135,426 research outputs found
The Total Acquisition Number of Random Graphs
Let be a graph in which each vertex initially has weight 1. In each step,
the weight from a vertex can be moved to a neighbouring vertex ,
provided that the weight on is at least as large as the weight on . The
total acquisition number of , denoted by , 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
such that with high probability, where
is a binomial random graph. We show that is a sharp threshold for this
property. We also show that almost all trees satisfy ,
confirming a conjecture of West.Comment: 18 pages, 1 figur
The Total Acquisition Number of Random Geometric Graphs
Let be a graph in which each vertex initially has weight 1. In each step,
the weight from a vertex to a neighbouring vertex can be moved,
provided that the weight on is at least as large as the weight on . The
total acquisition number of , denoted by , 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 with
vertices distributed u.a.r. in and two vertices being
adjacent if and only if their distance is at most . We show that
asymptotically almost surely for the
whole range of such that . By monotonicity,
asymptotically almost surely if , and
if
The Total Acquisition Number of the Randomly Weighted Path
There exists a significant body of work on determining the acquisition number
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 of the
-path when distinguishable "units" of integral weight, or chips, are
randomly distributed across its vertices between and . With
computer support, we improve it by showing that lies between
and . We then use subadditivity to show that the limiting
ratio 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
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
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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