188 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
Ten-tier and multi-scale supplychain network analysis of medical equipment: Random failure and intelligent attack analysis
Motivated by the COVID-19 pandemic, this paper explores the supply chain
viability of medical equipment, an industry whose supply chain was put under a
crucial test during the pandemic. This paper includes an empirical
network-level analysis of supplier reachability under Random Failure Experiment
(RFE) and Intelligent Attack Experiment (IAE). Specifically, this study
investigates the effect of RFA and IAE across multiple tiers and scales. The
global supply chain data was mined and analyzed from about 45,000 firms with
about 115,000 intertwined relationships spanning across 10 tiers of the
backward supply chain of medical equipment. This complex supply chain network
was analyzed at four scales, namely: firm, country-industry, industry, and
country. A notable contribution of this study is the application of a supply
chain tier optimization tool to identify the lowest tier of the supply chain
that can provide adequate resolution for the study of the supply chain pattern.
We also developed data-driven-tools to identify the thresholds for breakdown
and fragmentation of the medical equipment supply chain when faced with random
failures or different intelligent attack scenarios. The novel network analysis
tools utilized in the study can be applied to the study of supply chain
reachability and viability in other industries.Comment: 47 page
Escape times for subgraph detection and graph partitioning
We provide a rearrangement based algorithm for fast detection of subgraphs of
vertices with long escape times for directed or undirected networks.
Complementing other notions of densest subgraphs and graph cuts, our method is
based on the mean hitting time required for a random walker to leave a
designated set and hit the complement. We provide a new relaxation of this
notion of hitting time on a given subgraph and use that relaxation to construct
a fast subgraph detection algorithm and a generalization to -partitioning
schemes. Using a modification of the subgraph detector on each component, we
propose a graph partitioner that identifies regions where random walks live for
comparably large times. Importantly, our method implicitly respects the
directed nature of the data for directed graphs while also being applicable to
undirected graphs. We apply the partitioning method for community detection to
a large class of model and real-world data sets.Comment: 22 pages, 10 figures, 1 table, comments welcome!
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
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