The Early Signal

Simple contagion: exposure once is enough. Complex contagion: multiple exposures from different sources are required. The distinction determines whether a behavior spreads like a virus or stalls like an unpopular opinion.

Sharma and Singh model complex contagion as a stochastic Markov chain on preferential attachment networks. Notions propagate as high-dimensional vectors, and node activation depends on a decision function integrating propagation affinity (how similar the notion is to the node’s disposition), local influence (how many neighbors have adopted), and global influence (how widespread the notion already is).

The interaction between local and global influence produces a phase transition. Below a threshold, cascades die quickly. Above it, they capture a macroscopic fraction of the network. The transition is sharp — not a gradual increase in adoption but a discontinuous jump from failure to success.

The structural finding is in the early stages. Before the phase transition occurs, the growth pattern already signals which side the cascade will land on. Successful cascades and failing cascades are distinguishable in their first few time steps — not by their size, which is similar in both cases at early stages, but by the pattern of how activations distribute across the network. Cascades that will succeed spread across communities early; cascades that will fail concentrate locally.

The balanced interaction is the critical requirement: local reinforcement without global activation produces clusters that never merge; global activation without local reinforcement produces adoption too thin to sustain. The phase transition emerges at the balance point between these two failure modes.