Some features of the spread of epidemics and information on a random graph

Proceedings of the National Academy of Sciences of the United States of America

Volumn 107, Issue 10, 2010, Pages 4491-4498

  Durrett, Rick ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Complex networks; Contact process; Power law degree distributions; Random Boolean network; Voter model

Indexed keywords

ARTICLE; DISEASE TRANSMISSION; ECOLOGY; EPIDEMIC; INFORMATION; MATHEMATICAL MODEL; MODEL; POPULATION RESEARCH; PRIORITY JOURNAL; RESEARCH;

ALGORITHMS; ANIMALS; COMPUTER SIMULATION; EPIDEMIOLOGIC METHODS; HUMANS; INTERNET; MODELS, BIOLOGICAL; NEURAL NETWORKS (COMPUTER); PROBABILITY; REPRODUCIBILITY OF RESULTS;

EID: 77949647264     PISSN: 00278424     EISSN: 10916490     Source Type: Journal    
DOI: 10.1073/pnas.0914402107     Document Type: Article

References (66)

1. Bollobás B (2001) Random Graphs (Cambridge Univ Press, Cambridge, U.K.), 2nd Ed.
2. Janson S, Luczak T, Rucinski A (2000) Random Graphs (John Wiley and Sons, New York).
3. Faloutsos M, Faloutsos P, Faloutsos C (1999) On the power-law relationship of the internet topology. Proceedings of SIGCOMM '99 (Association for Computing Machinery, New York), pp 251-262.
4. Redner S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B 4:131-134.
5. Newman MEJ (2001) The structure of scientific collaboration networks. Proc Natl Acad Sci USA 98:404-409.
6. Newman MEJ (2001) Scientific collaboration networks. I and II.. Phys Rev E 64:016131-016132.
7. Liljeros F, Edling CR, Amaral LAN, Stanley HE, Aberg Y (2001) The web of human sexual contacts. Nature 411:907-908.
8. Barabási AL (2002) Linked. The New Science of Networks (Perseus, Cambridge, MA).
9. Watts DJ (2003) Six Degrees (W.W. Norton and Co., New York).
10. Chung F, Lu L (2006) Complex Graphs and Networks (Am Math. Society, Providence, RI).
11. Durrett R (2007) Random Graph Dynamics (Cambridge Univ Press, Cambridge, U.K.).
12. Watts DJ, Strogatz SH (1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.
13. Newman MEJ, Watts DJ (1999) Renormalization group analysis of the small-world network model. Physics Lett A 263:341-346.
14. Barabási AL, Albert R (1999) Emergence of scaling in random network. Science 286:509-512.
15. Krapivsky PL, Redner S, Leyvraz F (2000) Connectivity of growing random networks. Phys Rev Lett 85:4629-4632.
16. Cooper C, Frieze A (2003) A general model for web graphs. Random Struct Algor 22:311-335.
17. Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64:026118.
18. Newman MEJ, Strogatz SH, Watts DJ (2002) Random graph models of social networks. Proc Natl Acad Sci USA 99:2566-2572.
19. Moore C, Newman MEJ (2000) Exact solution of bond percolation on small-world networks. Phys Rev E 62:7059-7064.
20. Newman MEJ (2002) Spread of epidemic disease on networks. Phys Rev E 66:016128.
21. Harris TE (1974) Contact interactions on a lattice. Ann Probab 2:969-988.
22. Liggett TM (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, New York).
23. Bollobás B, Chung F (1988) The diameter of a cycle plus a random matching. SIAM J Discrete Math 1:328-333.
24. Durrett R, Jung P (2007) Two phase transitions for the contact on small worlds. Stoch Proc Appl 117:1910-1927.
25. Pemantle R (1992) The contact process on trees. Ann Probab 20:2089-2116.
26. Liggett TM(1996) Multiple transition points for the contact process on the binary tree. Ann Probab 24:1675-1710.
27. Stacey AM (1996) The existence of an intermediate phase for the contact process on trees. Ann Probab 24:1711-1726.
28. Durrett R, Liu X (1988) The contact process on a finite set. Ann Probab 16:1158-1173.
29. Mountford TS (1999) Existence of a constant for finite system extinction. J Stat Phys 96:1331-1341.
30. Pastor-Satorras R, Vespigniani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200-3203.
31. Pastor-Satorras R, Vespigniani A (2001) Epidemic dynamics and endemic states in complex networks. Phys Rev E 63:066117.
32. Pastor-Satorras R, Vespigniani A (2002) Epidemic dynamics in finite size scale-free networks. Phys Rev E 65:035108(R).
33. Gómez-Gardeñes J, Latora V, Moreno Y, Profumo E (2008) Spreading of sexually transmitted diseases in heterosexual populations. Proc Natl Acad Sci USA 105:1399-1404.
34. Chatterjee S, Durrett R (2009) Contact processes on random graphs with power law degree distributions have critical value 0. Ann Probab 37:2332-2356.
35. Berger N, Borgs C, Chayes JT, Saberi A (2005) On the spread of viruses on the internet. Proceedings of the 16th Symposium on Discrete Algorithms (Association for Computing Machinery, New York), pp 301-310.
36. Berger N, Borgs C, Chayes JT, Saberi A (2009) Weak local limits for preferential attachment graphs Preprint can be found at http://www.stanford. edu/saberi/.
37. Griffeath D (1978) Additive and Cancellative Interacting Particle Systems, Lecture Notes in Math (Springer, Berlin), 724.
38. Durrett R, Remenik D (2010) Chaos in a spatial epidemic model. Ann Appl Probab, in press.
39. Richards SA, Wilson WG, Socolar JES (1999) Selection for intermediate mortality and reproduction rates in a spatially structured population. Proc R Soc Lond B 266:2383-2388.
40. Socolar JES, Richards S, Wilson WG (2001) Evolution in spatially structure populations subject to rare epidemics. Phys Rev E 63:041908.
41. Nusse HE, Ott E, Yorke JA (1994) Border-collision bifurcations: An explanation for observed bifurcation phenomena. Phys Rev E 49:1073-1076.
42. Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82:985-992.
43. Lasota A, Yorke JA (1973) On the existence of invariant measures for piecewise monotonic transformations. Trans Amer Math Soc 186:481-488.
44. Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437-467.
45. Kadanoff LP, Coppersmith S, Aldana M (2002) Boolean dynamics with random couplings. arXiv:nlin.AO/0204062.
46. Flyvbjerg H, Kjaer NJ (1988) Exact solution of Kaufmann's model with connectivity one. J Phys A 21:1695-1718.
47. Derrida B, Pomeau Y (1986) Random networks of automata: a simplified annealed approximation. Europhys Lett 1:45-49.
48. Kauffman SA (1993) Origins of Order: Self-Organization and Selection in Evolution (Oxford Univ Press, New York).
49. Shmulevich I, Kauffman SA, Aldana M (2005) Eukaryotic cells are dynamically ordered or critical but not chaotic. Proc Natl Acad Sci USA 102:13439-13444.
50. Nykter M, et al. (2008) Gene expression dynamics in the macrophage exhibit criticality. Proc Natl Acad Sci USA 105:1897-1900.
51. Chatterjee S, Durrett R (2010) Persistence of activity in random Boolean networks. Random Struct Algor, in press.
52. Clifford P, Sudbury A (1973) A model for spatial conflict. Biometrika 60:581-588.
53. Holley R, Liggett T (1976) Ergodic theorems for weakly interacting systems and the voter model. Ann Probab 4:195-228.
54. d . Ann Probab 17:1333-1366.
55. Durrett R (2005) Probability: Theory and Examples (Duxbury Press, Belmont, CA).
56. Aldous D (1989) The Poisson Clumping Heuristic (Springer, New York).
57. Aldous D, Fill J (2002) Chap 3. Reversible Markov Chains and Random Walks on Graphs.
58. Sood V, Redner S (2005) Voter model on heterogeneous graphs. Phys Rev Lett 94:178701.
59. van der Hofstad R, Hooghiemstra G, and Znamenski D (2005) Distances in random graphs with infinite mean degrees. Extremes 8:111-141.
60. Suchecki K, Eguuíluz VM, Miguel MS (2005) Conservation laws for the voter model in complex networks. Europhys Lett 69:228-234.
61. Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312-316.
62. Antal T, Redner S, Sood V (2006) Evolutionary dynamics on degree-heterogeneous graphs. Phys Rev Lett 96:188104.
63. Antal T, Redner S, Sood V (2008) Voter models on heterogeneous graphs. Phys Rev E 77:041121.
64. Maruyama T (1970) The effective number of alleles in a subdivided population. Theor Popul Biol 1:273-306.
65. Slatkin M (1981) Fixation probabilities and fixation times in a subdivided population. Evolution 35:477-488.
66. Barrat A, Bathélemy M, Vespignani A (2008) Dynamical Processes on Complex Networks (Cambridge Univ Press, Cambridge, U.K.).