Abstract
We analyze the existence and the size of the giant component in the stationary state of a Markovian model for bipartite multigraphs, in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero-range process, the degrees being static on the other set. The analysis is based on approximations by independent variables and on the results of Molloy and Reed for graphs with prescribed degree sequences. The possible types of phase diagrams are identified by studying the behavior below the zero-range condensation point. As a specific example, we consider the so-called Evans interaction. In particular, we examine the values of a critical exponent, describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold. Rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero-range condensation. These results, supplemented with conjectures supported by Monte Carlo simulations, suggest that the phenomenological Landau theory for percolation on graphs is not broken by the fluctuations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
06 November 2023
A Correction to this paper has been published: https://doi.org/10.1007/s10955-023-03192-6
References
B. Bollobás (1985) Random Graphs Academic Press London
S. N. Dorogovtsev J. F. F. Mendes (2002) Adv. Phys. 51 1079 Occurrence Handle10.1080/00018730110112519
A.-L. Barabási R. Albert (1999) ArticleTitleEmergence of scaling in random networks Science 286 509–512 Occurrence Handle10.1126/science.286.5439.509 Occurrence Handle10521342 Occurrence HandleMR2091634
D. J. de S. Price A general (1976) ArticleTitletheory of bibliometric and other cumulative advantage processes J. Am. Soc. Inform. Sci. 27 292–306
F. Spitzer (1970) Adv. Math. 5 246–290 Occurrence Handle10.1016/0001-8708(70)90034-4
G. Palla I. Derényi I. Farkas T. Vicsek (2004) ArticleTitleStatistical mechanics of topological phase transitions in networks Phys. Rev. E69 046117
B. Pittel W. A. Woyczynski J. A. Mann (1990) ArticleTitleRandom tree-type partitions as a model for acyclic polymerization: Holtsmark (3/2-stable) distribution of the supercritical gel Ann. Prob. 18 319–341
B. Pittel W. A. Woyczynski A graph-valued (1990) ArticleTitleMarkov process as rings-allowed polymerization model: Subcritical behavior SIAM J. Appl. Math. 50 1200–1220 Occurrence Handle10.1137/0150073
S. N. Dorogovtsev J. F. F. Mendes A. N. Samukhin (2003) ArticleTitlePrinciples of statistical mechanics of uncorrelated random networks Nucl. Phys. B 666 396–416 Occurrence Handle10.1016/S0550-3213(03)00504-2
M. R. Evans, Brazilian J. Phys. 30: 42 (2000), e-print cond-mat/0007293.
M. E. J. Newman S. H. Strogatz D. J. Watts (2001) ArticleTitleRandom graphs with arbitrary degree distributions and their applications Phys. Rev. E 64 026118
M. Molloy B. Reed (1995) ArticleTitleA critical point for random graphs with a given degree sequence Random Struct. Algorithms 6 161
M. Molloy B. Reed (1998) ArticleTitleThe size of the giant component of a random graph with a given degree sequence Combinatorics Probab. Comput. 7 295 Occurrence Handle10.1017/S0963548398003526
We say that an event B happens with high probability if \(\lim_{L\rightarrow \infty} P(B)=1\).
S. Großkinsky G. M. Schütz H. Spohn (2003) J. Stat. Phys. 113 IssueID3 389–410 Occurrence Handle10.1023/A:1026008532442
I. Jeon P. March B. Pittel (2000) Ann. Probab. 28 1162–1194 Occurrence Handle10.1214/aop/1019160330
R. Arratia S. Tavaré (1994) ArticleTitleIndependent process approximations for random combinatorial structures Adv. Math. 104 90–154 Occurrence Handle10.1006/aima.1994.1022
S. Corteel B. Pittel C. D. Savage H. S. Wilf (1999) ArticleTitleOn the multiplicity of parts in a random partition, Random Struct Algorithms 14 185–197
P. Brémaud (1999) Markov Chains: Gibbs Fields, Monte-Carlo Simulation, and Queues Springer New York
B. V. Gnedenko A. N. Kolmogorov Limit Distributions for Sums of Independent Random Variables (1968) revised edition Addison-Wesley Reading
C. Kipnis C. Landim (1999) Scaling Limits of Interacting Particle Systems Springer Berlin
T. M. Liggett (1985) Interacting Particle Systems Springer Berlin
T. M. Liggett (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Springer Berlin
B. Derrida M. R. Evans in Nonequilibrium Statistical Mechanics in One Dimension V. Privman (1997) ed. Cambridge University Press Cambridge
M. Abramowitz I. A. Stegun (1972) Handbook of Mathematical Functions Dover New York
N. Alon J. H. Spencer P. Erdös (1991) The Probabilistic Method Wiley New York
Our main theoretical results and conjectures were tested by performing Monte-Carlo simulations, where the random number generator Mersenne twister was utilized. For the less interesting small values of $\rho$ the simulations were easier than for high density, where to actually observe the leading order critical behavior (and the exponent) would require simulations with of the order of hundred-thousand vertices and even a larger number of particles on $W$. Note that the parameter $M_2$ should be a ‘‘macroscopic’’ number, which becomes a problem for $r=2M_2/N$ approaching zero as in the case of Fig. \ref{fig3}.
R. Cohen D. ben-Avraham S. Havlin (2002) ArticleTitlePercolation critical exponents in scale-free networks Phys. Rev. E 66 036113
A. V. Goltsev S. N. Dorogovtsev J. F. F. Mendes (2003) ArticleTitleCritical phenomena in networks Phys. Rev. E 67 026123
C. Godrèche (2003) ArticleTitleDynamics of condensation in zero-range processes J. Phys. A: Math. Gen. 36 IssueID23 6313–6328 Occurrence Handle10.1088/0305-4470/36/23/303
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pulkkinen, O., Merikoski, J. Phase Transitions on Markovian Bipartite Graphs—an Application of the Zero-range Process. J Stat Phys 119, 881–907 (2005). https://doi.org/10.1007/s10955-005-3011-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-3011-7