Edge centrality via the Holevo quantity

In the study of complex networks, vertex centrality measures are used to identify the most important vertices within a graph. A related problem is that of measuring the centrality of an edge. In this paper, we propose a novel edge centrality index rooted in quantum information. More specifically, we measure the importance of an edge in terms of the contribution that it gives to the Von Neumann entropy of the graph. We show that this can be computed in terms of the Holevo quantity, a well known quantum information theoretical measure. While computing the Von Neumann entropy and hence the Holevo quantity requires computing the spectrum of the graph Laplacian, we show how to obtain a simplified measure through a quadratic approximation of the Shannon entropy. This in turns shows that the proposed centrality measure is strongly correlated with the negative degree centrality on the line graph. We evaluate our centrality measure through an extensive set of experiments on real-world as well as synthetic networks, and we compare it against commonly used alternative measures.

The average mixing matrix signature

Laplacian-based descriptors, such as the Heat Kernel Signature and the Wave Kernel Signature, allow one to embed the vertices of a graph onto a vectorial space, and have been successfully used to find the optimal matching between a pair of input graphs. While the HKS uses a heat di↵usion process to probe the local structure of a graph, the WKS attempts to do the same through wave propagation. In this paper, we propose an alternative structural descriptor that is based on continuoustime quantum walks. More specifically, we characterise the structure of a graph using its average mixing matrix. The average mixing matrix is a doubly-stochastic matrix that encodes the time-averaged behaviour of a continuous-time quantum walk on the graph. We propose to use the rows of the average mixing matrix for increasing stopping times to develop a novel signature, the Average Mixing Matrix Signature (AMMS). We perform an extensive range of experiments and we show that the proposed signature is robust under structural perturbations of the original graphs and it outperforms both the HKS and WKS when used as a node descriptor in a graph matching task.

Quantum kernels for unattributed graphs using discrete-time quantum walks

In this paper, we develop a new family of graph kernels where the graph structure is probed by means of a discrete-time quantum walk. Given a pair of graphs, we let a quantum walk evolve on each graph and compute a density matrix with each walk. With the density matrices for the pair of graphs to hand, the kernel between the graphs is defined as the negative exponential of the quantum Jensen–Shannon divergence between their density matrices. In order to cope with large graph structures, we propose to construct a sparser version of the original graphs using the simplification method introduced in Qiu and Hancock (2007). To this end, we compute the minimum spanning tree over the commute time matrix of a graph. This spanning tree representation minimizes the number of edges of the original graph while preserving most of its structural information. The kernel between two graphs is then computed on their respective minimum spanning trees. We evaluate the performance of the proposed kernels on several standard graph datasets and we demonstrate their effectiveness and efficiency.

Towards person-centered neuroimaging markers for resilience and vulnerability in Bipolar Disorder

Improved clinical care for Bipolar Disorder (BD) relies on the identification of diagnostic markers that can reliably detect disease-related signals in clinically heterogeneous populations. At the very least, diagnostic markers should be able to differentiate patients with BD from healthy individuals and from individuals at familial risk for BD who either remain well or develop other psychopathology, most commonly Major Depressive Disorder (MDD). These issues are particularly pertinent to the development of translational applications of neuroimaging as they represent challenges for which clinical observation alone is insufficient. We therefore applied pattern classification to task-based functional magnetic resonance imaging (fMRI) data of the n-back working memory task, to test their predictive value in differentiating patients with BD (n=30) from healthy individuals (n=30) and from patients' relatives who were either diagnosed with MDD (n=30) or were free of any personal lifetime history of psychopathology (n=30). Diagnostic stability in these groups was confirmed with 4-year prospective follow-up. Task-based activation patterns from the fMRI data were analyzed with Gaussian Process Classifiers (GPC), a machine learning approach to detecting multivariate patterns in neuroimaging datasets. Consistent significant classification results were only obtained using data from the 3-back versus 0-back contrast. Using contrast, patients with BD were correctly classified compared to unrelated healthy individuals with an accuracy of 83.5%, sensitivity of 84.6% and specificity of 92.3%. Classification accuracy, sensitivity and specificity when comparing patients with BD to their relatives with MDD, were respectively 73.1%, 53.9% and 94.5%. Classification accuracy, sensitivity and specificity when comparing patients with BD to their healthy relatives were respectively 81.8%, 72.7% and 90.9%. We show that significant individual classification can be achieved using whole brain pattern analysis of task-based working memory fMRI data. The high accuracy and specificity achieved by all three classifiers suggest that multivariate pattern recognition analyses can aid clinicians in the clinical care of BD in situations of true clinical uncertainty regarding the diagnosis and prognosis.