  We analyze, by means of statistical mechanics, a sparse network with random
competitive interactions among dichotomic variables pasted on the nodes, namely
a Viana-Bray model. The model is described by an infinite series of order
parameters (the multi-overlaps) and has two tunable degrees of freedom: the
noise level and the connectivity (the averaged number of links). We show that
there are no multiple transition lines, one for every order parameter, as a
naive approach would suggest, but just one corresponding to ergodicity
breaking. We explain this scenario within a novel and simple mathematical
technique via a driving mechanism such that, as the first order parameter (the
two replica overlap) becomes different from zero due to a real second order
phase transition (with properly associated diverging rescaled fluctuations), it
enforces all the other multi-overlaps toward positive values thanks to the
strong correlations which develop among themselves and the two replica overlap
at the critical line.
