Abstract
We study the large-time dynamics of a Markov process whose states are finite directed graphs. The number of the vertices is described by a supercritical branching process, and the edges follow a certain mean-field dynamics determined by the rates of appending and deleting. We find sufficient conditions under which asymptotically a.s. the order of the largest component is proportional to the order of the graph. A lower bound for the length of the longest directed path in the graph is provided as well. We derive an explicit formula for the limit as time goes to infinity, of the expected number of cycles of a given finite length. Finally, we study the phase diagram.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. H. Strogatz, Exploring complex networks, Nature 410:268-276 (2001).
V. A. Malyshev, Random graphs and grammars on graphs, Discrete Math. Appl. 8: 247-262 (1998).
D. J. Watts and S. H. Strogatz, Collective dynamics of “small-world” networks, Nature 393:440-442 (1998).
A. D. Barbour and G. Reinert, Small worlds, Random Structures Algorithms 19:54-74 (2001).
A.-L. Barabási, R. Albert, and H. Jeong, Mean-field theory for scale-free random networks, Phys. A 272:173-187 (1999).
B. Bollobás, O. Riordan, J. Spencer, and G. Tusnády, The degree sequence of a scale-free random graph process, Random Structures Algorithms 18:279-290 (2001).
D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman, and S. H. Strogatz, Are randomly grown graphs really random? Phys. Rev. E 64:041902(2001).
P. Erdös, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53:292-294 (1947).
B. Bollobás, Random Graphs (Academic Press, 1985).
S. Janson, T. Łuczak, and A. Rucińnski, Random Graphs (Wiley-Interscience, New York, 2000).
T. S. Turova, Dynamical random graphs with memory, Phys. Rev. E 65:066102(2002).
K. B. Athreya and P. E. Ney, Branching Processes, Die Grundlehren der mathematischen Wissenschaften, Band 196 (Springer-Verlag, New York/Heidelberg, 1972).
E. M. Jin, M. Girvan, and M. E. J. Newman, Structure of growing social networks, Phys. Rev. E 64:046132(2001).
J. Xing and G. L. Gerstein, Networks with lateral connectivity I, II, III, J. Neurophysiology 75:184-231 (1996).
Y. Prut, E. Vaadia, H. Bergman, I. Haalman, H. Slovin, and M. Abeles, Spatiotemporal structure of cortical activity: Properties and behavioral relevance, J. Neurophysiology 79:2857-2874 (1998).
S. N. Dorogovtsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: continuous approach, Phys. Rev. E 63:056125(2001).
R. Albert, H. Jeong, and A.-L. Barabási, Error and attack tolerance of complex networks, Nature 406:378-382 (2000).
T. S. Turova, Neural networks through the hourglass, BioSystems 58:159-165 (2000).
J. Jonasson, The random triangle model, J. Appl. Probab. 36:852-867 (1999).
O. Häggström and T. S. Turova, A strict inequality for the random triangle model, J. Statist. Phys. 104:471-482 (2001).
M. Krikun, Height of a random tree, Markov Process. Related Fields 6:135-146 (2000).
Y. Peres, Probability on trees: An introductory climb, Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Math. 1717:193-280 (Springer, Berlin, 1999).
T. S. Turova, Long paths and cycles in the dynamical graphs, preprints in Math. Sciences 2001:13(2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Turova, T.S. Long Paths and Cycles in Dynamical Graphs. Journal of Statistical Physics 110, 385–417 (2003). https://doi.org/10.1023/A:1021035131946
Issue Date:
DOI: https://doi.org/10.1023/A:1021035131946