在這篇論文當中，我們使用一個網路生成模型來形成社群網路(Social network)。首先，我們以 個節點(Vertices)來形成一個完全連接(Fully-connected)的網路。在每一個時間點，都會有一個新的vertex加入到網路當中，然後它會在等機率的情況下，隨機挑選在前一個時間點時的網路中的任一個vertex與其連接。再來，被選中的那個vertex它所相鄰的節點(Neighbors)，會有一個機率a與該新加入的vertex形成連接。我們稱這個方法為”triadic attachment(TA)”。在進行我的研究之前，先前已經有先行做了一些網路分析，分別是平均節點分支度(mean degree)與群聚係數(clustering coefficient)。在這篇論文當中，我們依據前面所做的Model與分析過的數據。來去更進一步的探討在該網路模型上的Pearson Correlation Coefficient(皮爾遜積差相關係數)。

In this paper we derive a first-order linear differential equation
for the Pearson degree correlation function of a duplication model.
The solution of the differential equation shows that the degree
correlation coefficient function is negative at early times, but
tends to be positive at late times.

Contents---------------------------------------1
List of Figures--------------------------------2
1. Introduction--------------------------------3
2. Additional Notations and Previous Results---5
3. Degree Correlation--------------------------7
4. Moments of Degrees--------------------------9
5. Products of Degrees------------------------14
6. Degree sum of second neighbors-------------21
7. Numerical and Simulation Results-----------26
8. Conclusions--------------------------------31
Bibliography----------------------------------32

[1] A. Bhan, D. J. Galas, and T. G. Deway. A duplication growth model of gene expression networks. Bioinformatics , 18(11):1486-1493, 2002.
[2] F. Chung and L. Lu. Complex graphs and network. In Regional Conference Series in Mathematics , number 107. American Mathematical Society, 2004.
[3] F. Chung and L. Lu. and T. G. Dewey. Duplication models for biological networks. Journal of Computational Biology , 10(5):677-687, 2003.
[4] I. Ispolatov, P. L. Krapivsky, I. Mazo, and A. Yuryev. Cliques and duplication-divergence network growth. New Journal of Physics , 2005.
[5] I. Ispolatov, P. L. Krapivsky, and A. Yuryev. Duplication-divergence model of protein interaction network. Phys. Rev. E , 71:061911, Jun 2005.
[6] D.-S. Lee, C.-S. Chang, W.-G. Ye. and M.-C. Cheng. Formation of social networks by random triadic attachment. unpublished, 2012.
[7] M. Newman. Networks: An Introduction. Oxford, 2010.
[8] R. V. Sole, R. Pastor-Satorras, E. Smith, and T. B. Kepler. A model of large-scale proteome evolution. Advances in Complex Systems , July 2002.
[9] D. Zhao, Z.-R. Liu, and J.-Z. Wang. Duplication: a mechanism producing disassortative mixing networks in biology. Chin. Phys. Lett. , 24(10):2766-2768, 2007.