755 research outputs found
Large Graph Analysis in the GMine System
Current applications have produced graphs on the order of hundreds of
thousands of nodes and millions of edges. To take advantage of such graphs, one
must be able to find patterns, outliers and communities. These tasks are better
performed in an interactive environment, where human expertise can guide the
process. For large graphs, though, there are some challenges: the excessive
processing requirements are prohibitive, and drawing hundred-thousand nodes
results in cluttered images hard to comprehend. To cope with these problems, we
propose an innovative framework suited for any kind of tree-like graph visual
design. GMine integrates (a) a representation for graphs organized as
hierarchies of partitions - the concepts of SuperGraph and Graph-Tree; and (b)
a graph summarization methodology - CEPS. Our graph representation deals with
the problem of tracing the connection aspects of a graph hierarchy with sub
linear complexity, allowing one to grasp the neighborhood of a single node or
of a group of nodes in a single click. As a proof of concept, the visual
environment of GMine is instantiated as a system in which large graphs can be
investigated globally and locally
On the typical and atypical solutions to the Kuramoto equations
The Kuramoto model is a dynamical system that models the interaction of
coupled oscillators. There has been much work to effectively bound the number
of equilibria to the Kuramoto model for a given network. By formulating the
Kuramoto equations as a system of algebraic equations, we first relate the
complex root count of the Kuramoto equations to the combinatorics of the
underlying network by showing that the complex root count is generically equal
to the normalized volume of the corresponding adjacency polytope of the
network. We then give explicit algebraic conditions under which this bound is
strict and show that there are conditions where the Kuramoto equations have
infinitely many equilibria.Comment: 28 page
Fisher Metric, Geometric Entanglement and Spin Networks
Starting from recent results on the geometric formulation of quantum
mechanics, we propose a new information geometric characterization of
entanglement for spin network states in the context of quantum gravity. For the
simple case of a single-link fixed graph (Wilson line), we detail the
construction of a Riemannian Fisher metric tensor and a symplectic structure on
the graph Hilbert space, showing how these encode the whole information about
separability and entanglement. In particular, the Fisher metric defines an
entanglement monotone which provides a notion of distance among states in the
Hilbert space. In the maximally entangled gauge-invariant case, the
entanglement monotone is proportional to a power of the area of the surface
dual to the link thus supporting a connection between entanglement and the
(simplicial) geometric properties of spin network states. We further extend
such analysis to the study of non-local correlations between two non-adjacent
regions of a generic spin network graph characterized by the bipartite
unfolding of an Intertwiner state. Our analysis confirms the interpretation of
spin network bonds as a result of entanglement and to regard the same spin
network graph as an information graph, whose connectivity encodes, both at the
local and non-local level, the quantum correlations among its parts. This gives
a further connection between entanglement and geometry.Comment: 29 pages, 3 figures, revised version accepted for publicatio
Description of Quantum Entanglement with Nilpotent Polynomials
We propose a general method for introducing extensive characteristics of
quantum entanglement. The method relies on polynomials of nilpotent raising
operators that create entangled states acting on a reference vacuum state. By
introducing the notion of tanglemeter, the logarithm of the state vector
represented in a special canonical form and expressed via polynomials of
nilpotent variables, we show how this description provides a simple criterion
for entanglement as well as a universal method for constructing the invariants
characterizing entanglement. We compare the existing measures and classes of
entanglement with those emerging from our approach. We derive the equation of
motion for the tanglemeter and, in representative examples of up to four-qubit
systems, show how the known classes appear in a natural way within our
framework. We extend our approach to qutrits and higher-dimensional systems,
and make contact with the recently introduced idea of generalized entanglement.
Possible future developments and applications of the method are discussed.Comment: 40 pages, 7 figures, 1 table, submitted for publication. v2: section
II.E has been changed and the Appendix on "Four qubit sl-entanglement
measure" has been removed. There are changes in the notation of section IV.
Typos and language mistakes has been corrected. A figure has been added and a
figure has been replaced. The references have been update
RiskNet: neural risk assessment in networks of unreliable resources
We propose a graph neural network (GNN)-based method to predict the distribution of penalties induced by outages in communication networks, where connections are protected by resources shared between working and backup paths. The GNN-based algorithm is trained only with random graphs generated on the basis of the Barabási–Albert model. However, the results obtained show that we can accurately model the penalties in a wide range of existing topologies. We show that GNNs eliminate the need to simulate complex outage scenarios for the network topologies under study—in practice, the entire time of path placement evaluation based on the prediction is no longer than 4 ms on modern hardware. In this way, we gain up to 12 000 times in speed improvement compared to calculations based on simulations.This work was supported by the Polish Ministry of Science and Higher Education with the subvention funds of the Faculty of Computer Science, Electronics and Telecommunications of AGH University of Science and Technology (P.B., P.C.) and by the PL-Grid Infrastructure (K.R.).Peer ReviewedPostprint (published version
Tropical Positivity and Semialgebraic Sets from Polytopes
This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes.
Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part?
In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.
Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4.
In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane.
Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of , and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction
1. Background
2. Tropical Positivity and Determinantal Varieties
3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes
4. Combinatorics of Correlated Equilibri
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