How network structure impacts socially reinforced diffusion? by Jad Sassine M.S. Applicable Mathematics London School of Economics (2013) SUBMITTED TO THE SLOAN SCHOOL OF MANAGEMENT IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MANAGEMENT RESEARCH at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY MAY 2020 ©2020 Massachusetts Institute of Technology. All rights reserved. Signature of Author: __________________________________________________________ Department of Management April 17, 2020 Certified by: _________________________________________________________________ Hazhir Rahmandad Associate Professor of System Dynamics Thesis Supervisor Accepted by: _________________________________________________________________ Catherine Tucker Sloan Distinguished Professor of Management Professor, Marketing Faculty Chair, MIT Sloan PhD Program 2 How network structure impacts socially reinforced diffusion? by Jad Sassine Submitted to the Sloan School of Management on April 17, 2020 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Management Research Abstract Social scientists have long studied adoption choices that depend on the number of prior adopters. What is the effect of network structure on such adoption dynamics? The emerging consensus holds that when agents require a high reinforcement threshold for adoption, clustered networks are better conduits of social contagion than random ones. Using models with deterministic thresholds this argument formalizes the idea that transmission will get ‘stuck’ should the number of neighboring adopters fall below a threshold. In this paper, we explore the effect of stochastic thresholds on the diffusion races between random and clustered networks. We show that even low probabilities of adoption upon a single contact would tilt the balance in favor of random networks, a tendency that is reinforced with the size of the network. Moreover, if repeated signals from the same adopter can reinforce a message, random networks are further promoted. However, we also show that clustered networks can still be preferred over random networks if adopters become ‘inactive’ – i.e. they stop sending messages - with high probability. These findings refocus our theoretical understanding of how network structure moderates social influence, and raises new questions on contagion phenomena that benefit from clustered networks. Thesis Supervisor: Hazhir Rahmandad Title: Associate Professor of System Dynamics 3 Introduction Social influence is at the heart of social sciences. From joining an online social network, to adopting a social norm, participating in a riot or buying a novel product, many important choices are strongly influenced by the existence and the number of other adopters (Karsai, Iniguez, Kaski, & Kertész, 2014; Scott, Konstantin, & Milan, 2016; Ugander, Backstrom, Marlow, & Kleinberg., 2012). These adoption dynamics have historically been modeled using behavioral thresholds, where a threshold represents the minimum number of social connections who should have signaled their adoption before the focal actor is convinced to adopt (Granovetter, 1978). In understanding social influence, a central question of theoretical and practical relevance relates to the type of network that best facilitates diffusion: Would cohesive communities promote the adoption of new social norms? What about the impact of random boundary spanning ties on diffusion of infectious diseases, or social movements? In promoting new organizational practices should intra-organizational social networks be seeded with cohesive clusters or cross-cutting relationships? Communication networks are rarely fully connected, and even neighboring adopters do not always broadcast their choices. Earlier studies highlighted the central role of random ties connecting disparate parts of social networks in promoting diffusion (Granovetter, 1973). However, threshold models identified an important value of repetition: unless located in a highly clustered network, high threshold agents may not receive sufficient reinforcement to adopt, thus breaking the diffusion dynamics in a population of high- threshold individuals. It is this intuition that led Morris (2000), Centola & Macy (2007), and Montanari & Saberi (2010) to argue that for an important class of diffusion dynamics clustered networks dominate random networks, i.e. they lead to more or faster diffusion, because they decrease the fraction of isolated agents. From a policy perspective, this insight has implied that maintaining social cohesion increases the speed of diffusion when adoption involves risk, complementarity or normative acceptance (Centola, 2018). The emerging consensus has thus separated social diffusion into “simple” and “complex” alternatives, where the simple diffusion is 4 enhanced by more random networks and the corresponding weak ties, but clustered networks engendering significant reinforcement are needed for enabling complex contagion. The argument in favor of clustered networks relies on the idea that transmission will get ‘stuck’ should the number of neighboring adopters fall below the adoption threshold. This argument is elegant; however, it has only been formally shown using deterministic thresholds where adoption is impossible below the threshold reinforcement. Considering (the arguably more realistic) stochastic activation functions, where transmission likelihood is non-zero for any level of reinforcement but becomes significantly more likely at a given threshold, could change the calculus. The transmission would not be stuck indefinitely for any actor, unless additional mechanisms are invoked. As a result, a tradeoff emerges between the cost of being temporarily slowed down due to low reinforcement and benefits of seeding the contagion in distant parts of a social network. The resulting tradeoff is intricately dependent on the state of diffusion. As the diffusion spreads in a random network, the frontier connecting susceptible to adopters expands creating an exponential growth dynamics not observed in clustered networks. Moreover, later in diffusion susceptible nodes in a random network become increasingly likely to be connected to more than one adopter as the network saturates. Therefore in the later stages random networks may actually facilitate complex contagion by exposing all susceptible nodes to a large level of reinforcement simultaneously. The implication of these mechanisms for the race between random and clustered networks has not been previously studied, and they may be of significant theoretical and practical relevance. If these mechanisms promote random networks significantly, we may need to revisit the dichotomy between simple and complex contagion, or seek other mechanisms that explain why clustered networks may promote diffusion under specific conditions. We develop a general model allowing us to study this question under a variety of probability activation functions. We build the model on the principle that two people may behave differently, even if they have the same adoption threshold and reinforcement levels. In other words, the thresholds are stochastic and not deterministic. We then consider that repeated signals from the same source may accumulate and reinforce adoption. After all, repetition is a very powerful tool of persuasion and 5 experimental evidence shows that agents generally overweight repeated information (Enke & Zimmermann, 2019). By comparing the behavior of this model on different network structures, we can identify the conditions under which clustered networks dominate random ones. Our analysis suggests those conditions are rather narrow and strongly depend both on the shape of the probability activation function and the size of the network. As prior theory had suggested, deterministic thresholds promote clustered networks, but this regularity breaks down as soon as realistically stochastic activation functions are considered. Moreover, the larger the network, the stronger are the benefits of random networks for complex contagion. Finally, if repetition can engender adoption, then random networks are further strengthened against clustered diffusion. Overall, incorporating two behaviorally realistic features of social influence, stochastic thresholds and repetition, can significantly tilt the balance of diffusion speed in favor of random networks. We also explore how the horserace between random and clustered networks is impacted by heterogeneity in adopters’ motivation to share information. In the ‘The Strength of Weak ties’, Granovetter (1973) had already postulated that “if the motivation to spread [a] rumor is dampened a bit on each wave of retelling […] bridges will not be crossed”. In other words, the spread will get ‘stuck’. We confirm this insight and show that clustered networks can still be preferred over random networks over some parts of the space as the probability that an adopter becomes ‘inactive’ (i.e. stops sending messages) increases. Therefore, we show that the strength of clustered networks depends on two likelihoods – one for adoption upon a single contact, the other for adopters becoming silent. We end by discussing the policy implication of this insight, proposing that how the messages are sent may be as important as ‘what is to be diffused’. Specifically, if the environment limits repeated interactions, then maintaining social cohesion may speed up diffusion. Otherwise, from common social networks to the so-called conversational firms (Turco, 2016),