812 research outputs found
Spectral Graph Forge: Graph Generation Targeting Modularity
Community structure is an important property that captures inhomogeneities
common in large networks, and modularity is one of the most widely used metrics
for such community structure. In this paper, we introduce a principled
methodology, the Spectral Graph Forge, for generating random graphs that
preserves community structure from a real network of interest, in terms of
modularity. Our approach leverages the fact that the spectral structure of
matrix representations of a graph encodes global information about community
structure. The Spectral Graph Forge uses a low-rank approximation of the
modularity matrix to generate synthetic graphs that match a target modularity
within user-selectable degree of accuracy, while allowing other aspects of
structure to vary. We show that the Spectral Graph Forge outperforms
state-of-the-art techniques in terms of accuracy in targeting the modularity
and randomness of the realizations, while also preserving other local
structural properties and node attributes. We discuss extensions of the
Spectral Graph Forge to target other properties beyond modularity, and its
applications to anonymization
Controllability and observability of grid graphs via reduction and symmetries
In this paper we investigate the controllability and observability properties
of a family of linear dynamical systems, whose structure is induced by the
Laplacian of a grid graph. This analysis is motivated by several applications
in network control and estimation, quantum computation and discretization of
partial differential equations. Specifically, we characterize the structure of
the grid eigenvectors by means of suitable decompositions of the graph. For
each eigenvalue, based on its multiplicity and on suitable symmetries of the
corresponding eigenvectors, we provide necessary and sufficient conditions to
characterize all and only the nodes from which the induced dynamical system is
controllable (observable). We discuss the proposed criteria and show, through
suitable examples, how such criteria reduce the complexity of the
controllability (respectively observability) analysis of the grid
Method and system for detecting a failure or performance degradation in a dynamic system such as a flight vehicle
A method and system for detecting a failure or performance degradation in a dynamic system having sensors for measuring state variables and providing corresponding output signals in response to one or more system input signals are provided. The method includes calculating estimated gains of a filter and selecting an appropriate linear model for processing the output signals based on the input signals. The step of calculating utilizes one or more models of the dynamic system to obtain estimated signals. The method further includes calculating output error residuals based on the output signals and the estimated signals. The method also includes detecting one or more hypothesized failures or performance degradations of a component or subsystem of the dynamic system based on the error residuals. The step of calculating the estimated values is performed optimally with respect to one or more of: noise, uncertainty of parameters of the models and un-modeled dynamics of the dynamic system which may be a flight vehicle or financial market or modeled financial system
Incremental eigenpair computation for graph Laplacian matrices: theory and applications
The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications, the number of clusters or communities (say, K) is generally unknown a priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the Kth smallest eigenpair of the Laplacian matrix given a collection of all previously compute
Spectral partitioning of time-varying networks with unobserved edges
We discuss a variant of `blind' community detection, in which we aim to
partition an unobserved network from the observation of a (dynamical) graph
signal defined on the network. We consider a scenario where our observed graph
signals are obtained by filtering white noise input, and the underlying network
is different for every observation. In this fashion, the filtered graph signals
can be interpreted as defined on a time-varying network. We model each of the
underlying network realizations as generated by an independent draw from a
latent stochastic blockmodel (SBM). To infer the partition of the latent SBM,
we propose a simple spectral algorithm for which we provide a theoretical
analysis and establish consistency guarantees for the recovery. We illustrate
our results using numerical experiments on synthetic and real data,
highlighting the efficacy of our approach.Comment: 5 pages, 2 figure
Tensor Graphical Lasso (TeraLasso)
This paper introduces a multi-way tensor generalization of the Bigraphical
Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal
model for the precision matrix to parsimoniously model conditional dependence
relationships of matrix-variate data based on the Cartesian product of graphs.
We call this generalization the {\bf Te}nsor g{\bf ra}phical Lasso (TeraLasso).
We demonstrate using theory and examples that the TeraLasso model can be
accurately and scalably estimated from very limited data samples of high
dimensional variables with multiway coordinates such as space, time and
replicates. Statistical consistency and statistical rates of convergence are
established for both the BiGLasso and TeraLasso estimators of the precision
matrix and estimators of its support (non-sparsity) set, respectively. We
propose a scalable composite gradient descent algorithm and analyze the
computational convergence rate, showing that the composite gradient descent
algorithm is guaranteed to converge at a geometric rate to the global minimizer
of the TeraLasso objective function. Finally, we illustrate the TeraLasso using
both simulation and experimental data from a meteorological dataset, showing
that we can accurately estimate precision matrices and recover meaningful
conditional dependency graphs from high dimensional complex datasets.Comment: accepted to JRSS-
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