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A weighted directed network (WDN) is a directed graph in which each edge isassociated to a unique value called weight. These networks are very suitablefor modeling real-world social networks in which there is an assessment of onevertex toward other vertices. One of the main problems studied in this paper isprediction of edge weights in such networks. We introduce, for the first time,a metric geometry approach to studying edge weight prediction in WDNs. Wemodify a usual notion of WDNs, and introduce a new type of WDNs which we cointhe term \textit{almost-weighted directed networks} (AWDNs). AWDNs can capturethe weight information of a network from a given training set. We thenconstruct a class of metrics (or distances) for AWDNs which equips suchnetworks with a metric space structure. Using the metric geometry structure ofAWDNs, we propose modified $k$ nearest neighbors (kNN) methods and modifiedsupport-vector machine (SVM) methods which will then be used to predict edgeweights in AWDNs. In many real-world datasets, in addition to edge weights, onecan also associate weights to vertices which capture information of vertices;association of weights to vertices especially plays an important role in graphembedding problems. Adopting a similar approach, we introduce two new types ofdirected networks in which weights are associated to either a subset of originvertices or a subset of terminal vertices . We, for the first time, constructnovel classes of metrics on such networks, and based on these new metricspropose modified $k$NN and SVM methods for predicting weights of origins andterminals in these networks. We provide experimental results on severalreal-world datasets, using our geometric methodologies.

Weight Prediction for Variants of Weighted Directed Networks