#### Bernat Corominas-Murtra, Joaquin Goñi, Carlos Rodríguez-Caso, Ricard Solé

Paper #: 10-12-036

In this paper we define a hierarchical index for feedforward structures taking, as the starting point, three fundamental concepts underlying hierarchy: order, predictability and pyramidal structure. Our definition applies to the so-called causal graphs, i.e., connected, directed acyclic graphs in which the arrows depict a direct causal relation between two elements defining the nodes. The estimator of hierarchy is obtained by evaluating the complexity of causal paths against the uncertainty in recovering them from a given end point. This naturally leads us to a definition of mutual information which, properly normalized and weighted through the layered structure of the graph, results in suitable index of hierarchy with strong theoretical grounds.