Message-Passing Methods for Complex Contagions

Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number N of nodes becomes infinite or on specific finite-size networks. We review the message-passing approach and show how to derive it for configurationmodel networks using the methods of [1–3]. Using this approach, we explain for such networks how to determine an analytical expression for a “cascade condition”, which determines whether a global cascade will occur. We extend this approach to the message-passing methods for specific finite-size networks, and we derive a generalized cascade condition. Throughout this chapter, we illustrate these ideas using the Watts threshold model.