Detecting network anomalies using Forman-Ricci curvature and a case study for human brain networks.

We analyze networks of functional correlations between brain regions to identify changes in their structure caused by Attention Deficit Hyperactivity Disorder (ADHD). We express the task for finding changes as a network anomaly detection problem on temporal networks. We propose the use of a curvature measure based on the Forman-Ricci curvature, which expresses higher-order correlations among two connected nodes. Our theoretical result on comparing this Forman-Ricci curvature with another well-known notion of network curvature, namely the Ollivier-Ricci curvature, lends further justification to the assertions that these two notions of network curvatures are not well correlated and therefore one of these curvature measures cannot be used as an universal substitute for the other measure. Our experimental results indicate nine critical edges whose curvature differs dramatically in brains of ADHD patients compared to healthy brains. The importance of these edges is supported by existing neuroscience evidence. We demonstrate that comparative analysis of curvature identifies changes that more traditional approaches, for example analysis of edge weights, would not be able to identify.