Complexity stems from the interactions among elements of a system. The tool closest to a standard for the formal representation of complexity, the NK model, identifies the degree of complexity as the number of interactions among the elements. However, the literature has already shown that the simple count of interactions is a poor measure of complexity, potentially even leading to erroneous or misleading results. Two additional features of interactions must be taken into account in order to fully represent the true nature of complexity: the intensity and the structure of interactions. Making explicit these two further aspects of complexity leads to a far clearer account of known results, and allows to explore new application of complexity studies. This paper discusses the role of the intensity and the structure of interactions in a number of examples, some already known in the literature and some formalizing old conjectures. For this purpose we propose a modification of the NK model, called pseudo-NK (pNK) meant to extend the possibility of NK and removing some of its limitations, both technical and theoretical. Firstly, the new model is deterministic, removing the need to accumulate vast data sets as required by the stochastic NK. Secondly, pNK defines as search space a N-dimensional real-valued space, allowing for more intuitive applications in respect of the binary space assumed by NK. Thirdly, pNK permits to define not only presence-absence of interactions but also a continuum of intermediate levels, allowing to implement the common understanding of weaker and stronger interaction. We sustain that the new model is able to represent more clearly the same results as produced by NK, and to extend the potential analysis in areas where NK cannot be applied.