A kinetic approach to the random f-graph process. Paths, cycles and components

Balińska, Krystyna T.; Galina, Henryk; Quintas, Louis V.; Szymański, Jerzy

doi:10.1016/0166-218x(95)00008-f

Cited by 1 publication

(5 citation statements)

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“…However, in chemistry this is not a general rule. Note that if κ α = 1/( f − α) we recover equations equivalent to the kinetic differential equations for the R f -GP derived in refs and .…”

Section: Law Of Mass Actionmentioning

confidence: 64%

“…We shall derive differential equations for the proportions (relative to n) of the X i 's. At a crucial step in this derivation we shall, as in previous work, 3 The latter step follows from nx i (t) ) X i (t) ) X i (2s/fn) ) X i (s).…”

Section: Derivation Of Equationsmentioning

confidence: 98%

“…We shall derive differential equations for the proportions (relative to n ) of the X i 's. At a crucial step in this derivation we shall, as in previous work, replace Δ X = X ( s +1) − X ( s ), the change in X as s goes from s to s + 1, by the expected change E (Δ X ) = ∑ δ δ P (Δ X = δ).…”

Section: Derivation Of Equationsmentioning

confidence: 99%

“…This is also the case for the R* f -GP. In refs and we showed how a particular differential equation technique can be used to obtain the asymptotic degree distribution for a random graph generated by the R f -GP. The results are valid for any step in the R f -GP for which the asymptotic proportion of vertices of degree f is not unity.…”

Section: Introductionmentioning

confidence: 99%

“…Using the Law of Mass Action , chemists have derived the degree distribution of the vertices in a graph generated by the R* f -GP. Here we apply the method used in ref to obtain this degree distribution.…”

Section: Introductionmentioning

confidence: 99%

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A Kinetic Approach to the Random f-Graph Process with Nonuniform Edge Probabilities

Balińska¹,

Galina²,

Quintas³

et al. 1996

J. Chem. Inf. Comput. Sci.

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Starting with n labeled vertices and no edges, introduce edges, one at a time, so as to obtain a sequence of graphs each having no vertex of degree greater than f. The latter are called f-graphs. At each step the edge to be added is selected from among those edges whose addition would not violate the f-degree restriction and with probability proportional to the product of the respective numbers of available sites at the vertices of the potential edge. We call this procedure the Random f-Graph Process with nonuniform edge probabilities (R*f-GP) of order n. This is in contrast to the Random f-Graph Process (Rf-GP) of order n in which the edges that are added are selected with uniform probability. Using the Law of Mass Action, chemists have derived the degree distribution of the vertices in a graph generated by the R*f-GP. Here we apply a differential equation technique to obtain this degree distribution.

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Section: Law Of Mass Actionmentioning

confidence: 64%

Section: Derivation Of Equationsmentioning

confidence: 98%

Section: Derivation Of Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations