305,291 research outputs found
Random Graphs with Hidden Color
We propose and investigate a unifying class of sparse random graph models,
based on a hidden coloring of edge-vertex incidences, extending an existing
approach, Random graphs with a given degree distribution, in a way that admits
a nontrivial correlation structure in the resulting graphs.
The approach unifies a number of existing random graph ensembles within a
common general formalism, and allows for the analytic calculation of observable
graph characteristics.
In particular, generating function techniques are used to derive the size
distribution of connected components (clusters) as well as the location of the
percolation threshold where a giant component appears.Comment: 4 pages, no figures, RevTe
A Sard theorem for Tame Set-Valued mappings
If is a set-valued mapping from into with closed graph,
then is a critical value of if for some with ,
is not metrically regular at . We prove that the set of critical
values of a set-valued mapping whose graph is a definable (tame) set in an
-minimal structure containing additions and multiplications is a set of
dimension not greater than (resp. a porous set). As a corollary of this
result we get that the collection of asymptotically critical values of a
semialgebraic set-valued mapping has dimension not greater than , thus
extending to such mappings a corresponding result by Kurdyka-Orro-Simon for
semialgebraic mappings. We also give an independent proof of the fact
that a definable continuous real-valued function is constant on components of
the set of its subdifferentiably critical points, thus extending to all
definable functions a recent result of Bolte-Daniilidis-Lewis for globally
subanalytic functions.Comment: 23
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
Toward An Uncertainty Principle For Weighted Graphs
International audienceThe uncertainty principle states that a signal cannot be localized both in time and frequency. With the aim of extending this result to signals on graphs, Agaskar & Lu introduce notions of graph and spectral spreads. They show that a graph uncertainty principle holds for some families of unweighted graphs. This principle states that a signal cannot be simultaneously localized both in graph and spectral domains. In this paper, we aim to extend their work to weighted graphs. We show that a naive extension of their definitions leads to inconsistent results such as discontinuity of the graph spread when regarded as a function of the graph structure. To circumvent this problem, we propose another definition of graph spread that relies on an inverse similarity matrix. We also discuss the choice of the distance function that appears in this definition. Finally, we compute and plot uncertainty curves for families of weighted graphs
Separation conditions for iterated function systems with overlaps on Riemannian manifolds
We formulate the weak separation condition and the finite type condition for
conformal iterated function systems on Riemannian manifolds with nonnegative
Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in
[Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the
Hausdorff dimension of a self-similar set defined by an iterated function
system satisfying the finite type condition, generalizing a corresponding
result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv.
Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a formula
for the Hausdorff dimension of a graph self-similar set generated by a
graph-directed iterated function system satisfying the graph finite type
condition, extending a result by Ngai \textit{et al.} in [Nonlinearity 23
(2010), 2333--2350]
Acylindricity of the action of right-angled Artin groups on extension graphs
The action of a right-angled Artin group on its extension graph is known to
be acylindrical because the cardinality of the so-called -quasi-stabilizer
of a pair of distant points is bounded above by a function of . The known
upper bound of the cardinality is an exponential function of . In this paper
we show that the -quasi-stabilizer is a subset of a cyclic group and its
cardinality is bounded above by a linear function of . This is done by
exploring lattice theoretic properties of group elements, studying prefixes of
powers and extending the uniqueness of quasi-roots from word length to star
length. We also improve the known lower bound for the minimal asymptotic
translation length of a right angled Artin group on its extension graph
Monge extensions of cooperation and communication structures
Cooperation structures without any {\it a priori} assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for mar\-ginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the LovĂĄsz extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson's graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley's convexity model for classical cooperative games.
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