Scaling of differentiation in networks: nervous systems, organisms, ant colonies, ecosystems, businesses, universities, cities, electronic circuits, and Legos.

Nodes in networks are often of different types, and in this sense networks are differentiated. Here we examine the relationship between network differentiation and network size in networks under economic or natural selective pressure, such as electronic circuits (networks of electronic components), Legos (networks of Lego pieces), businesses (networks of employees), universities (networks of faculty), organisms (networks of cells), ant colonies (networks of ants), and nervous systems (networks of neurons). For each of these we find that (i) differentiation increases with network size, and (ii) the relationship is consistent with a power law. These results are explained by a hypothesis that, because nodes are costly to build and maintain in such "selected networks", network size is optimized, and from this the power-law relationship may be derived. The scaling exponent depends on the particular kind of network, and is determined by the degree to which nodes are used in a combinatorial fashion to carry out network-level functions. We find that networks under natural selection (organisms, ant colonies, and nervous systems) have much higher combinatorial abilities than the networks for which human ingenuity is involved (electronic circuits, Legos, businesses, and universities). A distinct but related optimization hypothesis may be used to explain scaling of differentiation in competitive networks (networks where the nodes themselves, rather than the entire network, are under selective pressure) such as ecosystems (networks of organisms).