Metrics for Graph Comparison: A Practitioner's Guide

Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and comparison of structures such as modular communities, rich clubs, hubs, and trees in data in these fields yields insight into the generative mechanisms and functional properties of the graph. Often, two graphs are compared via a pairwise distance measure, with a small distance indicating structural similarity and vice versa. Common choices include spectral distances (also known as $\lambda$ distances) and distances based on node affinities. However, there has of yet been no comparative study of the efficacy of these distance measures in discerning between common graph topologies and different structural scales. In this work, we compare commonly used graph metrics and distance measures, and demonstrate their ability to discern between common topological features found in both random graph models and empirical datasets. We put forward a multi-scale picture of graph structure, in which the effect of global and local structure upon the distance measures is considered. We make recommendations on the applicability of different distance measures to empirical graph data problem based on this multi-scale view. Finally, we introduce the Python library NetComp which implements the graph distances used in this work

Graph theory applied to neuroimaging data reveals key functional connectivity alterations in brain of behavioral variant Frontotemporal Dementia subjects

Brain functional architecture and anatomical structure have been intensively studied to generate efficient models of its complex mechanisms. Functional alterations and cognitive impairments are the most investigated aspects in the recent clinical research as distinctive traits of neurodegeneration. Although specific behaviours are clearly associated to neurodegeneration, information flow breakdown within the brain functional network, responsible to deeply affect cognitive skills, remains not completely understood. Behavioural variant Frontotemporal Dementia (bvFTD) is the most common type of Frontotemporal degeneration, marked by behavioural disturbances, social instabilities and impairment of executive functions. Mathematical modelling offers effective tools to inspect deviations from physiological cognitive functions and connectivity alterations. As a popular recent methodology, graph theoretical approaches applied to imaging data expanded our knowledge of neurodegenerative disorders, although the need for unbiased metrics is still an open issue. In this thesis, we propose an integrated analysis of functional features among brain areas in bvFTD patients, to assess global connectivity and topological network alterations respect to the healthy condition, using a minimum spanning tree (MST) based-model to resting state functional MRI (rs-fMRI) data. Contrary to several graph theoretical approaches, dependent to arbitrary criteria (e.g., correlation thresholds, network density or a priori distribution), MST represents an unambiguous modelling solution, ensuring full reproducibility and robustness in different conditions. Our MSTs were obtained from wavelet correlation matrices derived from mean time series intensities, extracted from 116 regions of interest (ROIs) of 41 bvFTD patients and 39 healthy controls (HC), which underwent rs-fMRI. The resulting graphs were tested for global connectivity and topological differences between the two groups, by applying a Wilcoxon rank sum test with a significance level at 0.05 (nonparametric median difference estimates with 95% confidence interval). The same test was applied for methodological comparison between MST and other common graph theory methods. After methodological comparisons, our MST model achieved the best bvFTD/HC separation performances, without a priori assumptions. Direct MST comparison between bvFTD and healty controls revealed key brain functional architecture differences. Diseased subjects showed a linear-shape network configuration tendency, with high distance between nodes, low centrality parameter values, and a low exchange information capacity (i.e., low network integration) in MST parameters. Moreover, edge-level and node-level features (i.e., superhighways, and node degree and betweenness centrality) indicated a more complex scenario, showing some of the key bvFTD dysfunctions observed in large scale resting-state functional networks (default-mode (DMN), salience (SN), and executive (EN) networks), suggesting an underlying involvement of the limbic system in the observed functional deterioration. Functional isolation has been observed as a generalized process affecting the entire bvFTD network, showing brain macro-regions isolation, with homogeneous functional distribution of brain areas, longer distances between hubs, and longer within-lobe superhighways. Conversely, the HC network showed marked functional integration, where superhighways serve as shortcuts to connect areas from different brain macro-regions. The combination of this theoretical model with rs-fMRI data constitutes an effective method to generate a clear picture of the functional divergence between bvFTD and HCs, providing possible insights on the effects of frontotemporal neurodegeneration and compensatory mechanisms underlying characteristic bvFTD cognitive, social, and executive impairments

Accounting for the complex hierarchical topology of EEG phase-based functional connectivity in network binarisation

Research into binary network analysis of brain function faces a methodological challenge in selecting an appropriate threshold to binarise edge weights. For EEG phase-based functional connectivity, we test the hypothesis that such binarisation should take into account the complex hierarchical structure found in functional connectivity. We explore the density range suitable for such structure and provide a comparison of state-of-the-art binarisation techniques, the recently proposed Cluster-Span Threshold (CST), minimum spanning trees, efficiency-cost optimisation and union of shortest path graphs, with arbitrary proportional thresholds and weighted networks. We test these techniques on weighted complex hierarchy models by contrasting model realisations with small parametric differences. We also test the robustness of these techniques to random and targeted topological attacks.We find that the CST performs consistenty well in state-of-the-art modelling of EEG network topology, robustness to topological network attacks, and in three real datasets, agreeing with our hypothesis of hierarchical complexity. This provides interesting new evidence into the relevance of considering a large number of edges in EEG functional connectivity research to provide informational density in the topology.Comment: Accepted for publication in PLOS One, 27th September 201

Sizing the length of complex networks

Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short

Topological network alignment uncovers biological function and phylogeny

Sequence comparison and alignment has had an enormous impact on our understanding of evolution, biology, and disease. Comparison and alignment of biological networks will likely have a similar impact. Existing network alignments use information external to the networks, such as sequence, because no good algorithm for purely topological alignment has yet been devised. In this paper, we present a novel algorithm based solely on network topology, that can be used to align any two networks. We apply it to biological networks to produce by far the most complete topological alignments of biological networks to date. We demonstrate that both species phylogeny and detailed biological function of individual proteins can be extracted from our alignments. Topology-based alignments have the potential to provide a completely new, independent source of phylogenetic information. Our alignment of the protein-protein interaction networks of two very different species--yeast and human--indicate that even distant species share a surprising amount of network topology with each other, suggesting broad similarities in internal cellular wiring across all life on Earth.Comment: Algorithm explained in more details. Additional analysis adde