76 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
An efficient threshold dynamics method for topology optimization for fluids
We propose an efficient threshold dynamics method for topology optimization
for fluids modeled with the Stokes equation. The proposed algorithm is based on
minimization of an objective energy function that consists of the dissipation
power in the fluid and the perimeter approximated by nonlocal energy, subject
to a fluid volume constraint and the incompressibility condition. We show that
the minimization problem can be solved with an iterative scheme in which the
Stokes equation is approximated by a Brinkman equation. The indicator functions
of the fluid-solid regions are then updated according to simple convolutions
followed by a thresholding step. We demonstrate mathematically that the
iterative algorithm has the total energy decaying property. The proposed
algorithm is simple and easy to implement. A simple adaptive time strategy is
also used to accelerate the convergence of the iteration. Extensive numerical
experiments in both two and three dimensions show that the proposed iteration
algorithm converges in much fewer iterations and is more efficient than many
existing methods. In addition, the numerical results show that the algorithm is
very robust and insensitive to the initial guess and the parameters in the
model.Comment: 23 pages, 24 figure
Dynamics and stationary configurations of heterogeneous foams
We consider the variational foam model, where the goal is to minimize the
total surface area of a collection of bubbles subject to the constraint that
the volume of each bubble is prescribed. We apply sharp interface methods to
develop an efficient computational method for this problem. In addition to
simulating time dynamics, we also report on stationary states of this flow for
<22 bubbles in two dimensions and <18 bubbles in three dimensions. For small
numbers of bubbles, we recover known analytical results, which we briefly
discuss. In two dimensions, we also recover the previous numerical results of
Cox et. al. (2003), computed using other methods. Particular attention is given
to locally optimal foam configurations and heterogeneous foams, where the
volumes of the bubbles are not equal. Configurational transitions are reported
for the quasi-stationary flow where the volume of one of the bubbles is varied
and, for each volume, the stationary state is computed. The results from these
numerical experiments are described and accompanied by many figures and videos.Comment: 19 pages, 11 figure
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