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Exact solutions to the ErdĆs-Rothschild problem
Let k := (k1,...,k2) be a sequence of natural numbers. For a graph G, let F (G;k) denote the number of colourings of the edges of G with colours 1,...,s such that, for every c â {1,...,s}, the edges of colour c contain no clique of order kc. Write F (n; k) to denote the maximum of F (G;k) over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by ErdĆs and Rothschild in 1974. We prove some new exact results for n â â:
(i) A sufficient condition on k which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.
(ii) Addressing the original question of ErdĆs and Rothschild, in the case k = (3,..., 3) of length 7, the unique extremal graph is the complete balanced 8-partite graph, with colourings coming from Hadamard matrices of order 8.
(iii) In the case k = (k+ 1, k), for which the sufficient condition in (i) does not hold, for 3 †k †10, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as
possible
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
A Look at Financial Dependencies by Means of Econophysics and Financial Economics
This is a review about financial dependencies which merges efforts in
econophysics and financial economics during the last few years. We focus on the
most relevant contributions to the analysis of asset markets' dependencies,
especially correlational studies, which in our opinion are beneficial for
researchers in both fields. In econophysics, these dependencies can be modeled
to describe financial markets as evolving complex networks. In particular we
show that a useful way to describe dependencies is by means of information
filtering networks that are able to retrieve relevant and meaningful
information in complex financial data sets. In financial economics these
dependencies can describe asset comovement and spill-overs. In particular,
several models are presented that show how network and factor model approaches
are related to modeling of multivariate volatility and asset returns
respectively. Finally, we sketch out how these studies can inspire future
research and how they contribute to support researchers in both fields to find
a better and a stronger common language
Reconfiguration of Digraph Homomorphisms
For a fixed graph H, the H-Recoloring problem asks whether, given two homomorphisms from a graph G to H, one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to H throughout. The most general algorithmic result for H-Recoloring so far has been proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph H. We show that the topological approach can be used to recover essentially all previous algorithmic results for H-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that H-Recoloring admits a polynomial-time algorithm i) if H is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and ii) if H is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0
Factor Graph Neural Networks
In recent years, we have witnessed a surge of Graph Neural Networks (GNNs),
most of which can learn powerful representations in an end-to-end fashion with
great success in many real-world applications. They have resemblance to
Probabilistic Graphical Models (PGMs), but break free from some limitations of
PGMs. By aiming to provide expressive methods for representation learning
instead of computing marginals or most likely configurations, GNNs provide
flexibility in the choice of information flowing rules while maintaining good
performance. Despite their success and inspirations, they lack efficient ways
to represent and learn higher-order relations among variables/nodes. More
expressive higher-order GNNs which operate on k-tuples of nodes need increased
computational resources in order to process higher-order tensors. We propose
Factor Graph Neural Networks (FGNNs) to effectively capture higher-order
relations for inference and learning. To do so, we first derive an efficient
approximate Sum-Product loopy belief propagation inference algorithm for
discrete higher-order PGMs. We then neuralize the novel message passing scheme
into a Factor Graph Neural Network (FGNN) module by allowing richer
representations of the message update rules; this facilitates both efficient
inference and powerful end-to-end learning. We further show that with a
suitable choice of message aggregation operators, our FGNN is also able to
represent Max-Product belief propagation, providing a single family of
architecture that can represent both Max and Sum-Product loopy belief
propagation. Our extensive experimental evaluation on synthetic as well as real
datasets demonstrates the potential of the proposed model.Comment: Accepted by JML
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
Subgroup discovery for structured target concepts
The main object of study in this thesis is subgroup discovery, a theoretical framework for finding subgroups in dataâi.e., named sub-populationsâ whose behaviour with respect to a specified target concept is exceptional when compared to the rest of the dataset. This is a powerful tool that conveys crucial information to a human audience, but despite past advances has been limited to simple target concepts. In this work we propose algorithms that bring this framework to novel application domains. We introduce the concept of representative subgroups, which we use not only to ensure the fairness of a sub-population with regard to a sensitive trait, such as race or gender, but also to go beyond known trends in the data. For entities with additional relational information that can be encoded as a graph, we introduce a novel measure of robust connectedness which improves on established alternative measures of density; we then provide a method that uses this measure to discover which named sub-populations are more well-connected. Our contributions within subgroup discovery crescent with the introduction of kernelised subgroup discovery: a novel framework that enables the discovery of subgroups on i.i.d. target concepts with virtually any kind of structure. Importantly, our framework additionally provides a concrete and efficient tool that works out-of-the-box without any modification, apart from specifying the Gramian of a positive definite kernel. To use within kernelised subgroup discovery, but also on any other kind of kernel method, we additionally introduce a novel random walk graph kernel. Our kernel allows the fine tuning of the alignment between the vertices of the two compared graphs, during the count of the random walks, while we also propose meaningful structure-aware vertex labels to utilise this new capability. With these contributions we thoroughly extend the applicability of subgroup discovery and ultimately re-define it as a kernel method.Der Hauptgegenstand dieser Arbeit ist die Subgruppenentdeckung (Subgroup Discovery), ein theoretischer Rahmen fĂŒr das Auffinden von Subgruppen in Datenâd. h. benannte Teilpopulationenâderen Verhalten in Bezug auf ein bestimmtes Targetkonzept im Vergleich zum Rest des Datensatzes auĂergewöhnlich ist. Es handelt sich hierbei um ein leistungsfĂ€higes Instrument, das einem menschlichen Publikum wichtige Informationen vermittelt. Allerdings ist es trotz bisherigen Fortschritte auf einfache Targetkonzepte beschrĂ€nkt. In dieser Arbeit schlagen wir Algorithmen vor, die diesen Rahmen auf neuartige Anwendungsbereiche ĂŒbertragen. Wir fĂŒhren das Konzept der reprĂ€sentativen Untergruppen ein, mit dem wir nicht nur die Fairness einer Teilpopulation in Bezug auf ein sensibles Merkmal wie Rasse oder Geschlecht sicherstellen, sondern auch ĂŒber bekannte Trends in den Daten hinausgehen können. FĂŒr EntitĂ€ten mit zusĂ€tzlicher relationalen Information, die als Graph kodiert werden kann, fĂŒhren wir ein neuartiges MaĂ fĂŒr robuste Verbundenheit ein, das die etablierten alternativen DichtemaĂe verbessert; anschlieĂend stellen wir eine Methode bereit, die dieses MaĂ verwendet, um herauszufinden, welche benannte Teilpopulationen besser verbunden sind. Unsere BeitrĂ€ge in diesem Rahmen gipfeln in der EinfĂŒhrung der kernelisierten Subgruppenentdeckung: ein neuartiger Rahmen, der die Entdeckung von Subgruppen fĂŒr u.i.v. Targetkonzepten mit praktisch jeder Art von Struktur ermöglicht. Wichtigerweise, unser Rahmen bereitstellt zusĂ€tzlich ein konkretes und effizientes Werkzeug, das ohne jegliche Modifikation funktioniert, abgesehen von der Angabe des Gramian eines positiv definitiven Kernels. FĂŒr den Einsatz innerhalb der kernelisierten Subgruppentdeckung, aber auch fĂŒr jede andere Art von Kernel-Methode, fĂŒhren wir zusĂ€tzlich einen neuartigen Random-Walk-Graph-Kernel ein. Unser Kernel ermöglicht die Feinabstimmung der Ausrichtung zwischen den Eckpunkten der beiden unter-Vergleich-gestelltenen Graphen wĂ€hrend der ZĂ€hlung der Random Walks, wĂ€hrend wir auch sinnvolle strukturbewusste Vertex-Labels vorschlagen, um diese neue FĂ€higkeit zu nutzen. Mit diesen BeitrĂ€gen erweitern wir die Anwendbarkeit der Subgruppentdeckung grĂŒndlich und definieren wir sie im Endeffekt als Kernel-Methode neu
Scalable Scheduling for Industrial Time-Sensitive Networking: A Hyper-flow Graph Based Scheme
Industrial Time-Sensitive Networking (TSN) provides deterministic mechanisms
for real-time and reliable flow transmission. Increasing attention has been
paid to efficient scheduling for time-sensitive flows with stringent
requirements such as ultra-low latency and jitter. In TSN, the fine-grained
traffic shaping protocol, cyclic queuing and forwarding (CQF), eliminates
uncertain delay and frame loss by cyclic traffic forwarding and queuing.
However, it inevitably causes high scheduling complexity. Moreover, complexity
is quite sensitive to flow attributes and network scale. The problem stems in
part from the lack of an attribute mining mechanism in existing frame-based
scheduling. For time-critical industrial networks with large-scale complex
flows, a so-called hyper-flow graph based scheduling scheme is proposed to
improve the scheduling scalability in terms of schedulability, scheduling
efficiency and latency & jitter. The hyper-flow graph is built by aggregating
similar flow sets as hyper-flow nodes and designing a hierarchical scheduling
framework. The flow attribute-sensitive scheduling information is embedded into
the condensed maximal cliques, and reverse maps them precisely to congestion
flow portions for re-scheduling. Its parallel scheduling reduces network scale
induced complexity. Further, this scheme is designed in its entirety as a
comprehensive scheduling algorithm GH^2. It improves the three criteria of
scalability along a Pareto front. Extensive simulation studies demonstrate its
superiority. Notably, GH^2 is verified its scheduling stability with a runtime
of less than 100 ms for 1000 flows and near 1/430 of the SOTA FITS method for
2000 flows
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