January  2014, 1(1): 111-134. doi: 10.3934/jcd.2014.1.111

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

Received  February 2012 Revised  October 2012 Published  April 2014

We propose and illustrate an approach to coarse-graining the dynamics of evolving networks, i.e., networks whose connectivity changes dynamically. The approach is based on the equation-free framework: short bursts of detailed network evolution simulations are coupled with lifting and restriction operators that translate between actual network realizations and their appropriately chosen coarse observables. This framework is used here to accelerate temporal simulations through coarse projective integration, and to implement coarse-grained fixed point algorithms through matrix-free Newton-Krylov. The approach is illustrated through a very simple network evolution example, for which analytical approximations to the coarse-grained dynamics can be independently obtained, so as to validate the computational results. The scope and applicability of the approach, as well as the issue of selection of good coarse observables are discussed.
Citation: Katherine A. Bold, Karthikeyan Rajendran, Balázs Ráth, Ioannis G. Kevrekidis. An equation-free approach to coarse-graining the dynamics of networks. Journal of Computational Dynamics, 2014, 1 (1) : 111-134. doi: 10.3934/jcd.2014.1.111
References:
[1]

R. Albert and A. L. Barabási, Statistical mechanics of complex networks,, Rev. Mod. Phys., 74 (2002), 47.  doi: 10.1103/RevModPhys.74.47.  Google Scholar

[2]

A. Arenas, A. Díaz-Guilera and C. J. Pérez-Vicente, Synchronization reveals topological scales in complex networks,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.114102.  Google Scholar

[3]

A. L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

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A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks,, Cambridge University Press, (2008).  doi: 10.1017/CBO9780511791383.  Google Scholar

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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.  doi: 10.1016/j.neunet.2009.07.011.  Google Scholar

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J. Blitzstein and P. Diaconis, A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees,, Technical report, (2006).  doi: 10.1080/15427951.2010.557277.  Google Scholar

[7]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics,, Physics Reports, 424 (2006), 175.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

[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.  doi: 10.1007/s00285-007-0086-6.  Google Scholar

[9]

B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs,, European J. Combin., 1 (1980), 311.  doi: 10.1016/S0195-6698(80)80030-8.  Google Scholar

[10]

C. Borgs, J. Chayes, L. Lovász, V. Sós and K. Vesztergombi, Topics in Discrete Mathematics: Algorithms and Combinatorics,, Springer, (2006).   Google Scholar

[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.  doi: 10.1016/j.jnnfm.2003.12.007.  Google Scholar

[12]

F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences,, Ann. Comb., 6 (2002), 125.  doi: 10.1007/PL00012580.  Google Scholar

[13]

S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, How to construct a correlated net,, eprint, ().   Google Scholar

[14]

S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks,, Adv. Phys., 51 (2002), 1079.  doi: 10.1093/acprof:oso/9780198515906.001.0001.  Google Scholar

[15]

M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology,, In SIGCOMM, (1999), 251.  doi: 10.1145/316194.316229.  Google Scholar

[16]

C. Gounaris, K. Rajendran, I. G. Kevrekidis and C. Floudas, Generation of networks with prescribed degree-dependent clustering,, Optim. Lett., 5 (2011), 435.  doi: 10.1007/s11590-011-0319-x.  Google Scholar

[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).  doi: 10.1209/0295-5075/82/38004.  Google Scholar

[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.  doi: 10.1137/0110037.  Google Scholar

[19]

V. Havel, A remark on the existence of finite graphs. (czech),, Casopis Pest. Mat., 80 (1955), 477.   Google Scholar

[20]

M. Ispány and G. Pap, A note on weak convergence of random step processes,, Acta Mathematica Hungarica, 126 (2010), 381.  doi: 10.1007/s10474-009-9099-5.  Google Scholar

[21]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, number 16 in Frontiers in Applied Mathematics, (1995).  doi: 10.1137/1.9781611970944.  Google Scholar

[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.  doi: 10.1002/aic.10106.  Google Scholar

[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.  doi: 10.4310/CMS.2003.v1.n4.a5.  Google Scholar

[24]

I. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications,, Annu. Rev. Phys. Chem., 60 (2009), 321.  doi: 10.1146/annurev.physchem.59.032607.093610.  Google Scholar

[25]

B. J. Kim, Performance of networks of artificial neurons: The role of clustering,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.045101.  Google Scholar

[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.  doi: 10.1109/TPAMI.2006.184.  Google Scholar

[27]

C. R. Laing, The dynamics of chimera states in heterogeneous kuramoto networks,, Physica D, 238 (2009), 1569.  doi: 10.1016/j.physd.2009.04.012.  Google Scholar

[28]

P. Li and Z. Yi, Synchronization of Kuramoto oscillators in random complex networks,, Physica A, 387 (2008), 1669.  doi: 10.1016/j.physa.2007.11.008.  Google Scholar

[29]

L. Lovász, Very large graphs,, eprint, ().   Google Scholar

[30]

L. Lovász and B. Szegedy, Limits of dense graph sequences,, J. Comb. Theory Ser. B, 96 (2006), 933.  doi: 10.1016/j.jctb.2006.05.002.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2006.12.018.  Google Scholar

[32]

F. Mori and T. Odagaki, Synchronization of coupled oscillators on small-world networks,, Physica D, 238 (2009), 1180.  doi: 10.1016/j.physd.2009.04.002.  Google Scholar

[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.  doi: 10.1016/j.acha.2005.07.004.  Google Scholar

[34]

M. E. J. Newman, The structure and function of complex networks,, SIAM Review, 45 (2003), 167.  doi: 10.1137/S003614450342480.  Google Scholar

[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.  doi: 10.1073/pnas.012582999.  Google Scholar

[36]

M. E. J. Newman, A. L. Barabási and D. J. Watts, The Structure and Dynamics of Networks,, Princeton University Press, (2006).   Google Scholar

[37]

K. Rajendran and I. G. Kevrekidis, Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.036708.  Google Scholar

[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.  doi: 10.1137/0907058.  Google Scholar

[39]

M. A. Serrano and M. Boguná, Tuning clustering in random networks with arbitrary degree distributions,, Phys. Rev. E, 72 (2005).   Google Scholar

[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.  doi: 10.1016/j.socnet.2009.06.004.  Google Scholar

[41]

S. V. N. Vishwanathan, K. M. Borgwardt, I. Risi Kondor and N. N. Schraudolph, Graph Kernels,, eprint, ().   Google Scholar

[42]

D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world' networks,, Nature, 393 (1998), 440.   Google Scholar

[43]

N. C. Wormald, Some problems in the enumeration of labelled graphs,, B. Aust. Math. Soc., 21 (1980), 159.  doi: 10.1017/S0004972700011436.  Google Scholar

show all references

References:
[1]

R. Albert and A. L. Barabási, Statistical mechanics of complex networks,, Rev. Mod. Phys., 74 (2002), 47.  doi: 10.1103/RevModPhys.74.47.  Google Scholar

[2]

A. Arenas, A. Díaz-Guilera and C. J. Pérez-Vicente, Synchronization reveals topological scales in complex networks,, Phys. Rev. Lett., 96 (2006).  doi: 10.1103/PhysRevLett.96.114102.  Google Scholar

[3]

A. L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

[4]

A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks,, Cambridge University Press, (2008).  doi: 10.1017/CBO9780511791383.  Google Scholar

[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.  doi: 10.1016/j.neunet.2009.07.011.  Google Scholar

[6]

J. Blitzstein and P. Diaconis, A Sequential Importance Sampling Algorithm for Generating Random Graphs with Prescribed Degrees,, Technical report, (2006).  doi: 10.1080/15427951.2010.557277.  Google Scholar

[7]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics,, Physics Reports, 424 (2006), 175.  doi: 10.1016/j.physrep.2005.10.009.  Google Scholar

[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.  doi: 10.1007/s00285-007-0086-6.  Google Scholar

[9]

B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs,, European J. Combin., 1 (1980), 311.  doi: 10.1016/S0195-6698(80)80030-8.  Google Scholar

[10]

C. Borgs, J. Chayes, L. Lovász, V. Sós and K. Vesztergombi, Topics in Discrete Mathematics: Algorithms and Combinatorics,, Springer, (2006).   Google Scholar

[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.  doi: 10.1016/j.jnnfm.2003.12.007.  Google Scholar

[12]

F. Chung and L. Lu, Connected components in random graphs with given expected degree sequences,, Ann. Comb., 6 (2002), 125.  doi: 10.1007/PL00012580.  Google Scholar

[13]

S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, How to construct a correlated net,, eprint, ().   Google Scholar

[14]

S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks,, Adv. Phys., 51 (2002), 1079.  doi: 10.1093/acprof:oso/9780198515906.001.0001.  Google Scholar

[15]

M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology,, In SIGCOMM, (1999), 251.  doi: 10.1145/316194.316229.  Google Scholar

[16]

C. Gounaris, K. Rajendran, I. G. Kevrekidis and C. Floudas, Generation of networks with prescribed degree-dependent clustering,, Optim. Lett., 5 (2011), 435.  doi: 10.1007/s11590-011-0319-x.  Google Scholar

[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).  doi: 10.1209/0295-5075/82/38004.  Google Scholar

[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.  doi: 10.1137/0110037.  Google Scholar

[19]

V. Havel, A remark on the existence of finite graphs. (czech),, Casopis Pest. Mat., 80 (1955), 477.   Google Scholar

[20]

M. Ispány and G. Pap, A note on weak convergence of random step processes,, Acta Mathematica Hungarica, 126 (2010), 381.  doi: 10.1007/s10474-009-9099-5.  Google Scholar

[21]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations,, number 16 in Frontiers in Applied Mathematics, (1995).  doi: 10.1137/1.9781611970944.  Google Scholar

[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.  doi: 10.1002/aic.10106.  Google Scholar

[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.  doi: 10.4310/CMS.2003.v1.n4.a5.  Google Scholar

[24]

I. G. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications,, Annu. Rev. Phys. Chem., 60 (2009), 321.  doi: 10.1146/annurev.physchem.59.032607.093610.  Google Scholar

[25]

B. J. Kim, Performance of networks of artificial neurons: The role of clustering,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.045101.  Google Scholar

[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.  doi: 10.1109/TPAMI.2006.184.  Google Scholar

[27]

C. R. Laing, The dynamics of chimera states in heterogeneous kuramoto networks,, Physica D, 238 (2009), 1569.  doi: 10.1016/j.physd.2009.04.012.  Google Scholar

[28]

P. Li and Z. Yi, Synchronization of Kuramoto oscillators in random complex networks,, Physica A, 387 (2008), 1669.  doi: 10.1016/j.physa.2007.11.008.  Google Scholar

[29]

L. Lovász, Very large graphs,, eprint, ().   Google Scholar

[30]

L. Lovász and B. Szegedy, Limits of dense graph sequences,, J. Comb. Theory Ser. B, 96 (2006), 933.  doi: 10.1016/j.jctb.2006.05.002.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2006.12.018.  Google Scholar

[32]

F. Mori and T. Odagaki, Synchronization of coupled oscillators on small-world networks,, Physica D, 238 (2009), 1180.  doi: 10.1016/j.physd.2009.04.002.  Google Scholar

[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.  doi: 10.1016/j.acha.2005.07.004.  Google Scholar

[34]

M. E. J. Newman, The structure and function of complex networks,, SIAM Review, 45 (2003), 167.  doi: 10.1137/S003614450342480.  Google Scholar

[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.  doi: 10.1073/pnas.012582999.  Google Scholar

[36]

M. E. J. Newman, A. L. Barabási and D. J. Watts, The Structure and Dynamics of Networks,, Princeton University Press, (2006).   Google Scholar

[37]

K. Rajendran and I. G. Kevrekidis, Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps,, Phys. Rev. E, 84 (2011).  doi: 10.1103/PhysRevE.84.036708.  Google Scholar

[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.  doi: 10.1137/0907058.  Google Scholar

[39]

M. A. Serrano and M. Boguná, Tuning clustering in random networks with arbitrary degree distributions,, Phys. Rev. E, 72 (2005).   Google Scholar

[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.  doi: 10.1016/j.socnet.2009.06.004.  Google Scholar

[41]

S. V. N. Vishwanathan, K. M. Borgwardt, I. Risi Kondor and N. N. Schraudolph, Graph Kernels,, eprint, ().   Google Scholar

[42]

D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world' networks,, Nature, 393 (1998), 440.   Google Scholar

[43]

N. C. Wormald, Some problems in the enumeration of labelled graphs,, B. Aust. Math. Soc., 21 (1980), 159.  doi: 10.1017/S0004972700011436.  Google Scholar

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