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Bordeu I, Amarteifio S, Garcia-Millan R, Walter B, Wei N, Pruessner G. Volume explored by a branching random walk on general graphs. Sci Rep 2019; 9:15590. [PMID: 31666539 PMCID: PMC6821755 DOI: 10.1038/s41598-019-51225-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2019] [Accepted: 09/25/2019] [Indexed: 11/09/2022] Open
Abstract
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.
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Affiliation(s)
- Ignacio Bordeu
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. .,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK. .,DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK. .,The Wellcome Trust/Cancer Research UK Gurdon Institute, University of Cambridge, Cambridge, CB2 1QN, UK.
| | - Saoirse Amarteifio
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK
| | - Rosalba Garcia-Millan
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK
| | - Benjamin Walter
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK
| | - Nanxin Wei
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK
| | - Gunnar Pruessner
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. .,Centre for Complexity Science, Imperial College London, London, SW7 2AZ, UK.
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Werner M, Sommer JU. Self-organized stiffness in regular fractal polymer structures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051802. [PMID: 21728562 DOI: 10.1103/physreve.83.051802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2011] [Indexed: 05/31/2023]
Abstract
We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.
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Affiliation(s)
- Marco Werner
- Leibniz-Institut für Polymerforschung Dresden eV, Dresden, Germany
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Abstract
Let
F
n
be the
n
th stage in the construction of the Sierpiński carpet. Let
R
n
be the electrical resistance of
F
n
when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant
ρ
such that ¼
ρ
n
≼
R
n
≼ 4
ρ
n
. The motivation for this result came from the problem of establishing (
a
) the existence and (
b
) the value of the ‘spectral dimension’ of the Sierpiński carpet. In this and a subsequent paper, we settle (
a
) and give bounds for (
b
).
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9
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Burioni R, Cassi D. Comment on "Critical dimensionalities of phase transitions on fractals". PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:3782-3783. [PMID: 9963068 DOI: 10.1103/physreve.51.3782] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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10
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Fujiwara S, Yonezawa F. Anomalous relaxation in fractal structures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:2277-2285. [PMID: 9962889 DOI: 10.1103/physreve.51.2277] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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