2,359 research outputs found

    PyDEC: Software and Algorithms for Discretization of Exterior Calculus

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    This paper describes the algorithms, features and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest order finite element exterior calculus and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability, removed redundant text, emphasized the role of matrix based algorithms, shortened length of pape

    Kolmogorov Random Graphs and the Incompressibility Method

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    We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of the maximum possible Kolmogorov complexity) and (ii) a new elementary proof for the number of unlabeled graphs.Comment: LaTeX 9 page

    Layered Label Propagation: A MultiResolution Coordinate-Free Ordering for Compressing Social Networks

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    We continue the line of research on graph compression started with WebGraph, but we move our focus to the compression of social networks in a proper sense (e.g., LiveJournal): the approaches that have been used for a long time to compress web graphs rely on a specific ordering of the nodes (lexicographical URL ordering) whose extension to general social networks is not trivial. In this paper, we propose a solution that mixes clusterings and orders, and devise a new algorithm, called Layered Label Propagation, that builds on previous work on scalable clustering and can be used to reorder very large graphs (billions of nodes). Our implementation uses overdecomposition to perform aggressively on multi-core architecture, making it possible to reorder graphs of more than 600 millions nodes in a few hours. Experiments performed on a wide array of web graphs and social networks show that combining the order produced by the proposed algorithm with the WebGraph compression framework provides a major increase in compression with respect to all currently known techniques, both on web graphs and on social networks. These improvements make it possible to analyse in main memory significantly larger graphs

    Improved opportunity cost algorithm for carrier selection in combinatorial auctions

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    Transportation costs constitute up to thirty percent of the total costs involved in a supply chain. Outsourcing the transportation service requirements to third party logistics providers have been widely adopted, as they are economically more rational than owning and operating a service. Transportation service procurement has been traditionally done through an auctioning process where the auctioneer (shipper) auctions lanes (distinct delivery routes) to bidders (carriers). Individual lanes were being auctioned separately disallowing the carriers to express complements and substitutes. Using combinatorial auctions mechanism to auction all available lanes together would allow the carriers to take advantage of the lane bundles, their existing service schedule, probability of securing other lanes and available capacity to offer services at lower rates and be more competitive. The winners of the auction are the set of non-overlapping bids that minimize the cost for the shippers. The winner determination problem to be solved in determining the optimal allocation of the services in such kind of combinatorial auctions is a NP-hard problem. Many heuristics like approximate linear programming, stochastic local search have proposed to find an approximate solution to the problem in a reasonable amount of time. Akcoglu et al [22] developed the opportunity cost algorithm using the “local ratio technique” to compute a greedy solution to the problem. A recalculation modification to the opportunity cost algorithm has been formulated where opportunity costs are recalculated every time for the set of remaining bids after eliminating the bid chosen to be a part of the winning solution and its conflicts have eliminated. Another method that formulates the winning solution based on the maximum total revenue values calculated for each bid using the opportunity cost algorithm has also been researched

    Evolving graphs: dynamical models, inverse problems and propagation

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    Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics

    Towards a Queueing-Based Framework for In-Network Function Computation

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    We seek to develop network algorithms for function computation in sensor networks. Specifically, we want dynamic joint aggregation, routing, and scheduling algorithms that have analytically provable performance benefits due to in-network computation as compared to simple data forwarding. To this end, we define a class of functions, the Fully-Multiplexible functions, which includes several functions such as parity, MAX, and k th -order statistics. For such functions we exactly characterize the maximum achievable refresh rate of the network in terms of an underlying graph primitive, the min-mincut. In acyclic wireline networks, we show that the maximum refresh rate is achievable by a simple algorithm that is dynamic, distributed, and only dependent on local information. In the case of wireless networks, we provide a MaxWeight-like algorithm with dynamic flow splitting, which is shown to be throughput-optimal

    Frequency reassignment in cellular phone networks

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    In cellular communications networks, cells use beacon frequencies to ensure the smooth operation of the network, for example in handling call handovers from one cell to another. These frequencies are assigned according to a frequency plan, which is updated from time to time, in response to evolving network requirements. The migration from one frequency plan to a new one proceeds in stages, governed by the network's base station controllers. Existing methods result in periods of reduced network availability or performance during the reassgnment process. The problem posed to the Study Group was to develop a dynamic reassignment algorithm for implementing a new frequency plan so that there is little or no disruption of the network's performance during the transition. This problem was naturally formulated in terms of graph colouring and an effective algorithm was developed based on a straightforward approach of search and random colouring

    Enumerating Maximal Bicliques from a Large Graph using MapReduce

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    We consider the enumeration of maximal bipartite cliques (bicliques) from a large graph, a task central to many practical data mining problems in social network analysis and bioinformatics. We present novel parallel algorithms for the MapReduce platform, and an experimental evaluation using Hadoop MapReduce. Our algorithm is based on clustering the input graph into smaller sized subgraphs, followed by processing different subgraphs in parallel. Our algorithm uses two ideas that enable it to scale to large graphs: (1) the redundancy in work between different subgraph explorations is minimized through a careful pruning of the search space, and (2) the load on different reducers is balanced through the use of an appropriate total order among the vertices. Our evaluation shows that the algorithm scales to large graphs with millions of edges and tens of mil- lions of maximal bicliques. To our knowledge, this is the first work on maximal biclique enumeration for graphs of this scale.Comment: A preliminary version of the paper was accepted at the Proceedings of the 3rd IEEE International Congress on Big Data 201
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