  We propose the use of the Distributional Zeta-Function (DZF) for constructing
a new set of Systemic Performance Measures (SPM). SPM have been proposed to
investigate network synthesis problems such as the growing of linear consensus
networks. The adoption of the DZF has shown interesting physical consequences
that in the usual replica method are still unclarified, i.e., the connection
between the spontaneous symmetry breaking mechanism and the structure of the
replica space in the disordered model. We relate topology of the network and
the partition function present in the DZF by using the spectral and the
Hamiltonian structure of the system. The studied objects are the generalized
partition funcion, the DZF, the Expected value of the replica partition
function, and the quenched free energy of a field network. We show that with
these objects we need few operations to increase the percentage of performance
enhancement of a network. Furthermore, we evalue the location of the optimal
added links for each new SPM and calculate the performance improvement of the
new network for each new SPM via the spectral zeta function,
$\mathcal{H}_{2}$-norm, and the communicability between nodes. We present the
advantages of this new set of SPM in the network synthesis and we propose other
methods for using the DZF to explore some issues such as disorder, critical
phenomena, finite-temperature, and finite-size effects on networks. Relevance
of the results are discussed.
