Basic concept of network centralities and our ranking method.

<p>(a) Three basic network centralities. The <i>Degree</i> is the number of node's neighbors; the shaded node in the most central. The <i>Eigenvalue Centrality</i> considers the quality of a connection, so that being connected to a central node raises one's centrality in turn; the larger shaded node is more central than the smaller shaded node, although their degrees are equal. The <i>Betweenness</i> quantifies the node's role in acting as an intermediary between nodes by measuring how often it sits on the geodesic (shortest) paths between two nodes; the shaded node, even though its degree is low, is the most central. (b) In PageRank, with probability a random walker follows a randomly chosen outgoing link (solid lines) to travel to another node, and with probability makes a random jump to any node in the network (red dotted lines). The nodes are ranked by their stationary occupation probability. (c) A competition network is a directed network with weighted directional edges, where the weights can represent the number of wins or the points scored by one node against another. Our ranking method is based on random walk where the edge weights define a gradient between nodes. We use the stationary occupation probability as the measure of node's strength or weakness, depending on the defined directionality of the gradient. (d) A high degree can unfairly favor and penalize a node, necessitating a degree-neutralizing procedure.