Parties and networks: Stochastic models for relationship networks

This paper argues that parties and other gatherings are important for the development of friendship networks. It proposes a stochastic model for the evolution of networks over time, the distinctive feature of which is the party event. A party event occurs when a person in the network has a gathering and invites all of his/her friends who then also become friends. The Party Models discussed are all based upon this simple assumption. After formulating basic assumptions, various differential equations describing Party Models are derived. Subsequently several concepts useful for model analysis are defined and briefly explored. These include the concepts of potential, equivalence class form, and degenerate models. The penultimate section considers models for three person networks in some detail and with numerical illustrations. All extended mathematical arguments are placed in the Mathematical Appendix.