scholarly journals Kleinberg Navigation on Anisotropic Lattices

2008 ◽  
Vol 2008 ◽  
pp. 1-4 ◽  
Author(s):  
J. M. Campuzano ◽  
J. P. Bagrow ◽  
D. ben-Avraham
Keyword(s):  
Long Range ◽  
Power Law ◽  
Finite Size ◽  
Small World ◽  
Small World Networks ◽  
Infinite Lattice ◽  
Preferred Direction ◽  
Lattice Size ◽  
Anisotropic Lattices ◽  
Anisotropy Strength

We study the Kleinberg problem of navigation in small-world networks when the underlying lattice is stretched along a preferred direction. Extensive simulations confirm that maximally efficient navigation is attained when the length r of long-range links is taken from the distribution P(r)∼r−α, when the exponent α is equal to 2, the dimension of the underlying lattice, regardless of the amount of anisotropy, but only in the limit of infinite lattice size, L→∞. For finite size lattices we find an optimal α(L) that depends strongly on L. The convergence to α=2 as L→∞ shows interesting power-law dependence on the anisotropy strength.

INFLUENCE OF THE INITIAL SOURCE OF EPIDEMIC AND PREVENTIVE VACCINATION ON THE SPREADING PHENOMENA IN A TWO-DIMENSIONAL LATTICE

2004 ◽  
Vol 15 (06) ◽  
pp. 755-765 ◽  
Author(s):  
R. A. KOSIŃSKI ◽  
Ł. ADAMOWSKI
Keyword(s):  
Long Range ◽  
Probabilistic Model ◽  
Small World ◽  
Two Dimensional ◽  
Small World Networks ◽  
Spreading Process ◽  
Dimensional Lattice ◽  
Epidemic Curve ◽  
Preventive Vaccination ◽  
Initial Source

The probabilistic model of epidemic in a two-dimensional lattice with an additional random, long range connections characteristic for the small world networks is presented. Relations describing the spreading process of epidemics, like epidemic curve or range of epidemic in time, were found. The influence of the borders of the lattice and the localization of the initial source of epidemic on the epidemic curve is found analytically. The application of the preventive vaccination in the population is discussed.

Diffusion of innovations in Axelrod’s model on small-world networks

2020 ◽  
Vol 31 (08) ◽  
pp. 2050116
Author(s):  
Sandro M. Reia
Keyword(s):  
Long Range ◽  
Diffusion Of Innovations ◽  
Computational Models ◽  
Single Agent ◽  
Finite Size ◽  
Small World ◽  
Single Feature ◽  
State Formula ◽  
New Ideas ◽  
Text Features

The interest in learning how innovations spread in our society has led to the development of a variety of theoretical-computational models to describe the mechanisms that govern the diffusion of new ideas among people. In this paper, the diffusion of innovations is addressed with the use of the Axelrod’s cultural model where an agent is represented by a cultural vector of [Formula: see text] features, in which each feature can take on [Formula: see text] integer states. The innovation or new idea is introduced in the population by setting a single feature of a single agent to a new state ([Formula: see text]) in the initial configuration. Particularly, we focus on the effect of the small-world topology on the dynamics of the innovation adoption. Our results indicate that the innovation spreads sublinearly ([Formula: see text]) in a regular one-dimensional lattice of connectivity [Formula: see text], whereas the innovation spreads linearly ([Formula: see text]) when a nonvanishing fraction of the short-ranged links are replaced by long-range ones. In addition, we find that the small-world topology prevents the emergence of complete order in the thermodynamic limit. For systems of finite size, however, the introduction of long range links causes the dynamics to reach a final ordered state much more rapidly than for the regular lattice.

Fractal aggregation on small-world networks with long-range deposition

Fractals ◽  
2020 ◽  
Author(s):  
Ren-Fei Wang ◽  
Sheng-Jun Wang ◽  
Zi-Gang Huang
Keyword(s):  
Long Range ◽  
Small World ◽  
Small World Networks

scholarly journals Finite-size effects in disordered λφ4 model

2016 ◽  
Vol 30 (30) ◽  
pp. 1650207 ◽  
Author(s):  
R. Acosta Diaz ◽  
N. F. Svaiter
Keyword(s):  
Phase Transition ◽  
Long Range ◽  
Power Law ◽  
Order Phase Transition ◽  
Size Effects ◽  
Finite Size ◽  
Second Order ◽  
Finite Size Effects ◽  
Power Law Decay ◽  
Second Order Phase Transition

We discuss finite-size effects in one disordered [Formula: see text] model defined in a [Formula: see text]-dimensional Euclidean space. We consider that the scalar field satisfies periodic boundary conditions in one dimension and it is coupled with a quenched random field. In order to obtain the average value of the free energy of the system, we use the replica method. We first discuss finite-size effects in the one-loop approximation in [Formula: see text] and [Formula: see text]. We show that in both cases, there is a critical length where the system develop a second-order phase transition, when the system presents long-range correlations with power-law decay. Next, we improve the above result studying the gap equation for the size-dependent squared mass, using the composite field operator method. We obtain again that the system present a second-order phase transition with long-range correlation with power-law decay.

scholarly journals Multistage Random Growing Small-World Networks with Power-Law Degree Distribution

2006 ◽  
Vol 23 (3) ◽  
pp. 746-749 ◽  
Author(s):  
Liu Jian-Guo ◽  
Dang Yan-Zhong ◽  
Wang Zhong-Tuo
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Small World ◽  
Small World Networks ◽  
Law Degree

EXISTENCE, COST AND ROBUSTNESS OF SPATIAL SMALL-WORLD NETWORKS

2007 ◽  
Vol 17 (07) ◽  
pp. 2331-2342 ◽  
Author(s):  
P. DE LOS RIOS ◽  
T. PETERMANN
Keyword(s):  
Euclidean Space ◽  
Long Range ◽  
Communication Systems ◽  
Large Scale ◽  
Probability Distributions ◽  
Flow Distribution ◽  
Small World ◽  
Electronic Circuits ◽  
Small World Networks ◽  
Network Cost

Small-world networks embedded in Euclidean space represent useful cartoon models for a number of real systems such as electronic circuits, communication systems, the large-scale brain architecture and others. Since the small-world behavior relies on the presence of long-range connections that are likely to have a cost which is a growing function of the length, we explore whether it is possible to choose suitable probability distributions for the shortcut lengths so as to preserve the small-world feature and, at the same time, to minimize the network cost. The flow distribution for such networks, and their robustness, are also investigated.

scholarly journals Greedy routing in small-world networks with power-law degrees

2014 ◽  
Vol 27 (4) ◽  
pp. 231-253 ◽  
Author(s):  
Pierre Fraigniaud ◽  
George Giakkoupis
Keyword(s):  
Power Law ◽  
Small World ◽  
Greedy Routing ◽  
Small World Networks

scholarly journals Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Wei Liu ◽  
Zhengxin Yan ◽  
Gaoliang Zhou
Keyword(s):  
Phase Transitions ◽  
Tricritical Point ◽  
Equilibrium Phase ◽  
Finite Size ◽  
Small World ◽  
Ferromagnetic Phase ◽  
Kawasaki Dynamics ◽  
Small World Networks ◽  
Competing Dynamics ◽  
Non Equilibrium

Abstract This article offers a detailed analysis of the Ising model in 2D small-world networks with competing Glauber and Kawasaki dynamics. The non-equilibrium stationary state phase transitions are obtained in these networks. The phase transitions are discussed, and the phase diagrams are obtained via Monte Carlo simulations and finite-size analyzing. We find that as the addition of links increases the phase transition temperature increases and the transition competing probability of tricritical point decreases. For the competition of the two dynamics, ferromagnetic to anti-ferromagnetic phase transitions and the critical endpoints are found in the small-world networks.

scholarly journals Asymptotic Behaviour of Gossip Processes and Small-World Networks

2013 ◽  
Vol 45 (4) ◽  
pp. 981-1010 ◽  
Author(s):  
A. D. Barbour ◽  
G. Reinert
Keyword(s):  
Long Range ◽  
Branching Process ◽  
Small World ◽  
Random Variable ◽  
Random Networks ◽  
Small World Networks ◽  
Transmission Of Information ◽  
Characteristic Behaviour ◽  
Typical Distances ◽  
Common Behaviour

Both small-world models of random networks with occasional long-range connections and gossip processes with occasional long-range transmission of information have similar characteristic behaviour. The long-range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.

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