  We develop a novel method based in the sparse random graph to account the
interplay between geometric frustration and disorder in cluster magnetism. Our
theory allows to introduce the cluster network connectivity as a controllable
parameter. Two types of inner cluster geometry are considered: triangular and
tetrahedral. The theory was developed for a general, non-uniform intra-cluster
interactions, but in the present paper the results presented correspond to
uniform, anti-ferromagnetic (AF) intra-clusters interactions $J_{0}/J$. The
clusters are represented by nodes on a finite connectivity random graph, and
the inter-cluster interactions are random Gaussian distributed. The graph
realizations are treated in replica theory using the formalism of order
parameter functions, which allows to calculate the distribution of local fields
and, as a consequence, the relevant observable. In the case of triangular
cluster geometry, there is the onset of a classical Spin Liquid state at a
temperature $T^{*}/J$ and then, a Cluster Spin Glass (CSG) phase at a
temperature $T_{f}/J$. The CSG ground state is robust even for very weak
disorder or large negative $J_{0}/J$. These results does not depend on the
network connectivity. Nevertheless, variations in the connectivity strongly
affect the level of frustration $f_{p}=-\Theta_{CW}/T_{f}$ for large $J_{0}/J$.
In contrast, for the non-frustrated tetrahedral cluster geometry, the CSG
ground state is suppressed for weak disorder or large negative $J_{0}/J$. The
CSG boundary phase presents a re-entrance which is dependent on the network
connectivity.
