18,369 research outputs found

    Minimal Connectivity

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    A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

    Organic Design of Massively Distributed Systems: A Complex Networks Perspective

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    The vision of Organic Computing addresses challenges that arise in the design of future information systems that are comprised of numerous, heterogeneous, resource-constrained and error-prone components or devices. Here, the notion organic particularly highlights the idea that, in order to be manageable, such systems should exhibit self-organization, self-adaptation and self-healing characteristics similar to those of biological systems. In recent years, the principles underlying many of the interesting characteristics of natural systems have been investigated from the perspective of complex systems science, particularly using the conceptual framework of statistical physics and statistical mechanics. In this article, we review some of the interesting relations between statistical physics and networked systems and discuss applications in the engineering of organic networked computing systems with predictable, quantifiable and controllable self-* properties.Comment: 17 pages, 14 figures, preprint of submission to Informatik-Spektrum published by Springe

    Distinguishing Infections on Different Graph Topologies

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    The history of infections and epidemics holds famous examples where understanding, containing and ultimately treating an outbreak began with understanding its mode of spread. Influenza, HIV and most computer viruses, spread person to person, device to device, through contact networks; Cholera, Cancer, and seasonal allergies, on the other hand, do not. In this paper we study two fundamental questions of detection: first, given a snapshot view of a (perhaps vanishingly small) fraction of those infected, under what conditions is an epidemic spreading via contact (e.g., Influenza), distinguishable from a "random illness" operating independently of any contact network (e.g., seasonal allergies); second, if we do have an epidemic, under what conditions is it possible to determine which network of interactions is the main cause of the spread -- the causative network -- without any knowledge of the epidemic, other than the identity of a minuscule subsample of infected nodes? The core, therefore, of this paper, is to obtain an understanding of the diagnostic power of network information. We derive sufficient conditions networks must satisfy for these problems to be identifiable, and produce efficient, highly scalable algorithms that solve these problems. We show that the identifiability condition we give is fairly mild, and in particular, is satisfied by two common graph topologies: the grid, and the Erdos-Renyi graphs

    The essence of P2P: A reference architecture for overlay networks

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    The success of the P2P idea has created a huge diversity of approaches, among which overlay networks, for example, Gnutella, Kazaa, Chord, Pastry, Tapestry, P-Grid, or DKS, have received specific attention from both developers and researchers. A wide variety of algorithms, data structures, and architectures have been proposed. The terminologies and abstractions used, however, have become quite inconsistent since the P2P paradigm has attracted people from many different communities, e.g., networking, databases, distributed systems, graph theory, complexity theory, biology, etc. In this paper we propose a reference model for overlay networks which is capable of modeling different approaches in this domain in a generic manner. It is intended to allow researchers and users to assess the properties of concrete systems, to establish a common vocabulary for scientific discussion, to facilitate the qualitative comparison of the systems, and to serve as the basis for defining a standardized API to make overlay networks interoperable

    Recent progress on the combinatorial diameter of polytopes and simplicial complexes

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    The Hirsch conjecture, posed in 1957, stated that the graph of a dd-dimensional polytope or polyhedron with nn facets cannot have diameter greater than n−dn - d. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but some try to shed light on the question by generalizing it to simplicial complexes. In particular, we include here our recent and previously unpublished proof that the maximum diameter of arbitrary simplicial complexes is in nTheta(d)n^{Theta(d)} and we summarize the main ideas in the polymath 3 project, a web-based collective effort trying to prove an upper bound of type nd for the diameters of polyhedra and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter of simplicial complexes and abstractions of them, in preparation
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