  We propose and analyse a general tensor-based framework for incorporating
second order features into network measures. This approach allows us to combine
traditional pairwise links with information that records whether triples of
nodes are involved in wedges or triangles. Our treatment covers classical
spectral methods and recently proposed cases from the literature, but we also
identify many interesting extensions. In particular, we define a
mutually-reinforcing (spectral) version of the classical clustering
coefficient. The underlying object of study is a constrained nonlinear
eigenvalue problem associated with a cubic tensor. Using recent results from
nonlinear Perron--Frobenius theory, we establish existence and uniqueness under
appropriate conditions, and show that the new spectral measures can be computed
efficiently with a nonlinear power method. To illustrate the added value of the
new formulation, we analyse the measures on a class of synthetic networks. We
also give computational results on centrality and link prediction for
real-world networks.
