The Wiener Polynomial of the k

We presented a formula for the Wiener polynomial of the kth power graph. We use this formula to find the Wiener polynomials of the kth power graphs of paths, cycles, ladder graphs, and hypercubes. Also, we compute the Wiener indices of these graphs

Graph-Facilitated Resonant Mode Counting in Stochastic Interaction Networks

Oscillations in a stochastic dynamical system, whose deterministic counterpart has a stable steady state, are a widely reported phenomenon. Traditional methods of finding parameter regimes for stochastically-driven resonances are, however, cumbersome for any but the smallest networks. In this letter we show by example of the Brusselator how to use real root counting algorithms and graph theoretic tools to efficiently determine the number of resonant modes and parameter ranges for stochastic oscillations. We argue that stochastic resonance is a network property by showing that resonant modes only depend on the squared Jacobian matrix $J^2$ , unlike deterministic oscillations which are determined by $J$. By using graph theoretic tools, analysis of stochastic behaviour for larger networks is simplified and chemical reaction networks with multiple resonant modes can be identified easily.Comment: 5 pages, 4 figure

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering

We study causal waveform estimation (tracking) of time-varying signals in a paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation probing. We use Kalman filtering, which optimally tracks known linear Gaussian stochastic processes, to estimate stochastic input signals that we generate by optical pumping. Comparing the known input to the estimates, we confirm the accuracy of the atomic statistical model and the reliability of the Kalman filter, allowing recovery of waveform details far briefer than the sensor's intrinsic time resolution. With proper filter choice, we obtain similar benefits when tracking partially-known and non-Gaussian signal processes, as are found in most practical sensing applications. The method evades the trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure

Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering

We study causal waveform estimation (tracking) of time-varying signals in a paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation probing. We use Kalman filtering, which optimally tracks known linear Gaussian stochastic processes, to estimate stochastic input signals that we generate by optical pumping. Comparing the known input to the estimates, we confirm the accuracy of the atomic statistical model and the reliability of the Kalman filter, allowing recovery of waveform details far briefer than the sensor's intrinsic time resolution. With proper filter choice, we obtain similar benefits when tracking partially-known and non-Gaussian signal processes, as are found in most practical sensing applications. The method evades the trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure

Forecasting Time Series with VARMA Recursions on Graphs

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This work provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., timevarying graph signals whose first and second order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with VAR and VARMA models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain: (i) the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate ARMA models and (ii) an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.Comment: submitted to the IEEE Transactions on Signal Processin

1/f3/21/f^{3/2} noise in heavy traffic M/M/1 queue

We study the power spectral density of continuous time Markov chains and explicit its relationship with the eigenstructure of the infinitesimal generator. This result helps us understand the dynamics of the number of customers for a M/M/1 queuing process in the heavy traffic regime.Closed-form relations for the power law scalings associated to the eigenspectrum of the M/M/1 queue generator are obtained, providing a detailed description of the power spectral density structure, which is shown to exhibit a $1/f^{3/2}$ noise.We confirm this result by numerical simulation.We also show that a continuous time random walk on a ring exhibits very similar behavior.It is remarkable than a complex behavior such as $1/f$ noise can emerge from the M/M/1 queue, which is the ''simplest'' queuing model

Advances in Distributed Graph Filtering

Graph filters are one of the core tools in graph signal processing. A central aspect of them is their direct distributed implementation. However, the filtering performance is often traded with distributed communication and computational savings. To improve this tradeoff, this work generalizes state-of-the-art distributed graph filters to filters where every node weights the signal of its neighbors with different values while keeping the aggregation operation linear. This new implementation, labeled as edge-variant graph filter, yields a significant reduction in terms of communication rounds while preserving the approximation accuracy. In addition, we characterize the subset of shift-invariant graph filters that can be described with edge-variant recursions. By using a low-dimensional parametrization the proposed graph filters provide insights in approximating linear operators through the succession and composition of local operators, i.e., fixed support matrices, which span applications beyond the field of graph signal processing. A set of numerical results shows the benefits of the edge-variant filters over current methods and illustrates their potential to a wider range of applications than graph filtering