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2
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0004219960
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Academic Press, New York
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B. Bollobás, Random Graphs (Academic Press, New York, 1985).
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(1985)
Random Graphs
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Bollobás, B.1
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4
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85036386740
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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
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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.
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5
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85036334059
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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
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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.
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6
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85036184810
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D. J. Watts, Ph.D. thesis, Cornell University, 1997
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D. J. Watts, Ph.D. thesis, Cornell University, 1997.
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8
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85036316794
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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
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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.
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9
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85036177688
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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
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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.
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11
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85036193446
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unpublished
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A. Barrat cond-mat/9903323 (unpublished).
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Barrat, A.1
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15
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85036156887
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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
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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.
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16
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85036259959
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We are indebted to Professor S. H. Strogatz for suggesting the use of a Padé approximant in this context
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We are indebted to Professor S. H. Strogatz for suggesting the use of a Padé approximant in this context.
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17
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85036342767
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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
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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.
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19
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3342916075
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T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, Phys. Rev. A 33, 1141 (1986).
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(1986)
Phys. Rev. A
, vol.33
, pp. 1141
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Halsey, T.C.1
Jensen, M.H.2
Kadanoff, L.P.3
Procaccia, I.4
Shraiman, B.I.5
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20
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85036288830
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Eur. Phys. J. B (to be published)
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A. Barrat and M. Weigt, Eur. Phys. J. B (to be published).
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Barrat, A.1
Weigt, M.2
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21
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85036337440
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Eur. Phys. J. B (to be published)
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R. Monasson, Eur. Phys. J. B (to be published).
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Monasson, R.1
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31
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85036276922
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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
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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.
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34
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0001716835
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J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, and L. V. Chayes, Phys. Rev. Lett. 75, 2792 (1995).
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(1995)
Phys. Rev. Lett.
, vol.75
, pp. 2792
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Machta, J.1
Choi, Y.S.2
Lucke, A.3
Schweizer, T.4
Chayes, L.V.5
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35
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0003919678
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D. Ferguson, J. I. Siepmann, D. G. Truhlar, Wiley, New York
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G. T. Barkema and M. E. J. Newman, in Monte Carlo Methods in Chemical Physics, edited by D. Ferguson, J. I. Siepmann, and D. G. Truhlar (Wiley, New York, 1999).
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(1999)
Monte Carlo Methods in Chemical Physics
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Barkema, G.T.1
Newman, M.E.J.2
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36
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85036246590
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A naive recursive cluster-finding algorithm by contrast takes time proportional to (Formula presented)
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A naive recursive cluster-finding algorithm by contrast takes time proportional to (Formula presented)
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