697 research outputs found
Preventing Unraveling in Social Networks Gets Harder
The behavior of users in social networks is often observed to be affected by
the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp}
introduced a formal mathematical model for user engagement in social networks
where each individual derives a benefit proportional to the number of its
friends which are engaged. Given a threshold degree the equilibrium for
this model is a maximal subgraph whose minimum degree is . However the
dropping out of individuals with degrees less than might lead to a
cascading effect of iterated withdrawals such that the size of equilibrium
subgraph becomes very small. To overcome this some special vertices called
"anchors" are introduced: these vertices need not have large degree. Bhawalkar
et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored -Core}
problem: Given a graph and integers and do there exist a set of
vertices such that and
every vertex has degree at least is the induced
subgraph . They showed that the problem is NP-hard for and gave
some inapproximability and fixed-parameter intractability results. In this
paper we give improved hardness results for this problem. In particular we show
that the \textsc{Anchored -Core} problem is W[1]-hard parameterized by ,
even for . This improves the result of Bhawalkar et al.
\cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by ) as our
parameter is always bigger since . Then we answer a question of
Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored
-Core} problem remains NP-hard on planar graphs for all , even if
the maximum degree of the graph is . Finally we show that the problem is
FPT on planar graphs parameterized by for all .Comment: To appear in AAAI 201
Parameterized Complexity of the Anchored k-Core Problem for Directed Graphs
We consider the Directed Anchored k-Core problem, where the task is for a given directed graph G and integers b, k and p, to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (the anchors) of H have in-degree at least k. For undirected graphs, this problem was introduced by Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012]. We undertake a
systematic analysis of the computational complexity of Directed Anchored k-Core and show that:
- The decision version of the problem is NP-complete for every k>=1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k+2.
- The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k=1, and W[1]-hard for k>=2.
- When the maximum degree of the graph is at most Delta, the problem is FPT parameterized by p+Delta if k>=Delta/2
A fit to the simultaneous broadband spectrum of Cygnus X-1 using the transition disk model
We have used the transition disk model to fit the simultaneous broad band
( keV) spectrum of Cygnus X-1 from OSSE and Ginga observations. In this
model, the spectrum is produced by saturated Comptonization within the inner
region of the accretion disk, where the temperature varies rapidly with radius.
In an earlier attempt, we demonstrated the viability of this model by fitting
the data from EXOSAT, XMPC balloon and OSSE observations, though these were not
made simultaneously. Since the source is known to be variable, however, the
results of this fit were not conclusive. In addition, since only once set of
observations was used, the good agreement with the data could have been a
chance occurrence. Here, we improve considerably upon our earlier analysis by
considering four sets of simultaneous observations of Cygnus X-1, using an
empirical model to obtain the disk temperature profile. The vertical structure
is then obtained using this profile and we show that the analysis is self-
consistent. We demonstrate conclusively that the transition disk spectrum is a
better fit to the observations than that predicted by the soft photon
Comptonization model. Since the temperature profile is obtained by fitting the
data, the unknown viscosity mechanism need not be specified. The disk structure
can then be used to infer the viscosity parameter , which appears to
vary with radius and luminosity. This behavior can be understood if
depends intrinsically on the local parameters such as density, height and
temperature. However, due to uncertainties in the radiative transfer,
quantitative statements regarding the variation of cannot yet be made.Comment: 8 figures. uses aasms4.sty, accepted by ApJ (Mar 98
Fourier Neural Operator Networks: A Fast and General Solver for the Photoacoustic Wave Equation
Simulation tools for photoacoustic wave propagation have played a key role in
advancing photoacoustic imaging by providing quantitative and qualitative
insights into parameters affecting image quality. Classical methods for
numerically solving the photoacoustic wave equation relies on a fine
discretization of space and can become computationally expensive for large
computational grids. In this work, we apply Fourier Neural Operator (FNO)
networks as a fast data-driven deep learning method for solving the 2D
photoacoustic wave equation in a homogeneous medium. Comparisons between the
FNO network and pseudo-spectral time domain approach demonstrated that the FNO
network generated comparable simulations with small errors and was several
orders of magnitude faster. Moreover, the FNO network was generalizable and can
generate simulations not observed in the training data
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