10,629 research outputs found
Symplectic fillings of Seifert fibered spaces
We give finiteness results and some classifications up to diffeomorphism of
minimal strong symplectic fillings of Seifert fibered spaces over S^2
satisfying certain conditions, with a fixed natural contact structure. In some
cases we can prove that all symplectic fillings are obtained by rational
blow-downs of a plumbing of spheres. In other cases, we produce new manifolds
with convex symplectic boundary, thus yielding new cut-and-paste operations on
symplectic manifolds containing certain configurations of symplectic spheres.Comment: 39 pages, 21 figures, v2 a few minor corrections and citations, v3
added clarifications in the proof of Lemma 2.8, plus some minor change
Neural Embeddings of Graphs in Hyperbolic Space
Neural embeddings have been used with great success in Natural Language
Processing (NLP). They provide compact representations that encapsulate word
similarity and attain state-of-the-art performance in a range of linguistic
tasks. The success of neural embeddings has prompted significant amounts of
research into applications in domains other than language. One such domain is
graph-structured data, where embeddings of vertices can be learned that
encapsulate vertex similarity and improve performance on tasks including edge
prediction and vertex labelling. For both NLP and graph based tasks, embeddings
have been learned in high-dimensional Euclidean spaces. However, recent work
has shown that the appropriate isometric space for embedding complex networks
is not the flat Euclidean space, but negatively curved, hyperbolic space. We
present a new concept that exploits these recent insights and propose learning
neural embeddings of graphs in hyperbolic space. We provide experimental
evidence that embedding graphs in their natural geometry significantly improves
performance on downstream tasks for several real-world public datasets.Comment: 7 pages, 5 figure
Algorithms to measure diversity and clustering in social networks through dot product graphs.
Social networks are often analyzed through a graph model of the network. The dot product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector a v represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a u · a v  ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph..
We consider two measures for diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is measured through the size of the largest independent set of the graph, and clustering is measured through the size of the largest clique. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-complete for d ≥ 3. We show that the clustering problem is polynomial-time solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs.
We also consider the situation when two individuals are connected if their preferences are not opposite. This leads to a variant of the standard dot product graph model by taking the threshold t to be zero. We prove in this case that the diversity problem is polynomial-time solvable for any fixed d
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
sl(3) link homology
We define a bigraded homology theory whose Euler characteristic is the
quantum sl(3) link invariant.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-45.abs.htm
Realizations of self branched coverings of the 2-sphere
For a degree d self branched covering of the 2-sphere, a notable
combinatorial invariant is an integer partition of 2d -- 2, consisting of the
multiplicities of the critical points. A finer invariant is the so called
Hurwitz passport. The realization problem of Hurwitz passports remain largely
open till today. In this article, we introduce two different types of finer
invariants: a bipartite map and an incident matrix. We then settle completely
their realization problem by showing that a map, or a matrix, is realized by a
branched covering if and only if it satisfies a certain balanced condition. A
variant of the bipartite map approach was initiated by W. Thurston. Our results
shed some new lights to the Hurwitz passport problem
Adversarial Sets for Regularising Neural Link Predictors
In adversarial training, a set of models learn together by pursuing competing
goals, usually defined on single data instances. However, in relational
learning and other non-i.i.d domains, goals can also be defined over sets of
instances. For example, a link predictor for the is-a relation needs to be
consistent with the transitivity property: if is-a(x_1, x_2) and is-a(x_2, x_3)
hold, is-a(x_1, x_3) needs to hold as well. Here we use such assumptions for
deriving an inconsistency loss, measuring the degree to which the model
violates the assumptions on an adversarially-generated set of examples. The
training objective is defined as a minimax problem, where an adversary finds
the most offending adversarial examples by maximising the inconsistency loss,
and the model is trained by jointly minimising a supervised loss and the
inconsistency loss on the adversarial examples. This yields the first method
that can use function-free Horn clauses (as in Datalog) to regularise any
neural link predictor, with complexity independent of the domain size. We show
that for several link prediction models, the optimisation problem faced by the
adversary has efficient closed-form solutions. Experiments on link prediction
benchmarks indicate that given suitable prior knowledge, our method can
significantly improve neural link predictors on all relevant metrics.Comment: Proceedings of the 33rd Conference on Uncertainty in Artificial
Intelligence (UAI), 201
On the Implicit Graph Conjecture
The implicit graph conjecture states that every sufficiently small,
hereditary graph class has a labeling scheme with a polynomial-time computable
label decoder. We approach this conjecture by investigating classes of label
decoders defined in terms of complexity classes such as P and EXP. For
instance, GP denotes the class of graph classes that have a labeling scheme
with a polynomial-time computable label decoder. Until now it was not even
known whether GP is a strict subset of GR. We show that this is indeed the case
and reveal a strict hierarchy akin to classical complexity. We also show that
classes such as GP can be characterized in terms of graph parameters. This
could mean that certain algorithmic problems are feasible on every graph class
in GP. Lastly, we define a more restrictive class of label decoders using
first-order logic that already contains many natural graph classes such as
forests and interval graphs. We give an alternative characterization of this
class in terms of directed acyclic graphs. By showing that some small,
hereditary graph class cannot be expressed with such label decoders a weaker
form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201
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