Complex fluctuations with correlations involving multiple scales appear in many physical, social and biological systems. In particular, in physiological systems the degree of complexity, measured in terms of the exponent of the time correlations of the fluctuations, is altered with disease and ageing. Here, we show that correlated fluctuations characterized by 1/f scaling of their power spectra can emerge from networks of simple signalling units. We analyse networks of simple signalling units where the type of scaling of the fluctuations is associated with ( i) a complex topology with a discrete and sparse number of random links between units, ( ii) a restricted set of nonlinear interaction rules, and ( iii) the presence of noise. Furthermore, we find that changes in one or more of these properties leads to degradation of the correlation properties. Moreover, changes in the microscopic construction of the model do not produce qualitative changes in the dynamical behaviour, showing hence the robustness of our findings.