Biological networks are many and varied, including food webs describing who eats whom in an ecosystem, gene regulation networks in which genes are connected through suppression and enhancement, or mutualistic networks in which plants are connected to their pollinators. What they have in common is that they are all large (containing thousands of nodes) and possess definite structure, deviating considerably from simple random graphs. In these networks the nodes interact with one another, but also with themselves: when the nodes represent species, a negative self-effect (self-regulation) generally arises from intraspecific competition for limited resources. One important question concerns the role of self-effects in stabilizing network dynamics. While it is easy to construct networks which can be stabilized by only a handful of nodes exhibiting self-regulation, their structure is highly special, and quite different from what is observed empirically. Here I show that, in general, the vast majority of nodes must exhibit substantial self-regulation in order to stabilize dynamics. The argument holds for both random and highly structured empirical networks. I show that in the case of competitive networks the arrangement of self-interactions leading to the most/least stabilized configurations can be justified analytically, and are confirmed by thorough simulations.