Andreas Loukas

Distributed Autoregressive Moving Average Graph Filters

We introduce the concept of autoregressive moving average (ARMA) filters on a graph and show how they can be implemented in a distributed fashion. Our graph filter design philosophy is independent of the particular graph, meaning that the filter coefficients are derived irrespective of the graph. In contrast to finite-impulse response (FIR) graph filters, ARMA graph filters are robust against changes in the signal and/or graph. In addition, when time-varying signals are considered, we prove that the proposed graph filters behave as ARMA filters in the graph domain and, depending on the implementation, as first or higher order ARMA filters in the time domain.

Lightweight neighborhood cardinality estimation in dynamic wireless networks

We address the problem of estimating the neighborhood cardinality of nodes in dynamic wireless networks. Different from previous studies, we consider networks with high densities (a hundred neighbors per node) and where all nodes estimate cardinality concurrently. Performing concurrent estimations on dense mobile networks is hard; we need estimators that are not only accurate, but also fast, asynchronous (due to mobility) and lightweight (due to concurrency and high density). To cope with these requirements, we propose Estreme, a neighborhood cardinality estimator with extremely low overhead that leverages the rendezvous time of low-power medium access control (MAC) protocols. We implemented Estreme on the Contiki OS and show a significant improvement over the state-of-the-art. With Estreme, 100 nodes can concurrently estimate their neighborhood cardinality with an error of ≈10%. State-of-the-art solutions provide a similar accuracy, but on networks consisting of a few tens of nodes and where only a fraction of nodes estimate the cardinality concurrently.

Think globally, act locally: on the reshaping of information landscapes

In large-scale resource-constrained systems, such as wireless sensor networks, global objectives should be ideally achieved through inexpensive local interactions. A technique satisfying these requirements is information potentials, in which distributed functions disseminate information about the process monitored by the network. Information potentials are usually computed through local aggregation or gossiping. These methods however, do not consider the topological properties of the network, such as node density, which could be exploited to enhance the performance of the system. This paper proposes a novel aggregation method with which a potential becomes sensitive to the network topology. Our method introduces the notion of affinity spaces, which allow us to uncover the deep connections between the aggregation scope (the radius of the extended neighborhood whose information is aggregated) and the networks Laplacian (which captures the topology of the connectivity graph). Our study provides two additional contributions: (i) It characterizes the convergence of information potentials for static and dynamic networks. Our analysis captures the impact of key parameters, such as node density, time-varying information, as well as of the addition (or removal) of links and nodes. (ii) It shows that information potentials are decomposed into wave-like eigenfunctions that depend on the aggregation scope. This result has important implications, for example it prevents greedy routing techniques from getting stuck by eliminating local-maxima. Simulations and experimental evaluation show that our main findings hold under realistic conditions, with unstable links and message loss.

How to identify global trends from local decisions? Event region detection on mobile networks

The decentralized detection of event regions is a fundamental building block for monitoring and reasoning about spatial phenomena. However, so far the problem has been studied almost exclusively for static networks. This study proposes a theoretical framework with which we can analyze event detection algorithms suitable for large-scale mobile networks. Our analysis builds on the following insight: the inherent trends of spatial events are well captured by the spectral domain of the network graph. Using this framework, we propose novel local algorithms that are location-free; that work with mobile nodes and dynamic events; that operate on 3D topologies; and that are simple to implement. We are not aware of event detection algorithms possessing all these traits. Simulations based on complex oil spill traces showcase the resilience and robustness of our methods. Additionally, we demonstrate their validity for practical scenarios by evaluating them on a 105 node testbed.

Autoregressive Moving Average Graph Filtering

One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogs of classical filters, but intended for signals defined on graphs. This paper brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which are able to approximate any desired graph frequency response, and give exact solutions for specific graph signal denoising and interpolation problems. The philosophy to design the ARMA coefficients independently from the underlying graph renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph topology are changing over time. We show that in case of a time-varying graph signal, our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domain. We also derive the graph filter behavior, as well as sufficient conditions for filter stability when the graph and signal are time varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.

Towards stationary time-vertex signal processing

Graph-based methods for signal processing have shown promise for the analysis of data exhibiting irregular structure, such as those found in social, transportation, and sensor networks. Yet, though these systems are often dynamic, state-of-the-art methods for graph signal processing ignore the time dimension. To address this shortcoming, this paper considers the statistical analysis of time-varying graph signals. We introduce a novel definition of joint (time-vertex) stationarity, which generalizes the classical definition of time stationarity and the recent definition appropriate for graphs. This gives rise to a scalable Wiener optimization framework for denoising, semi-supervised learning, or more generally inverting a linear operator, that is provably optimal. Experimental results on real weather data demonstrate that taking into account graph and time dimensions jointly can yield significant accuracy improvements in the reconstruction effort.

Separable autoregressive moving average graph-temporal filters

Despite their widespread use for the analysis of graph data, current graph filters are designed for graph signals that do not change over time, and thus they cannot simultaneously process time and graph frequency content in an adequate manner. This work presents ARMA2D, an autoregressive moving average graph-temporal filter that captures jointly the signal variations over the graph and time. By its unique nature, this filter is able to achieve a separable 2-dimensional frequency response, making it possible to approximate the filtering specifications along both the graph and temporal frequency domains. Numerical results show that the proposed solution outperforms the state of the art graph filters when the graph signal is time-varying.

On distributed computation of information potentials

A common task of mobile wireless ad-hoc networks is to distributedly extract information from a monitored process. We define process information as a measure that is sensed and computed by each mobile node in a network. For complex tasks, such as searching in a network and coordination of robotic swarms, we are typically interested in the spatial distribution of the process information. Spatial distributions can be thought of as information potentials that recursively consider the richness of information around each node. This paper describes a localized mechanism for determining the information potential on each node based on local process information and the potential of neighboring nodes. The mechanism allows us to distributedly generate a spectrum of possible information potentials between the extreme points of a local view and distributed averaging. In this work, we describe the mechanism, prove its exponential convergence, and characterize the spectrum of information potentials. Moreover, we use the mechanism to generate information potentials that are unimodal, i.e., feature a single extremum. Unimodality is a very valuable property for chemotactic search, which can be used in diverse application tasks such as directed search of information and rendezvous of mobile agents.

Graph scale-space theory for distributed peak and pit identification

Graph filters are a recent and powerful tool to process information in graphs. Yet despite their advantages, graph filters are limited. The limitation is exposed in a filtering task that is common, but not fully solved in sensor networks: the identification of a signals peaks and pits. Choosing the correct filter necessitates a-priori information about the signal and the network topology. Furthermore, in sparse and irregular networks graph filters introduce distortion, effectively rendering identification inaccurate, even when signal-specific information is available. Motivated by the need for a multi-scale approach, this paper extends classical results on scale-space analysis to graphs. We derive the family of scale-space kernels (or filters) that are suitable for graphs and show how these can be used to observe a signal at all possible scales: from fine to coarse. The gathered information is then used to distributedly identify the signals peaks and pits. Our graph scale-space approach diminishes the need for a-priori knowledge, and reduces the effects caused by noise, sparse and irregular topologies, exhibiting: (i) superior resilience to noise than the state-of-the-art, and (ii) at least 20% higher precision than the best graph filter, when evaluated on our testbed.

Stochastic graph filtering on time-varying graphs

We have recently seen a surge of work on distributed graph filters, extending classical results to the graph setting. State of the art filters have however only been examined from a deterministic standpoint, ignoring the impact of stochasticity in the computation (e.g., temporal fluctuation of links) and input (e.g., the value of each node is a random process). Initiating the study of stochastic graph signal processing, this paper shows that a prominent class of graph filters, namely autoregressive moving average (ARMA) filters, are suitable for the stochastic setting. In particular, we prove that an ARMA filter that operates on a stochastic signal over a stochastic graph is equivalent, in the mean, to the same filter operating on the expected signal over the expected graph. We also characterize the variance of the output and we provide an upper bound for its average value among different nodes. Our results are validated by numerical simulations.