The Tutte polynomial of an infinite family of outerplanar, small-world and self-similar graphs

2013 ◽  
Vol 392 (19) ◽  
pp. 4584-4593 ◽  
Author(s):  
Yunhua Liao ◽  
Aixiang Fang ◽  
Yaoping Hou
Keyword(s):  
Infinite Family ◽  
Tutte Polynomial ◽  
Small World ◽  
Self Similar

scholarly journals The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs

2013 ◽  
Vol 392 (12) ◽  
pp. 2803-2806 ◽  
Author(s):  
Francesc Comellas ◽  
Alícia Miralles ◽  
Hongxiao Liu ◽  
Zhongzhi Zhang
Keyword(s):  
Spanning Trees ◽  
Infinite Family ◽  
Small World ◽  
Self Similar ◽  
Number Of Spanning Trees

SMALL-WORLD AND SCALE-FREE PROPERTIES OF FRACTAL NETWORKS MODELED ON n-DIMENSIONAL SIERPINSKI PYRAMID

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750057 ◽  
Author(s):  
CHENG ZENG ◽  
MENG ZHOU
Keyword(s):  
Small World ◽  
The Self ◽  
Scale Free ◽  
Evolving Networks ◽  
Self Similar

In this paper, we construct evolving networks based on the construction of the [Formula: see text]-dimensional Sierpinski pyramid by the self-similar structure. We show that such networks have scale-free and small-world effects.

The Shortest Path to Network Geometry

2021 ◽  
Author(s):  
M. Ángeles Serrano ◽  
Marián Boguñá
Keyword(s):  
Routing Protocols ◽  
Hyperbolic Plane ◽  
Network Science ◽  
Small World ◽  
The Self ◽  
Geometric Description ◽  
Apparent Lack ◽  
Network Geometry ◽  
Connectivity Patterns ◽  
Self Similar

Real networks comprise from hundreds to millions of interacting elements and permeate all contexts, from technology to biology to society. All of them display non-trivial connectivity patterns, including the small-world phenomenon, making nodes to be separated by a small number of intermediate links. As a consequence, networks present an apparent lack of metric structure and are difficult to map. Yet, many networks have a hidden geometry that enables meaningful maps in the two-dimensional hyperbolic plane. The discovery of such hidden geometry and the understanding of its role have become fundamental questions in network science giving rise to the field of network geometry. This Element reviews fundamental models and methods for the geometric description of real networks with a focus on applications of real network maps, including decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks by geometric renormalization.

scholarly journals Tutte polynomial of a small-world Farey graph

2013 ◽  
Vol 104 (3) ◽  
pp. 38001 ◽  
Author(s):  
Yunhua Liao ◽  
Yaoping Hou ◽  
Xiaoling Shen
Keyword(s):  
Tutte Polynomial ◽  
Small World

scholarly journals Dynamics and processing in finite self-similar networks

2012 ◽  
Vol 9 (74) ◽  
pp. 2131-2144 ◽  
Author(s):  
Simon DeDeo ◽  
David C. Krakauer
Keyword(s):  
Time Scales ◽  
Biological Networks ◽  
Regulatory Networks ◽  
Multiple Scales ◽  
Social Systems ◽  
Small World ◽  
System Size ◽  
Low Noise ◽  
Trophic Networks ◽  
Self Similar

A common feature of biological networks is the geometrical property of self-similarity. Molecular regulatory networks through to circulatory systems, nervous systems, social systems and ecological trophic networks show self-similar connectivity at multiple scales. We analyse the relationship between topology and signalling in contrasting classes of such topologies. We find that networks differ in their ability to contain or propagate signals between arbitrary nodes in a network depending on whether they possess branching or loop-like features. Networks also differ in how they respond to noise, such that one allows for greater integration at high noise, and this performance is reversed at low noise. Surprisingly, small-world topologies, with diameters logarithmic in system size, have slower dynamical time scales, and may be less integrated (more modular) than networks with longer path lengths. All of these phenomena are essentially mesoscopic, vanishing in the infinite limit but producing strong effects at sizes and time scales relevant to biology.

TRAPPING ON WEIGHTED TETRAHEDRON KOCH NETWORKS WITH SMALL-WORLD PROPERTY

Fractals ◽  
2014 ◽  
Vol 22 (01n02) ◽  
pp. 1450006 ◽  
Author(s):  
MEIFENG DAI ◽  
QI XIE ◽  
LIFENG XI
Keyword(s):  
Small World ◽  
Clustering Coefficient ◽  
Weight Factor ◽  
First Passage ◽  
Mean First Passage Times ◽  
Weighted Clustering ◽  
Self Similar ◽  
Small World Property ◽  
Analytical Expressions ◽  
Passage Times

In this paper, we present weighted tetrahedron Koch networks depending on a weight factor. According to their self-similar construction, we obtain the analytical expressions of the weighted clustering coefficient and average weighted shortest path (AWSP). The obtained solutions show that the weighted tetrahedron Koch networks exhibits small-world property. Then, we calculate the average receiving time (ART) on weighted-dependent walks, which is the sum of mean first-passage times (MFPTs) for all nodes absorpt at the trap located at a hub node. We find that the ART exhibits a sublinear or linear dependence on network order.

THE SCALE-FREE AND SMALL-WORLD PROPERTIES OF COMPLEX NETWORKS ON SIERPINSKI-TYPE HEXAGON

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050054
Author(s):  
KUN CHENG ◽  
DIRONG CHEN ◽  
YUMEI XUE ◽  
QIAN ZHANG
Keyword(s):  
Power Law ◽  
Path Length ◽  
Small World ◽  
The Self ◽  
Renewal Theorem ◽  
Self Similarity ◽  
Scale Free ◽  
Power Law Exponent ◽  
Self Similar ◽  
Encoding Method

In this paper, a network is generated from a Sierpinski-type hexagon by applying the encoding method in fractal. The criterion of neighbor is established to quantify the relationships among the nodes in the network. Based on the self-similar structures, we verify the scale-free and small-world effects. The power-law exponent on degree distribution is derived to be [Formula: see text] and the average clustering coefficients are shown to be larger than [Formula: see text]. Moreover, we give the bounds of the average path length of our proposed network from the renewal theorem and self-similarity.

scholarly journals Almost-automorphisms of trees, cloning systems and finiteness properties

2019 ◽  
pp. 1-46
Author(s):  
Rachel Skipper ◽  
Matthew C. B. Zaremsky
Keyword(s):  
Free Group ◽  
Infinite Family ◽  
Finitely Generated ◽  
Natural Conditions ◽  
Type Formula ◽  
System A ◽  
Automorphisms Of Trees ◽  
Self Similar ◽  
Virtually Free Group ◽  
Finiteness Properties

We prove that the group of almost-automorphisms of the infinite rooted regular [Formula: see text]-ary tree [Formula: see text] arises naturally as the Thompson-like group of a so-called [Formula: see text]-ary cloning system. A similar phenomenon occurs for any Röver–Nekrashevych group [Formula: see text], for [Formula: see text] a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for [Formula: see text], is of type [Formula: see text]. Namely, we find some natural conditions on subgroups of [Formula: see text] to ensure that [Formula: see text] is of type [Formula: see text] and, in particular, we prove this for all [Formula: see text] in the infinite family of Šunić groups. We also prove that if [Formula: see text] is itself of type [Formula: see text], then so is [Formula: see text], and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group [Formula: see text] yields a type [Formula: see text] Röver–Nekrashevych group [Formula: see text].

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