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

2013 ◽  
Vol 45 (04) ◽  
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.

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.

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.

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 Evolution of Equity Norms in Small-World Networks

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
José I. Santos ◽  
David J. Poza ◽  
José M. Galán ◽  
Adolfo López-Paredes
Keyword(s):  
Regular Ring ◽  
Finite Population ◽  
Evolutionary Game ◽  
Small World ◽  
Nash Bargaining ◽  
Random Networks ◽  
Small World Networks ◽  
Nash Bargaining Game ◽  
Regular Networks ◽  
Topology Of Interactions

The topology of interactions has been proved very influential in the results of models based on learning and evolutionary game theory. This paper is aimed at investigating the effect of structures ranging from regular ring lattices to random networks, including small-world networks, in a model focused on property distribution norms. The model considers a fixed and finite population of agents who play the Nash bargaining game repeatedly. Our results show that regular networks promote the emergence of the equity norm, while less-structured networks make possible the appearance of fractious regimes. Additionally, our analysis reveals that the speed of adoption can also be affected by the network structure.

scholarly journals Revisiting the Small-World Phenomenon

2016 ◽  
Vol 20 (1) ◽  
pp. 149-173 ◽  
Author(s):  
Tore Opsahl ◽  
Antoine Vernet ◽  
Tufool Alnuaimi ◽  
Gerard George
Keyword(s):  
Significant Variation ◽  
Random Network ◽  
Small World ◽  
Random Networks ◽  
Significant Heterogeneity ◽  
Small World Network ◽  
Small World Networks ◽  
Firm Outcomes ◽  
Baseline Levels

Research has explored how embeddedness in small-world networks influences individual and firm outcomes. We show that there remains significant heterogeneity among networks classified as small-world networks. We develop measures of the efficiency of a network, which allow us to refine predictions associated with small-world networks. A network is classified as a small-world network if it exhibits a distance between nodes that is comparable to the distance found in random networks of similar sizes—with ties randomly allocated among nodes—in addition to containing dense clusters. To assess how efficient a network is, there are two questions worth asking: (a) What is a compelling random network for baseline levels of distance and clustering? and (b) How proximal should an observed value be to the baseline to be deemed comparable? Our framework tests properties of networks, using simulation, to further classify small-world networks according to their efficiency. Our results suggest that small-world networks exhibit significant variation in efficiency. We explore implications for the field of management and organization.

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.

RELEVANT CYCLES IN CHEMICAL REACTION NETWORKS

2001 ◽  
Vol 04 (02n03) ◽  
pp. 207-226 ◽  
Author(s):  
PETRA M. GLEISS ◽  
PETER F. STADLER ◽  
ANDREAS WAGNER ◽  
DAVID A. FELL
Keyword(s):  
Metabolic Network ◽  
Chemical Reaction ◽  
Small World ◽  
Chemical Reaction Networks ◽  
Reaction Networks ◽  
Random Networks ◽  
Small World Networks ◽  
External Perturbations ◽  
Transition Times ◽  
Law Degree

We characterize the distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution. Compared with three classes of random networks, including Erdős–Rényi random graphs and synthetic small world networks of the same connectivity, both the metabolic network and models for the chemical reaction networks of planetary atmospheres have a particularly large number of triangles and a deficit in large cycles. Short cycles reduce the length of detours when a connection is clipped, so we propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood of oscillations in response to external perturbations.

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.

scholarly journals Breaking Apart Contact Networks with Vaccination

2020 ◽  
Author(s):  
Gianrocco Lazzari ◽  
Marcel Salathé
Keyword(s):  
Disease Outbreaks ◽  
Small World ◽  
Contact Network ◽  
Random Networks ◽  
Constant Number ◽  
Small World Networks ◽  
Transmission Potential ◽  
Social Mixing ◽  
Node Removal ◽  
The Continuum

ABSTRACTInfectious diseases can cause large disease outbreaks due to their transmission potential from one individual to the next. Vaccination is an effective way of cutting off possible chains of transmission, thereby mitigating the outbreak potential of a disease in a population. From a contact network perspective, vaccination effectively removes nodes from the network, thereby breaking apart the contact network into a much smaller network of susceptible individuals on which the disease can spread. Here, we look at the continuum of small world networks to random networks, and find that vaccination breaks apart networks in ways that can dramatically influence the maximum outbreak size. In particular, after the removal of a constant number of nodes (representing vaccination coverage), the more clustered small world networks more readily fall apart into many disjoint and small susceptible sub-networks, thus preventing large outbreaks, while more random networks remain largely connected even after node removal through vaccination. We further develop a model of social mixing that moves small world networks closer to the random regime, thereby facilitating larger disease outbreaks after vaccination. Our results show that even when vaccination is entirely random, social mixing can lead to contact network structures that strongly influence outbreak sizes. We find the largest effects to be in the regime of relatively high vaccination coverages of around 80%, where despite vaccination being random, outbreak sizes can vary by a factor of 20.

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