Exact solutions for models of evolving networks with addition and deletion of nodes

TL;DR

This paper develops exact solutions for evolving network models that include both addition and deletion of nodes, revealing how growth rates influence degree distribution exponents, with implications for understanding real-world networks.

Contribution

It introduces models of network evolution with node addition and deletion and provides exact solutions, extending prior models that only considered growth.

Findings

Networks with preferential attachment have power-law degree distributions.

The degree distribution exponent diverges as the growth rate approaches zero.

Slower growth in networks leads to higher degree distribution exponents.

Abstract

There has been considerable recent interest in the properties of networks, such as citation networks and the worldwide web, that grow by the addition of vertices, and a number of simple solvable models of network growth have been studied. In the real world, however, many networks, including the web, not only add vertices but also lose them. Here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest. For the case of net growth and so-called preferential attachment -- in which newly appearing vertices attach to previously existing ones in proportion to vertex degree -- we show that the resulting networks have power-law degree distributions, but with an exponent that diverges as the growth rate vanishes. We conjecture that the low exponent values observed in real-world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal. Were growth to slow in the future, for instance in a more mature future version of the web, we would expect to see exponents increase, potentially without bound.