Mark Newman
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805090
- eISBN:
- 9780191843235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805090.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are ...
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A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are randomly removed from a network, and of the percolation phase transition at which a giant percolating cluster forms. The properties of percolation on configuration model networks are studied, including networks with power-law degree distributions, and including both uniform and non-uniform removal of nodes. Computer algorithms for simulating percolation on real-world networks are also discussed, and numerical results are given for several example networks, including the internet and a social network.Less
A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are randomly removed from a network, and of the percolation phase transition at which a giant percolating cluster forms. The properties of percolation on configuration model networks are studied, including networks with power-law degree distributions, and including both uniform and non-uniform removal of nodes. Computer algorithms for simulating percolation on real-world networks are also discussed, and numerical results are given for several example networks, including the internet and a social network.
Sergey N. Dorogovtsev
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to ...
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This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to the percolation properties of these networks. The value of the percolation threshold for various degree distributions is estimated, and the resilience of scale-free networks against random failures is described. The hierarchical organization of k-cores in these networks is discussed. A few basic epidemic models on complex networks are introduced, and the evolution of diseases in networks is described.Less
This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to the percolation properties of these networks. The value of the percolation threshold for various degree distributions is estimated, and the resilience of scale-free networks against random failures is described. The hierarchical organization of k-cores in these networks is discussed. A few basic epidemic models on complex networks are introduced, and the evolution of diseases in networks is described.
Paul Charbonneau
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691176840
- eISBN:
- 9781400885497
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691176840.003.0004
- Subject:
- Computer Science, Programming Languages
This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In ...
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This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In its more abstract form, percolation is an exemplar of criticality, a concept in statistical physics related to phase transitions. A classic example of criticality is liquid water boiling into water vapor, or freezing into ice. The chapter first provides an overview of percolation in one and two dimensions before discussing the use of a tagging algorithm for identifying and sizing clusters. It then considers fractal clusters on a lattice at the percolation threshold, scale invariance of power-law behavior, and critical behavior of natural systems. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.Less
This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In its more abstract form, percolation is an exemplar of criticality, a concept in statistical physics related to phase transitions. A classic example of criticality is liquid water boiling into water vapor, or freezing into ice. The chapter first provides an overview of percolation in one and two dimensions before discussing the use of a tagging algorithm for identifying and sizing clusters. It then considers fractal clusters on a lattice at the percolation threshold, scale invariance of power-law behavior, and critical behavior of natural systems. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.
Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199666515
- eISBN:
- 9780191748639
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666515.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where ...
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This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where rocks are necessarily fractured at various scales by the slow but constant motion of continental masses. This book is situated between three disciplines. First, geology and geophysics provide most of the data and most applications; a characteristic feature is that one never has a complete knowledge of the studied objects — such as an oil reservoir — in contrast with a laboratory experiment where every quantity can be measured. Second, engineering develops the main tools of analysis for one- and two-phase flows, and calculations of permeability (absolute and relative). Third, statistical physics plays a major role in concepts such as the excluded volume, dimensionless density, percolation threshold and power laws. In view of this interdisciplinary character, the general results presented in this book may have unexpected applications in many different domains. This book is based on courses which have been taught in several countries at Master and Ph.D. levels in universities, in research centers and at conferences. It should provide, in a compact form, all the necessary tools to achieve the general objective. The mathematical level in this book has been kept as low as possible. The interested reader can always go further, thanks to the references that are provided, where the mathematical level is not restricted. The colloquial aspect of a course has been preserved where one tries to explain abstract concepts in simple terms.Less
This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where rocks are necessarily fractured at various scales by the slow but constant motion of continental masses. This book is situated between three disciplines. First, geology and geophysics provide most of the data and most applications; a characteristic feature is that one never has a complete knowledge of the studied objects — such as an oil reservoir — in contrast with a laboratory experiment where every quantity can be measured. Second, engineering develops the main tools of analysis for one- and two-phase flows, and calculations of permeability (absolute and relative). Third, statistical physics plays a major role in concepts such as the excluded volume, dimensionless density, percolation threshold and power laws. In view of this interdisciplinary character, the general results presented in this book may have unexpected applications in many different domains. This book is based on courses which have been taught in several countries at Master and Ph.D. levels in universities, in research centers and at conferences. It should provide, in a compact form, all the necessary tools to achieve the general objective. The mathematical level in this book has been kept as low as possible. The interested reader can always go further, thanks to the references that are provided, where the mathematical level is not restricted. The colloquial aspect of a course has been preserved where one tries to explain abstract concepts in simple terms.
