66,116 research outputs found
Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups
We examine two criteria for balance of a gain graph, one based on binary
cycles and one on circles. The graphs for which each criterion is valid depend
on the set of allowed gain groups. The binary cycle test is invalid, except for
forests, if any possible gain group has an element of odd order. Assuming all
groups are allowed, or all abelian groups, or merely the cyclic group of order
3, we characterize, both constructively and by forbidden minors, the graphs for
which the circle test is valid. It turns out that these three classes of groups
have the same set of forbidden minors. The exact reason for the importance of
the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in
Journal of Graph Theor
Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology
Topological data analysis is an emerging area in exploratory data analysis
and data mining. Its main tool, persistent homology, has become a popular
technique to study the structure of complex, high-dimensional data. In this
paper, we propose a novel method using persistent homology to quantify
structural changes in time-varying graphs. Specifically, we transform each
instance of the time-varying graph into metric spaces, extract topological
features using persistent homology, and compare those features over time. We
provide a visualization that assists in time-varying graph exploration and
helps to identify patterns of behavior within the data. To validate our
approach, we conduct several case studies on real world data sets and show how
our method can find cyclic patterns, deviations from those patterns, and
one-time events in time-varying graphs. We also examine whether
persistence-based similarity measure as a graph metric satisfies a set of
well-established, desirable properties for graph metrics
Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrisation
We present a technique for reducing the computational requirements by several
orders of magnitude in the evaluation of semidefinite relaxations for bounding
the set of quantum correlations arising from finite-dimensional Hilbert spaces.
The technique, which we make publicly available through a user-friendly
software package, relies on the exploitation of symmetries present in the
optimisation problem to reduce the number of variables and the block sizes in
semidefinite relaxations. It is widely applicable in problems encountered in
quantum information theory and enables computations that were previously too
demanding. We demonstrate its advantages and general applicability in several
physical problems. In particular, we use it to robustly certify the
non-projectiveness of high-dimensional measurements in a black-box scenario
based on self-tests of -dimensional symmetric informationally complete
POVMs.Comment: A. T. and D. R. contributed equally for this projec
Groups not presentable by products
In this paper we study obstructions to presentability by products for
finitely generated groups. Along the way we develop both the concept of
acentral subgroups, and the relations between presentability by products on the
one hand, and certain geometric and measure or orbit equivalence invariants of
groups on the other. This leads to many new examples of groups not presentable
by products, including all groups with infinitely many ends, the (outer)
automorphism groups of free groups, Thompson's groups, and even some elementary
amenable groups.Comment: 25 pages, minor changes, to appear in Groups, Geometry, and Dynamic
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