scholarly journals On the Critical Value for ‘Percolation’ of Minimum-Weight Trees in the Mean-Field Distance Model

1998 ◽  
Vol 7 (1) ◽  
pp. 1-10 ◽  
Author(s):  
DAVID ALDOUS
Keyword(s):  
Fixed Point ◽  
Probability Distributions ◽  
Minimum Weight ◽  
Mean Field ◽  
Critical Value ◽  
Edge Weight ◽  
Maximal Size ◽  
Distance Model ◽  
The Mean

Consider the complete n-graph with independent exponential (mean n) edge-weights. Let M(c, n) be the maximal size of subtree for which the average edge-weight is at most c. It is shown that M(c, n) makes the transition from o(n) to Ω(n) around some critical value c(0), which can be specified in terms of a fixed point of a mapping on probability distributions.

scholarly journals Insensitivity of the mean field limit of loss systems under SQ(d) routeing

2019 ◽  
Vol 51 (4) ◽  
pp. 1027-1066
Author(s):  
Thirupathaiah Vasantam ◽  
Arpan Mukhopadhyay ◽  
Ravi R. Mazumdar
Keyword(s):  
Fixed Point ◽  
Service Time ◽  
Mean Field ◽  
Service Time Distribution ◽  
Field Limit ◽  
Mean Field Limit ◽  
Mean Field Equations ◽  
The Mean ◽  
Loss Systems ◽  
Exponential Case

AbstractIn this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

Performance Analysis of Work Stealing in Large-scale Multithreaded Computing

2021 ◽  
Vol 6 (2) ◽  
pp. 1-28
Author(s):  
Nikki Sonenberg ◽  
Grzegorz Kielanski ◽  
Benny Van Houdt
Keyword(s):  
Fixed Point ◽  
Large Scale ◽  
Field Model ◽  
Unique Fixed Point ◽  
Mean Field ◽  
Response Time Distribution ◽  
Mean Field Model ◽  
Work Stealing ◽  
Poisson Arrivals ◽  
The Mean

Randomized work stealing is used in distributed systems to increase performance and improve resource utilization. In this article, we consider randomized work stealing in a large system of homogeneous processors where parent jobs spawn child jobs that can feasibly be executed in parallel with the parent job. We analyse the performance of two work stealing strategies: one where only child jobs can be transferred across servers and the other where parent jobs are transferred. We define a mean-field model to derive the response time distribution in a large-scale system with Poisson arrivals and exponential parent and child job durations. We prove that the model has a unique fixed point that corresponds to the steady state of a structured Markov chain, allowing us to use matrix analytic methods to compute the unique fixed point. The accuracy of the mean-field model is validated using simulation. Using numerical examples, we illustrate the effect of different probe rates, load, and different child job size distributions on performance with respect to the two stealing strategies, individually, and compared to each other.

scholarly journals Noise and disorder: Phase transitions and universality in a model of opinion formation

2016 ◽  
Vol 27 (06) ◽  
pp. 1650060 ◽  
Author(s):  
Nuno Crokidakis
Keyword(s):  
Phase Transitions ◽  
Probability Distributions ◽  
Mean Field ◽  
Universality Class ◽  
Formation Model ◽  
Opinion Formation ◽  
Discrete Probability ◽  
Wide Range ◽  
The Mean ◽  
Discrete Probability Distributions

In this work, we study a three-state opinion formation model considering two distinct mechanisms, namely independence and conviction. Independence is introduced in the model as a noise by means of a probability of occurrence q. On the other hand, conviction acts as a disorder in the system, and it is introduced by two discrete probability distributions. We analyze the effects of such two mechanisms on the phase transitions of the model, and we found that the critical exponents are universal over the order–disorder frontier, presenting the same universality class of the mean-field Ising model. In addition, for one of the probability distributions, the transition may be eliminated for a wide range of the parameters.

scholarly journals Non-perturbative effects in spin glasses

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Michele Castellana ◽  
Giorgio Parisi
Keyword(s):  
Fixed Point ◽  
Ising Model ◽  
Spin Glasses ◽  
Numerical Study ◽  
Mean Field ◽  
Critical Dimension ◽  
Upper Critical Dimension ◽  
Ising Spin ◽  
Ferromagnetic Ising Model ◽  
The Mean

Abstract We present a numerical study of an Ising spin glass with hierarchical interactions—the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the "Equation missing"-expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.

scholarly journals COLLISIONLESS RELATIVISTIC SHOCKS: CURRENT DRIVEN TURBULENCE AND PARTICLE ACCELERATION

2014 ◽  
Vol 28 ◽  
pp. 1460193
Author(s):  
GUY PELLETIER ◽  
MARTIN LEMOINE ◽  
LAURENT GREMILLET ◽  
ILLYA PLOTNIKOV
Keyword(s):  
Particle Acceleration ◽  
Gamma Ray ◽  
Mean Field ◽  
Critical Value ◽  
Gamma Ray Bursts ◽  
Incoming Flow ◽  
Buffer Effect ◽  
Relativistic Shocks ◽  
Onset Of Turbulence ◽  
The Mean

The physics of collisionless relativistic shocks with a moderate magnetization is presented. Micro-physics is relevant to explain the most energetic radiative phenomena of Nature, namely that of the termination shock of Gamma Ray Bursts. A transition towards Fermi process occurs for decreasing magnetization around a critical value which turns out to be the condition for the scattering to break the mean field inhibition. Scattering is produced by magnetic micro-turbulence driven by the current carried by returning particles, which had not been considered till now, but turns out to be more intense than Weibel's one around the transition. The current is also responsible for a buffer effect on the motion of the incoming flow, on which the threshold for the onset of turbulence depends.

Variational Migdal–Kadanoff Approach to Molecular Ordering on a Lattice

1997 ◽  
Vol 11 (18) ◽  
pp. 2183-2193 ◽  
Author(s):  
Fabio Siringo
Keyword(s):  
Fixed Point ◽  
Molecular Orientation ◽  
Mean Field ◽  
Real Space ◽  
Group Method ◽  
Renormalization Group Method ◽  
Formal Equivalence ◽  
Real Space Renormalization Group ◽  
Molecular Ordering ◽  
The Mean

A variational extension of the Migdal–Kadanoff real space renormalization group method is applied to a recently proposed d-dimensional lattice model describing some aspects of molecular orientation in solids. As expected the method yields the exact fixed point for d=2, given the formal equivalence to the two-dimensional Ising model. For d=3 such equivalence breaks down and an approximate estimate of the fixed point is explicitly recovered and compared to the mean-field prediction. The possible existence of an intermediate symmetric phase is discussed.

scholarly journals Entropy of Simulated Liquids Using Multiscale Cell Correlation

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 750
Author(s):  
Hafiz Saqib Ali ◽  
Jonathan Higham ◽  
Richard H. Henchman
Keyword(s):  
Degrees Of Freedom ◽  
Probability Distributions ◽  
Mean Field ◽  
Vibrational Entropy ◽  
Amber Force Field ◽  
Equal Importance ◽  
The Mean ◽  
Dynamics Simulations ◽  
Consistent Treatment ◽  
Better Than

Accurately calculating the entropy of liquids is an important goal, given that many processes take place in the liquid phase. Of almost equal importance is understanding the values obtained. However, there are few methods that can calculate the entropy of such systems, and fewer still to make sense of the values obtained. We present our multiscale cell correlation (MCC) method to calculate the entropy of liquids from molecular dynamics simulations. The method uses forces and torques at the molecule and united-atom levels and probability distributions of molecular coordinations and conformations. The main differences with previous work are the consistent treatment of the mean-field cell approximation to the approriate degrees of freedom, the separation of the force and torque covariance matrices, and the inclusion of conformation correlation for molecules with multiple dihedrals. MCC is applied to a broader set of 56 important industrial liquids modeled using the Generalized AMBER Force Field (GAFF) and Optimized Potentials for Liquid Simulations (OPLS) force fields with 1.14*CM1A charges. Unsigned errors versus experimental entropies are 8.7 J K − 1 mol − 1 for GAFF and 9.8 J K − 1 mol − 1 for OPLS. This is significantly better than the 2-Phase Thermodynamics method for the subset of molecules in common, which is the only other method that has been applied to such systems. MCC makes clear why the entropy has the value it does by providing a decomposition in terms of translational and rotational vibrational entropy and topographical entropy at the molecular and united-atom levels.

scholarly journals Quantitative Analysis of a Transient Dynamics of a Gene Regulatory Network

2018 ◽  
Author(s):  
JaeJun Lee ◽  
Julian Lee
Keyword(s):  
Fixed Point ◽  
Quantitative Analysis ◽  
Rate Equation ◽  
Regulatory Network ◽  
Mean Field ◽  
Extinction Time ◽  
Stochastic Noise ◽  
Stable Fixed Point ◽  
The Mean ◽  
Gene Regulatory

AbstractIn a stochastic process, noise often modifies the picture offered by the mean field dynamics. In particular, when there is an absorbing state, the noise erases a stable fixed point of the mean field equation from the stationary distribution, and turns it into a transient peak. We make a quantitative analysis of this effect for a simple genetic regulatory network with positive feedback, where the proteins become extinct in the presence of stochastic noise, contrary to the prediction of the deterministic rate equation that the protein number converges to a non-zero value. We show that the transient peak appears near the stable fixed point of the rate equation, and the extinction time diverges exponentially as the stochastic noise approaches zero. We also show how the baseline production from the inactive gene ameliorates the effect of the stochastic noise, and interpret the opposite effects of the noise and the baseline production in terms of the position shift of the unstable fixed point. The order of magnitude estimates using biological parameters suggest that for a real gene regulatory network, the stochastic noise is sufficiently small so that not only is the extinction time much larger than biologically relevant time-scales, but also the effect of the baseline production dominates over that of the stochastic noise, leading to the protection from the catastrophic rare event of protein extinction.

scholarly journals Dynamically Weighted Clique Evolution Model in Clique Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Zhang-Wei Li ◽  
Xu-Hua Yang ◽  
Feng-Ling Jiang ◽  
Guang Chen ◽  
Guo-Qing Weng ◽  
...  
Keyword(s):  
Real World ◽  
Mean Field Theory ◽  
Mean Field ◽  
Strength Distribution ◽  
Evolution Model ◽  
Edge Weight ◽  
Time Step ◽  
Complete Subgraph ◽  
Scale Free ◽  
The Mean

This paper proposes a weighted clique evolution model based on clique (maximal complete subgraph) growth and edge-weight driven for complex networks. The model simulates the scheme of real-world networks that the evolution of networks is likely to be driven by the flow, such as traffic or information flow needs, as well as considers that real-world networks commonly consist of communities. At each time step of a network’s evolution progress, an edge is randomly selected according to a preferential scheme. Then a new clique which contains the edge is added into the network while the weight of the edge is adjusted to simulate the flow change brought by the new clique addition. We give the theoretical analysis based on the mean field theory, as well as some numerical simulation for this model. The result shows that the model can generate networks with scale-free distributions, such as edge weight distribution and node strength distribution, which can be found in many real-world networks. It indicates that the evolution rule of the model may attribute to the formation of real-world networks.

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