Volume explored by a branching random walk on general graphs

Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding—the branching random walk (BRW)—is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs’ dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks.

[1]  Krishna P. Gummadi,et al.  On the evolution of user interaction in Facebook , 2009, WOSN '09.

[2]  Eytan Domany,et al.  Introduction to the renormalization group and to critical phenomena , 1977 .

[3]  S. N. Dorogovtsev,et al.  Laplacian spectra of, and random walks on, complex networks: are scale-free architectures really important? , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Spectral dimension of a wire network , 1985 .

[5]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[6]  A. M. Berezhkovskii,et al.  Wiener sausage volume moments , 1989 .

[7]  R. Albert.  Scale-free networks in cell biology , 2005, Journal of Cell Science.

[8]  S. Havlin,et al.  Scaling theory of transport in complex biological networks , 2007, Proceedings of the National Academy of Sciences.

[9]  B. Durhuus,et al.  On the spectral dimension of random trees , 2006 .

[10]  R. Dasgupta,et al.  Scaling exponents for random walks on Sierpinski carpets and number of distinct sites visited: a new algorithm for infinite fractal lattices , 1999 .

[11]  J. L. Gall,et al.  Random trees and applications , 2005 .

[12]  Alberto Rosso,et al.  Spatial extent of an outbreak in animal epidemics , 2013, Proceedings of the National Academy of Sciences.

[13]  M. Doi.  Second quantization representation for classical many-particle system , 1976 .

[14]  Aravind Srinivasan,et al.  Modelling disease outbreaks in realistic urban social networks , 2004, Nature.

[15]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[16]  B. M. Fulk.  MATH , 1992 .

[17]  S. Majumdar,et al.  Spatial extent of branching Brownian motion. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Shen Lin,et al.  The range of tree-indexed random walk in low dimensions , 2014, 1401.7830.

[19]  Boming Yu,et al.  SOME FRACTAL CHARACTERS OF POROUS MEDIA , 2001 .

[20]  On the growth of bounded trees , 2001, cond-mat/0112394.

[21]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[22]  Q. Sattentau,et al.  Avoiding the void: cell-to-cell spread of human viruses , 2008, Nature Reviews Microbiology.

[23]  G. Pruessner,et al.  Field-theoretic approach to the universality of branching processes , 2018, Physical Review E.

[24]  Wei Wang,et al.  Unification of theoretical approaches for epidemic spreading on complex networks , 2016, Reports on progress in physics. Physical Society.

[25]  L. Peliti.  Path integral approach to birth-death processes on a lattice , 1985 .

[26]  R. Guimerà,et al.  The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[28]  J. Cardy,et al.  Non-Equilibrium Statistical Mechanics and Turbulence , 2009 .

[29]  A. Barabasi,et al.  Scale-free characteristics of random networks: the topology of the world-wide web , 2000 .

[30]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

[31]  Uwe C. Täuber,et al.  Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior , 2014 .

[32]  M. Vidal,et al.  Effect of sampling on topology predictions of protein-protein interaction networks , 2005, Nature Biotechnology.

[33]  Mason A. Porter,et al.  Random walks and diffusion on networks , 2016, ArXiv.

[34]  Jean-François Le Gall,et al.  THE RANGE OF TREE-INDEXED RANDOM WALK , 2013, Journal of the Institute of Mathematics of Jussieu.

[35]  D. Cassi,et al.  Random walks on graphs: ideas, techniques and results , 2005 .

[36]  Carsten Wiuf,et al.  Subnets of scale-free networks are not scale-free: sampling properties of networks. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Felix M. Schmidt,et al.  The critical Casimir effect in films for generic non-symmetry-breaking boundary conditions , 2011, 1110.1241.

[38]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[39]  K. Sneppen,et al.  Diffusion on complex networks: a way to probe their large-scale topological structures , 2003, cond-mat/0312476.

[40]  S. Sawyer,et al.  Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[41]  G. Pruessner,et al.  A Field-Theoretic Approach to the Wiener Sausage , 2014, 1407.7419.