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International institutions strive to achieve their own objectives while operating within a complex network of interdependencies. These interdependencies create an extensive web of relationships that serve as potential pathways for broader institutional impacts. The actions taken by individual institutions, their mutual influences, and the pattern of connectivity collectively shape the overall performance of institutional complexes. Understanding the performance of these complexes is crucial, yet we currently lack a methodology to assess it. To address this gap, we have developed a conceptual framework that integrates three distinct areas of study on three different scales: institutional effectiveness, institutional interlinkages, and institutional networks. This framework enables us to consider what we call the latent net effectiveness, or collective problem-solving potential, of a group of interconnected institutions. To put this framework into practice, we have devised a heuristic model, drawing from the extensive literature on international environmental institutions. As an illustrative example, we have applied this model to a network of 378 multilateral environmental agreements with 810 known issue linkages. Our work underscores the relevance of considering the system-level properties of institutional complexes and emphasizes the need for timely network-based governance and policy interventions to enhance the overall effectiveness of institutional complexes. This article is part of the theme issue 'A complexity science approach to law and governance'. [ABSTRACT FROM AUTHOR]