Speeding up computation of the reliability polynomial coefficients for a random graph

2011 ◽  
Vol 72 (7) ◽  
pp. 1474-1486 ◽  
Author(s):  
A. S. Rodionov
Keyword(s):  
Random Graph ◽  
Polynomial Coefficients ◽  
Reliability Polynomial

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

2021 ◽  
Vol 383 ◽  
pp. 107669
Author(s):  
Anshul Adve ◽  
Colleen Robichaux ◽  
Alexander Yong
Keyword(s):  
Efficient Algorithm ◽  
Schubert Polynomial ◽  
Polynomial Coefficients

scholarly journals A Note on the Majority Dynamics in Inhomogeneous Random Graphs

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graph ◽  
Initial State ◽  
Time Step ◽  
Dynamic Monopoly ◽  
Inhomogeneous Random Graphs ◽  
Assignment Probability ◽  
Edge Probability ◽  
Consensus Reaching Process ◽  
Consensus Reaching ◽  
Inhomogeneous Random Graph

AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

scholarly journals The Rabi problem with elliptical polarization

2020 ◽  
Vol 75 (11) ◽  
pp. 937-962
Author(s):  
Heinz-Jürgen Schmidt
Keyword(s):  
Floquet Theory ◽  
Equation Of Motion ◽  
Quantum Spin ◽  
Heun Equation ◽  
Elliptical Polarization ◽  
Third Order ◽  
Polynomial Coefficients ◽  
Resonance Frequencies ◽  
Classical Quantum ◽  
Elliptically Polarized

AbstractWe consider the solution of the equation of motion of a classical/quantum spin subject to a monochromatical, elliptically polarized external field. The classical Rabi problem can be reduced to third-order differential equations with polynomial coefficients and hence solved in terms of power series in close analogy to the confluent Heun equation occurring for linear polarization. Application of Floquet theory yields physically interesting quantities like the quasienergy as a function of the problem’s parameters and expressions for the Bloch–Siegert shift of resonance frequencies. Various limit cases are thoroughly investigated.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

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