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Volumn 60, Issue 6, 1999, Pages 7332-7342

Scaling and percolation in the small-world network model

Author keywords

[No Author keywords available]

Indexed keywords

ARTIFICIAL NEURAL NETWORK; BIOLOGICAL MODEL; DISEASE COURSE; EPIDEMIC; HUMAN; NERVE TRACT; REACTION TIME; STATISTICAL MODEL; STATISTICS AND NUMERICAL DATA;

EID: 0033487710     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.60.7332     Document Type: Article
Times cited : (1080)

References (37)
  • 4
    • 85036386740 scopus 로고    scopus 로고
    • previous work the letter p has been used to denote the density of random connections, rather than (Formula presented) We use (Formula presented) here, however, to avoid confusion with the percolation probability introduced in Sec. VI, which is also conventionally denoted p
    • In previous work the letter p has been used to denote the density of random connections, rather than (Formula presented) We use (Formula presented) here, however, to avoid confusion with the percolation probability introduced in Sec. VI, which is also conventionally denoted p.
  • 5
    • 85036334059 scopus 로고    scopus 로고
    • Watts and Strogatz used the letter k to refer to the average coordination number, but we will find it convenient to distinguish between the coordination number, which we call z, and the range of the bonds. In one dimension (Formula presented) and in general (Formula presented) for networks based on d-dimensional lattices
    • Watts and Strogatz used the letter k to refer to the average coordination number, but we will find it convenient to distinguish between the coordination number, which we call z, and the range of the bonds. In one dimension (Formula presented) and in general (Formula presented) for networks based on d-dimensional lattices.
  • 6
    • 85036184810 scopus 로고    scopus 로고
    • D. J. Watts, Ph.D. thesis, Cornell University, 1997
    • D. J. Watts, Ph.D. thesis, Cornell University, 1997.
  • 8
    • 85036316794 scopus 로고    scopus 로고
    • The exact definition of (Formula presented) depends on how you measure lengths in the model. The definition given here is appropriate if (Formula presented) is measured in terms of the lattice constant of the underlying lattice. It would, however, be reasonable to measure it in terms of the number of bonds traversed between the ends of two shortcuts. Since we are measuring lattice size L in terms of the underlying lattice constant rather than number of bonds, the present definition is the more appropriate one in our case, but it would be perfectly consistent to define both (Formula presented) and L to be a factor of k smaller; all the physical results would work out the same
    • The exact definition of (Formula presented) depends on how you measure lengths in the model. The definition given here is appropriate if (Formula presented) is measured in terms of the lattice constant of the underlying lattice. It would, however, be reasonable to measure it in terms of the number of bonds traversed between the ends of two shortcuts. Since we are measuring lattice size L in terms of the underlying lattice constant rather than number of bonds, the present definition is the more appropriate one in our case, but it would be perfectly consistent to define both (Formula presented) and L to be a factor of k smaller; all the physical results would work out the same.
  • 9
    • 85036177688 scopus 로고    scopus 로고
    • a system of finite size the average distance between the ends of two shortcuts cannot be larger than (Formula presented) so we cannot observe this divergence once (Formula presented) is larger than this
    • In a system of finite size the average distance between the ends of two shortcuts cannot be larger than (Formula presented) so we cannot observe this divergence once (Formula presented) is larger than this.
  • 11
    • 85036193446 scopus 로고    scopus 로고
    • unpublished
    • A. Barrat cond-mat/9903323 (unpublished).
    • Barrat, A.1
  • 15
    • 85036156887 scopus 로고    scopus 로고
    • D. S. Gaunt and A. J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic Press, London, 1974), Vol. 3
    • D. S. Gaunt and A. J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic Press, London, 1974), Vol. 3.
  • 16
    • 85036259959 scopus 로고    scopus 로고
    • We are indebted to Professor S. H. Strogatz for suggesting the use of a Padé approximant in this context
    • We are indebted to Professor S. H. Strogatz for suggesting the use of a Padé approximant in this context.
  • 17
    • 85036342767 scopus 로고    scopus 로고
    • We use the capital letter D to denote the dimension here, to distinguish it from the dimension d of the underlying lattice defined in Sec. II
    • We use the capital letter D to denote the dimension here, to distinguish it from the dimension d of the underlying lattice defined in Sec. II.
  • 20
    • 85036288830 scopus 로고    scopus 로고
    • Eur. Phys. J. B (to be published)
    • A. Barrat and M. Weigt, Eur. Phys. J. B (to be published).
    • Barrat, A.1    Weigt, M.2
  • 21
    • 85036337440 scopus 로고    scopus 로고
    • Eur. Phys. J. B (to be published)
    • R. Monasson, Eur. Phys. J. B (to be published).
    • Monasson, R.1
  • 31
    • 85036276922 scopus 로고    scopus 로고
    • A closely related issue is that of disease spreading when transmission does not take place with 100% probability along every edge in the graph. This can be represented by placing random two-state variables on the bonds of the graph to denote whether a bond will transmit the disease. Although we will not go through the calculation in detail, an approximate figure for the point at which an epidemic occurs in this bond percolation system can be calculated by a method very similar to the one presented here for the site percolation case
    • A closely related issue is that of disease spreading when transmission does not take place with 100% probability along every edge in the graph. This can be represented by placing random two-state variables on the bonds of the graph to denote whether a bond will transmit the disease. Although we will not go through the calculation in detail, an approximate figure for the point at which an epidemic occurs in this bond percolation system can be calculated by a method very similar to the one presented here for the site percolation case.
  • 36
    • 85036246590 scopus 로고    scopus 로고
    • A naive recursive cluster-finding algorithm by contrast takes time proportional to (Formula presented)
    • A naive recursive cluster-finding algorithm by contrast takes time proportional to (Formula presented)


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