Computing Shapley Values in the Plane

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points

Computing Shapley Values for Mean Width in 3-D

The Shapley value is a common tool in game theory to evaluate the importance of a player in a cooperative setting. In a geometric context, it provides a way to measure the contribution of a geometric object in a set towards some function on the set. Recently, Cabello and Chan (SoCG 2019) presented algorithms for computing Shapley values for a number of functions for point sets in the plane. More formally, a coalition game consists of a set of players $N$ and a characteristic function $v: 2^N \to \mathbb{R}$ with $v(\emptyset) = 0$. Let $\pi$ be a uniformly random permutation of $N$, and $P_N(\pi, i)$ be the set of players in $N$ that appear before player $i$ in the permutation $\pi$. The Shapley value of the game is defined to be $\phi(i) = \mathbb{E}_\pi[v(P_N(\pi, i) \cup \{i\}) - v(P_N(\pi, i))]$. More intuitively, the Shapley value represents the impact of player $i$'s appearance over all insertion orders. We present an algorithm to compute Shapley values in 3-D, where we treat points as players and use the mean width of the convex hull as the characteristic function. Our algorithm runs in $O(n^3\log^2{n})$ time and $O(n)$ space. Our approach is based on a new data structure for a variant of the dynamic convolution problem $(u, v, p)$, where we want to answer $u\cdot v$ dynamically. Our data structure supports updating $u$ at position $p$, incrementing and decrementing $p$ and rotating $v$ by $1$. We present a data structure that supports $n$ operations in $O(n\log^2{n})$ time and $O(n)$ space. Moreover, the same approach can be used to compute the Shapley values for the mean volume of the convex hull projection onto a uniformly random $(d - 2)$-subspace in $O(n^d\log^2{n})$ time and $O(n)$ space for a point set in $d$-dimensional space ($d \geq 3$)

The Structure of the Outer Halo of the Galaxy and its Relationship to Nearby Large-Scale Structure

We present evidence to support an earlier indication that the Galaxy is embedded in an extended, highly inclined, triaxial halo outlined by the spatial distribution of companion galaxies to the Milky Way. Signatures of this spatial distribution are seen in 1) the angular variation of the radial-velocity dispersion of the companion galaxies, 2) the spatial distribution of the M~31 sub-group of galaxies, 3) the spatial distribution of the isolated, mainly dwarf irregular, galaxies of the Local Group, 4) the velocity anisotropy quadrupole of a sub-group of high-velocity clouds, and 5) the spatial distribution of galaxies in the Coma-Sculptor cloud. Tidal effects of M~31 and surrounding galaxies on the Galaxy are not strong enough to have affected the observed structure. We conclude that this distribution is a reflection of initial conditions. A simple galaxy formation scenario is proposed which ties together the results found here with those of Holmberg (1969) and Zaritsky et al. (1997) on the peculiar distribution of satellites around a large sample of spiral galaxies.Comment: Accepted for publication in the Astron J., March 2000, 12 pages with 1 figur

Photometric Properties of Lyman-break Galaxies at z=3 in Cosmological SPH Simulations

We study the photometric properties of Lyman-break galaxies (LBGs) formed by redshift z=3 in a set of large cosmological smoothed-particle hydrodynamics simulations of the Lambda cold dark matter (CDM) model. Our numerical simulations include radiative cooling and heating with a uniform UV background, star formation, supernova feedback, and a phenomenological model for galactic winds. Analysing a series of simulations of varying boxsize and particle number allows us to isolate the impact of numerical resolution on our results. We compute spectra of simulated galaxies using a population synthesis model, and derive colours and luminosity functions of galaxies at z=3 after applying local dust extinction and absorption by the intergalactic medium (IGM). We find that the simulated galaxies have U-G and G-R colours consistent with observations, provided that intervening absorption by the IGM is applied. The observed properties of LBGs, including their number density, colours, and luminosity functions, can be explained if LBGs are identified with the most massive galaxies at z=3, having typical stellar mass of M_{star} ~ 1e10 Msun/h, a conclusion broadly consistent with earlier studies based on hydrodynamic simulations of the Lamda CDM model. We also find that most simulated LBGs were continuously forming stars at a high rate for more than one Gyr up until z=3, but with numerous starbursts lying on top of the continuous component. Interestingly, our simulations suggest that more than 50% of the total stellar mass and star formation rate in the Universe are accounted for by galaxies that are not detected in the current generation of LBG surveys.Comment: 12 pages, 8 figures, Error in AB magnitude calculation corrected. Figures in the original published version in MNRAS contain error except Fig.5 & 6, but the basic conclusions are unchanged. Higher resolution version available at http://cfa-www.harvard.edu/~knagamine/lbg.ps.g

Escape of Ionizing Radiation from High Redshift Galaxies

We model the escape of ionizing radiation from high-redshift galaxies using high-resolution Adaptive Mesh Refinement N-body + hydrodynamics simulations. Our simulations include time-dependent and spatially-resolved transfer of ionizing radiation in three dimensions, including effects of dust absorption. For galaxies of total mass M > 10^11 Msun and star formation rates SFR ~ 1-5 Msun/yr, we find angular averaged escape fractions of 0.01-0.03 over the entire redshift interval studied (3<z<9). In addition, we find that the escape fraction varies by more than an order of magnitude along different lines-of-sight within individual galaxies, from the largest values near galactic poles to the smallest along the galactic disk. The escape fraction declines steeply at lower masses and SFR. We show that the low values of escape fractions are due to a small fraction of young stars located just outside the edge of HI disk. We compare our predicted escape fraction of ionizing photons with previous results, and find a general agreement with both other simulation results and available direct detection measurements at z ~ 3. We also compare our simulations with a novel method to estimate the escape fraction in galaxies from the observed distribution of neutral hydrogen column densities along the lines of sights to long duration gamma-ray bursts. Using this method we find escape fractions of the GRB host galaxies of 2-3%, consistent with our theoretical predictions. [abridged]Comment: submitted to Ap

Axiomatic Attribution for Multilinear Functions

We study the attribution problem, that is, the problem of attributing a change in the value of a characteristic function to its independent variables. We make three contributions. First, we propose a formalization of the problem based on a standard cost sharing model. Second, we show that there is a unique attribution method that satisfies Dummy, Additivity, Conditional Nonnegativity, Affine Scale Invariance, and Anonymity for all characteristic functions that are the sum of a multilinear function and an additive function. We term this the Aumann-Shapley-Shubik method. Conversely, we show that such a uniqueness result does not hold for characteristic functions outside this class. Third, we study multilinear characteristic functions in detail; we describe a computationally efficient implementation of the Aumann-Shapley-Shubik method and discuss practical applications to pay-per-click advertising and portfolio analysis.Comment: 21 pages, 2 figures, updated version for EC '1

Cosmic rays can drive strong outflows from gas-rich high-redshift disk galaxies

We present simulations of the magnetized interstellar medium (ISM) in models of massive star forming (40 Msun / yr) disk galaxies with high gas surface densities (~100 Msun / pc^2) similar to observed star forming high-redshift disks. We assume that type II supernovae deposit 10 per cent of their energy into the ISM as cosmic rays and neglect the additional deposition of thermal energy or momentum. With a typical Galactic diffusion coefficient for CRs (3e28 cm^2 / s) we demonstrate that this process alone can trigger the local formation of a strong low density galactic wind maintaining vertically open field lines. Driven by the additional pressure gradient of the relativistic fluid the wind speed can exceed 1000 km/s, much higher than the escape velocity of the galaxy. The global mass loading, i.e. the ratio of the gas mass leaving the galactic disk in a wind to the star formation rate becomes of order unity once the system has settled into an equilibrium. We conclude that relativistic particles accelerated in supernova remnants alone provide a natural and efficient mechanism to trigger winds similar to observed mass-loaded galactic winds in high-redshift galaxies. These winds also help explaining the low efficiencies for the conversion of gas into stars in galaxies as well as the early enrichment of the intergalactic medium with metals. This mechanism can be at least of similar importance than the traditionally considered momentum feedback from massive stars and thermal and kinetic feedback from supernova explosions.Comment: 5 pages, 5 figures, accepted in ApJL; corrected titl