3,422 research outputs found
The Complexity of Rationalizing Network Formation
We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to be stable. From this we would like to infer some unobserved data about the system: edge prices, or how much each vertex values short paths to each other vertex. We study two rationalization problems arising from the network formation model of Jackson and Wolinsky [14]. When the goal is to infer edge prices, we observe that the rationalization problem is easy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find prices that maximize the stability of the system. In contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex by each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a close connection between rationalization problems and the Inequality-SAT (I-SAT) problem. Finally and most significantly, we prove that an approximation version of this NP-complete rationalization problem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial algorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input graphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible assuming P ≠ NP To do this we prove a tight (1/2 + δ) -approximation hardness for a variant of I-SAT in which all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LlN_(R_+) (linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification of the recent result of Guruswami and Raghavendra [10] which achieved tight hardness for this problem without the non-negativity constraint. Our technical contributions regarding the hardness of I-SAT and MAX-LIN_(R_+) may be of independent interest, given the generality of these problem
Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods
There are very few results on mixed finite element methods on surfaces. A
theory for the study of such methods was given recently by Holst and Stern,
using a variational crimes framework in the context of finite element exterior
calculus. However, we are not aware of any numerical experiments where mixed
finite elements derived from discretizations of exterior calculus are used for
a surface domain. This short note shows results of our preliminary experiments
using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed
form) on surfaces. We demonstrate two numerical methods. One is derived from
the primal-dual Discrete Exterior Calculus and the other from lowest order
finite element exterior calculus. The programming was done in the language
Python, using the PyDEC package which makes the code very short and easy to
read. The qualitative convergence studies seem to be promising.Comment: 14 pages, 11 figure
Optimal Bandwidth Choice for the Regression Discontinuity Estimator
We investigate the problem of optimal choice of the smoothing parameter (bandwidth) for the regression discontinuity estimator. We focus on estimation by local linear regression, which was shown to be rate optimal (Porter, 2003). Investigation of an expected-squared-error-loss criterion reveals the need for regularization. We propose an optimal, data dependent, bandwidth choice rule. We illustrate the proposed bandwidth choice using data previously analyzed by Lee (2008), as well as in a simulation study based on this data set. The simulations suggest that the proposed rule performs well.optimal bandwidth selection, local linear regression, regression discontinuity designs
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
Subcellular optogenetic activation of Cdc42 controls local and distal signaling to drive immune cell migration
Migratory immune cells use intracellular signaling networks to generate and orient spatially polarized responses to extracellular cues. The monomeric G protein Cdc42 is believed to play an important role in controlling the polarized responses, but it has been difficult to determine directly the consequences of localized Cdc42 activation within an immune cell. Here we used subcellular optogenetics to determine how Cdc42 activation at one side of a cell affects both cell behavior and dynamic molecular responses throughout the cell. We found that localized Cdc42 activation is sufficient to generate polarized signaling and directional cell migration. The optically activated region becomes the leading edge of the cell, with Cdc42 activating Rac and generating membrane protrusions driven by the actin cytoskeleton. Cdc42 also exerts long-range effects that cause myosin accumulation at the opposite side of the cell and actomyosin-mediated retraction of the cell rear. This process requires the RhoA-activated kinase ROCK, suggesting that Cdc42 activation at one side of a cell triggers increased RhoA signaling at the opposite side. Our results demonstrate how dynamic, subcellular perturbation of an individual signaling protein can help to determine its role in controlling polarized cellular responses
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