Grand Canonical Approach to an Interacting Network


We consider a network composed of an arbitrary number of directed links. We employ a grand canonical partition function to study the statistical averages of the network in equilibrium. The Hamiltonian is composed of two parts: a “free” Hamiltonian H0 attributing a constant energy E to each link, and an interacting Hamiltonian Hint involving terms quadratic in the number of links. A Gaussian integration leads to a reformulated Hamiltonian, where now the number of links appears linearly. The reformulated Hamiltonian allows obtaining the exact behavior in limiting cases. At high temperatures the system reproduces the behavior of the free model, while at low temperatures the thermodynamic behavior is obtained by using a renormalized chemical potential, μeff = μ + l, where l is the strength of the interaction. We also resort to a mean field approximation, describing accurately the system over the entire range of all dynamical parameters. A detailed Monte-Carlo simulation verifies our theoretical expectations. We indicate that our model may serve as a prototype model to address a number of different systems.

Share and Cite:

Nicolaidis, A. , Kosmidis, K. and Kiosses, V. (2015) Grand Canonical Approach to an Interacting Network. Journal of Modern Physics, 6, 472-482. doi: 10.4236/jmp.2015.64051.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Peirce, C.S. (1870) Memoirs of the American Academy of Sciences, 9, 317-378.
[2] Peirce, C.S. (1880) American Journal of Mathematics, 3, 15-57.
[3] Nicolaidis, A. (2009) International Journal of Modern Physics A, 24, 1175-1183.
[4] Misner, C.W., Kip, S. and Thorne, J.A. (1973) Gravitation. W. H. Freeman, San Francisco.
[5] Wheeler, J.A. (1980) Pregeometry: Motivations and Prospects. In: Marlow, A.R., Ed., Quantum Theory and Gravitation, Academic Press, New York, 1-11.
[6] Penrose, R. (1971) Angular Momentum: An Approach to Combinatorial Space-Time. In: Bastin, T., Ed., Quantum Theory and Beyond, Cambridge University Press, Cambridge.
[7] Penrose, R. (1972) On the Nature of Quantum Geometry. In: Klauder, J., Ed., Magic Without Magic, Freeman, San Francisco, 333-354.
[8] Konopka, T., Markopoulou, F. and Smolin, L. Quantum Graphity [arXiv:hep-th/0611197].
[9] Konopka, T., Markopoulou, F. and Severini, S. (2008) Physical Review D, 77, Article ID: 104029.
[10] Ambjorn, J., Jurkiewicz, J. and Loll, R. (2004) Physical Review Letters, 93, Article ID: 131301.
[11] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguna, M. (2010) Physical Review E, 82, Article ID: 036106.
[12] Strogatz, S.H. (2001) Nature, 410, 268-276.
[13] Albert, R. and Albert-László, B. (2002) Reviews of Modern Physics, 74, 47-97.
[14] Dorogovtsev, S.N. and Mendes, J.F.F. (2002) Advances in Physics, 51, 1079-1187.
[15] Newman, M.E.J. (2003) SIAM Review, 45, 167-256.
[16] Barrat, A., Barthelemy, M. and Vespignani, A. (2008) Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge.
[17] Nicolaidis, A., Kosmidis, K. and Argyrakis, P. (2009) Journal of Statistical Mechanics: Theory and Experiment, 2009, Article ID: P12008.
[18] Cohen, R. and Havlin, S. (2010) Complex Networks: Structure, Robustness and Function. Cambridge University Press, Cambridge.
[19] Childs, A.M., Gosset, D. and Webb, Z. (2013) Science, 339, 791-794.
[20] Faccin, M., Johnson, T., Biamonte, J., Kais, S. and Migda, P. (2013) Physical Review X, 3, Article ID: 041007.
[21] Park, J. and Newman, M.E.J. (2004) Physical Review E, 70, Article ID: 066117.
[22] Park, J. and Newman, M.E.J. (2004) Physical Review E, 70, Article ID: 066146.
[23] Binder, K. and Heermann, D. (2010) Monte Carlo Simulation in Statistical Physics: An Introduction. Springer, Berlin.

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.