Structural trade-offs can predict rewiring in shrinking social networks.

There is growing evidence that organisms can respond to declining population sizes by adapting their interactions with others. Regulating connections with others could underpin resilience of biological networks spanning from social groups to ecological communities. However, our ability to predict the dynamics of shrinking social networks remains limited. Network regulation involves several trade-offs. Removing nodes (and therefore their connections) from networks reduces the number of connections among remaining nodes. Responding by forming new connections then impacts other network properties. A simple way to minimize the impact of up-regulating network connections is to form new connections or to strengthen connections, between nodes that share a lost connection with a recently removed node. I propose a simple 'second-degree rewiring' rule as a biologically plausible regulatory mechanism in shrinking social networks. I argue that two individuals that have lost a connection with a common removed individual will both be more likely, or more willing, to form a new, or strengthen an existing, connection among themselves. I then show that such second-degree rewiring has less impact on important structural properties of the network than forming random new connections. For example, in a network with phenotypic assortment, second-degree nodes are more likely to be similar than any random pair of nodes, and connecting these will better maintain assortativity. This simple rule can therefore maintain network properties without individuals having any knowledge of the global structure of the network or the relative properties of the nodes within it. In this paper, I outline an algorithm for second-degree rewiring. I demonstrate how second-degree rewiring can have less impact than adding new, or increasing the strength of, random connections on both the individual and whole network properties. That is, relative to randomly adding or strengthening connections, second-degree rewiring has less impact on mean degree, assortativity, clustering and network density. I then demonstrate empirically, using social networks of great tits (Parus major), that individuals that previously shared connections to a removed conspecific were more likely to form a new connection or to strengthen their connection, relative to other individuals in the same population. This study highlights how developing a better mechanistic understanding of the structural properties of networks, and the consequences of adding new connections, can provide useful insights into how organisms are likely to regulate their interactions in shrinking populations.