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Volumn 67, Issue 2, 2003, Pages 13-

Mixing patterns in networks

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EID: 84876159694     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.67.026126     Document Type: Article
Times cited : (246)

References (78)
  • 13
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    • J. M. Kleinberg, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Association of Computing Machinery, New York, 2000), pp. 163–170
    • J. M. Kleinberg, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Association of Computing Machinery, New York, 2000), pp. 163–170.
  • 23
    • 85037217242 scopus 로고    scopus 로고
    • We find it convenient even on a undirected graph to consider the ends of the edges to be distinguishable—each edge has a unique A-end and B-end, which are marked in some way. We can think of one of the ends as having a dot or other identifying feature on it. This makes the counting of edges simpler: the matrix element (Formula presented) is defined as the probability that a randomly chosen edge is connected to a vertex of type i at its A end and type j at its B end. Thus every edge, whether it joins unlike vertices or like ones, appears only once in the matrix—no edge appears both above and below the diagonal. It is possible to construct a theory in which the ends of undirected edges are indistinguishable, but in this case each edge that joins unlike vertices appears twice in the matrix, both above and below the diagonal, and edges joining like vertices appear only once. This necessitates the introduction of an extra factor of 2 into the off-diagonal terms. This approach is adopted for example in Ref. 43
    • We find it convenient even on a undirected graph to consider the ends of the edges to be distinguishable—each edge has a unique A-end and B-end, which are marked in some way. We can think of one of the ends as having a dot or other identifying feature on it. This makes the counting of edges simpler: the matrix element (Formula presented) is defined as the probability that a randomly chosen edge is connected to a vertex of type i at its A end and type j at its B end. Thus every edge, whether it joins unlike vertices or like ones, appears only once in the matrix—no edge appears both above and below the diagonal. It is possible to construct a theory in which the ends of undirected edges are indistinguishable, but in this case each edge that joins unlike vertices appears twice in the matrix, both above and below the diagonal, and edges joining like vertices appear only once. This necessitates the introduction of an extra factor of 2 into the off-diagonal terms. This approach is adopted for example in Ref. 43.
  • 25
    • 85037184836 scopus 로고    scopus 로고
    • M. Morris, in Epidemic Models: Their Structure and Relation to Data, edited by D. Mollison (Cambridge University Press, Cambridge, 1995), pp. 302–322
    • M. Morris, in Epidemic Models: Their Structure and Relation to Data, edited by D. Mollison (Cambridge University Press, Cambridge, 1995), pp. 302–322.
  • 28
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    • J. L. Fleiss, Statistical Methods for Rates and Proportions (Wiley, New York, 1981)
    • J. L. Fleiss, Statistical Methods for Rates and Proportions (Wiley, New York, 1981).
  • 31
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    • T. Łuczak, in Proceedings of the Symposium on Random Graphs, Poznań 1989, edited by A. M. Frieze and T. Łuczak (Wiley, New York, 1992), pp. 165–182
    • T. Łuczak, in Proceedings of the Symposium on Random Graphs, Poznań 1989, edited by A. M. Frieze and T. Łuczak (Wiley, New York, 1992), pp. 165–182.
  • 34
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    • W. Aiello, F. Chung, and L. Lu, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Association of Computing Machinery, New York, 2000), pp. 171–180
    • W. Aiello, F. Chung, and L. Lu, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (Association of Computing Machinery, New York, 2000), pp. 171–180.
  • 37
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    • M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999)
    • M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).
  • 42
  • 43
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    • One can also calculate a value for r by simply ignoring the directed nature of the edges in a directed network. This approach, which we adopted in Ref. 22, will in general give a different figure from that given by Eq. (25). While Eq. (25) will normally give a more meaningful result for a directed network, there may be cases in which ignoring direction is the correct thing to do. For example, in a food web one might only be interested in which species have tropic relations with with others, and not in which direction that relation lies in terms of energy or carbon flow
    • One can also calculate a value for r by simply ignoring the directed nature of the edges in a directed network. This approach, which we adopted in Ref. 22, will in general give a different figure from that given by Eq. (25). While Eq. (25) will normally give a more meaningful result for a directed network, there may be cases in which ignoring direction is the correct thing to do. For example, in a food web one might only be interested in which species have tropic relations with with others, and not in which direction that relation lies in terms of energy or carbon flow.
  • 46
    • 85037179564 scopus 로고    scopus 로고
    • G. F. Davis, M. Yoo, and W. E. Baker (unpublished)
    • G. F. Davis, M. Yoo, and W. E. Baker (unpublished).
  • 47
    • 85037185207 scopus 로고    scopus 로고
    • P. S. Bearman, J. Moody, and K. Stovel (unpublished)
    • P. S. Bearman, J. Moody, and K. Stovel (unpublished).
  • 49
    • 85037240371 scopus 로고    scopus 로고
    • Q. Chen, H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Willinger, in Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Comput. Soc., Los Alanitos, CA, 2002)
    • Q. Chen, H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Willinger, in Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Comput. Soc., Los Alanitos, CA, 2002).
  • 57
    • 85037220447 scopus 로고    scopus 로고
    • T.A.B. Snijders, J. Soc. Struct. 2(2002)
    • T.A.B. Snijders, J. Soc. Struct. 2(2002).
  • 58
    • 85037184677 scopus 로고    scopus 로고
    • Strictly these probabilities are only correct in a “canonical ensemble” of graphs in which the degree distribution is fixed rather than the degree sequence. This ensemble and the fixed-degree-sequence one studied here, however, become equivalent in the limit of large graph size; the error introduced here by substituting one for the other is of the order of (Formula presented) and is small compared with other sources of error in our simulations
    • Strictly these probabilities are only correct in a “canonical ensemble” of graphs in which the degree distribution is fixed rather than the degree sequence. This ensemble and the fixed-degree-sequence one studied here, however, become equivalent in the limit of large graph size; the error introduced here by substituting one for the other is of the order of (Formula presented) and is small compared with other sources of error in our simulations.
  • 64
    • 85037205165 scopus 로고    scopus 로고
    • Not all graphs with (Formula presented) are without degree correlations. A measurement of (Formula presented) implies only that the mean degree correlation is zero when averaged over all degrees. The grown graph model of Barabási and Albert 6provides an example of a network that possesses degree correlations although it has (Formula presented) 22
    • Not all graphs with (Formula presented) are without degree correlations. A measurement of (Formula presented) implies only that the mean degree correlation is zero when averaged over all degrees. The grown graph model of Barabási and Albert 6 provides an example of a network that possesses degree correlations although it has (Formula presented) 22.
  • 66
    • 85037255374 scopus 로고    scopus 로고
    • The degree is not recalculated after each removal. Removal is in the order of vertices’ starting degree in the network before any deletion has taken place
    • The degree is not recalculated after each removal. Removal is in the order of vertices’ starting degree in the network before any deletion has taken place.
  • 76
    • 85037245711 scopus 로고    scopus 로고
    • N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications (Hafner Press, New York, 1975)
    • N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications (Hafner Press, New York, 1975).


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.