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An equation-free approach to coarse-graining the dynamics of networks
1. | Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, New Jersey 08544, United States |
2. | Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States |
3. | Institute of Mathematics, Budapest University of Technology (BME), H-1111 Budapest, Hungary |
4. | Department of Chemical and Biological Engineering, and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544 |
References:
[1] |
R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), 47-97.
doi: 10.1103/RevModPhys.74.47. |
[2] |
A. Arenas, A. Díaz-Guilera and C. J. Pérez-Vicente, Synchronization reveals topological scales in complex networks, Phys. Rev. Lett., 96 (2006), 114102.
doi: 10.1103/PhysRevLett.96.114102. |
[3] |
A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.
doi: 10.1126/science.286.5439.509. |
[4] |
A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, 2008.
doi: 10.1017/CBO9780511791383. |
[5] |
T. Binzegger, R. J. Douglas and K. A. C. Martin, Topology and dynamics of the canonical circuit of cat v1, Neural Networks, 22 (2009), 1071-1078.
doi: 10.1016/j.neunet.2009.07.011. |
[6] |
J. Blitzstein and P. Diaconis, A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees, Technical report, 2006. Available from: http://www.people.fas.harvard.edu/~blitz/BlitzsteinDiaconisGraphAlgorithm.pdf.
doi: 10.1080/15427951.2010.557277. |
[7] |
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Physics Reports, 424 (2006), 175-308.
doi: 10.1016/j.physrep.2005.10.009. |
[8] |
K. A. Bold, Y. Zou, I. G. Kevrekidis and M. A. Hensonevrekidis, An equation-free approach to analyzing heterogeneous cell population dynamics, J. Math. Biol., 55 (2007), 331-352.
doi: 10.1007/s00285-007-0086-6. |
[9] |
B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, European J. Combin., 1 (1980), 311-316.
doi: 10.1016/S0195-6698(80)80030-8. |
[10] |
C. Borgs, J. Chayes, L. Lovász, V. Sós and K. Vesztergombi, Topics in Discrete Mathematics: Algorithms and Combinatorics, Springer, Berlin, 2006. |
[11] |
L. Chen, P. G. Debenedetti, C. W. Gear and I. G. Kevrekidis, From molecular dynamics to coarse self-similar solutions: a simple example using equation-free computation, J. Non-Newton Fluid, 120 (2004), 215-223.
doi: 10.1016/j.jnnfm.2003.12.007. |
[12] |
F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences, Ann. Comb., 6 (2002), 125-145.
doi: 10.1007/PL00012580. |
[13] |
S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, How to construct a correlated net, eprint, arXiv:cond-mat/0206131. |
[14] |
S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks, Adv. Phys., 51 (2002), 1079-1187.
doi: 10.1093/acprof:oso/9780198515906.001.0001. |
[15] |
M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology, In SIGCOMM, (1999), 251-262.
doi: 10.1145/316194.316229. |
[16] |
C. Gounaris, K. Rajendran, I. G. Kevrekidis and C. Floudas, Generation of networks with prescribed degree-dependent clustering, Optim. Lett., 5 (2011), 435-451.
doi: 10.1007/s11590-011-0319-x. |
[17] |
T. Gross and I. G. Kevrekidis, Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated moment closure, Europhys. Lett., 82 (2008), 38004, 6 pp.
doi: 10.1209/0295-5075/82/38004. |
[18] |
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Ind. Appl. Math., 10 (1962), 496-506.
doi: 10.1137/0110037. |
[19] |
V. Havel, A remark on the existence of finite graphs. (czech), Casopis Pest. Mat., 80 (1955), 477-480. |
[20] |
M. Ispány and G. Pap, A note on weak convergence of random step processes, Acta Mathematica Hungarica, 126 (2010), 381-395.
doi: 10.1007/s10474-009-9099-5. |
[21] |
C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, 1995.
doi: 10.1137/1.9781611970944. |
[22] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems, AIChE Journal, 50 (2004), 1346-1355.
doi: 10.1002/aic.10106. |
[23] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), 715-762.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[24] |
I. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications, Annu. Rev. Phys. Chem., 60 (2009), 321-344.
doi: 10.1146/annurev.physchem.59.032607.093610. |
[25] |
B. J. Kim, Performance of networks of artificial neurons: The role of clustering, Phys. Rev. E, 69 (2004), 045101.
doi: 10.1103/PhysRevE.69.045101. |
[26] |
S. Lafon and A. B. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning and data set parameterization, IEEE T. Pattern Anal., 28 (2006), 1393-1403.
doi: 10.1109/TPAMI.2006.184. |
[27] |
C. R. Laing, The dynamics of chimera states in heterogeneous kuramoto networks, Physica D, 238 (2009), 1569-1588.
doi: 10.1016/j.physd.2009.04.012. |
[28] |
P. Li and Z. Yi, Synchronization of Kuramoto oscillators in random complex networks, Physica A, 387 (2008), 1669-1674.
doi: 10.1016/j.physa.2007.11.008. |
[29] |
L. Lovász, Very large graphs, eprint, arXiv:0902.0132. |
[30] |
L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Comb. Theory Ser. B, 96 (2006), 933-957.
doi: 10.1016/j.jctb.2006.05.002. |
[31] |
S. J. Moon, B. Nabet, N. E. Leonard, S. A. Levin and I. G. Kevrekidis, Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach, J. Theor. Biol., 246 (2007), 100-112.
doi: 10.1016/j.jtbi.2006.12.018. |
[32] |
F. Mori and T. Odagaki, Synchronization of coupled oscillators on small-world networks, Physica D, 238 (2009), 1180-1185.
doi: 10.1016/j.physd.2009.04.002. |
[33] |
B. Nadler, S. Lafon, R. R. Coifman and I. G. Kevrekidis, Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmon. A., 21 (2006), 113-127.
doi: 10.1016/j.acha.2005.07.004. |
[34] |
M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[35] |
M. E. J. Newman, D. J. Watts and S. H. Strogatz, Random graph models of social networks, Proc. Natl. Acad. Sci., 1 (2002), 2566-2572.
doi: 10.1073/pnas.012582999. |
[36] |
M. E. J. Newman, A. L. Barabási and D. J. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006. |
[37] |
K. Rajendran and I. G. Kevrekidis, Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps, Phys. Rev. E, 84 (2011), 036708.
doi: 10.1103/PhysRevE.84.036708. |
[38] |
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), 856-869.
doi: 10.1137/0907058. |
[39] |
M. A. Serrano and M. Boguná, Tuning clustering in random networks with arbitrary degree distributions, Phys. Rev. E, 72 (2005), 036133. |
[40] |
R. Toivonen, L. Kovanen, M. Kivelä, J. Onnela, J. Saramäki and K. Kaski, A comparative study of social network models: Network evolution models and nodal attribute models, Soc. Networks, 31 (2009), 240-254.
doi: 10.1016/j.socnet.2009.06.004. |
[41] |
S. V. N. Vishwanathan, K. M. Borgwardt, I. Risi Kondor and N. N. Schraudolph, Graph Kernels, eprint, arXiv:0807.0093. |
[42] |
D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world' networks, Nature, 393 (1998), 440-442. |
[43] |
N. C. Wormald, Some problems in the enumeration of labelled graphs, B. Aust. Math. Soc., 21 (1980), 159-160.
doi: 10.1017/S0004972700011436. |
show all references
References:
[1] |
R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), 47-97.
doi: 10.1103/RevModPhys.74.47. |
[2] |
A. Arenas, A. Díaz-Guilera and C. J. Pérez-Vicente, Synchronization reveals topological scales in complex networks, Phys. Rev. Lett., 96 (2006), 114102.
doi: 10.1103/PhysRevLett.96.114102. |
[3] |
A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.
doi: 10.1126/science.286.5439.509. |
[4] |
A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, 2008.
doi: 10.1017/CBO9780511791383. |
[5] |
T. Binzegger, R. J. Douglas and K. A. C. Martin, Topology and dynamics of the canonical circuit of cat v1, Neural Networks, 22 (2009), 1071-1078.
doi: 10.1016/j.neunet.2009.07.011. |
[6] |
J. Blitzstein and P. Diaconis, A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees, Technical report, 2006. Available from: http://www.people.fas.harvard.edu/~blitz/BlitzsteinDiaconisGraphAlgorithm.pdf.
doi: 10.1080/15427951.2010.557277. |
[7] |
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Physics Reports, 424 (2006), 175-308.
doi: 10.1016/j.physrep.2005.10.009. |
[8] |
K. A. Bold, Y. Zou, I. G. Kevrekidis and M. A. Hensonevrekidis, An equation-free approach to analyzing heterogeneous cell population dynamics, J. Math. Biol., 55 (2007), 331-352.
doi: 10.1007/s00285-007-0086-6. |
[9] |
B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, European J. Combin., 1 (1980), 311-316.
doi: 10.1016/S0195-6698(80)80030-8. |
[10] |
C. Borgs, J. Chayes, L. Lovász, V. Sós and K. Vesztergombi, Topics in Discrete Mathematics: Algorithms and Combinatorics, Springer, Berlin, 2006. |
[11] |
L. Chen, P. G. Debenedetti, C. W. Gear and I. G. Kevrekidis, From molecular dynamics to coarse self-similar solutions: a simple example using equation-free computation, J. Non-Newton Fluid, 120 (2004), 215-223.
doi: 10.1016/j.jnnfm.2003.12.007. |
[12] |
F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences, Ann. Comb., 6 (2002), 125-145.
doi: 10.1007/PL00012580. |
[13] |
S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, How to construct a correlated net, eprint, arXiv:cond-mat/0206131. |
[14] |
S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks, Adv. Phys., 51 (2002), 1079-1187.
doi: 10.1093/acprof:oso/9780198515906.001.0001. |
[15] |
M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology, In SIGCOMM, (1999), 251-262.
doi: 10.1145/316194.316229. |
[16] |
C. Gounaris, K. Rajendran, I. G. Kevrekidis and C. Floudas, Generation of networks with prescribed degree-dependent clustering, Optim. Lett., 5 (2011), 435-451.
doi: 10.1007/s11590-011-0319-x. |
[17] |
T. Gross and I. G. Kevrekidis, Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated moment closure, Europhys. Lett., 82 (2008), 38004, 6 pp.
doi: 10.1209/0295-5075/82/38004. |
[18] |
S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, J. Soc. Ind. Appl. Math., 10 (1962), 496-506.
doi: 10.1137/0110037. |
[19] |
V. Havel, A remark on the existence of finite graphs. (czech), Casopis Pest. Mat., 80 (1955), 477-480. |
[20] |
M. Ispány and G. Pap, A note on weak convergence of random step processes, Acta Mathematica Hungarica, 126 (2010), 381-395.
doi: 10.1007/s10474-009-9099-5. |
[21] |
C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, 1995.
doi: 10.1137/1.9781611970944. |
[22] |
I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems, AIChE Journal, 50 (2004), 1346-1355.
doi: 10.1002/aic.10106. |
[23] |
I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), 715-762.
doi: 10.4310/CMS.2003.v1.n4.a5. |
[24] |
I. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications, Annu. Rev. Phys. Chem., 60 (2009), 321-344.
doi: 10.1146/annurev.physchem.59.032607.093610. |
[25] |
B. J. Kim, Performance of networks of artificial neurons: The role of clustering, Phys. Rev. E, 69 (2004), 045101.
doi: 10.1103/PhysRevE.69.045101. |
[26] |
S. Lafon and A. B. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning and data set parameterization, IEEE T. Pattern Anal., 28 (2006), 1393-1403.
doi: 10.1109/TPAMI.2006.184. |
[27] |
C. R. Laing, The dynamics of chimera states in heterogeneous kuramoto networks, Physica D, 238 (2009), 1569-1588.
doi: 10.1016/j.physd.2009.04.012. |
[28] |
P. Li and Z. Yi, Synchronization of Kuramoto oscillators in random complex networks, Physica A, 387 (2008), 1669-1674.
doi: 10.1016/j.physa.2007.11.008. |
[29] |
L. Lovász, Very large graphs, eprint, arXiv:0902.0132. |
[30] |
L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Comb. Theory Ser. B, 96 (2006), 933-957.
doi: 10.1016/j.jctb.2006.05.002. |
[31] |
S. J. Moon, B. Nabet, N. E. Leonard, S. A. Levin and I. G. Kevrekidis, Heterogeneous animal group models and their group-level alignment dynamics: An equation-free approach, J. Theor. Biol., 246 (2007), 100-112.
doi: 10.1016/j.jtbi.2006.12.018. |
[32] |
F. Mori and T. Odagaki, Synchronization of coupled oscillators on small-world networks, Physica D, 238 (2009), 1180-1185.
doi: 10.1016/j.physd.2009.04.002. |
[33] |
B. Nadler, S. Lafon, R. R. Coifman and I. G. Kevrekidis, Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmon. A., 21 (2006), 113-127.
doi: 10.1016/j.acha.2005.07.004. |
[34] |
M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[35] |
M. E. J. Newman, D. J. Watts and S. H. Strogatz, Random graph models of social networks, Proc. Natl. Acad. Sci., 1 (2002), 2566-2572.
doi: 10.1073/pnas.012582999. |
[36] |
M. E. J. Newman, A. L. Barabási and D. J. Watts, The Structure and Dynamics of Networks, Princeton University Press, 2006. |
[37] |
K. Rajendran and I. G. Kevrekidis, Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps, Phys. Rev. E, 84 (2011), 036708.
doi: 10.1103/PhysRevE.84.036708. |
[38] |
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), 856-869.
doi: 10.1137/0907058. |
[39] |
M. A. Serrano and M. Boguná, Tuning clustering in random networks with arbitrary degree distributions, Phys. Rev. E, 72 (2005), 036133. |
[40] |
R. Toivonen, L. Kovanen, M. Kivelä, J. Onnela, J. Saramäki and K. Kaski, A comparative study of social network models: Network evolution models and nodal attribute models, Soc. Networks, 31 (2009), 240-254.
doi: 10.1016/j.socnet.2009.06.004. |
[41] |
S. V. N. Vishwanathan, K. M. Borgwardt, I. Risi Kondor and N. N. Schraudolph, Graph Kernels, eprint, arXiv:0807.0093. |
[42] |
D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world' networks, Nature, 393 (1998), 440-442. |
[43] |
N. C. Wormald, Some problems in the enumeration of labelled graphs, B. Aust. Math. Soc., 21 (1980), 159-160.
doi: 10.1017/S0004972700011436. |
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