The paper applies information theory and the theory of dissipative systems to discuss the emergence of complexity in an innovation system, as a result of its adaptation to an uneven distribution of the cognitive distance between its members. By modelling, on one hand, cognitive distance as noise, and, on the other hand, the inefficiencies linked to a bad flow of information as costs, we propose a model of the dynamics by which a horizontal network evolves into a hierarchical network, with some members emerging as intermediaries in the transfer of knowledge between seekers and problem-solvers. Our theoretical model contributes to the understanding of the evolution of an innovation system by explaining how the increased complexity of the system can be thermodynamically justified by purely internal factors. Complementing previous studies, we demonstrate mathematically that the complexity of an innovation system can increase not only to address the complexity of the problems that the system has to solve, but also to improve the performance of the system in transferring the knowledge needed to find a solution. © Springer Nature Switzerland AG 2020.