391 | Node 17 | 6.1957 | 4.8912 |

Node 8 | 5.4548 | 3.8740 | Node 18 | 7.1057 | 6.1104 |

Node 9 | 6.2895 | 4.6118 | Node 19 | 6.9367 | 5.4700 |

Node 10 | 5.8114 | 5.0576 | Node 20 | 4.3388 | 3.1913 |

Table

The top 20 nodes of Dolphin social network.

Serial number | Without mass | With mass | Serial number | Without mass | With mass |
---|---|---|---|---|---|

1 | Node 15 | Node 15 | 11 | Node 39 | Node 51 |

2 | Node 38 | Node 38 | 12 | Node 18 | Node 39 |

3 | Node 46 | Node 46 | 13 | Node 2 | Node 14 |

4 | Node 34 | Node 34 | 14 | Node 19 | Node 44 |

5 | Node 21 | Node 52 | 15 | Node 44 | Node 2 |

6 | Node 41 | Node 30 | 16 | Node 58 | Node 19 |

7 | Node 30 | Node 18 | 17 | Node 16 | Node 37 |

8 | Node 52 | Node 21 | 18 | Node 14 | Node 22 |

9 | Node 37 | Node 41 | 19 | Node 22 | Node 10 |

10 | Node 51 | Node 58 | 20 | Node 9 | Node 25 |

Dolphin social network contains two communities: the community

As can be seen from Figure

There emerge some new boundary nodes for the “with mass” scheme, such as number 24 node and number 60 node. In Figure

Topology potential theory is a new community detection theory for complex network. At present, almost all topology-potential-based community detection methods assume that network nodes have the same mass. This hypothesis leads to inaccuracy of topology potential calculation and then decreases the precision of community detection. Inspired by the idea of PageRank algorithm, this paper puts forward a novel mass calculation method for complex network node. A node’s mass obtained by our method can effectively reflect its importance and influence in complex network. The more important a node is, the bigger its mass is. Simulation experiment results showed that, after taking node mass into consideration, the topology potential of node is more accurate, the distribution of topology potential is more reasonable, and the results of community detection are more precise.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (2014QNB23).