Skip to Main Content

Access Provided by Xinjiang University

Physical Review E

Access by Xinjiang University

Identifying communities within energy landscapes

Claire P. Massen and Jonathan P. K. Doye^*

University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom

^*Electronic address: jpkd1@cam.ac.uk

Phys. Rev. E 71, 046101 – Published 1 April, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.046101

Abstract

Potential energy landscapes can be represented as a network of minima linked by transition states. The community structure of such networks has been obtained for a series of small Lennard-Jones (LJ) clusters. This community structure is compared to the concept of funnels in the potential energy landscape. Two existing algorithms have been used to find community structure, one involving removing edges with high betweenness, the other involving optimization of the modularity. The definition of the modularity has been refined, making it more appropriate for networks such as these where multiple edges and self-connections are not included. The optimization algorithm has also been improved, using Monte Carlo methods with simulated annealing and basin hopping, both often used successfully in other optimization problems. In addition to the small clusters, two examples with known heterogeneous landscapes, the 13-atom cluster $({LJ}_{13})$ with one labeled atom and the 38-atom cluster $({LJ}_{38})$ , were studied with this approach. The network methods found communities that are comparable to those expected from landscape analyses. This is particularly interesting since the network model does not take any barrier heights or energies of minima into account. For comparison, the network associated with a two-dimensional hexagonal lattice is also studied and is found to have high modularity, thus raising some questions about the interpretation of the community structure associated with such partitions.

Article Text

References (54)

R. Albert and A. L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
M. E. J. Newman, SIAM Rev. 45, 167 (2003).
S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003).
M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commun. Rev. 29, 251 (1999).
R. Albert, H. Jeong, and A. L. Barabási, Nature (London) 401, 130 (1999).
H. Jeong, S. P. Mason, A. L. Barabási, and Z. N. Oltvai, Nature (London) 411, 41 (2001).
A. Wagner, Mol. Biol. Evol. 18, 1283 (2001).
A. L. Barabási and R. Albert, Science 286, 509 (1999).
R. Albert and A. L. Barabási, Phys. Rev. Lett. 85, 5234 (2000).
M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 98, 404 (2001).
D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998).
M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002).
M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002).
M. T. Gastner and M. E. J. Newman, e-print cond-mat∕0409702.
A. F. Rozenfeld, R. Cohen, D. ben Avraham, and S. Havlin, Phys. Rev. Lett. 89, 218701 (2002).
See Eur. Phys. J. B 38, 143 (2004).
D. J. Wales, Energy Landscapes (Cambridge University Press, Cambridge, 2004).
F. H. Stillinger and T. A. Weber, Science 225, 4666 (1984).
T. B. Schrøder, S. Sastry, J. C. Dyre, and S. C. Glotzer, J. Chem. Phys. 112, 9834 (2000).
J. P. K. Doye and D. J. Wales, J. Chem. Phys. 102, 9659 (1995).
F. Sciortino, W. Kob, and P. Tartaglia, Phys. Rev. Lett. 83, 3214 (1999).
S. Büchner and A. Heuer, Phys. Rev. E 60, 6507 (1999).
J. P. K. Doye, Phys. Rev. Lett. 88, 238701 (2002).
J. P. K. Doye and C. P. Massen, J. Chem. Phys.122, 084105 (2005).
J. R. Tyler, D. M. Wilkinson, and B. A. Huberman, http://www.hpl.hp.com/research/idl/papers/email/ (2003).
A. Capocci, V. D. P. Servedio, G. Caldarelli, and F. Colaiori, e-print cond-mat∕0402499.
M. E. J. Newman, Eur. Phys. J. B 38, 321 (2004).
M. A. Miller, J. P. K. Doye, and D. J. Wales, J. Chem. Phys. 110, 328 (1999).
J. P. K. Doye, M. A. Miller, and D. J. Wales, J. Chem. Phys. 111, 8417 (1999).
M. A. Miller, J. P. K. Doye, and D. J. Wales, Phys. Rev. E 60, 3701 (1999).
J. P. K. Doye, M. A. Miller, and D. J. Wales, J. Chem. Phys. 110, 6896 (1999).
D. J. Wales, Mol. Phys. 102, 891 (2004).
P. Radicchi, C. Castellano, F. Cecconi, V. Loreto, and D. Parisi, Proc. Natl. Acad. Sci. U.S.A. 101, 2658 (2004).
H. Zhou, Phys. Rev. E 67, 061901 (2003).
J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004).
I. Simonsen, K. A. Eriksec, S. Maslov, and K. Sneppen, Physica A 336, 163 (2004).
F. Wu and B. A. Huberman, Eur. Phys. J. B 38, 331 (2004).
M. Latapy and P. Pons, e-print cond-mat∕0412368.
M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 (2004).
M. E. J. Newman, Phys. Rev. E 69, 066133 (2004).
D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101, 5111 (1997).
A. Clauset, M. E. J. Newman, and C. Moore, Phys. Rev. E 70, 066111 (2004).
S. Maslov, K. Sneppen, and A. Zaliznyak, Physica A 333, 529 (2004).
R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, and U. Alon, e-print cond-mat∕0312028.
R. Guimerá, L. Danon, A. Diaz-Guilera, F. Giralt, and A. Arenas, Phys. Rev. E 68, 065103(R) (2003).
D. Wilkinson and B. A. Huberman, Proc. Natl. Acad. Sci. U.S.A. 101, 5241 (2004).
J. Park and M. E. J. Newman, Phys. Rev. E 68, 026112 (2003).
A. Grönlund and P. Holme, Phys. Rev. E 70, 036108 (2004).
R. Guimerá, M. Sales-Pardo, and L. A. N. Amaral, Phys. Rev. E 70, 025101(R) (2004).
J. P. K. Doye and C. P. Massen, Phys. Rev. E 71, 016128 (2005).
M. E. J. Newman and J. Park, Phys. Rev. E 68, 036122 (2003).
M. E. J. Newman, Phys. Rev. E 68, 026121 (2003).
J. P. K. Doye, D. J. Wales, and M. A. Miller, J. Chem. Phys. 109, 8143 (1998).
J. P. K. Doye and D. J. Wales, J. Chem. Phys. 116, 3777 (2002).