Immersions in manifolds of positive weights

1978 ◽  
pp. 88-92 ◽  
Author(s):  
Henry Glover ◽  
Bill Homer ◽  
Guido Mislin
Keyword(s):  
Positive Weights

scholarly journals Weighted approximate Bayesian computation via Sanov’s theorem

2021 ◽  
Author(s):  
Cecilia Viscardi ◽  
Michele Boreale ◽  
Fabio Corradi
Keyword(s):  
Large Deviations ◽  
Posterior Distribution ◽  
Approximate Bayesian Computation ◽  
Bayesian Computation ◽  
Information Theoretic ◽  
Discrete Random Variables ◽  
Positive Weights ◽  
Approximate Bayesian ◽  
Information Theoretic Method ◽  
Computational Resources

AbstractWe consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation.

scholarly journals Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

2021 ◽  
Vol 182 (3) ◽  
pp. 219-242
Author(s):  
Mostafa Haghir Chehreghani ◽  
Albert Bifet ◽  
Talel Abdessalem
Keyword(s):  
Betweenness Centrality ◽  
Directed Graphs ◽  
Randomized Algorithm ◽  
Exact Algorithm ◽  
Directed Network ◽  
Weighted Graphs ◽  
Single Vertex ◽  
Small Constant ◽  
Positive Weights ◽  
Real World Datasets

Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network G and a vertex r ∈ V (G), we propose an exact algorithm to compute betweenness score of r. Our algorithm pre-computes a set ℛ𝒱(r), which is used to prune a huge amount of computations that do not contribute to the betweenness score of r. Time complexity of our algorithm depends on |ℛ𝒱(r)| and it is respectively Θ(|ℛ𝒱(r)| · |E(G)|) and Θ(|ℛ𝒱(r)| · |E(G)| + |ℛ𝒱(r)| · |V(G)| log |V(G)|) for unweighted graphs and weighted graphs with positive weights. |ℛ𝒱(r)| is bounded from above by |V(G)| – 1 and in most cases, it is a small constant. Then, for the cases where ℛ𝒱(r) is large, we present a simple randomized algorithm that samples from ℛ𝒱(r) and performs computations for only the sampled elements. We show that this algorithm provides an (ɛ, δ)-approximation to the betweenness score of r. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

2011 ◽  
Vol 03 (03) ◽  
pp. 323-336 ◽  
Author(s):  
FANICA GAVRIL
Keyword(s):  
Minimum Weight ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Circle Graphs ◽  
Intersection Model ◽  
The Family ◽  
Positive Weights ◽  
Vertex Set ◽  
Vertex Sets ◽  
Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

Graph matching

2020 ◽  
pp. 291-358
Author(s):  
Rob H. Bisseling
Keyword(s):  
Approximation Algorithm ◽  
Parallel Algorithm ◽  
Graph Matching ◽  
Linear Time ◽  
Total Weight ◽  
Mathematical Representation ◽  
Optimal Weight ◽  
Graph Problems ◽  
Positive Weights ◽  
Local Dominance

This chapter explores parallel algorithms for graph matching. Here, a graph is the mathematical representation of a network, with vertices representing the nodes of the network and edges representing their connections. The edges have positive weights, and the aim is to find a matching with maximum total weight. The chapter first presents a sequential, parallelizable approximation algorithm based on local dominance that guarantees attaining at least half the optimal weight in near-linear time. This algorithm, coupled with a vertex partitioning, is the basis for developing a parallel algorithm. The BSP approach is shown to be especially advantageous for graph problems, both in developing a parallel algorithm and in proving it correct. The basic parallel algorithm is enhanced by giving preference to local matches when breaking ties and by adding a load-balancing mechanism. The scalability of the parallel algorithm is put to the test using graphs of up to 150 million edges.

Next-To-Interpolatory Approximation on Sets with Multiplicities

1966 ◽  
Vol 18 ◽  
pp. 1196-1211 ◽  
Author(s):  
T. S. Motzkin ◽  
A. Sharma
Keyword(s):  
Real Numbers ◽  
Sign Changes ◽  
Positive Weights ◽  
The Difference ◽  
Unique Polynomial

It is known that given a set X of m (⩾n) distinct real numbers and a real-valued function f denned on X, there exists a unique polynomial pn-1,f,x of degree n — 1 or less which approximates best to f(x) on X, that is, which minimizes the deviation δ = δ(f, p) defined by the αth-power metric (α < 1) with positive weights, or by the positively weighted maximum of |f — p| on X; these deviations shall be denoted by δα and δβ. The polynomial pn-1,f,x has the property that f — pn-1,f,x has at least n strong sign changes; in other words, there are at least n + 1 points in X where the difference takes alternatingly positive and negative values.

scholarly journals Generating Nested Quadrature Rules with Positive Weights based on Arbitrary Sample Sets

2020 ◽  
Vol 8 (1) ◽  
pp. 139-169 ◽  
Author(s):  
Laurent van den Bos ◽  
Benjamin Sanderse ◽  
Wim Bierbooms ◽  
Gerard van Bussel
Keyword(s):  
Quadrature Rules ◽  
Positive Weights

GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM

2012 ◽  
Vol 15 (03) ◽  
pp. 1250016 ◽  
Author(s):  
FRANCO FAGNOLA ◽  
VERONICA UMANITÀ
Keyword(s):  
Small Time ◽  
Jump Processes ◽  
Time Interval ◽  
Markov Semigroup ◽  
Cycle Decomposition ◽  
Markov Jump ◽  
Partial Isometries ◽  
Small Time Interval ◽  
Positive Weights ◽  
The Difference

A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.

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