10 research outputs found

    Beyond Flatland : exploring graphs in many dimensions

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    Societies, technologies, economies, ecosystems, organisms, . . . Our world is composed of complex networks—systems with many elements that interact in nontrivial ways. Graphs are natural models of these systems, and scientists have made tremendous progress in developing tools for their analysis. However, research has long focused on relatively simple graph representations and problem specifications, often discarding valuable real-world information in the process. In recent years, the limitations of this approach have become increasingly apparent, but we are just starting to comprehend how more intricate data representations and problem formulations might benefit our understanding of relational phenomena. Against this background, our thesis sets out to explore graphs in five dimensions: descriptivity, multiplicity, complexity, expressivity, and responsibility. Leveraging tools from graph theory, information theory, probability theory, geometry, and topology, we develop methods to (1) descriptively compare individual graphs, (2) characterize similarities and differences between groups of multiple graphs, (3) critically assess the complexity of relational data representations and their associated scientific culture, (4) extract expressive features from and for hypergraphs, and (5) responsibly mitigate the risks induced by graph-structured content recommendations. Thus, our thesis is naturally situated at the intersection of graph mining, graph learning, and network analysis.Gesellschaften, Technologien, Volkswirtschaften, Ökosysteme, Organismen, . . . Unsere Welt besteht aus komplexen Netzwerken—Systemen mit vielen Elementen, die auf nichttriviale Weise interagieren. Graphen sind natürliche Modelle dieser Systeme, und die Wissenschaft hat bei der Entwicklung von Methoden zu ihrer Analyse große Fortschritte gemacht. Allerdings hat sich die Forschung lange auf relativ einfache Graphrepräsentationen und Problemspezifikationen beschränkt, oft unter Vernachlässigung wertvoller Informationen aus der realen Welt. In den vergangenen Jahren sind die Grenzen dieser Herangehensweise zunehmend deutlich geworden, aber wir beginnen gerade erst zu erfassen, wie unser Verständnis relationaler Phänomene von intrikateren Datenrepräsentationen und Problemstellungen profitieren kann. Vor diesem Hintergrund erkundet unsere Dissertation Graphen in fünf Dimensionen: Deskriptivität, Multiplizität, Komplexität, Expressivität, und Verantwortung. Mithilfe von Graphentheorie, Informationstheorie, Wahrscheinlichkeitstheorie, Geometrie und Topologie entwickeln wir Methoden, welche (1) einzelne Graphen deskriptiv vergleichen, (2) Gemeinsamkeiten und Unterschiede zwischen Gruppen multipler Graphen charakterisieren, (3) die Komplexität relationaler Datenrepräsentationen und der mit ihnen verbundenen Wissenschaftskultur kritisch beleuchten, (4) expressive Merkmale von und für Hypergraphen extrahieren, und (5) verantwortungsvoll den Risiken begegnen, welche die Graphstruktur von Inhaltsempfehlungen mit sich bringt. Damit liegt unsere Dissertation naturgemäß an der Schnittstelle zwischen Graph Mining, Graph Learning und Netzwerkanalyse

    On a generalization of Abelian equivalence and complexity of infinite words

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    In this paper we introduce and study a family of complexity functions of infinite words indexed by k in Z^+ U {+infinity}. Let k in Z^+ U {+infinity} and A be a finite non-empty set. Two finite words u and v in A* are said to be k-Abelian equivalent if for all x in A* of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations sim_k on A*, bridging the gap between the usual notion of Abelian equivalence (when k = 1) and equality (when k = +infinity). We show that the number of k-Abelian equivalence classes of words of length n grows polynomially, although the degree is exponential in k. Given an infinite word omega in A^N, we consider the associated complexity function P^(k)_omega : N -> N which counts the number of k-Abelian equivalence classes of factors of omega of length n. We show that the complexity function P_k is intimately linked with periodicity. More precisely we define an auxiliary function q^k : N -> N and show that if P^(k)_omega(n) < q^k(n) for some k in Z^+ U {+infinity} and n >= 0, then omega is ultimately periodic. Moreover if omega is aperiodic, then P^(k)_omega(n) = q^k(n) if and only if omega is Sturmian. We also study k-Abelian complexity in connection with repetitions in words. Using Szemeredi's theorem, we show that if omega has bounded k-Abelian complexity, then for every D subset of N with positive upper density and for every positive integer N, there exists a k-Abelian N-power occurring in omega at some position j in D

    Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

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    In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k >= 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = infinity). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: Each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

    Acta Cybernetica : Volume 23. Number 1.

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    Variations of the Morse-Hedlund theorem for k-abelian equivalence

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    In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k ≥ 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = ∞). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

    On a generalization of Abelian equivalence and complexity of infinite words

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    In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and AA be a finite non-empty set. Two finite words uu and vv in A∗A^* are said to be kk-Abelian equivalent if for all x∈A∗x\in A^* of length less than or equal to k,k, the number of occurrences of xx in uu is equal to the number of occurrences of xx in v.v. This defines a family of equivalence relations ∼k\thicksim_k on A∗,A^*, bridging the gap between the usual notion of Abelian equivalence (when k=1k=1) and equality (when k=+∞).k=+\infty). We show that the number of kk-Abelian equivalence classes of words of length nn grows polynomially, although the degree is exponential in k.k. Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of kk-Abelian equivalence classes of factors of ω\omega of length n.n. We show that the complexity function P(k)\mathcal {P}^{(k)} is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if Pω(k)(n)<qk(n)\mathcal {P}^{(k)}_{\omega}(n)<q^k(n) for some k \in \ints ^+ \cup {+\infty} and n≥0,n\geq 0, the ω\omega is ultimately periodic. Moreover if ω\omega is aperiodic, then Pω(k)(n)=qk(n)\mathcal {P}^{(k)}_{\omega}(n)=q^k(n) if and only if ω\omega is Sturmian. We also study kk-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if ω\omega has bounded kk-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer N,N, there exists a kk-Abelian NN power occurring in ω\omega at some position $j\in D.

    Dagstuhl Annual Report January - December 2011

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    The International Conference and Research Center for Computer Science is a non-profit organization. Its objective is to promote world-class research in computer science and to host research seminars which enable new ideas to be showcased, problems to be discussed and the course to be set for future development in this field. The work being done to run this informatics center is documented in this report for the business year 2011

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081)

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    Sequences form the most basic and natural data structure. They occur whenever information is electronically transmitted (as bit streams), when natural language text is spoken or written down (as words over, for example, the latin alphabet), in the process of heredity transmission in living cells (as DNA sequence) or the protein synthesis (as sequence of amino acids), and in many more different contexts. Given this universal form of representing information, the need to process strings is apparent and actually a core purpose of computer use. Algorithms to efficiently search through, analyze, (de-)compress, match, learn, and encode/decode strings are therefore of chief interest. Combinatorial problems about strings lie at the core of such algorithmic questions. Many such combinatorial problems are common in the string processing efforts in the different fields of application. Scientists working in the fields of Combinatorics on Words, Computational Biology, Stringology, Natural Computing, and Machine Learning were invited to consider the seminar\u27s topic from a~wide range of perspectives. This report documents the program and the outcomes of Dagstuhl Seminar 11081 ``Combinatorial and Algorithmic Aspects of Sequence Processing\u27\u27
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