24,285 research outputs found

    Edge- and Node-Disjoint Paths in P Systems

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    In this paper, we continue our development of algorithms used for topological network discovery. We present native P system versions of two fundamental problems in graph theory: finding the maximum number of edge- and node-disjoint paths between a source node and target node. We start from the standard depth-first-search maximum flow algorithms, but our approach is totally distributed, when initially no structural information is available and each P system cell has to even learn its immediate neighbors. For the node-disjoint version, our P system rules are designed to enforce node weight capacities (of one), in addition to edge capacities (of one), which are not readily available in the standard network flow algorithms.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

    Minimum Input Selection for Structural Controllability

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    Given a linear system x˙=Ax\dot{x} = Ax, where AA is an n×nn \times n matrix with mm nonzero entries, we consider the problem of finding the smallest set of state variables to affect with an input so that the resulting system is structurally controllable. We further assume we are given a set of "forbidden state variables" FF which cannot be affected with an input and which we have to avoid in our selection. Our main result is that this problem can be solved deterministically in O(n+mn)O(n+m \sqrt{n}) operations

    Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

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    The nn-dimensional hypercube network QnQ_n is one of the most popular interconnection networks since it has simple structure and is easy to implement. The nn-dimensional locally twisted cube, denoted by LTQnLTQ_n, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as QnQ_n. One advantage of LTQnLTQ_n is that the diameter is only about half of the diameter of QnQ_n. Recently, some interesting properties of LTQnLTQ_n were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube LTQnLTQ_n, for any integer n⩾4n\geqslant 4. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.Comment: 7 pages, 4 figure

    Walking Through Waypoints

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    We initiate the study of a fundamental combinatorial problem: Given a capacitated graph G=(V,E)G=(V,E), find a shortest walk ("route") from a source s∈Vs\in V to a destination t∈Vt\in V that includes all vertices specified by a set W⊆V\mathscr{W}\subseteq V: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

    Controlling edge dynamics in complex networks

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    The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the last decade, with considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics. Evaluation of real-world networks indicates that most of them are more controllable than their randomized counterparts. We also find that transcriptional regulatory networks are particularly easy to control. Analytic calculations show that networks with scale-free degree distributions have better controllability properties than uncorrelated networks, and positively correlated in- and out-degrees enhance the controllability of the proposed dynamics.Comment: Preprint. 24 pages, 4 figures, 2 tables. Source code available at http://github.com/ntamas/netctr

    Maximum flow and topological structure of complex networks

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    The problem of sending the maximum amount of flow qq between two arbitrary nodes ss and tt of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between ss and tt. The average of qq over all node pairs with smaller degree kmink_{\rm min} is kmin≃ckmin_{k_{\rm min}} \simeq c k_{\rm min} for large kmink_{\rm min} with cc a constant implying that the statistics of qq is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and qq can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient cc. The applicability of our results to real world networks is tested for the Internet at the autonomous system level.Comment: 7 pages, 4 figure
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