#### Chapter 3 – The Theory of Complex Contagions

To see what effect, if any, the complexity of a contagion has on the process of network diffusion, the small-world model from chapter 2 can be used to test the hypothesis from the strength of weak ties. Like the computational experiments in the previous chapter, diffusion can be studied first on a clustered network and then observed as long ties are introduced and the degrees of separation in the network are reduced. In the book, the conclusion of this chapter also explores what happens when we relax the simplifying assumptions of the small-world model to include more realistic features of empirical social networks. However, to start with, a proper test of the strength of weak ties is to keep everything the same as it was before, with one exception. This time, let’s assume that the contagion is complex. Thus, instead of one social contact being sufficient to transmit the contagion, each person requires confirmation from a second source of activation before being willing to adopt. In every other way, the computational experiments are the same as before.

**Figure 3.3: Diffusion in a Large World**

The first experiment is shown in figure 3.3. It uses the same clustered network that we began with last time, in which every person has four neighbors, two on the right and two on the left. The nodes shown in gray indicate the actors who have not yet adopted the behavior. The two individuals shown in black are the seeds who introduce the contagion into the population.

In the clustered network, the complex contagion spreads much like it did before. It cascades from neighborhood to neighborhood across the population. But notice that even though the behavior spreads to the entire population, it spreads more slowly than the simple contagion did. This time, it takes twenty-six days to spread because each person must wait for confirmation from a second source before they are willing to adopt the behavior themselves.

The “slow but successful” pattern of diffusion in the clustered network seems to lend credence to the strength of weak ties hypothesis. Because the complex contagion spreads so slowly through the spatial network, a few changes to the network topology will probably help speed things up. Based on the slow rate of diffusion in figure 3.3, there’s a lot of room for improvement. Reducing the size of the world by adding a few long ties should be able to help out quite a bit—perhaps even more than it did for the simple contagion. Figure 3.4 shows what happens when the network is rewired with a few weak ties.

**Figure 3.4: Diffusion with Weak Ties**

Instead of spreading faster, the contagion slows down. How can reducing the redundancy of network ties slow down a diffusion process? This seems like a paradoxical result—reducing the degrees of separation in a network should increase the rate of diffusion.

Before worrying too much about this odd result, we can here take advantage of the fact that we have a counterfactual model. We can explore the dynamics of diffusion a bit more and see what happens when some additional weak ties are added. If we’re lucky, we can use the model to push past this bump in the road and see if a little more rewiring solves the problem.

**Figure 3.5: ****Diffusion with More Weak Ties**

To explain this puzzling result, we need to understand why long-distance ties would not help diffusion. For simple contagions, each long-distance link presents an opportunity for the contagion to jump across the network and discover new targets of activation. However, for a complex contagion, a signal that travels across a long tie arrives alone, without any social reinforcement. Consequently, the first problem with adding long ties to the network is that they do not create useful pathways for complex contagions to diffuse. But there is also a second problem, which is worse; in addition to not helping, they also hurt diffusion.