906 research outputs found
RiffleScrambler - a memory-hard password storing function
We introduce RiffleScrambler: a new family of directed acyclic graphs and a
corresponding data-independent memory hard function with password independent
memory access. We prove its memory hardness in the random oracle model.
RiffleScrambler is similar to Catena -- updates of hashes are determined by a
graph (bit-reversal or double-butterfly graph in Catena). The advantage of the
RiffleScrambler over Catena is that the underlying graphs are not predefined
but are generated per salt, as in Balloon Hashing. Such an approach leads to
higher immunity against practical parallel attacks. RiffleScrambler offers
better efficiency than Balloon Hashing since the in-degree of the underlying
graph is equal to 3 (and is much smaller than in Ballon Hashing). At the same
time, because the underlying graph is an instance of a Superconcentrator, our
construction achieves the same time-memory trade-offs.Comment: Accepted to ESORICS 201
Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model
In the mean field (or random link) model there are points and inter-point
distances are independent random variables. For and in the
limit, let (maximum number of steps
in a path whose average step-length is ). The function
is analogous to the percolation function in percolation theory:
there is a critical value at which becomes
non-zero, and (presumably) a scaling exponent in the sense
. Recently developed probabilistic
methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi)
provides a simple albeit non-rigorous way of writing down such functions in
terms of solutions of fixed-point equations for probability distributions.
Solving numerically gives convincing evidence that . A parallel
study with trees instead of paths gives scaling exponent . The new
exponents coincide with those found in a different context (comparing optimal
and near-optimal solutions of mean-field TSP and MST) and reinforce the
suggestion that these scaling exponents determine universality classes for
optimization problems on random points.Comment: 19 page
Routed Planar Networks
Modeling a road network as a planar graph seems very natural. However, in studying continuum limits of such networks it is useful to take {\em routes} rather than {\em edges} as primitives. This article is intended to introduce the relevant (discrete setting) notion of {\em routed network} to graph theorists. We give a naive classification of all 71 topologically different such networks on 4 leaves, and pose a variety of challenging research questions
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
Optimal spatial transportation networks where link-costs are sublinear in link-capacity
Consider designing a transportation network on vertices in the plane,
with traffic demand uniform over all source-destination pairs. Suppose the cost
of a link of length and capacity scales as for fixed
. Under appropriate standardization, the cost of the minimum cost
Gilbert network grows essentially as , where on and on . This quantity is an upper bound in
the worst case (of vertex positions), and a lower bound under mild regularity
assumptions. Essentially the same bounds hold if we constrain the network to be
efficient in the sense that average route-length is only times
average straight line length. The transition at corresponds to
the dominant cost contribution changing from short links to long links. The
upper bounds arise in the following type of hierarchical networks, which are
therefore optimal in an order of magnitude sense. On the large scale, use a
sparse Poisson line process to provide long-range links. On the medium scale,
use hierachical routing on the square lattice. On the small scale, link
vertices directly to medium-grid points. We discuss one of many possible
variant models, in which links also have a designed maximum speed and the
cost becomes .Comment: 13 page
Spectral coarse graining for random walk in bipartite networks
Many real-world networks display a natural bipartite structure, while
analyzing or visualizing large bipartite networks is one of the most
challenges. As a result, it is necessary to reduce the complexity of large
bipartite systems and preserve the functionality at the same time. We observe,
however, the existing coarse graining methods for binary networks fail to work
in the bipartite networks. In this paper, we use the spectral analysis to
design a coarse graining scheme specifically for bipartite networks and keep
their random walk properties unchanged. Numerical analysis on artificial and
real-world bipartite networks indicates that our coarse graining scheme could
obtain much smaller networks from large ones, keeping most of the relevant
spectral properties. Finally, we further validate the coarse graining method by
directly comparing the mean first passage time between the original network and
the reduced one.Comment: 7 pages, 3 figure
Feller property and infinitesimal generator of the exploration process
We consider the exploration process associated to the continuous random tree
(CRT) built using a Levy process with no negative jumps. This process has been
studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is
a useful tool to study CRT as well as super-Brownian motion with general
branching mechanism. In this paper we prove this process is Feller, and we
compute its infinitesimal generator on exponential functionals and give the
corresponding martingale
Dynamic critical exponents of Swendsen-Wang and Wolff algorithms by nonequilibrium relaxation
With a nonequilibrium relaxation method, we calculate the dynamic critical
exponent z of the two-dimensional Ising model for the Swendsen-Wang and Wolff
algorithms. We examine dynamic relaxation processes following a quench from a
disordered or an ordered initial state to the critical temperature T_c, and
measure the exponential relaxation time of the system energy. For the
Swendsen-Wang algorithm with an ordered or a disordered initial state, and for
the Wolff algorithm with an ordered initial state, the exponential relaxation
time fits well to a logarithmic size dependence up to a lattice size L=8192.
For the Wolff algorithm with a disordered initial state, we obtain an effective
dynamic exponent z_exp=1.19(2) up to L=2048. For comparison, we also compute
the effective dynamic exponents through the integrated correlation times. In
addition, an exact result of the Swendsen-Wang dynamic spectrum of a
one-dimension Ising chain is derived.Comment: 13 pages, 6 figure
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
Empires and Percolation: Stochastic Merging of Adjacent Regions
We introduce a stochastic model in which adjacent planar regions merge
stochastically at some rate , and observe analogies with the
well-studied topics of mean-field coagulation and of bond percolation. Do
infinite regions appear in finite time? We give a simple condition on
for this {\em hegemony} property to hold, and another simple condition for it
to not hold, but there is a large gap between these conditions, which includes
the case . For this case, a non-rigorous analytic
argument and simulations suggest hegemony.Comment: 13 page
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