The main object of this paper is to present a network formation model for social network. To start with, we create a fully connected graph that has m0 vertices. At every time step, one vertex v is introduced into the network. The new vertex v randomly attaches to one existing vertex with equal probability. Then, each neighbor of the attached vertex forms a connection to the new vertex v with probability a. We call this operation triad formation. We derive the mean degree, degree distribution and the clustering coefficient for this model. We show that the clustering coefficient is non-vanishing as the network grows. The expected degree and expected number of triangles are tunable simply by changing the control parameter a.

[1] B. Kirman, S. Lawson and C. Linehan, Gaming on and off the Social Graph: The Social
Structure of Facebook Games,2009 International Conference on Computational
Science and Engineering.
[2] S. Milgram, “The Small World Problem,” Psychol. Today, 2, 60-67, 1967.
[3] M.E.J. Newman, “Networks: An Introduction,” Oxford, 2010.
[4] D. J. Watts and S. H. Strogatz, “Collective dynamics of ’small-world’ networks,”
Nature (London), 393-409, 1998.
[5] A.-L. Barab´asi and R. Albert, “Emergence of Scaling in Random Networks,” Science,
286, 509-512, 1999.
[6] C.C. Foster, A. Rapoport and C. J. Orwant, “A Study of Large Sociogram II; Elimination
of Free Parameters,” Behav. Sci., 8, 56, 1963.
[7] M.E.J. Newman, “Clustering and Preferential Attachment in Growing Networks,”
Phys. Rev. E 64, 025102, 2001.
[8] P. Holme and B. J. Kim, “Growing Scale-Free Networks with Tunable Clustering,”
Physical Review E, Vol. 65, 026107, 2002.
[9] G. Szab´o, M. Alava, and J. Kertesz, Structural Transitions in Scale-Free Networks,
Physical Review E, Vol. 67, 056102, 2003.
[10] Gabor Csardi, Tamas Nepusz: “The igraph software package for complex network
research.” Inter Journal Complex Systems, 1695, 2006.
[11] A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, B. Bhattacharjee,“
Measurement and Analysis of Online Social Networks,” Proceedings of the 5th
ACM/Usenix Internet Measurement Conference, San Diego, CA, 2007.
[12] B. Viswanath, A. Mislove, M. Cha and K.P. Gummadi, “On the Evolution of User
Interaction in Facebook,” Proceedings of the 2nd ACM SIGCOMM Workshop on
Social Networks (WOSN’09), 2009.
[13] C.Y. Huang, “Sampling and Data Analysis of Online Social Networks”