  We propose a method of analysis of dynamical networks based on a recent
measure of Granger causality between time series, based on kernel methods. The
generalization of kernel Granger causality to the multivariate case, here
presented, shares the following features with the bivariate measures: (i) the
nonlinearity of the regression model can be controlled by choosing the kernel
function and (ii) the problem of false-causalities, arising as the complexity
of the model increases, is addressed by a selection strategy of the
eigenvectors of a reduced Gram matrix whose range represents the additional
features due to the second time series. Moreover, there is no {\it a priori}
assumption that the network must be a directed acyclic graph. We apply the
proposed approach to a network of chaotic maps and to a simulated genetic
regulatory network: it is shown that the underlying topology of the network can
be reconstructed from time series of node's dynamics, provided that a
sufficient number of samples is available. Considering a linear dynamical
network, built by preferential attachment scheme, we show that for limited data
use of bivariate Granger causality is a better choice w.r.t methods using $L1$
minimization. Finally we consider real expression data from HeLa cells, 94
genes and 48 time points. The analysis of static correlations between genes
reveals two modules corresponding to well known transcription factors; Granger
analysis puts in evidence nineteen causal relationships, all involving genes
related to tumor development.
