Some asymptotic properties of duplication graphs

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 68, Issue 6 2, 2003, Pages 661191-6611910

  Raval, Alpan ^a,b  

^a KECK GRADUATE INSTITUTE   (United States)

^b CLAREMONT GRADUATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; DYNAMICS; GENES; PROBABILITY; PROTEINS;

DUPLICATION GRAPHS; PROTEIN-PROTEIN INTERACTION NETWORKS;

GRAPH THEORY;

EID: 17044458532     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (17)

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11. Note that the distinction between the dynamics of a single realization and that of an ensemble is necessary only for models that exhibit a lack of "self-averaging." For models such as the scale-free preferential attachment model [1] this distinction is not an issue.
12. An asymptotic probability distribution that is zero everywhere is possible if the random variable in question (here, the degree) has an infinite range.
13. 0 - 1) - Y]〉, where θ(x) is the usual Heaviside function, and 〈〉 denotes expected value with respect to the hypergeometric distribution.
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16. -2β/(1 - δ) is divergent at x = 1 for δ≤ 1/2, φ ( 1 ) is still finite. However, this is not sufficient to guarantee normalizability of the probability distribution. The meaning of this inconsistency is that the expansion (23) breaks down at x = 1 for δ≤ 1/2.
17. It has been shown [4] that, for δ≤1/2, the mean degree grows without bound as t → ∞ instead of approaching a finite value. While, for this model, the unbounded growth of the mean degree coincides with the breakdown of stationarity, this is not generally true. The following section discusses a model where the mean degree grows without bound, yet the asymptotic degree distribution is stationary.