Centola & Macy 2007

Complex Contagions and the Weakness of Long Ties

Abstract

Principle

The strength of weak ties is that they tend to be long—they connect socially distant locations, allowing information to diffuse rapidly.

Question

Whether this “strength of weak ties” generalizes from simple to complex contagions.

# Simple, tie to tie

Test

Complex contagions require social affirmation from multiple sources. 

Ex. the spread of high-risk social movements, avant-garde fashions, and unproven technologies

Result

As adoption thresholds increase, long ties can impede diffusion.

Complex contagions depend primarily on the width of the bridges across a network, not just their length.

Wide bridges are a characteristic feature of many spatial networks, which may account in part for the widely observed tendency for social movements to diffuse spatially.

Introduction

“Complex contagions”

Many collective behaviors also spread through social contact, but when these behaviors are costly, risky, or controversial, the willingness to participate may require independent affirmation or reinforcement from multiple sources.

Successful transmission depends upon interaction with multiple carriers

FROM WEAK TIES TO SMALL WORLDS

“Strong” and “weak” have a double meaning in Granovetter’s usage. One meaning is relational (at the dyadic level), the other is structural (at the population level).

1) The relational meaning refers to the strength of the tie as a conduit of information.

2) The structural strength of a tie refers to the ability of a tie to facilitate diffusion, cohesion, and integration of a social network by linking otherwise distant nodes.

Granovetter’s insight: ties that are weak in the relational sense— that the relations are less salient or frequent—are often strong in the structural sense—that they provide shortcuts across the social topology.

Conversely, strong social relations also have a structural weakness— transitivity

# Clique

Watts and Strogatz (1998)

Not only do weak ties facilitate diffusion when they provide “shortcuts” between remote clusters, but it takes only a small fraction of these long ties to give even highly clustered networks the “degrees of separation” characteristic of a random network.

Granovetter (1978) and Schelling (1978) modeled this contagion process as a threshold effect in which a small number of “seeds” can trigger a chain reaction of adoption, leading to a population-wide cascade of participation in collective behavior.

FROM SIMPLE TO COMPLEX CONTAGIONS

# What is the weakness of the long tie?

Generalization of the small world principle from the spread of information and disease to the spread of collective behavior

The problem is that while all contagions have a minimum threshold of one, the range of nonzero thresholds can be quite large.

However, many collective behaviors involve complex contagions that require social affirmation or reinforcement from multiple sources.

The distinction between simple and complex refers to the number of sources of exposure required for activation, not the number of exposures.

The distinction between multiple exposures and exposure to multiple sources is subtle and easily overlooked, but it turns out to be decisively important for understanding the weakness of long ties

The probability of transmission: P

The probability of contracting the disease after E exposures = 1 - (1 - P)^E

Mechanisms of Complex Contagion

1. Strategic complementarity: The positive externalities of each prior participant’s contribution

2. Credibility: The need for confirmation

3. Legitimacy

4. Emotional contagion: Expressive and symbolic impulses

The Structural Weakness of Long Ties

For simple contagions, what matters is the length of the bridge between otherwise distant nodes. 

For complex contagions, the effect of bridges depends not only on their length (the range that is spanned) but also on their width. 

Hence, we can measure a bridge not only by its length (the range that is spanned by the bridge) but also its width (the number of ties it contains).

The structural weakness of long ties is that they form bridges that are too narrow for complex contagions to pass.

EFFECTS OF LONG TIES ON A RING LATTICE

Begin with the original Watts and Strogatz (1998) small world model

Long ties can also be introduced by adding random ties to the network (Newman and Watts 1999), but this increases both density and the fraction of long ties, which confounds the effects of randomization with the effects of densification.

Thresholds can be expressed in two ways— as the number (Granovetter 1978) or the fraction (Watts 2002) of neighbors that need to be activated.

The threshold t as a fraction t p a/z, where a is the number of activated nodes and z is the number of neighbors.

One of the main findings of our study is that there is a qualitative dif- ference between a p 1 and a 1 1, even when the proportions are identical.

To review, the analysis of the ring lattice reveals two qualitative differences between simple and complex contagions:

1. While a single random tie is sufficient to promote the spread of simple contagions, complex contagions require more rewiring in order to benefit from randomization. The number of ties that need to be randomly rewired increases exponentially with the number required to form a bridge (Wc), and the number of ties needed to form a bridge in turn increases exponentially with the required number of activated neighbors (a).

2. As the ring becomes increasingly randomized, the width of the bridges that make up the lattice structure may be eroded below the critical width required for the contagion to spread. Simple contagions can propagate on a connected network even if every tie is random, and the rate of propagation increases monotonically with the proportion of random ties. In contrast, there is a critical upper limit of randomization above which complex contagions cannot propagate. As thresholds increase, this critical value decreases.

LONG TIES ON HIGHER DIMENSIONAL NETWORKS 

1. Complex contagions fail to benefit from low levels of randomization, as shown by the initial failure of propagation rates to improve as p increases above zero.

2. Increasing p has a nonmonotonic effect on complex contagions, exhibiting a U-shaped effect, in which randomization starts to help—but ultimately impedes—propagation. 

3. As p exceeds a critical upper limit, complex contagions entirely fail to propagate.

TESTS FOR ROBUSTNESS OF THE SIMPLIFYING ASSUMPTIONS

Threshold Heterogeneity

Heterogeneity of Influence

Strong and Weak Ties

Heterogeneity of Degree 

DISCUSSION

Network topologies that make it easy for everyone to know about something do not necessarily make it likely that people will change their behavior

Our study reveals a structural property of spatial networks—wide bridges—that has received far less attention.

While spatial proximity can make the connection relationally strong, it is the width of the bridge that makes the connection structurally strong for the propagation of complex contagions.

Thus, consistent with McAdam (1986), our results show that the optimal networks for coordinating action will depend upon the costs and risks of participation.

Our results suggest that efforts to change behavioral norms through peer influence may reach greater numbers with greater speed by targeting tightly knit residential networks rather than the complex networks through which disease is more rapidly transmitted (like acquaintance or employment networks).

Further research on the social and structural conditions that allow contagions to spread most effectively as thresholds increase.

CONCLUSION

Long ties do not always facilitate the spread of complex contagions and can even preclude dif- fusion entirely if nodes have too few common neighbors to provide multiple sources of confirmation or reinforcement