Maximal colourings for graphs

We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them

There is no going back: Properties of the non-backtracking Laplacian

We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking graphs are isomorphic, and we investigate properties of various classes of non-backtracking Laplacian eigenfunctions, such as symmetric and antisymmetric eigenfunctions. Moreover, we introduce and study circularly partite graphs as a generalization of bipartite graphs, and we use this notion to state a sharp upper bound for the spectral gap from $1$. We also investigate the singular values of the non-backtracking Laplacian in relation to independence numbers, and we use them to bound the moduli of the eigenvalues

Developing Robust Models, Algorithms, Databases and Tools With Applications to Cybersecurity and Healthcare

As society and technology becomes increasingly interconnected, so does the threat landscape. Once isolated threats now pose serious concerns to highly interdependent systems, highlighting the fundamental need for robust machine learning. This dissertation contributes novel tools, algorithms, databases, and models—through the lens of robust machine learning—in a research effort to solve large-scale societal problems affecting millions of people in the areas of cybersecurity and healthcare. (1) Tools: We develop TIGER, the first comprehensive graph robustness toolbox; and our ROBUSTNESS SURVEY identifies critical yet missing areas of graph robustness research. (2) Algorithms: Our survey and toolbox reveal existing work has overlooked lateral attacks on computer authentication networks. We develop D2M, the first algorithmic framework to quantify and mitigate network vulnerability to lateral attacks by modeling lateral attack movement from a graph theoretic perspective. (3) Databases: To prevent lateral attacks altogether, we develop MALNET-GRAPH, the world’s largest cybersecurity graph database—containing over 1.2M graphs across 696 classes—and show the first large-scale results demonstrating the effectiveness of malware detection through a graph medium. We extend MALNET-GRAPH by constructing the largest binary-image cybersecurity database—containing 1.2M images, 133×more images than the only other public database—enabling new discoveries in malware detection and classification research restricted to a few industry labs (MALNET-IMAGE). (4) Models: To protect systems from adversarial attacks, we develop UNMASK, the first model that flags semantic incoherence in computer vision systems, which detects up to 96.75% of attacks, and defends the model by correctly classifying up to 93% of attacks. Inspired by UNMASK’s ability to protect computer visions systems from adversarial attack, we develop REST, which creates noise robust models through a novel combination of adversarial training, spectral regularization, and sparsity regularization. In the presence of noise, our method improves state-of-the-art sleep stage scoring by 71%—allowing us to diagnose sleep disorders earlier on and in the home environment—while using 19× less parameters and 15×less MFLOPS. Our work has made significant impact to industry and society: the UNMASK framework laid the foundation for a multi-million dollar DARPA GARD award; the TIGER toolbox for graph robustness analysis is a part of the Nvidia Data Science Teaching Kit, available to educators around the world; we released MALNET, the world’s largest graph classification database with 1.2M graphs; and the D2M framework has had major impact to Microsoft products, inspiring changes to the product’s approach to lateral attack detection.Ph.D

An Initial Framework Assessing the Safety of Complex Systems

Trabajo presentado en la Conference on Complex Systems, celebrada online del 7 al 11 de diciembre de 2020.Atmospheric blocking events, that is large-scale nearly stationary atmospheric pressure patterns, are often associated with extreme weather in the mid-latitudes, such as heat waves and cold spells which have significant consequences on ecosystems, human health and economy. The high impact of blocking events has motivated numerous studies. However, there is not yet a comprehensive theory explaining their onset, maintenance and decay and their numerical prediction remains a challenge. In recent years, a number of studies have successfully employed complex network descriptions of fluid transport to characterize dynamical patterns in geophysical flows. The aim of the current work is to investigate the potential of so called Lagrangian flow networks for the detection and perhaps forecasting of atmospheric blocking events. The network is constructed by associating nodes to regions of the atmosphere and establishing links based on the flux of material between these nodes during a given time interval. One can then use effective tools and metrics developed in the context of graph theory to explore the atmospheric flow properties. In particular, Ser-Giacomi et al. [1] showed how optimal paths in a Lagrangian flow network highlight distinctive circulation patterns associated with atmospheric blocking events. We extend these results by studying the behavior of selected network measures (such as degree, entropy and harmonic closeness centrality)at the onset of and during blocking situations, demonstrating their ability to trace the spatio-temporal characteristics of these events.This research was conducted as part of the CAFE (Climate Advanced Forecasting of sub-seasonal Extremes) Innovative Training Network which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 813844