63 research outputs found
A complete characterization of plateaued Boolean functions in terms of their Cayley graphs
In this paper we find a complete characterization of plateaued Boolean
functions in terms of the associated Cayley graphs. Precisely, we show that a
Boolean function is -plateaued (of weight ) if and only
if the associated Cayley graph is a complete bipartite graph between the
support of and its complement (hence the graph is strongly regular of
parameters ). Moreover, a Boolean function is
-plateaued (of weight ) if and only if the associated
Cayley graph is strongly -walk-regular (and also strongly
-walk-regular, for all odd ) with some explicitly given
parameters.Comment: 7 pages, 1 figure, Proceedings of Africacrypt 201
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
The spectra of lifted digraphs
We present a method to derive the complete spectrum of the lift \mathrm{\Gamma\alpha} of a base digraph \mathrm{\Gamma}, with voltage assignment α on a (finite) group . The method is based on assigning to \mathrm{\Gamma} a quotient-like matrix whose entries are elements of the group algebra \mathds{C}[], which fully represents \mathrm{\Gamma\alpha}. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovász and Babai concerning the spectra of Cayley digraphs
Simple, Fast and Accurate Implementation of the Diffusion Approximation Algorithm for Stochastic Ion Channels with Multiple States
The phenomena that emerge from the interaction of the stochastic opening and
closing of ion channels (channel noise) with the non-linear neural dynamics are
essential to our understanding of the operation of the nervous system. The
effects that channel noise can have on neural dynamics are generally studied
using numerical simulations of stochastic models. Algorithms based on discrete
Markov Chains (MC) seem to be the most reliable and trustworthy, but even
optimized algorithms come with a non-negligible computational cost. Diffusion
Approximation (DA) methods use Stochastic Differential Equations (SDE) to
approximate the behavior of a number of MCs, considerably speeding up
simulation times. However, model comparisons have suggested that DA methods did
not lead to the same results as in MC modeling in terms of channel noise
statistics and effects on excitability. Recently, it was shown that the
difference arose because MCs were modeled with coupled activation subunits,
while the DA was modeled using uncoupled activation subunits. Implementations
of DA with coupled subunits, in the context of a specific kinetic scheme,
yielded similar results to MC. However, it remained unclear how to generalize
these implementations to different kinetic schemes, or whether they were faster
than MC algorithms. Additionally, a steady state approximation was used for the
stochastic terms, which, as we show here, can introduce significant
inaccuracies. We derived the SDE explicitly for any given ion channel kinetic
scheme. The resulting generic equations were surprisingly simple and
interpretable - allowing an easy and efficient DA implementation. The algorithm
was tested in a voltage clamp simulation and in two different current clamp
simulations, yielding the same results as MC modeling. Also, the simulation
efficiency of this DA method demonstrated considerable superiority over MC
methods.Comment: 32 text pages, 10 figures, 1 supplementary text + figur
Behavioral Modernity and the Cultural Transmission of Structured Information: The Semantic Axelrod Model
Cultural transmission models are coming to the fore in explaining increases
in the Paleolithic toolkit richness and diversity. During the later
Paleolithic, technologies increase not only in terms of diversity but also in
their complexity and interdependence. As Mesoudi and O'Brien (2008) have shown,
selection broadly favors social learning of information that is hierarchical
and structured, and multiple studies have demonstrated that teaching within a
social learning environment can increase fitness. We believe that teaching also
provides the scaffolding for transmission of more complex cultural traits.
Here, we introduce an extension of the Axelrod (1997} model of cultural
differentiation in which traits have prerequisite relationships, and where
social learning is dependent upon the ordering of those prerequisites. We
examine the resulting structure of cultural repertoires as learning
environments range from largely unstructured imitation, to structured teaching
of necessary prerequisites, and we find that in combination with individual
learning and innovation, high probabilities of teaching prerequisites leads to
richer cultural repertoires. Our results point to ways in which we can build
more comprehensive explanations of the archaeological record of the Paleolithic
as well as other cases of technological change.Comment: 24 pages, 7 figures. Submitted to "Learning Strategies and Cultural
Evolution during the Paleolithic", edited by Kenichi Aoki and Alex Mesoudi,
and presented at the 79th Annual Meeting of the Society for American
Archaeology, Austin TX. Revised 5/14/1
Cycles in graphs
This volume deals with a variety of problems involving cycles in graphs and circuits in digraphs. Leading researchers in this area present here 3 survey papers and 42 papers containing new results. There is also a collection of unsolved problems
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