366,747 research outputs found
Undirected Graphs of Entanglement Two
Entanglement is a complexity measure of directed graphs that origins in fixed
point theory. This measure has shown its use in designing efficient algorithms
to verify logical properties of transition systems. We are interested in the
problem of deciding whether a graph has entanglement at most k. As this measure
is defined by means of games, game theoretic ideas naturally lead to design
polynomial algorithms that, for fixed k, decide the problem. Known
characterizations of directed graphs of entanglement at most 1 lead, for k = 1,
to design even faster algorithms. In this paper we present an explicit
characterization of undirected graphs of entanglement at most 2. With such a
characterization at hand, we devise a linear time algorithm to decide whether
an undirected graph has this property
Fixed-point adiabatic quantum search
Fixed-point quantum search algorithms succeed at finding one of M target items among N total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property—the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.American Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipMIT-Harvard Center for Ultracold Atoms MIT International Science and Technology InitiativeNational Science Foundation (U.S.) (RQCC Project 1111337)Massachusetts Institute of Technology. Undergraduate Research Opportunities Program (Paul E. Gray Endowed Fund
Sequential Estimation of Structural Models with a Fixed Point Constraint
This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze the conditions under which the nested pseudo-likelihood (NPL) algorithm achieves convergence and derive its convergence rate. We find that the NPL algorithm may not necessarily converge when the fixed point mapping does not have a local contraction property. To address the issue of non-convergence, we propose alternative sequential estimation procedures that can achieve convergence even when the NPL algorithm does not. Upon convergence, some of our proposed estimation algorithms produce more efficient estimators than the NPL estimator.contraction, dynamic games, nested pseudo likelihood, recursive projection method
Sequential Estimation of Structural Models with a Fixed Point Constraint
This paper considers the estimation problem of structural models for which empirical restrictions are characterized by a fixed point constraint, such as structural dynamic discrete choice models or models of dynamic games. We analyze the conditions under which the nested pseudo-likelihood (NPL) algorithm achieves convergence and derive its convergence rate. We find that the NPL algorithm may not necessarily converge when the fixed point mapping does not have a local contraction property. To address the issue of non-convergence, we propose alternative sequential estimation procedures that can achieve convergence even when the NPL algorithm does not. Upon convergence, some of our proposed estimation algorithms produce more efficient estimators than the NPL estimator.contraction, dynamic games, nested pseudo likelihood, recursive projection method
A Scalable Null Model for Directed Graphs Matching All Degree Distributions: In, Out, and Reciprocal
Degree distributions are arguably the most important property of real world
networks. The classic edge configuration model or Chung-Lu model can generate
an undirected graph with any desired degree distribution. This serves as a good
null model to compare algorithms or perform experimental studies. Furthermore,
there are scalable algorithms that implement these models and they are
invaluable in the study of graphs. However, networks in the real-world are
often directed, and have a significant proportion of reciprocal edges. A
stronger relation exists between two nodes when they each point to one another
(reciprocal edge) as compared to when only one points to the other (one-way
edge). Despite their importance, reciprocal edges have been disregarded by most
directed graph models.
We propose a null model for directed graphs inspired by the Chung-Lu model
that matches the in-, out-, and reciprocal-degree distributions of the real
graphs. Our algorithm is scalable and requires random numbers to
generate a graph with edges. We perform a series of experiments on real
datasets and compare with existing graph models.Comment: Camera ready version for IEEE Workshop on Network Science; fixed some
typos in tabl
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