22,376 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Less is More: Real-time Failure Localization in Power Systems
Cascading failures in power systems exhibit non-local propagation patterns
which make the analysis and mitigation of failures difficult. In this work, we
propose a distributed control framework inspired by the recently proposed
concepts of unified controller and network tree-partition that offers strong
guarantees in both the mitigation and localization of cascading failures in
power systems. In this framework, the transmission network is partitioned into
several control areas which are connected in a tree structure, and the unified
controller is adopted by generators or controllable loads for fast timescale
disturbance response. After an initial failure, the proposed strategy always
prevents successive failures from happening, and regulates the system to the
desired steady state where the impact of initial failures are localized as much
as possible. For extreme failures that cannot be localized, the proposed
framework has a configurable design, that progressively involves and
coordinates more control areas for failure mitigation and, as a last resort,
imposes minimal load shedding. We compare the proposed control framework with
Automatic Generation Control (AGC) on the IEEE 118-bus test system. Simulation
results show that our novel framework greatly improves the system robustness in
terms of the N-1 security standard, and localizes the impact of initial
failures in majority of the load profiles that are examined. Moreover, the
proposed framework incurs significantly less load loss, if any, compared to
AGC, in all of our case studies
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Random Forests and Networks Analysis
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning trees
and more generally weighted trees or forests spanning a given graph. This
algorithm provides a powerful tool in analyzing structures on networks and
along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we
focused on applications of spanning rooted forests on finite graphs. The
resulting main conclusions are reviewed in this paper by collecting related
theorems, algorithms, heuristics and numerical experiments. A first
foundational part on determinantal structures and efficient sampling procedures
is followed by four main applications: 1) a random-walk-based notion of
well-distributed points in a graph 2) how to describe metastable dynamics in
finite settings by means of Markov intertwining dualities 3) coarse graining
schemes for networks and associated processes 4) wavelets-like pyramidal
algorithms for graph signals.Comment: Survey pape
Local multiresolution order in community detection
Community detection algorithms attempt to find the best clusters of nodes in
an arbitrary complex network. Multi-scale ("multiresolution") community
detection extends the problem to identify the best network scale(s) for these
clusters. The latter task is generally accomplished by analyzing community
stability simultaneously for all clusters in the network. In the current work,
we extend this general approach to define local multiresolution methods, which
enable the extraction of well-defined local communities even if the global
community structure is vaguely defined in an average sense. Toward this end, we
propose measures analogous to variation of information and normalized mutual
information that are used to quantitatively identify the best resolution(s) at
the community level based on correlations between clusters in
independently-solved systems. We demonstrate our method on two constructed
networks as well as a real network and draw inferences about local community
strength. Our approach is independent of the applied community detection
algorithm save for the inherent requirement that the method be able to identify
communities across different network scales, with appropriate changes to
account for how different resolutions are evaluated or defined in a particular
community detection method. It should, in principle, easily adapt to
alternative community comparison measures.Comment: 19 pages, 11 figure
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