135,404 research outputs found

    The Total Acquisition Number of Random Graphs

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    Let GG be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex uu can be moved to a neighbouring vertex vv, provided that the weight on vv is at least as large as the weight on uu. The total acquisition number of GG, denoted by at(G)a_t(G), is the minimum possible size of the set of vertices with positive weight at the end of the process. LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of p=p(n)p=p(n) such that at(G(n,p))=1a_t(\mathcal{G}(n,p)) = 1 with high probability, where G(n,p)\mathcal{G}(n,p) is a binomial random graph. We show that p=log2nn1.4427 lognnp = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n} is a sharp threshold for this property. We also show that almost all trees TT satisfy at(T)=Θ(n)a_t(T) = \Theta(n), confirming a conjecture of West.Comment: 18 pages, 1 figur

    The Total Acquisition Number of Random Geometric Graphs

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    Let GG be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex uu to a neighbouring vertex vv can be moved, provided that the weight on vv is at least as large as the weight on uu. The total acquisition number of GG, denoted by at(G)a_t(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs G(n,r)G(n,r) with nn vertices distributed u.a.r. in [0,n]2[0,\sqrt{n}]^2 and two vertices being adjacent if and only if their distance is at most rr. We show that asymptotically almost surely at(G(n,r))=Θ(n/(rlgr)2)a_t(G(n,r)) = \Theta( n / (r \lg r)^2) for the whole range of r=rn1r=r_n \ge 1 such that rlgrnr \lg r \le \sqrt{n}. By monotonicity, asymptotically almost surely at(G(n,r))=Θ(n)a_t(G(n,r)) = \Theta(n) if r<1r < 1, and at(G(n,r))=Θ(1)a_t(G(n,r)) = \Theta(1) if rlgr>nr \lg r > \sqrt{n}

    The Total Acquisition Number of the Randomly Weighted Path

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    There exists a significant body of work on determining the acquisition number at(G)a_t(G) of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number E(at(Pn))E(a_t(P_n)) of the nn-path when nn distinguishable "units" of integral weight, or chips, are randomly distributed across its vertices between 0.242n0.242n and 0.375n0.375n. With computer support, we improve it by showing that E(at(Pn))E(a_t(P_n)) lies between 0.29523n0.29523n and 0.29576n0.29576n. We then use subadditivity to show that the limiting ratio limE(at(Pn))/n\lim E(a_t(P_n))/n exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set.Comment: 19 page

    Bayesian Semi-supervised Learning with Graph Gaussian Processes

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    We propose a data-efficient Gaussian process-based Bayesian approach to the semi-supervised learning problem on graphs. The proposed model shows extremely competitive performance when compared to the state-of-the-art graph neural networks on semi-supervised learning benchmark experiments, and outperforms the neural networks in active learning experiments where labels are scarce. Furthermore, the model does not require a validation data set for early stopping to control over-fitting. Our model can be viewed as an instance of empirical distribution regression weighted locally by network connectivity. We further motivate the intuitive construction of the model with a Bayesian linear model interpretation where the node features are filtered by an operator related to the graph Laplacian. The method can be easily implemented by adapting off-the-shelf scalable variational inference algorithms for Gaussian processes.Comment: To appear in NIPS 2018 Fixed an error in Figure 2. The previous arxiv version contains two identical sub-figure

    Comparison of Randomized Multifocal Mapping and Temporal Phase Mapping of Visual Cortex for Clinical Use

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    fMRI is becoming an important clinical tool for planning and guidance of surgery to treat brain tumors, arteriovenous malformations, and epileptic foci. For visual cortex mapping, the most popular paradigm by far is temporal phase mapping, although random multifocal stimulation paradigms have drawn increased attention due to their ability to identify complex response fields and their random properties. In this study we directly compared temporal phase and multifocal vision mapping paradigms with respect to clinically relevant factors including: time efficiency, mapping completeness, and the effects of noise. Randomized, multifocal mapping accurately decomposed the response of single voxels to multiple stimulus locations and made correct retinotopic assignments as noise levels increased despite decreasing sensitivity. Also, multifocal mapping became less efficient as the number of stimulus segments (locations) increased from 13 to 25 to 49 and when duty cycle was increased from 25% to 50%. Phase mapping, on the other hand, activated more extrastriate visual areas, was more time efficient in achieving statistically significant responses, and had better sensitivity as noise increased, though with an increase in systematic retinotopic mis-assignments. Overall, temporal phase mapping is likely to be a better choice for routine clinical applications though random multifocal mapping may offer some unique advantages for selected applications
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