1,503 research outputs found
Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations
Human brain anatomy and function display a combination of modular and
hierarchical organization, suggesting the importance of both cohesive
structures and variable resolutions in the facilitation of healthy cognitive
processes. However, tools to simultaneously probe these features of brain
architecture require further development. We propose and apply a set of methods
to extract cohesive structures in network representations of brain connectivity
using multi-resolution techniques. We employ a combination of soft
thresholding, windowed thresholding, and resolution in community detection,
that enable us to identify and isolate structures associated with different
weights. One such mesoscale structure is bipartivity, which quantifies the
extent to which the brain is divided into two partitions with high connectivity
between partitions and low connectivity within partitions. A second,
complementary mesoscale structure is modularity, which quantifies the extent to
which the brain is divided into multiple communities with strong connectivity
within each community and weak connectivity between communities. Our methods
lead to multi-resolution curves of these network diagnostics over a range of
spatial, geometric, and structural scales. For statistical comparison, we
contrast our results with those obtained for several benchmark null models. Our
work demonstrates that multi-resolution diagnostic curves capture complex
organizational profiles in weighted graphs. We apply these methods to the
identification of resolution-specific characteristics of healthy weighted graph
architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom
Community detection in temporal multilayer networks, with an application to correlation networks
Networks are a convenient way to represent complex systems of interacting
entities. Many networks contain "communities" of nodes that are more densely
connected to each other than to nodes in the rest of the network. In this
paper, we investigate the detection of communities in temporal networks
represented as multilayer networks. As a focal example, we study time-dependent
financial-asset correlation networks. We first argue that the use of the
"modularity" quality function---which is defined by comparing edge weights in
an observed network to expected edge weights in a "null network"---is
application-dependent. We differentiate between "null networks" and "null
models" in our discussion of modularity maximization, and we highlight that the
same null network can correspond to different null models. We then investigate
a multilayer modularity-maximization problem to identify communities in
temporal networks. Our multilayer analysis only depends on the form of the
maximization problem and not on the specific quality function that one chooses.
We introduce a diagnostic to measure \emph{persistence} of community structure
in a multilayer network partition. We prove several results that describe how
the multilayer maximization problem measures a trade-off between static
community structure within layers and larger values of persistence across
layers. We also discuss some computational issues that the popular "Louvain"
heuristic faces with temporal multilayer networks and suggest ways to mitigate
them.Comment: 42 pages, many figures, final accepted version before typesettin
Incompatibility boundaries for properties of community partitions
We prove the incompatibility of certain desirable properties of community
partition quality functions. Our results generalize the impossibility result of
[Kleinberg 2003] by considering sets of weaker properties. In particular, we
use an alternative notion to solve the central issue of the consistency
property. (The latter means that modifying the graph in a way consistent with a
partition should not have counterintuitive effects). Our results clearly show
that community partition methods should not be expected to perfectly satisfy
all ideally desired properties.
We then proceed to show that this incompatibility no longer holds when
slightly relaxed versions of the properties are considered, and we provide in
fact examples of simple quality functions satisfying these relaxed properties.
An experimental study of these quality functions shows a behavior comparable to
established methods in some situations, but more debatable results in others.
This suggests that defining a notion of good partition in communities probably
requires imposing additional properties.Comment: 17 pages, 3 figure
Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
Dynamic programming on tree decompositions is a frequently used approach to
solve otherwise intractable problems on instances of small treewidth. In recent
work by Bodlaender et al., it was shown that for many connectivity problems,
there exist algorithms that use time, linear in the number of vertices, and
single exponential in the width of the tree decomposition that is used. The
central idea is that it suffices to compute representative sets, and these can
be computed efficiently with help of Gaussian elimination.
In this paper, we give an experimental evaluation of this technique for the
Steiner Tree problem. A comparison of the classic dynamic programming algorithm
and the improved dynamic programming algorithm that employs the table reduction
shows that the new approach gives significant improvements on the running time
of the algorithm and the size of the tables computed by the dynamic programming
algorithm, and thus that the rank based approach from Bodlaender et al. does
not only give significant theoretical improvements but also is a viable
approach in a practical setting, and showcases the potential of exploiting the
idea of representative sets for speeding up dynamic programming algorithms
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
A Modular Programmable CMOS Analog Fuzzy Controller Chip
We present a highly modular fuzzy inference analog CMOS chip architecture with on-chip digital programmability. This chip consists of the interconnection of parameterized instances of two different kind of blocks, namely label blocks and rule blocks. The architecture realizes a lattice partition of the universe of discourse, which at the hardware level means that the fuzzy labels associated to every input (realized by the label blocks) are shared among the rule blocks. This reduces the area and power consumption and is the key point for chip modularity. The proposed architecture is demonstrated through a 16-rule two input CMOS 1-μm prototype which features an operation speed of 2.5 Mflips (2.5×10^6 fuzzy inferences per second) with 8.6 mW power consumption. Core area occupation of this prototype is of only 1.6 mm 2 including the digital control and memory circuitry used for programmability. Because of the architecture modularity the number of inputs and rules can be increased with any hardly design effort.This work was
supported in part by the Spanish C.I.C.Y.T under Contract TIC96-1392-C02-
02 (SIVA)
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