196,016 research outputs found
Tensor Spectral Clustering for Partitioning Higher-order Network Structures
Spectral graph theory-based methods represent an important class of tools for
studying the structure of networks. Spectral methods are based on a first-order
Markov chain derived from a random walk on the graph and thus they cannot take
advantage of important higher-order network substructures such as triangles,
cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering
(TSC) algorithm that allows for modeling higher-order network structures in a
graph partitioning framework. Our TSC algorithm allows the user to specify
which higher-order network structures (cycles, feed-forward loops, etc.) should
be preserved by the network clustering. Higher-order network structures of
interest are represented using a tensor, which we then partition by developing
a multilinear spectral method. Our framework can be applied to discovering
layered flows in networks as well as graph anomaly detection, which we
illustrate on synthetic networks. In directed networks, a higher-order
structure of particular interest is the directed 3-cycle, which captures
feedback loops in networks. We demonstrate that our TSC algorithm produces
large partitions that cut fewer directed 3-cycles than standard spectral
clustering algorithms.Comment: SDM 201
Directed Network Laplacians and Random Graph Models
We consider spectral methods that uncover hidden structures in directed
networks. We develop a general framework that allows us to associate methods
based on optimization formulations with maximum likelihood problems on random
graphs. We focus on two existing spectral approaches that build and analyse
Laplacian-style matrices via the minimization of frustration and trophic
incoherence. These algorithms aim to reveal directed periodic and linear
hierarchies, respectively. We show that reordering nodes using the two
algorithms, or mapping them onto a specified lattice, is associated with new
classes of directed random graph models. Using this random graph setting, we
are able to compare the two algorithms on a given network and quantify which
structure is more likely to be present. We illustrate the approach on synthetic
and real networks, and discuss practical implementation issues
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Isolated effective coherence (iCoh): causal information flow excluding indirect paths
A problem of great interest in real world systems, where multiple time series
measurements are available, is the estimation of the intra-system causal
relations. For instance, electric cortical signals are used for studying
functional connectivity between brain areas, their directionality, the direct
or indirect nature of the connections, and the spectral characteristics (e.g.
which oscillations are preferentially transmitted). The earliest spectral
measure of causality was Akaike's (1968) seminal work on the noise contribution
ratio, reflecting direct and indirect connections. Later, a major breakthrough
was the partial directed coherence of Baccala and Sameshima (2001) for direct
connections. The simple aim of this study consists of two parts: (1) To expose
a major problem with the partial directed coherence, where it is shown that it
is affected by irrelevant connections to such an extent that it can
misrepresent the frequency response, thus defeating the main purpose for which
the measure was developed, and (2) To provide a solution to this problem,
namely the "isolated effective coherence", which consists of estimating the
partial coherence under a multivariate auto-regressive model, followed by
setting all irrelevant associations to zero, other than the particular
directional association of interest. Simple, realistic, toy examples illustrate
the severity of the problem with the partial directed coherence, and the
solution achieved by the isolated effective coherence. For the sake of
reproducible research, the software code implementing the methods discussed
here (using lazarus free-pascal "www.lazarus.freepascal.org"), including the
test data as text files, are freely available at:
https://sites.google.com/site/pascualmarqui/home/icoh-isolated-effective-coherenceComment: 2014-02-21 pre-print, technical report, KEY Institute for Brain-Mind
Research, University of Zurich, et a
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