All characterizations of non degenerate multivariate tail dependence structure are both functional and non finite dimensional, as the stable tail dependence function $\ell$. Taking advantage of the Hoeffding-Sobol decomposition of $\ell$, we derive new measures to summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph in which vertices represent components of the vector of interest and where edge weights are proportional to the tail superset importance coefficients. For inference, as soon as an estimator of is given, it yields an estimation of the tail dependograph. In particular, the empirical tail superset importance coefficient is rank-based statistics and its asymptotic behavior is stated. These new concepts are illustrated with several examples including theoretical models through simulation and real data, which shows that our methodology works well in practice.