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It is important to establish relations between the network reconstruction and the topological dynamical structure of networks. In this article, we quantify the effect for two types of network topologies on the performance of network reconstruction. First, we generate two network modes with variable clustering coefficient based on Holme-Kim model and Newman-Watts small-world model, then we reconstruct the artificial networks by using a novel framework called

Network reconstruction has attracted much attention for various collective dynamical behaviors [

When dealing with the network reconstruction problem, we should think about the reliability of network data since the accessible data may be fragmentary and limited in consideration of network size. Efficient approaches to solve the reconstruction problem with low data requirements are mainly obtained from methods as Link prediction [

Prior studies about network reconstruction mainly focused on analyzing the collective dynamics complex networks with evolutionary game data. Santos et al. [

In this paper, first we generate a series networks by Holme–Kim model and Newman-Watts small-world model, then we reveal the structure of network with evolutionary-game data by using

We address the mechanism of uncovering two types network topologies with evolutionary PD game time series data based on

CS is first used for finding solutions to underdetermine linear systems through processing signal to acquire and remodeling signal. The advantage of CS [

Each agent possesses one node in the evolutionary game. Assume that the number of nodes in the game network is

In a similar fashion, through compressive sensing method, the remaining agents yield payoff from their neighbor-connection; the overall network adjacency matrix can be expressed as

A network can be denoted as an adjacent matrix

We validated our method by using PD games data occurring in Holme–Kim networks. Contrary to BA model, Holme–Kim model adds Triad Formation process (TF process), in order to change the principle when a new node is attached to an existing node by the principle called Preferential Attachment process (PA process). By this way, it not only makes the network growth mode more flexible, but also increases the clustering coefficient. In the Holme–Kim model, the evolution process of network has three driver factors: growth, preference attachment, and triad formation. The algorithm of the Holme–Kim model is shown as follows [

Correlation between the

Another PD game data occurring in Newman-Watts small-world network (homogeneous small-world network) exhibits a homogeneous connectivity distribution, in the sense that the number of connections for all nodes is the same; the algorithm steps can be described as

First, initializing a network with

Then adding an edge between the unconnected nodes with probability

Dunring the process, there will be multiple edges between any pair of nodes. All nodes will have no self-loops; the clustering coefficients of the network are denoted as

The PD game is simulated on two types of networks, Holme–Kim networks and Newman-Watts small-world. To test the efficiency in reconstructing a network with our method, we first generate an artificial scale-free network with

PD game is implemented to investigate the performance of the network diffusion dynamics and structures; after recording the measurement matrix

Figure

Success rate (SR) of networks with tunalbe clustering coefficients

Furthermore, in order to validate

As a control group, we investigate the relationship between SR and clustering coefficient

(a) Correlation between the clustering coefficients

The methods used to solve sparse approximation problems are available in a variety of ways. As we present our

The parameter for different network.

Type | clustering coefficient | modularity |
---|---|---|

B-A | big | big |

W-S | bigger | small |

To achieve high accuracy (85%) the convergence rate was performed on scale-free network with two methods.

time/s | ||||||||
---|---|---|---|---|---|---|---|---|

N=100 | N=150 | N=200 | N=250 | |||||

data length | | OMP | | OMP | | OMP | | OMP |

| 784 | 653 | 1075 | 853 | 1283 | 1058 | 1462 | 1107 |

| 923 | 842 | 1382 | 1191 | 1596 | 1227 | 1686 | 1340 |

| 1080 | 818 | 1652 | 1458 | 1789 | 1503 | 2074 | 1773 |

| 1425 | 1096 | 1893 | 1578 | 2056 | 1631 | 2261 | 1916 |

To achieve high accuracy (90%) the convergence rate was performed on scale-free network with two methods.

time/s | ||||||||
---|---|---|---|---|---|---|---|---|

N=100 | N=150 | N=200 | N=250 | |||||

data length | | OMP | | OMP | | OMP | | OMP |

| 803 | 676 | 1104 | 876 | 1309 | 1087 | 1497 | 1130 |

| 945 | 869 | 1398 | 1209 | 1616 | 1250 | 1686 | 1378 |

| 1109 | 840 | 1678 | 1483 | 1809 | 1528 | 2098 | 1799 |

| 1445 | 1108 | 1914 | 1596 | 2076 | 1651 | 2279 | 1934 |

To achieve high accuracy (80%) the convergence rate was performed on small-world network with two methods.

time/s | ||||||||
---|---|---|---|---|---|---|---|---|

N=20 | N=50 | N=70 | N=100 | |||||

data length | | OMP | | OMP | | OMP | | OMP |

| 976 | 820 | 1180 | 984 | 1375 | 1100 | 1498 | 1247 |

| 1195 | 1012 | 1416 | 1151 | 1564 | 1348 | 1207 | 1040 |

| 1401 | 1120 | 1692 | 1421 | 1689 | 1407 | 1002 | 871 |

| 1672 | 1416 | 1795 | 1504 | 1879 | 1527 | 864 | 701 |

To achieve high accuracy (85%) the convergence rate was performed on small-world network with two methods.

time/s | ||||||||
---|---|---|---|---|---|---|---|---|

N=20 | N=50 | N=70 | N=100 | |||||

data length | | OMP | | OMP | | OMP | | OMP |

| 998 | 832 | 1203 | 997 | 1397 | 1117 | 1512 | 1264 |

| 1215 | 1032 | 1431 | 1171 | 1584 | 1360 | 1221 | 1062 |

| 1420 | 1141 | 1709 | 1445 | 1700 | 1427 | 1029 | 901 |

| 1692 | 1438 | 1815 | 1528 | 1899 | 1557 | 896 | 728 |

(a) SR of networks with tunalbe clustering coefficients

Correlation between the clustering coefficients

Performance comparison of

Performance comparison of

In our study, we investigate the effect on the accuracy of the network reconstruction by

Our method, mentioned in the paper, can be adopted to reconstruct sparse networks among the real-world networks for the reason that small-world phenomenon and scale-free characteristics are two typical complex network characteristics. Our contribution to the current literature can be summarized as follows: the method metioned above needs time series data; usually the topology structure of the network cannot be abtained. From this point of view, if we do not know about the topology structure, we may first choose

Meanwhile, there exist some shortcomings in this paper, which may influence the further research for network reconstruction. First, the two methods mentioned in this paper are used to recover sparse network; better methods are worth exploring [

Generally speaking, this paper puts forward two methods to deal with the problem of network reconstruction through two different manners. It provides directions for us to reconstruct complex network; yet it is expected to make efforts to pursue better approaches. All these deserved to be explored.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work is supported by the National Natural Science Foundation of China (Grant No. 71271126), doctoral Fund of Ministry of Education of China (No. 20120078110002), and Fundamental Research Funds for the Central Universities, supported by the GIFSUFE (Grant No. CXJJ-2016-414).