1,703 research outputs found
Common adversaries form alliances: modelling complex networks via anti-transitivity
Anti-transitivity captures the notion that enemies of enemies are friends,
and arises naturally in the study of adversaries in social networks and in the
study of conflicting nation states or organizations. We present a simplified,
evolutionary model for anti-transitivity influencing link formation in complex
networks, and analyze the model's network dynamics. The Iterated Local
Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step,
and joins anti-clones to the parent node's non-neighbor set. The graphs
generated by ILAT exhibit familiar properties of complex networks such as
densification, short distances (bounded by absolute constants), and bad
spectral expansion. We determine the cop and domination number for graphs
generated by ILAT, and finish with an analysis of their clustering
coefficients. We interpret these results within the context of real-world
complex networks and present open problems
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index
{Let be a -dimensional fractional Brownian motion
with Hurst index , or more generally a Gaussian process whose paths
have the same local regularity. Defining properly iterated integrals of is
a difficult task because of the low H\"older regularity index of its paths. Yet
rough path theory shows it is the key to the construction of a stochastic
calculus with respect to , or to solving differential equations driven by
.
We intend to show in a series of papers how to desingularize iterated
integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure
defined by a limit in law procedure. Convergence is proved by using "standard"
tools of constructive field theory, in particular cluster expansions and
renormalization. These powerful tools allow optimal estimates, and call for an
extension of Gaussian tools such as for instance the Malliavin calculus.
After a first introductory paper \cite{MagUnt1}, this one concentrates on the
details of the constructive proof of convergence for second-order iterated
integrals, also known as L\'evy area
Shearer's point process, the hard-sphere model and a continuum Lov\'asz Local Lemma
A point process is R-dependent, if it behaves independently beyond the
minimum distance R. This work investigates uniform positive lower bounds on the
avoidance functions of R-dependent simple point processes with a common
intensity. Intensities with such bounds are described by the existence of
Shearer's point process, the unique R-dependent and R-hard-core point process
with a given intensity. This work also presents several extensions of the
Lov\'asz Local Lemma, a sufficient condition on the intensity and R to
guarantee the existence of Shearer's point process and exponential lower
bounds. Shearer's point process shares combinatorial structure with the
hard-sphere model with radius R, the unique R-hard-core Markov point process.
Bounds from the Lov\'asz Local Lemma convert into lower bounds on the radius of
convergence of a high-temperature cluster expansion of the hard-sphere model.
This recovers a classic result of Ruelle on the uniqueness of the Gibbs measure
of the hard-sphere model via an inductive approach \`a la Dobrushin
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