Poking Holes in Structural Holes
Rethinking the Orthodoxy of Network Topology
Reading Length: 10 minutes
Keywords: Network Theory, Quantitative Sociology, Epidemiology
How do plagues spread? The Covid pandemic reignited interest in contagion models that previously resided in the dull domain of academia and public-health circles. Statisticians and epidemiologists repeatedly made the front page of national newspapers with their predictions.1 These sophisticated mathematical models, however, presuppose the existence of social ties to begin with. In other words, beyond the interesting question about how quickly a contagion will spread, there is the deeper question about the human behavior that creates social networks. The structure of these social networks, formally called network topology, can engender and impede the spread of contagions. Now we have entered the domain of quantitative sociologists.
Strong Weak Ties
In 1973, the sociologist Granovetter published a seminal paper titled ‘The Strength of Weak Ties’ that still sets the tone for network sociologists today. A social network is just a formalization of human relations. We can characterize relations between humans as either ‘weak’ or ‘strong’.2 A weak tie is basically a loose acquaintance, whereas a strong tie is a close friend or family member.
It should be noted here that while a pandemic is the most vivid example of a contagion, the concept covers almost any type of human behavior. For example, fashion fads, political ideas, and technological innovations spread in much the same way as sicknesses through a population.
Before Granovetter it was an open question whether weak ties or strong ties were more likely to precipitate the diffusion of sicknesses and other human behaviors. On the one hand, it was reasonable to assume that strong ties were responsible for the spread of contagions. We spend the most time with our strong ties, and in more intimate settings. Therefore many sociologists assumed that a ‘dense’ network with many such strong ties was more susceptible to diffusions.
Granovetter, as his paper suggests, showed on the contrary that it was actually the weak ties that were more likely to lead to contagions:
Whatever is to be diffused can reach a larger number of people, and traverse a greater social distance, when passed through weak ties rather than strong.
The rationale for his argument is that our strong ties are more likely to be related to other strong ties that also have ourselves as a connection. I have a strong tie with my brother, who has a strong tie with my sister, who also has a strong tie with me.3 Loose ties, on the other hand, connect disparate social groups together. Even though I associate less with my weak tie, the weak ties other connections are less likely to be correlated to mine. Therefore weak ties ‘open up’ the spread of a contagion to more vast amount of people. This is why weak ties are, counterintuitively, stronger than strong ties.
Small Worlds
The physicists Watts and Strogratz formalized this way of looking at networks as small worlds.4 The physicists borrowed the term small world from the psychologist Milgram (of the infamous Milgram experiments), who in turn got it from the colloquial expression ‘it’s a small world after all’ when people run into unexpected connections.5
Watts and Strogratz translated the ideas of Granovetter into mathematical models that could be stressed and used to predict behaviors in experiments and the real world. Confirming the strong weak ties hypothesis, they found that the small-world networks connected by loose ties were most likely to see contagions spread across the population.
How prevalent are small worlds? The authors claim that small worlds represent such diverse contexts as the neural network of a worm, the power grid of the western United States, the collaboration of film actors, food webs in ecosystems, and even online communities across the Internet.
Enter Structural Holes
The organizational sociologist Ronald Burt became interested in these theories and how it could relate to human capital in an organization. Organizations can of course be conceptualized as a social network of strong and weak ties. If contagions, such as ideas and innovations, spread more quickly across weak ties, then what are the implications for the individuals inside the organization?
In 2004 Burt published ‘Structural Holes and Good Ideas’ to study this question. A structural hole is basically a gap between two different groups of people. An individual who spans this gap fills the hole and becomes a ‘broker’. As an example, consider two departments in an organization, marketing and manufacturing. If there are no social connections between the two departments, there is a structural hole that implies a lack of communication and collaboration. An individual that bridges this gap becomes a valuable go-between between the two departments.
Visually, we can see that individual A brokers connections between three clusters of people, whereas individual B clearly plays no such role.
All this may seem fairly obvious. Burt’s achievement was to quantify and prove empirically that information brokers do in fact benefit from their position:6
Compensation, positive performance evaluations, promotions, and good ideas are disproportionately in the hands of people whose networks span structural holes.
An Empirical Puzzle
Despite the success of the weak tie theory, the small worlds model, and structural holes, sociologist began to notice that some real world situations were not always following the predictions expected by the theory. The sociologist Damon Centola states simply:
It is puzzling, then, that social movements often do not follow the path predicted by the theory of weak ties…studies of collective action consistently report that mobilization spreads spatially, like a wave front, propagating through clustered networks, rather than exploding globally, like a virus, jumping across long distances.7
Centola cites the Paris Communes revolts of 1789, the trade unionization movement in Europe at the turn of the 20th century, and the Freedom Summer movement of 1964 as contagions that did not spread according to the predictions of the small world model.
Complex Contagions
Centola centered this puzzle as the research project of his career, and formulated a theory of complex contagions to reconcile the empirical results with theory.
Small world models with weak ties were so successful in epidemiology because the transmission mechanisms of sicknesses are usually simple. You only need to come into contact with one person, and only for a moment, to be vulnerable to transmission.
Many behaviors, however, are more complex because they require contact with multiple ‘infected’ people. If there is political turmoil and I am considering joining a revolution, I will probably have to come in contact with many likeminded individuals before I am ‘infected’ and convinced to participate. This can be called social confirmation, social proof, or social reinforcement.
It quickly becomes obvious that most human behaviors are in fact complex. Centola lists “investing in a market, choosing a career, selecting a neighborhood, adopting a high cost technology, choosing a method of contraception, joining a church, and voting” as examples.
Centola posits that complex behaviors are characterized as either high risk, highly complementary, or both. We can create a matrix to represent this as:
Complementarity means that the attractiveness or virality of the behavior depends on how many others also participate. In the example listed above, it is only worth it to join a social media platform if your friends have as well.
In this schema, three out of four possible categories are complex contagions, so it is not surprising that we should expect complex behavior to make up the bulk of examples that manifest in the real world.
Despite this abundance of complex behaviors, however, Centola was the first one to classify and model the implications of complex contagions.
Wide Bridges against Weak Ties
If weak ties are insufficient to spread complex contagions, which structural features of network topology will? Centola noticed that networks with so-called wide bridges will exhibit the diffusion of complex contagions.8
What is a wide bridge? In the above graphic we can see that Ashley is connected indirectly to Jordan, not just through Elif but also through Sephira. The ‘bridge’ between Ashley and Jordan is ‘wide’, because it contains not just Sephira (who would otherwise be a weak tie), but also Elif.9
Through computational modelling and experimental studies, Centola has proved that when a contagion is complex, weak ties will actual hinder diffusion while bridge ties will accommodate spread.10
Deflating Structural Holes
Returning to the organizational context of information brokers serving as ‘information arbitrageurs’ between clusters of people, we can see now that a problem has arisen.
Many behaviors that are beneficial to organizations can clearly be labeled as complex. From implementing a new software system to running a pricing experiment, there is good reason to posit that these are both high-risk and high-complementarity behaviors.
Contrary to Burt, we see now that structural holes can be disadvantageous to organizations. Rather than promoting information brokers and giving them positive performance appraisals, organizations should be going out of their way to eliminate the possibility of becoming a broker in the first place by fostering wide bridges.
The interests of the information broker now run contrary to that of the organization:
Because an information broker controls the flow of information between two groups, she is in a position to exploit both groups in order to extract resources for herself. If coworkers attempt to create additional links across groups, it is in the broker’s interest to prevent these ties from forming. Her individual gains are maximized when she is the only channel through which information can flow.
To be clear, Centola is not disputing Burt’s thesis that information brokers are in an advantageous position. What is disputed, however is Burt’s statement that:
Organizations with management and collaboration networks that bridge structural holes in their markets seem to learn faster and be more productively creative.
Centola’s mathematical simulations and experiments show, on the other hand, that plugging structural holes with loose ties can actually impede the progress of new ideas and innovations.
Centola’s Complexity
Beyond organizations, the theory of complex contagion and wide bridges has aided researchers in understanding everything from why the AIDS prophylactic PrEP has failed to spread quickly to swarming robot behavior.11 Some pathogens may even be complex from an epidemiological standpoint. This is especially noticeable when considering co-infections, such as the coincidence of influenza and the bacteria Streptococcus that commonly leads to pneumonia.12
Centola challenged the prevailing orthodoxy of network topology, following the scent of empirical failures that ultimately led to a classic Kuhnian paradigm shift.
“Unanticipated novelty, the new discovery, can emerge only to the extent that his anticipations about nature and his instruments prove wrong.”13
For example, this article. Dr. Shaman is, coincidentally, the son of one of my statistics professors from graduate school, Paul Shaman.
This schematization is called a ‘dyadic tie’.
A fancy term for this is ‘triadic closure’.
Watts, Strogatz, ‘Collective dynamics of ‘small-world’ networks’, 1998. See also the book Small Worlds by Watts.
Milgram, ‘The Small World Problem’, 1967.
The causal direction of Burt’s claim is debated. It could be that information brokers are in the position to span structural holes specifically because they outperform.
Centola, ‘How Behavior Spreads’, 2018.
The analysis is covered in his book, but for a more technical expositon see Centola’s paper ‘Complex Contagions and the Weakness of Long Ties’, 2007. Centola also wrote a less-technical book for the popular audience titled ‘Change: How to Make Big Things Happen’.
The graphic is from How Behavior Spreads, page 44.
The research is covered in ‘How Behavior Spreads’, but see also Centola, ‘The Spread of Behavior in an Online Social Network Experiment’ and ‘The Social Origins of Networks and Diffusion’.
For PrEP, see der Straten et al., ‘Perspectives on use of oral and vaginal antiretrovirals for HIV prevention’, 2014. For robot swarms, see Horsevad et al., ‘Transition from simple to complex contagion in collective decision-making’, 2022. For a broader literature review of complex contagions, see here.
See Hébert-Dufresne & Althouse, ‘Complex dynamics of synergistic coinfections on realistically clustered networks’, 2015.
Kuhn, ‘The Structure of Scientific Revolutions’, 1962. Although the phrase paradigm shift is now cliché, it was Kuhn who first popularized the term.