Fractality of Substitution Networks

Xi, Lifeng; Sun, Bingbin; Yao, Jialing

doi:10.1142/s0218348x19500348

Cited by 11 publications

(2 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They also showed that certain of these colored substitution networks have the fractality property. Similarly, substitutions of nodes, rather then arcs, were considered in 15,16 . By constructing self-similar networks, Yao et al 16 proved that node substitution networks have the fractality property; see also [17][18][19] .…”

Section: Fig 1 a Substitution Networkmentioning

confidence: 99%

On the scale-freeness of random colored substitution networks

Li,

Britz

2021

Preprint

View full text Add to dashboard Cite

Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced.Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.

show abstract

Section: Fig 1 a Substitution Networkmentioning

confidence: 99%

On the scale-freeness of random colored substitution networks

Li,

Britz

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We emphasize that it is difficult to obtain the above results if one utilizes the Newhouse thickness theorem [45]. The main reason is that it is not easy to calculate the thickness of K as there are very complicated overlaps in K. For the Assouad dimension of K and the geodesic distance on K × K, we refer to [36,38,51,50,37,56,47,49,48,46,53,35,52]. For the average weighted receiving time on the complex networks or on K, the readers can find results in the papers [23,6,7].…”

Section: Introductionmentioning

confidence: 99%

Multiple Representations of Real Numbers on Self-Similar Sets With Overlaps

et al. 2019

View full text Add to dashboard Cite

Let K be the attractor of the following IFSThe main results of this paper are as follows:By virtue of Lemma 2.3, it follows thatLemma 2.12. If c ≥ (1 − λ) 2 , then K + K = [0, 2]. Proof. By Lemmas 2.1 and 2.11. Take I = J = [c − λ, c] ∪ [1 − λ, 1]. Therefore, for f (x, y) = x + y, we have f (I, J) = [2(c − λ), 2c] ∪ [c + 1 − 2λ, 1 + c] ∪ [2(1 − λ), 2]. By Lemma 2.3, we conculde that 2c − (c + 1 − 2λ) = c + 2λ − 1 ≥ 0, 1 + c − (2(1 − λ)) = c + 2λ − 1 ≥ 0.

show abstract