1,773 research outputs found
Three-phase point in a binary hard-core lattice model?
Using Monte Carlo simulation, Van Duijneveldt and Lekkerkerker [Phys. Rev.
Lett. 71, 4264 (1993)] found gas-liquid-solid behaviour in a simple
two-dimensional lattice model with two types of hard particles. The same model
is studied here by means of numerical transfer matrix calculations, focusing on
the finite size scaling of the gaps between the largest few eigenvalues. No
evidence for a gas-liquid transition is found. We discuss the relation of the
model with a solvable RSOS model of which the states obey the same exclusion
rules. Finally, a detailed analysis of the relation with the dilute three-state
Potts model strongly supports the tricritical point rather than a three-phase
point.Comment: 17 pages, LaTeX2e, 13 EPS figure
Causal Fourier Analysis on Directed Acyclic Graphs and Posets
We present a novel form of Fourier analysis, and associated signal processing
concepts, for signals (or data) indexed by edge-weighted directed acyclic
graphs (DAGs). This means that our Fourier basis yields an eigendecomposition
of a suitable notion of shift and convolution operators that we define. DAGs
are the common model to capture causal relationships between data values and in
this case our proposed Fourier analysis relates data with its causes under a
linearity assumption that we define. The definition of the Fourier transform
requires the transitive closure of the weighted DAG for which several forms are
possible depending on the interpretation of the edge weights. Examples include
level of influence, distance, or pollution distribution. Our framework is
different from prior GSP: it is specific to DAGs and leverages, and extends,
the classical theory of Moebius inversion from combinatorics. For a
prototypical application we consider DAGs modeling dynamic networks in which
edges change over time. Specifically, we model the spread of an infection on
such a DAG obtained from real-world contact tracing data and learn the
infection signal from samples assuming sparsity in the Fourier domain.Comment: 13 pages, 11 figure
SciTech News Volume 71, No. 1 (2017)
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Lie Group Algebra Convolutional Filters
In this paper we propose a framework to leverage Lie group symmetries on
arbitrary spaces exploiting \textit{algebraic signal processing} (ASP). We show
that traditional group convolutions are one particular instantiation of a more
general Lie group algebra homomorphism associated to an algebraic signal model
rooted in the Lie group algebra for given Lie group . Exploiting
this fact, we decouple the discretization of the Lie group convolution
elucidating two separate sampling instances: the filter and the signal. To
discretize the filters, we exploit the exponential map that links a Lie group
with its associated Lie algebra. We show that the discrete Lie group filter
learned from the data determines a unique filter in , and we show how
this uniqueness of representation is defined by the bandwidth of the filter
given a spectral representation. We also derive error bounds for the
approximations of the filters in with respect to its learned
discrete representations. The proposed framework allows the processing of
signals on spaces of arbitrary dimension and where the actions of some elements
of the group are not necessarily well defined. Finally, we show that multigraph
convolutional signal models come as the natural discrete realization of Lie
group signal processing models, and we use this connection to establish
stability results for Lie group algebra filters. To evaluate numerically our
results, we build neural networks with these filters and we apply them in
multiple datasets, including a knot classification problem
Learning in Networks: a survey
This paper presents a survey of research on learning with a special focus on the structure of interaction between individual entities. The structure is formally modelled as a network: the nodes of the network are individuals while the arcs admit a variety of interpretations (ranging from information channels to social and economic ties). I first examine the nature of learning about optimal actions for a given network architecture. I then discuss learning about optimal links and actions in evolving networks.
How can innovation economics benefit from complex network analysis?
There is a deficit in economics of theories and empirical data on complex networks, though mathematicians, physicists, biologists, computer scientists, and sociologists are actively engaged in their study. This paper offers a focused review of prominent concepts in contemporary thinking in network research that may motivate further theoretical research and stimulate interest of economists. Possible avenues for modelling innovation, considered the driving force behind economic change, have been explored. A transition is needed from the analysis in economics of the transaction to the explicit examination of market structure and how it processes, or is processed by, innovation.Network; statistics; economy; innovation; modelling
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