Geodesic
Approximate Master Equations for Dynamical Processes on Graphs
N. Nagy1 ; I.Z. Kiss2 ; P.L. Simon1
1 Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary
2 School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 2, pp. 43-57.

Voir la notice de l'article provenant de la source EDP Sciences

We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.
DOI : 10.1051/mmnp/20149203

N. Nagy 1 ; I.Z. Kiss 2 ; P.L. Simon 1

1 Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary
2 School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK 
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N. Nagy; I.Z. Kiss; P.L. Simon. Approximate Master Equations for Dynamical Processes on Graphs. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 2, pp. 43-57. doi : 10.1051/mmnp/20149203. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149203/

[1] A. Bátkai, I.Z. Kiss, E. Sikolya, P.L. Simon Networks and Heterogeneous Media 2012 43 58

[2] B. Bollobás. Random graphs. Cambridge University Press, Cambridge, (2001).

[3] L. Decreusefond, J.-S. Dhersin, P. Moyal, V. C. Tran Ann. Appl. Probab. 2012 541 575

[4] K.T.D. Eames, M.J. Keeling PNAS 2002 13330 13335

[5] D. M. Green, I. Z. Kiss Journal of Biological Dynamics 2010 431 445

[6] T. House, M.J. Keeling PLoS Comput. Biol. 2010 e1000721

[7] T. House, M. J. Keeling J. Roy. Soc. Interface 2011 67 73

[8] M.J. Keeling Proc. R. Soc. Lond. B 1999 859 867

[9] M.J. Keeling, K.T.D. Eames J. Roy. Soc. Interface 2005 295 307

[10] J. Lindquist, J. Ma, P. Van Den Driessche, F.H. Willeboordse J. Math. Biol. 2011 143 164

[11] V. Marceau, P.-A. Noël, L. Hébert-Dufresne, A. Allard, L. J. Dubé Phys. Rev. E. 2010 036116

[12] J. C. Miller, A. C. Slim, E. M. Volz J. R. Soc. Interface 2012 890 906

[13] M. Roy, M. Pascual Ecol. Complexity 2006 80 90

[14] P. L. Simon, I. Z. Kiss IMA J. Appl. Math. 2013 945 964

[15] P.L. Simon, M. Taylor, I.Z. Kiss J. Math. Biol. 2012 479 508

[16] M. Taylor, P. L. Simon, D. M. Green, T. House, I. Z. Kiss J. Math. Biol. 2012 1021 1042

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