Random Deletion in a Scale-Free Random Graph Process

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut . The neighbours of ut are chosen with probability proportional to degree. With probability α -α1 ≥ 0 we add m random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability 1-α-α0 we delete a random vertex, if there are vertices left to delete. With probability α0 we delete m random edges. Assuming that α + α1 + α0 > 1 and α0 is sufficently small, we show that for large k, t, the expected number of vertices of degree k is approximately dkt where as k → 8, dk ~ Ck -1-β where and C > 0 is a constant. Note that β can take any value greater than 1.

[1]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[2]  Béla Bollobás,et al.  Coupling Scale-Free and Classical Random Graphs , 2004, Internet Math..

[3]  HighWire Press.  Philosophical Transactions of the Royal Society of London , 1781, The London Medical Journal.

[4]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[5]  Christos H. Papadimitriou,et al.  On the Eigenvalue Power Law , 2002, RANDOM.

[6]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[7]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[8]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[9]  Fan Chung Graham,et al.  Coupling Online and Offline Analyses for Random Power Law Graphs , 2004, Internet Math..

[10]  Fan Chung Graham,et al.  Random evolution in massive graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[11]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[12]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..

[13]  Béla Bollobás,et al.  Mathematical results on scale‐free random graphs , 2005 .

[14]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[15]  E. N.,et al.  The Calculus of Finite Differences , 1934, Nature.

[16]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

[17]  Alan M. Frieze,et al.  A General Model of Undirected Web Graphs , 2001, ESA.

[18]  B. Hayes.  Graph Theory in Practice: Part II , 2000, American Scientist.

[19]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[20]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[21]  Jie Wu,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2003 .

[22]  G. Yule,et al.  A Mathematical Theory of Evolution Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[23]  Béla Bollobás,et al.  Robustness and Vulnerability of Scale-Free Random Graphs , 2004, Internet Math..