Self-reinforcing cascades: A spreading model for beliefs or products of varying intensity or quality

Models of how things spread often assume that transmission mechanisms are fixed over time. However, social contagions–the spread of ideas, beliefs, innovations–can lose or gain in momentum as they spread: ideas can get reinforced, beliefs strengthened, products refined. We study the impacts of such self-reinforcement mechanisms in cascade dynamics. We use different mathematical modeling techniques to capture the recursive, yet changing nature of the process. We find a critical regime with a range of power-law cascade size distributions with varying scaling exponents. This regime clashes with classic models, where criticality requires fine tuning at a precise critical point. Self-reinforced cascades produce critical-like behavior over a wide range of parameters, which may help explain the ubiquity of power-law distributions in empirical social data.

Imagine a cascading product like a meme, conspiracy theory, rumor, or a piece of software spreading in a population of agents. At every transmission step in the cascade, the product has the chance to independently improve with probability p or get worse with probability 1 − p. This process can stop for two reasons: either the quality of the product drops to zero, or the agents sharing it cannot find others to pass it on to (see Fig. 1).

Although cascade models vary, the vast majority of them use fixed mechanisms such that the same rules apply at every step of the cascade. For example, a new case of a disease produces infections through the same mechanism as the previous cases do. However, cascades of beliefs and ideas might be different. Beliefs can be reinforced and strengthened when instilled by a passionate teacher. Ideas or products can be refined as they are transmitted from one person to the next.

we provide an exact recursive solution, closed-form expressions for the expected cascade sizes and their critical point, and offer other new avenues of mathematical analyses.

The self-reinforcing cascade process presents key features that makes it particularly appealing for modeling contagions observed in socio-technical systems. It is a parsimonious model to capture the fact that the strength of individual beliefs or the quality of products may vary and influence the ability of an individual to further transmit the cascade. This variability is aligned with real-world phenomena, where not all individuals or contents are equally influential in the transmission of ideas or behaviors. With this simple mechanism, the SRC model can produce a wide range of scaling behaviors for cascade size distributions, whereas classic percolation is constrained by a unique and universal scaling exponent obtained only at a precise critical point.