scholarly journals Single-seed cascades on clustered networks

2020 ◽  
pp. 1-14
Author(s):  
John K. McSweeney
Keyword(s):  
Branching Process ◽  
Dynamic Network ◽  
Average Degree ◽  
Random Networks ◽  
Threshold Condition ◽  
Cascade Process ◽  
Active Node ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Clustered Networks

Abstract We consider a dynamic network cascade process developed by Duncan Watts applied to a class of random networks, developed independently by Newman and Miller, which allows a specified amount of clustering (short loops). We adapt existing methods for locally tree-like networks to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a formula that has appeared in various forms in works by Hackett et al. and Miller which provides a threshold condition for certain extinction of the cascade. We find that clustering impedes cascade propagation for networks of low average degree by reducing the connectivity of the network, but for networks with high average degree, the presence of small cycles makes cascades more likely.

scholarly journals The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

2011 ◽  
Vol 48 (1) ◽  
pp. 173-188
Author(s):  
Simon E. F. Spencer ◽  
Philip D. O‘Neill
Keyword(s):  
Infectious Disease ◽  
Multiple Solutions ◽  
Branching Process ◽  
Specific Model ◽  
Process Models ◽  
Multitype Branching Process ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Disease Epidemics ◽  
Special Case

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

scholarly journals The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

2011 ◽  
Vol 48 (01) ◽  
pp. 173-188
Author(s):  
Simon E. F. Spencer ◽  
Philip D. O‘Neill
Keyword(s):  
Infectious Disease ◽  
Multiple Solutions ◽  
Branching Process ◽  
Specific Model ◽  
Process Models ◽  
Multitype Branching Process ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Disease Epidemics ◽  
Special Case

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

scholarly journals Best Proximity Point Results in Complex Valued Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Binayak S. Choudhury ◽  
Nikhilesh Metiya ◽  
Pranati Maity
Keyword(s):  
Minimum Distance ◽  
Metric Spaces ◽  
Optimal Solution ◽  
Global Optimal Solution ◽  
Complex Valued Metric Space ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Global Optimal ◽  
Metric Function ◽  
Complex Valued

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples.

A Model for Integrated Inventory and Assortment Planning

2021 ◽  
Author(s):  
Victor Martínez-de-Albéniz ◽  
Sumit Kunnumkal
Keyword(s):  
Closed Form Solution ◽  
Optimal Solution ◽  
Integer Program ◽  
Form Solution ◽  
Assortment Planning ◽  
Dimensional Parameter ◽  
Planning Decisions ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Random Instances

Integrating inventory and assortment planning decisions is a challenging task that requires comparing the value of demand expansion through broader choice for consumers with the value of higher in-stock availability. We develop a stockout-based substitution model for trading off these values in a setting with inventory replenishment, a feature missing in the literature. Using the closed form solution for the single-product case, we develop an accurate approximation for the multiproduct case. This approximated formulation allows us to optimize inventory decisions by solving a fractional integer program with a fixed point equation constraint. When products have equal margins, we solve the integer program exactly by bisection over a one-dimensional parameter. In contrast, when products have different margins, we propose a fractional relaxation that we can also solve by bisection and that results in near-optimal solutions. Overall, our approach provides solutions within 0.1% of the optimal policy and finds the optimal solution in 80% of the random instances we generate. This paper was accepted by David Simchi-Levi, optimization.

Analysis of Crossword Puzzle Difficulty Using a Random Graph Process

2015 ◽  
Author(s):  
John K. McSweeney
Keyword(s):  
New York ◽  
Stochastic Process ◽  
Fixed Point ◽  
Network Structure ◽  
New York Times ◽  
Final Solution ◽  
Crossword Puzzle ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Network Properties

This chapter quantifies the dynamics of a crossword puzzle by using a network structure to model it. Specifically, the chapter determines how the interaction between the structure of cells in the puzzle and the difficulty of the clues affects the puzzle's solvability. It first builds an iterative stochastic process that exactly describes the solution and obtains its deterministic approximation, which gives a very simple fixed-point equation to solve for the final solution proportion. The chapter then shows via simulation on actual crosswords from the Sunday edition of The New York Times that certain network properties inherent to actual crossword networks are important predictors of the final solution size of the puzzle.

A Characterization of the Compound-Exponential Type Distributions

2011 ◽  
Vol 54 (3) ◽  
pp. 464-471
Author(s):  
Tea-Yuan Hwang ◽  
Chin-Yuan Hu
Keyword(s):  
Fixed Point ◽  
Exponential Type ◽  
Existence And Uniqueness ◽  
Fixed Point Equation ◽  
Point Equation

AbstractIn this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.

scholarly journals A conformal fixed-point equation for the effective average action

2019 ◽  
Vol 34 (05) ◽  
pp. 1950027 ◽  
Author(s):  
Oliver J. Rosten
Keyword(s):  
Fixed Point ◽  
Conformal Invariance ◽  
Legendre Transform ◽  
Fixed Point Equation ◽  
Average Action ◽  
Point Equation

A Legendre transform of the recently discovered conformal fixed-point equation is constructed, providing an unintegrated equation encoding full conformal invariance within the framework of the effective average action.

scholarly journals Modified SOR-Like Method for Absolute Value Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Cui-Xia Li ◽  
Shi-Liang Wu
Keyword(s):  
Fixed Point ◽  
Nonlinear Equation ◽  
Computational Efficiency ◽  
Numerical Experiments ◽  
Absolute Value Equations ◽  
Convergence Conditions ◽  
Absolute Value ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Better Than

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like method are presented. The computational efficiency of the modified SOR-like method is better than that of the SOR-like method by some numerical experiments.

scholarly journals A Relaxed Self-Adaptive Projection Algorithm for Solving the Multiple-Sets Split Equality Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Haitao Che ◽  
Haibin Chen
Keyword(s):  
Fixed Point ◽  
Equation System ◽  
Projection Algorithm ◽  
Step Size ◽  
Split Equality Problem ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Equality Problem ◽  
Adaptive Step ◽  
Self Adaptive

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly, we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system is established. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equation system. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage of the self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove the convergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.

scholarly journals On exact sampling of stochastic perpetuities

2011 ◽  
Vol 48 (A) ◽  
pp. 165-182 ◽  
Author(s):  
Jose H. Blanchet ◽  
Karl Sigman
Keyword(s):  
Fixed Point ◽  
Net Present Value ◽  
Positive Density ◽  
Simulation Algorithm ◽  
Present Value ◽  
The Past ◽  
Coupling From The Past ◽  
Fixed Point Equation ◽  
Point Equation ◽  
The Right

A stochastic perpetuity takes the formD∞=∑n=0∞exp(Y1+⋯+Yn)Bn, whereYn:n≥0) and (Bn:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively byDn+1=AnDn+Bn,n≥0, whereAn=eYn;D∞then satisfies the stochastic fixed-point equationD∞D̳AD∞+B, whereAandBare independent copies of theAnandBn(and independent ofD∞on the right-hand side). In our framework, the quantityBn, which represents a random reward at timen, is assumed to be positive, unbounded with EBnp<∞ for somep>0, and have a suitably regular continuous positive density. The quantityYnis assumed to be light tailed and represents a discount rate from timenton-1. The RVD∞then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples ofD∞. Our method is a variation ofdominated coupling from the pastand it involves constructing a sequence of dominating processes.

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