2,528 research outputs found
The infinite random simplicial complex
We study the Fraisse limit of the class of all finite simplicial complexes.
Whilst the natural model-theoretic setting for this class uses an infinite
language, a range of results associated with Fraisse limits of structures for
finite languages carry across to this important example. We introduce the
notion of a local class, with the class of finite simplicial complexes as an
archetypal example, and in this general context prove the existence of a 0-1
law and other basic model-theoretic results. Constraining to the case where all
relations are symmetric, we show that every direct limit of finite groups, and
every metrizable profinite group, appears as a subgroup of the automorphism
group of the Fraisse limit. Finally, for the specific case of simplicial
complexes, we show that the geometric realisation is topologically surprisingly
simple: despite the combinatorial complexity of the Fraisse limit, its
geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page
Limits of Ordered Graphs and their Applications
The emerging theory of graph limits exhibits an analytic perspective on
graphs, showing that many important concepts and tools in graph theory and its
applications can be described more naturally (and sometimes proved more easily)
in analytic language. We extend the theory of graph limits to the ordered
setting, presenting a limit object for dense vertex-ordered graphs, which we
call an \emph{orderon}. As a special case, this yields limit objects for
matrices whose rows and columns are ordered, and for dynamic graphs that expand
(via vertex insertions) over time. Along the way, we devise an ordered
locality-preserving variant of the cut distance between ordered graphs, showing
that two graphs are close with respect to this distance if and only if they are
similar in terms of their ordered subgraph frequencies. We show that the space
of orderons is compact with respect to this distance notion, which is key to a
successful analysis of combinatorial objects through their limits.
We derive several applications of the ordered limit theory in extremal
combinatorics, sampling, and property testing in ordered graphs. In particular,
we prove a new ordered analogue of the well-known result by Alon and Stav
[RS\&A'08] on the furthest graph from a hereditary property; this is the first
known result of this type in the ordered setting. Unlike the unordered regime,
here the random graph model with an ordering over the vertices is
\emph{not} always asymptotically the furthest from the property for some .
However, using our ordered limit theory, we show that random graphs generated
by a stochastic block model, where the blocks are consecutive in the vertex
ordering, are (approximately) the furthest. Additionally, we describe an
alternative analytic proof of the ordered graph removal lemma [Alon et al.,
FOCS'17].Comment: Added a new application: An Alon-Stav type result on the furthest
ordered graph from a hereditary property; Fixed and extended proof sketch of
the removal lemma applicatio
The Widom-Rowlinson Model on the Delaunay Graph
We establish phase transitions for continuum Delaunay multi-type particle
systems (continuum Potts or Widom-Rowlinson models) with a repulsive
interaction between particles of different types. Our interaction potential
depends solely on the length of the Delaunay edges. We show that a phase
transition occurs for sufficiently large activities and for sufficiently large
potential parameter proving an old conjecture of Lebowitz and Lieb extended to
the Delaunay structure. Our approach involves a Delaunay random-cluster
representation analogous to the Fortuin-Kasteleyn representation of the Potts
model. The phase transition manifests itself in the mixed site-bond percolation
of the corresponding random-cluster model. Our proofs rely mainly on geometric
properties of Delaunay tessellations in and on recent studies
[DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is
a uniform bound on the number of connected components in the Delaunay graph
which provides a novel approach to Delaunay Widom Rowlinson models based on
purely geometric arguments. The interaction potential ensures that shorter
Delaunay edges are more likely to be open and thus offsets the possibility of
having an unbounded number of connected components.Comment: 36 pages, 11 figure
Every Minor-Closed Property of Sparse Graphs is Testable
Suppose is a graph with degrees bounded by , and one needs to remove
more than of its edges in order to make it planar. We show that in
this case the statistics of local neighborhoods around vertices of is far
from the statistics of local neighborhoods around vertices of any planar graph
with the same degree bound. In fact, a similar result is proved for any
minor-closed property of bounded degree graphs.
As an immediate corollary of the above result we infer that many well studied
graph properties, like being planar, outer-planar, series-parallel, bounded
genus, bounded tree-width and several others, are testable with a constant
number of queries, where the constant may depend on and , but not
on the graph size. None of these properties was previously known to be testable
even with queries
Topics in social network analysis and network science
This chapter introduces statistical methods used in the analysis of social
networks and in the rapidly evolving parallel-field of network science.
Although several instances of social network analysis in health services
research have appeared recently, the majority involve only the most basic
methods and thus scratch the surface of what might be accomplished.
Cutting-edge methods using relevant examples and illustrations in health
services research are provided
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