159 research outputs found
Simplified Energy Landscape for Modularity Using Total Variation
Networks capture pairwise interactions between entities and are frequently
used in applications such as social networks, food networks, and protein
interaction networks, to name a few. Communities, cohesive groups of nodes,
often form in these applications, and identifying them gives insight into the
overall organization of the network. One common quality function used to
identify community structure is modularity. In Hu et al. [SIAM J. App. Math.,
73(6), 2013], it was shown that modularity optimization is equivalent to
minimizing a particular nonconvex total variation (TV) based functional over a
discrete domain. They solve this problem, assuming the number of communities is
known, using a Merriman, Bence, Osher (MBO) scheme.
We show that modularity optimization is equivalent to minimizing a convex
TV-based functional over a discrete domain, again, assuming the number of
communities is known. Furthermore, we show that modularity has no convex
relaxation satisfying certain natural conditions. We therefore, find a
manageable non-convex approximation using a Ginzburg Landau functional, which
provably converges to the correct energy in the limit of a certain parameter.
We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et
al. and which is 7 times faster at solving the associated diffusion equation
due to the fact that the underlying discretization is unconditionally stable.
Our numerical tests include a hyperspectral video whose associated graph has
2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper
of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat
Presbyterian Imitation Practices in Zachary Boydâs Nebuchadnezzars Fierie Furnace
The university administrator, preacher and poet Zachary Boyd (1585â1653) relied heavily on epithets and similes borrowed from Josuah Sylvester's poetry when composing his scriptural versifications Zion's Flowers(c. 1640?). The composition of Boyd's adaptation of Daniel 3, Nebuchadnezzars Fierie Furnace, provides an unusually lucid example of the reading and imitation practices of a mid-seventeenth-century Scottish Presbyterian in the years preceding civil war. This article begins by re-considering a manuscript transcription of Fierie Furnace held at the British Library previously described as an anonymous playtext from the early 1610s, then establishes the nature of Boyd's reliance on Sylvester by analyzing holograph manuscripts held at Glasgow University Library, a sermon Boyd wrote on the same theme, and the copy of Sylvester's Devine Weekes, and Workes that Boyd probably used.Arts and Humanities Research
Counci
Stochastic Block Models are a Discrete Surface Tension
Networks, which represent agents and interactions between them, arise in
myriad applications throughout the sciences, engineering, and even the
humanities. To understand large-scale structure in a network, a common task is
to cluster a network's nodes into sets called "communities", such that there
are dense connections within communities but sparse connections between them. A
popular and statistically principled method to perform such clustering is to
use a family of generative models known as stochastic block models (SBMs). In
this paper, we show that maximum likelihood estimation in an SBM is a network
analog of a well-known continuum surface-tension problem that arises from an
application in metallurgy. To illustrate the utility of this relationship, we
implement network analogs of three surface-tension algorithms, with which we
successfully recover planted community structure in synthetic networks and
which yield fascinating insights on empirical networks that we construct from
hyperspectral videos.Comment: to appear in Journal of Nonlinear Scienc
A metric on directed graphs and Markov chains based on hitting probabilities
The shortest-path, commute time, and diffusion distances on undirected graphs
have been widely employed in applications such as dimensionality reduction,
link prediction, and trip planning. Increasingly, there is interest in using
asymmetric structure of data derived from Markov chains and directed graphs,
but few metrics are specifically adapted to this task. We introduce a metric on
the state space of any ergodic, finite-state, time-homogeneous Markov chain
and, in particular, on any Markov chain derived from a directed graph. Our
construction is based on hitting probabilities, with nearness in the metric
space related to the transfer of random walkers from one node to another at
stationarity. Notably, our metric is insensitive to shortest and average walk
distances, thus giving new information compared to existing metrics. We use
possible degeneracies in the metric to develop an interesting structural theory
of directed graphs and explore a related quotienting procedure. Our metric can
be computed in time, where is the number of states, and in
examples we scale up to nodes and edges on a desktop
computer. In several examples, we explore the nature of the metric, compare it
to alternative methods, and demonstrate its utility for weak recovery of
community structure in dense graphs, visualization, structure recovering,
dynamics exploration, and multiscale cluster detection.Comment: 26 pages, 9 figures, for associated code, visit
https://github.com/zboyd2/hitting_probabilities_metric, accepted at SIAM J.
Math. Data Sc
Effect of microstructure on the internal hydriding behavior of uranium
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