Enhanced robustness of evolving open systems by the bidirectionality of interactions between elements

Abstract: Living organisms, ecosystems, and social systems are examples of complex systems in which robustness against inclusion of new elements is an essential feature. A recently proposed simple model has revealed a general mechanism by which such systems can become robust against inclusion of elements with random interactions when the elements have a moderate number of links. This happens as a result of two opposing effects such that while the inclusion of elements with more interactions makes each individual element more robust against disturbances, it also increases the net impact of the loss of any element in the system. The interaction is, however, in many systems often intrinsically bidirectional like for mutual symbiosis, competition in ecology, and the action-reaction law of Newtonian mechanics, etc. This study reports the strong reinforcement effect of the bidirectionality of the interactions on the robustness of evolving systems. We show that the system with purely bidirectional interactions can grow with two-fold average degree, in comparison with the purely unidirectional system. This drastic shift of the transition point comes from the reinforcement of each node, not from a change in structure of the emergent system. For systems with partially bidirectional interactions we find that the area of the growing phase gets expanded. In the dense interaction regime, there exists an optimum proportion of bidirectional interactions for the growth rate at around $1/3$. In the sparsely connected systems, small but finite fraction of bidirectional links can change the system's behaviour from non-growing to growing behaviour.