Abstract Preferential attachment is an important Classical Normal Tension mechanism in the structural evolution of complex networks.However, though resources on a network propagate and have an effect beyond a direct relationship, growth by preferential attachment based on indirectly propagated resources has not been systematically investigated.Here, we propose a mathematical model of an evolving network in which preference is proportional to a utility function reflecting direct utility from directly connected nodes and indirect utility from indirectly connected nodes beyond the directly connected nodes.Our analysis showed that preferential attachment involving indirect utility forms a converged and hierarchical structure, thereby significantly increasing the indirect utility across the entire network.
Further, we found that the structures are formed by mutual growth between adjacent nodes, which promotes a scaling exponent of 1.5 between the number of indirect and direct links.Lastly, by examining several real networks, we found evidence of mutual growth, especially in social networks.Our findings demonstrate a growth mechanism emerging in evolving networks ORG SPIRULINA POWDER with preference for indirect utility, and provide a foundation for systematically investigating the role of preference for indirect utility in the structural and functional evolution of large-scale social networks.