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. Author manuscript; available in PMC: 2024 Oct 18.
Published in final edited form as: J Comput Soc Sci. 2021 Apr 8;5(1):47–68. doi: 10.1007/s42001-021-00115-x

Network diffusion of competing behaviors

Yuan Hsiao 1
PMCID: PMC11487607  NIHMSID: NIHMS2011080  PMID: 39430418

Abstract

Research indicates that network structure affects the diffusion of a single behavior. However, in many social settings, two or more behaviors may compete for adoption, as in the case of religious competition, social movements and counter-movements, or conflicting rumors. Lessons from one-behavior diffusion cannot be easily applied because the outcome can take the form of one-behavior domination, two behaviors splitting the network, both behaviors occupying a small fraction of the network, or no diffusion. This article tests how three well-known factors of single-behavior diffusion — network transitivity, adoption threshold, and connectedness of early adopters — apply to scenarios of competitive diffusion. Results show that minor differences in initial adopter size tend to magnify, creating a significant “head-start advantage.” Nevertheless, the degree of this advantage depends on the interaction between network transitivity, adoption threshold, and connectedness of initial adopters. The article describes the conditions under which countervailing ties may (or may not) create inequality in behavioral diffusion.

Keywords: Competitive diffusion, behavioral adoption, social networks, simulations

1. INTRODUCTION

In 2004, Mark Zuckerberg launched a social media platform named “Facebook.” Envisioning that people could connect via the platform, Zuckerberg aimed to accrue as many users as possible. However, there was a major problem—other firms were seeking to do the same. People could choose other social media like Friendster or Myspace. If Zuckerberg were to be successful, Facebook would need to overpower its competitors (Forbes 2018).

Facebook did eventually succeed. The number of people using Friendster or Myspace declined, while Facebook users ballooned. Currently, Facebook is the social media of top popularity, boasting more than 2,375 million users (Statistica 2019).

The rise of Facebook is relevant in itself, as Facebook now has the potential to influence millions on political values, commercial advertisements, or romantic relationships. Nonetheless, more generally the social media battle is a case of competition for behavioral adoption—whether people adopt the behavior of using Facebook or using Friendster. We can reframe the question of which social media wins the market into a general one: When multiple organizations compete for behavioral adoption, which organization would prevail? 1

Such cases of competition for the adoption of different behaviors are prevalent. In the realm of technology adoption, Yahoo and Google competed for users to adopt their product until Google finally claimed the crown (Vise 2007). Social movements and counter-movements compete for the behavior of participation from bystanders (Andrews 2002). When civil conflict arises, rebels and government forces compete for the support of civilians (Kalyvas 2006). In the political arena, Republicans and Democrats compete for citizens to vote for them. Among religions, multiple sects compete for individuals to practice their religion (Iannaccone 1991). Each case consists of different behaviors, but the common concern is how to determine the number of adopters for each behavior.

How can we explain which behavior acquires more adopters in competitive settings? This paper aims to explain the outcome of competitive diffusion through the lens of network diffusion, especially by examining the structure of the network. One way to explain how one behavior spreads is through understanding social networks. People seek reference to others before action, and many encouragements from others increase the probability that a person would adopt a certain behavior (Gould 1991; Granovetter 1978; Valente 1996; Zhang and Centola 2019). If a person has many friends who use Facebook, the person would more likely sign up for Facebook. Through social networks, a behavior like using Facebook diffuses from one person to another.

The structure of who is connected to whom is critical to network diffusion (Kim and Bearman 1997; Heckathorn 1993; Macy 1990; Marwell and Oliver 1993; Siegel 2009; Valente 1996). As a simplified example, if Eddy and Eric both use Facebook and they have a common friend Emma, then Emma receives encouragement from two people to use Facebook. In contrast, if Eddy is friends with Emma but Eric is instead friends with Skylar, then Emma receives only one encouragement. The network structure of who is friends with whom determines the number of encouragements each person receives.

The example of when Eddy and Eric are friends but also both friends with Emma helped spread the use of Facebook because it created a network structure of shared friendship. This degree of shared friends is called the “transitivity” of the network (Wasserman and Faust 1994). High transitivity networks indicate a lot of overlapping ties that foster encouragement from multiple people, as when Eddy and Eric influenced Emma (Centola and Macy 2007). However, high transitivity networks also make it difficult for information to spread from one social circle to another (Wasserman and Faust 1994; Watts 1999). For instance, suppose that in addition to the Eddy-Eric-Emma triangle, there is another George-Galen-Gina triangle. The network is a high transitivity one since if two people are friends, they share a common friend. However, even if Eddy, Eric, and Emma all use Facebook, the lack of shared ties to the George-Galen-Gina circle makes it difficult to spread Facebook adoption to that social circle. Transitivity creates social reinforcement at the expense of informational reach.

Given the tradeoff between social reinforcement and informational reach, whether transitivity benefits diffusion depends on the cost of behavior and consequently the “threshold of adoption” (Granovetter 1978; Schelling 1978; Valente 1996). Some behaviors, such as using a new emoji, have negligible cost and need only a nudge from another person to be adopted. Some behaviors, such as joining a rebellion, incur significant costs and require many people to persuade the focal person to be adopted. In other words, some behaviors require many people to activate (high threshold), while others need just a few (low threshold). High-transitivity networks benefit the diffusion of high-threshold behavior since transitivity generates the social reinforcement required for adoption. On the other hand, low-transitivity networks benefit the diffusion of low-threshold behavior because such networks reach more people and broaden the scope of diffusion (Centola and Macy 2007).

In addition to network transitivity and adoption threshold, one should also consider how initial adopters of a behavior are connected (Barash, Cameron, and Macy 2012; Centola 2013; Marwell and Oliver 1993; Centola 2018).2 Network diffusion processes often start very slowly, then speed of dramatically. Suppose each user of Facebook can persuade two friends, then two users will become four, then eight, then sixteen. However, when the behavior is of high-threshold, how to overcome the initial stage of generating enough adopters becomes a formidable challenge. If initial adopters are connected, they can form coalitions and co-influence their common friends, so overcoming the “start-up problem.”

Network transitivity, adoption threshold, and connectedness of initial adopters are three factors that affect the diffusion of a single behavior. However, when it comes to cases of competitive diffusion, such as firms competing for product usage or movements competing for participation, it is not clear how the three factors affect diffusion. In competitive scenarios, diffusion of a focal behavior implies contraction for the competing behavior. Such tug-of-war situations render outcomes distinct from one-behavior diffusion, and thus the effects of the three factors are unclear. The outcome is not simply whether a behavior diffuses, but encompasses a spectrum of possibilities, including—(1) one behavior wins the whole market; (2) two or more behaviors split the network, with very few non-adopters; (3) each behavior gains some adopters, but most remain non-adopters; or (4) all behaviors fail to attract adopters.

The different outcomes have sociological implications. In the example of religious competition, if one religion dominates a country it may become the state religion and oppress alternative religions (Mahmood 2012). If two or more religions split the market, we may expect factions that can lead to religious conflict (Wimmer 2003). If non-believers constitute the majority, each religion may focus on recruitment rather than attacking other religions, thus de-escalating tensions (Frederiks 2010).

This paper extends research on the diffusion of a single behavior to the scenario of competitive diffusion. The goal is to test how (1) network transitivity, (2) adoption threshold, and (3) connectedness of early adopters affect the outcome in competitive diffusion. To achieve this goal, I use computer experiments and vary the three factors in the case of two competing behaviors. Furthermore, I assign a tiny difference in the number of initial adopters, which creates a “head-start” for one of the behaviors. However, it is very possible that this tiny “head-start” can snowball into large discrepancies in the diffusion process (David 1985; Petersen et al. 2011; Salganik et al. 2006). The question is straightforward—how do the three factors affect the discrepancy in market share between the “head-start behavior” and the “lagging behavior?”

To preview the results, even a tiny head-start can snowball into total dominance for the head-start behavior. However, the extent to which this happened is contingent on the interaction between the three factors. For low-threshold behaviors, low network transitivity lead to total dominance by the head-start behavior. On the other hand, high transitivity networks benefited the lagging behavior and shrank the discrepancy in adopter numbers.

For high-threshold behaviors, low network transitivity indicated that both behaviors did not diffuse, and all remained non-adopters. High transitivity networks allowed both behaviors to obtain some adopters, although the head-start behavior obtained a larger market share.

Connectedness among initial adopters was also critical in determining the outcome. Compared to scattering initial adopters, connecting them allowed both behaviors to diffuse for high-threshold behaviors. Furthermore, connecting initial adopters shrank the market share discrepancy between the head-start behavior and the lagging behavior.

The article shows that factors that affect diffusion of a single behavior are relevant for competitive diffusion. However, such factors affect diffusion in a different way. The factors can either benefit one behavior, benefit both behaviors, inhibit both behaviors, or benefit one but not the other. These effects then affect the discrepancy between the “head-start behavior” and the “lagging behavior.” By showing the network conditions that exacerbate or minimize group inequalities, this article has implications for technological adoption, firm competition, collective action, religious recruitment, and digital networks.

The structure of the paper is as follows. Section 2. reviews the literature on network diffusion processes regarding the spread of a single behavior. Section 3. discusses the limitations on applying current network diffusion theories to scenarios where multiple behaviors are spreading, and how a network perspective can help us understand such scenarios of “competitive diffusion.” Section 4. documents the computation simulation setup and the computational experiment design. Section 5. presents the results, while Section 6. and Section 7. discuss theoretical implications and directions for future research.

2. THREE FACTORS OF NETWORK DIFFUSION FOR A SINGLE BEHAVIOR: TRANSITIVITY, ADOPTION THRESHOLD, AND CONNECTEDNESS OF EARLY ADOPTERS

How can we explain why people signup for a Facebook account, practice a religion, or join a movement? These are cases of behavioral adoption, and a network perspective provides a useful tool to explain why they occur. Connections between people encourage behavioral adoption, ranging from the emergence of social norms (Centola, Willer and Macy 2005), the advent of new technologies (Coleman, Katz and Menzel 1966), to the rise of social movements (Gould 1991).

In network processes of diffusion, the extent to which a behavior spreads depends on the structure of the network (Kim and Bearman 1997; Heckathorn 1993; Macy 1990; Marwell and Oliver 1993; Siegel 2009). The significance of “weak ties” is a well-known example. A few weak ties that span across relatively isolated social circles can speed up information dissemination and, in turn, behavioral diffusion. Scholars have shown that weak ties accelerate diffusion in cases of job information flow (Granovetter 1973), usage of new technologies (Rogers 1995), or coordination of collective action (Macy 1990). Watts and colleagues (Newman and Watts 1999; Watts 1999; Watts and Strogatz 1998) mathematically derived this network structure by showing how randomly replacing a few ties with new ones in a high transitivity network can create such a phenomenon.3 This network structure is called the “small world” network structure, as people can meet anyone through a few intermediate connections and feel that the world is small. With lower network transitivity, a small-world network structure eases information flow.

Nevertheless, small world structures do not always benefit diffusion. Sometimes rather than broad informational reach, repeated encouragement is needed for diffusion. The necessity of encouragement from many is most evident for costly behaviors, such as joining a rebellion. Persuasion from one friend is often insufficient to convince a person to participate in a rebellion, but persuasion from five friends might prompt to person to join. Phrased differently, compared to low-cost behaviors, high-cost behaviors have a higher threshold of adoption (Granovetter 1978; Schelling 1978; Valente 1996).

The adoption threshold determines whether network transitivity facilitates diffusion. High transitivity networks facilitate the spread of high-threshold behaviors, as they create the necessary social support for persuasion; at the same time, such networks also inhibit diffusion of low-threshold behaviors, as it slows down the speed of information flow (Centola and Macy 2007). Empirical evidence has also supported the relationship between network transitivity and adoption threshold (Aral and Van Alstyne 2011; Centola 2010; Centola 2018; González-Bailón et al. 2010). Centola (2010) performed an online experiment confirming how clustered social networks assist the spread of health-related behavior, as behavioral adoption requires multiple contacts. González-Bailón et al. (2010) showed how protest behavior on Twitter networks required multiple reinforcements to spread.

Connections between early adopters are also critical to the spread of high-threshold behaviors. One common difficulty for the diffusion of high-threshold behavior is insufficient early adopters. Many encouragements from adopters are needed for more people to adopt, but in the early diffusion stage there are few adopters to generate the necessary reinforcement. The diffusion of high-threshold behavior thus often follows an “S-shaped curve” (Heckathorn 1996; Marwell and Oliver 1993; Oliver et al. 1985; Schelling 1978), with a slow start followed by dramatic acceleration once enough people adopt. How to overcome this “start-up problem” is thus a great challenge for high-threshold behaviors (Centola 2013). One solution may be to generate “clusters of adoption” by connecting early adopters (Barash, Cameron, and Macy 2012; Centola 2013; Centola 2018). Barash, Cameron, and Macy (2012) indicated that whether there are sufficient early adopters linked to one another affects whether diffusion of high-threshold behavior takes off. Centola (2013) showed that even if not all adopters are connected, small coalitions among a few can eventually form a critical mass that ushers the diffusion process into the acceleration phase.

3. HOW NETWORK DIFFUSION AFFECTS OUTCOMES OF COMPETING BEHAVIORS

Section 2. pointed out how three outlined factors (i.e., network transitivity, threshold of adoption, and connectedness of early adopters) affect the spread of a single behavior in the network (Centola and Macy 2007; Heckathorn 1993; Marwell, Oliver and Prahl 1988; Siegel 2009). However, it is often the case that multiple behaviors are competing for adoption, which I name as “competitive diffusion.” In this section, I discuss why research that assume the spread of a single behavior may inadequately capture many empirical scenarios (subsection 3.1), followed by how one can use network factors to explain the tug-of-war in scenarios of competitive diffusion (subsection 3.2).

3.1. Empirical significance of a framework for competitive diffusion

Rather than a single behavior spread in the network, countless cases correspond to scenarios of competitive diffusion. The clash of search engine behemoths Yahoo and Google lasted for years (Vise 2007). In the Mau Mau rebellion during colonial Kenya, rebels and governmental forces tried to persuade villagers to join their side (Barnett and Njama 1966). The competition between Muslim and Christian religions led to tensions in multiple states for more than 1300 years (Karabell 2008).

Among sociologists, cases of competitive diffusion have attracted widespread interest in terms of theoretical explanations for conflict behavior (Collins 2012), political polarization (McVeigh et al. 2014), civic organization affiliation (McPherson 1983), rising religions (Kim and Pfaff 2012), and competition among social movements (Soule and King 2008).

A network perspective can help us explain scenarios of competitive diffusion, but it is unclear how factors that affect single-behavior diffusion apply to such situations. In single-behavior diffusion, the outcome is how much the behavior spreads in the network; but when there are two or more competing behaviors, how much one behavior diffuses also depends on how much the competing behavior spreads (Kim and Bearman 1997; Kitts 2000; McPherson 1983; McPherson 2004). When opposite forces clash, the interdependence of competing behavior entails a broad spectrum of outcomes, including: (1) one behavior dominates the network; (2) the two behaviors equally split the network; (3) both behaviors diffuse to some extent, but most remain non-adopters/bystanders; or (4) both behaviors fail to diffuse, and all actors are non-adopters.

The outcome types have social implications. Organizations are embedded in an ecology of competition, and the state of the ecology influences the action of organizations (McPherson 1983). In the case of technological adoption, if two firms equally split the network, one technological firm may try to steal customers from their rivals by developing new differentiation strategies. However, if many non-adopters exist, firms should broaden their informational reach to tap market potential. In cases of collective action, if a movement fails to diffuse, the issue for a social movement organization would be how to achieve the critical mass for the movement to “take off” (Barash, Cameron, and Macy 2012). However, if a strong countermovement splits the network, the goal might instead be to develop effective counteracting strategies to the opponent (Andrews 2002).

3.2. A network model in explaining competitive diffusion

Recent research, especially in the fields of computer science and mathematics, has proposed general models to account for countervailing forces of diffusion (Apt and Markakis 2011; Borodin et al. 2010; Fazeli et al. 2017; Hu 2017). For example, Borodin et al. (2010) used extensions of threshold models in one-behavior diffusion to show how to choose early adopters in the network that maximizes diffusion in competitive settings. Fazeli et al. (2017) studied the tradeoff between investing in the quality of a product and seeding in a social network. They further demonstrated how simple network structures, such as a star network or a balanced network, can affect which strategy the organization should choose. This nascent literature contributed to our understanding of competitive diffusion by proposing rules on how individuals decide, and further elaborated optimal strategies organizations should deploy.

The present study studies competitive diffusion at a different angle. Instead of focusing on strategies for organizations, I follow a tradition that investigates macro outcomes of diffusion: what proportion in the network adopts behaviors (Centola and Macy 2007; Heckathorn 1993; Kim and Bearman 1997; Marwell, Oliver and Prahl 1988; Siegel 2009). For example, this study is not concerned with what strategies Facebook deploys, but instead what market share Facebook achieves.

The micro-macro distinction is well illustrated by Baldassarri and Bearman (2007). In their study of attitude polarization, they used a dynamic model to demonstrate how political values of a person are influenced by connected others, while network ties also evolve according to political values. They showed why people might perceive attitude homogeneity in their immediate networks, but at the same time the network is characterized by attitude heterogeneity. The present study concurs with the perspective that investigating competing influences is significant, but differs from their study as well. Most importantly, this article studies a different phenomenon. Whereas Baldassarri and Bearman (2007) studied attitudes as a spectrum of continuous possibilities, this article is concerned with behavioral adoption—to adopt or not to adopt (Centola and Macy; Granovetter 1978; Schelling 1978; Marwell, Oliver and Prahl 1988). Second, instead of evolving networks, this study is concerned with how network structures affect diffusion outcomes, both in terms of network transitivity and connectedness of early adopters. Most importantly, the study examines different explanatory factors — the interaction of the three factors.

Research on competitive diffusion opens up a large set of intriguing questions, as competing behaviors can embody different characteristics. We can then test how network transitivity, adoption threshold, and connectedness of early adopters, and the interaction between the three factors, benefit each behavior according to these behavioral characteristics. There are numerous ways to assign behavioral differences. I considered quality of behavior, resource of organizations, interest of adopters, inertia of behavior, but chose to examine a sociologically relevant characteristic—number of initial adopters. In cases of competitive diffusion, gaining an initial upper hand may eventually transfer into enormous advantages (David 1985; Petersen et al. 2011; Salganik et al. 2006). For example, even though competitors had better designs, the QWERTY keyboard prevailed due to its early market advantage (David 1985). Diffusion is a path-dependent process (Mahoney 2000), and once one behavior amasses enough adopters to initiate widespread diffusion, it can easily overpower opponents. The process is similar to the start-up problem of one-behavior diffusion (Heckathorn 1996; Marwell and Oliver 1993; Oliver et al. 1985; Schelling 1978), but instead that multiple behaviors are racing to cross the start-up barrier. Nevertheless, the degree by which this “head-start” effect occurs may depend on the three factors outlined above.

For the interested reader, all the other ways to assign behavioral differences are straightforward, and in the online supplementary I document how to incorporate such differences in the model. Nevertheless, testing all such possibilities is beyond the scope of this study.

4. DESIGN OF THE COMPUTER EXPERIMENTS

The present study tests how the interaction between the three factors affects the “head start advantage.” In the perfect world, one would find many different groups with the same number of actors, map all the relationships between the actors, then longitudinally record the number of adopters for each behavior. However, conducting such social experiments is nearly impossible. Groups are rarely equal in size, and even if they were, the number of ties would often be different, confounding the ability to make causal claims. Furthermore, one would need to consider actor differences in race, gender, age, social status, and many other characteristics. Instead, scholars have drawn insight via computer simulations (Baldassarri & Bearman, 2007; Centola & Macy, 2007; Kim & Bearman, 1997; Marwell et al., 1998). Computer simulations permit the researcher to make networks exactly equal except for a few characteristics, which allows the researcher to isolate causal effects.

Drawing from this tradition of computer simulations, I model the diffusion of two competing behaviors. I experiment with how transitivity, adoption threshold, and connectedness of early adopters affect the “head-start advantage.”

In the model, each actor can decide to adopt one behavior, adopt the other behavior, or remain a bystander in each iteration. The decision is based on which behavior(s) are adopted by the focal actor’s connected others, which in turn depends on the adoption threshold (i.e., how many connected others need to adopt for the focal actor to adopt) and the network structure (i.e., who is connected to whom). This general model can be used to explore many empirical scenarios, such as what happens when conflicting rumors spread on social media, when political candidates compete in village networks, and what medical practices dominate workplace relationships in hospitals. To better anchor the model to social settings, Table 1 highlights a few examples that correspond to different parameter spaces.4

Table 1.

Some empirical scenarios involving varied parameters

Diffusion scenario Adoption threshold Transitivity Early adopters
Competing memes on social media
(Bennett and Segerberg 2012)		 Low Low Unconnected
Conflicting rumors in cities
(Thomas 2007)		 Low Low Unconnected
Organized rebellions in rural villages
(Kalyvas 2006)		 High High Connected
Product adoption through social media
(Taylor et al. 2011)		 Depends on product cost Low Unconnected
Fashion competition in high schools
(Kim et al. 2008)		 Low High Unconnected
Religious conflict in metropolitan areas
(Kim and Pfaff 2012)		 High Low Connected
Social movement (and countermovement) mobilization through organizations’ online accounts
(Mercea 2013)		 High High Connected
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Before delving into the parameters, I should note that the study examines discrete behaviors that are mutually exclusive, such as technological adoption (adopt/not adopt) or protest participation (participate/not participate). The model does not apply to scenarios where the competing behaviors lead to hybrid or innovative outcomes (Aral and Van Alstyne 2011; Burt 2001). It also does not apply to situations where a spectrum of other possibilities exists between the two behaviors, as for in the case of belief systems, which may be strongly conservative, slightly conservative, slightly liberal, or strongly liberal. It is possible to incorporate such possibilities in the model, and I detail modifications to the model in the online supplementary. However, since this study already experiments with three factors and their interactions, I leave these endeavors to future research.

4.1. Model parameters

In line with earlier computer experiments (Baldassarri and Bearman 2007; Heckathorn 1993; Kim and Bearman 1997), I simulated the diffusion process while varying the parameters governing behavior adoption. Macro outcomes, such as the proportion of actors in the network adopting a given behavior, reflect their interdependent decisions—if one actor adopts a behavior, it affects other actors to behave similarly, which further affects others. However, whether an actor adopted a behavior in the first place depends on which actors it was tied to. The advantage of computer simulation is that it can trace the consequences of these interdependent processes, which cannot be solved mathematically.

In the present study, the decision to adopt a behavior is understood as a function of the following parameters.

(1). Threshold:

Threshold is defined as how many connected others are needed to adopt a behavior for the focal actor to adopt. For instance, if the threshold is one, only one neighbor needs to adopt the behavior before the focal actor follows suit; this is often the case for low-cost behavior such as spreading a rumor. However, if the threshold is five, at least five neighbors must adopt before the focal actor does so; these cases of high-cost behavior, such as joining a protest, require multiple encouragements for activation.

(2). Network Transitivity:

In a network graph, transitivity is calculated by the ratio between the observed number of closed triplets and the maximum possible number of closed triplets in the graph (Wasserman and Faust 1994). A triplet consists of three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. Formally,

Transitivity=numberofclosedtripletsnumberofalltriplets(openandclosed).

Intuitively, it measures the extent to which actors that share a neighbor are connected—such as when friends of friends are also friends. High transitivity indicates that actors form social circles, as for instance in small towns. In contrast, low transitivity means that people have many opportunities to connect to a wide range of others, as in metropolitan areas.

(3). Connectedness of early adopters:

There are different ways to connect early adopters. In this study, I binarize connectedness of early adopters as either connected as one component, or randomly scattered. Fully connected is defined as all the early adopters are connected as one component in the network graph, and each early adopter has at least one other adopter that it is linked to. Randomly scattered indicates that the position of the early adopters in the graph are random.

Whether early adopters are connected may have implications for diffusion. Early adopters form local coalitions that create the social reinforcement required for high-threshold behaviors. Such scenarios are more likely when organizations form before initiating diffusion, as in the case of a secret society before a rebellion (Perry 1980).

The parameter space of the computer experiment is summarized below:

Threshold includes five possibilities: {1, 2, 3, 4, 5}

Transitivity is a continuous range of possibilities: [0, 0.71]

Connectedness of early adopters includes two possibilities: {Early adopters are randomly scattered, Early adopters are connected}

And the outcome of the computer experiment is the proportion that adopts each behavior:

Proportion that adopts one behavior: [0, 1]

Proportion that adopts the other behavior: [0, 1]

Proportion of non-adopters: [0, 1]

The three proportions sum up to one.

4.2. Model algorithm

I examine the diffusion of two competing behaviors, denoted as Behavior Ahead and Behavior Behind. Behavior Ahead is the behavior that gained a small upper hand in the number of initial adopters (i.e., a “head-start” advantage), while Behavior Behind had fewer initial adopters. I adopted the following procedure to simulate the diffusion process.

  1. Following Watts (1999) and Centola and Macy (2007), I constructed a “small world network” by rewiring ties from a ring network of 400 actors, each connected to 10 neighbors, resulting in a density of 0.025. The network structure varied in terms of rewiring probability,5 which changes network transitivity.

  2. About 6% of the network actors were assigned to Behavior Ahead and about 4% to Behavior Behind. For these early adopters, the connectedness of early adopters was either

    1. randomly scattered in the network, or

    2. connected to one another. For each behavior, I assigned one actor and its neighbors to adopt, then to select another actor and its neighbors to adopt. I repeated the process until the criterion of 6% or 4% was reached.

    The rationale was that a small proportion of actors (roughly 5%) are initial adopters. I opted for a very small difference of 2% between the two to examine the conditions in which a small “head-start” difference creates large differences in diffusion. As the behaviors were otherwise substitutable, this was the only difference between the two behaviors. Ceteris paribus, then, Behavior Ahead will always have an advantage over Behavior Behind, although the extent of that advantage will vary according to the three experimental factors.

  3. For each iteration, whether the focal actor adopts/changes a behavior depends on the behavior of its neighbors and the threshold of adoption6. Specifically, for each actor,7

    1. if the number of Behavior Ahead neighbors is larger than the number of Behavior Behind neighbors, and the difference is larger than threshold, then adopt Behavior Ahead;

    2. if the number of Behavior Behind neighbors is larger than the number of Behavior Ahead neighbors, and the difference is larger than threshold, then adopt Behavior Behind;

    3. otherwise, become Neutral (bystanders).

  4. For each simulation, I ran the simulation until no actors changed their adoption status. I then recorded the final proportion of each behavior in the network—that is, the proportion of adopters of Behavior Ahead, of Behavior Behind, and of Neutral actors. Together, the three account for the entire set of actors.

  5. I reran 1000 simulations and took the average final proportions for each behavior as the final result.

5. RESULTS

The outcome of competitive diffusion depends on the interaction of the three parameters (transitivity, adoption threshold, connectedness of early adopters). As the results are somewhat abstract, I also describe an illustrative empirical setting for each to clarify the scenario and outcomes (similar to Table 1 previously).

5.1. When early adopters are randomly scattered

These are scenarios where early adopters are not coordinated or organized—for example, when there is a source of mass influence, and individuals initially react spontaneously. For instance, when many citizens are enraged by a political event to protest but did not coordinate beforehand.

Figure 1 presents the results. I first examine how transitivity affects the outcome for low-threshold behaviors (i.e., threshold is one or two). The varied parameters are transitivity and adoption threshold. The x-axis represents network transitivity, and the y-axis represents the final proportion of each behavior at equilibrium—that is, the outcome of the last iteration (I do not plot the process for each iteration, and the x-axis represents network transitivity and not the iteration number). The red line (circles) indicates the final proportion of Behavior Ahead; the green line (triangles) indicates the final proportion of Behavior Behind; and the blue line (squares) indicates the final proportion of non-adopters. In each subplot, results differ by threshold; the left plot shows results for a threshold of one.

Figure 1.

Figure 1.

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Effect of transitivity when early adopters are scattered.

The results demonstrate that network transitivity contributes to the magnitude of the “head-start advantage.” For low-threshold behaviors, low transitivity means that “head-start” behavior (Behavior Ahead) dominates completely. The red line reaches almost 1 when transitivity is 0.3 or smaller while the blue and green lines are close to 0. Although the initial difference between Behavior Ahead and Behavior Behind was only 2%, the final difference increases to almost 100%. Small differences in initial adoption promote large differences in diffusion when the network structure exhibits few overlapping ties. The magnifying effect is because low transitivity extends the reach of behavioral diffusion, and in cases of low-threshold behaviors, diffusion scales with number of adopters. Behavior Ahead therefore snowballs much faster than Behavior Behind, and eventually acquires enough adopters to influence adopts of Behavior Behind to switch to Behavior Ahead. An example is the diffusion of rumors through online networks; once a rumor attracts enough believers, it can quickly grow to reach many people and so dispel any conflicting rumor.

On the other hand, when transitivity is high, the difference between Behavior Ahead and Behavior Behind shrinks. When transitivity is 0.6, the difference is only 10%. In such scenarios of low-threshold behavior, transitivity is beneficial for Behavior Behind but detrimental for Behavior Ahead. In the case of cultural norms in school networks, different cliques of friends may emphasize different behaviors are “cool,” with some valuing academic performance and some valuing outdoor activities. In such cases, because the threshold is low and transitivity is high, small circles of peers quickly converge to one norm. However, spread to another peer circle is much more difficult, leading to factions.

However, the results are different for high-threshold behaviors (i.e., when threshold is three or higher). When the threshold is too high (i.e., 4 or 5), neither behavior diffuses because there is not enough social support to overcome the “start-up problem” (Heckathorn 1996; Marwell and Oliver 1993; Oliver et al. 1985; Schelling 1978).

At a medium threshold of three, transitivity is beneficial for diffusion of both behaviors. Both the red and green lines increase with transitivity, but the slope of the red line is greater. Unlike the detrimental effect of transitivity for Behavior Ahead in low-threshold behaviors (see Figure 1), transitivity is beneficial for Behavior Ahead in cases of high-threshold behaviors. These processes occur, for example, when politicians in villages with few grassroots aim for political support. Because of the high level of network transitivity, scattered supporters may still be able to connect and to persuade others to support their candidate. However, the high threshold means that they may only convince a few, and diffusion quickly ceases. Where social or geographical boundaries confine small groups of supporters, neither candidate is likely to win a majority.

5.2. When early adopters are connected

In these scenarios, early adopters were connected rather than randomly scattered. Such scenarios often arise when the diffusion process is initiated by organizations, such as social movement organizations or marketing groups. These initial adopters first coalesce and then spark the diffusion process. Figure 2 shows the results.

Figure 2.

Figure 2.

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Effect of transitivity when early adopters are connected.

For low-threshold behavior, similar to the results when early adopters were scattered, transitivity diminishes the number of adopters of Behavior Ahead and increases the market share for Behavior Behind. However, one can also notice that the magnitude of the “head-start advantage” shrinks (compare with Figure 1). Behavior Ahead does not completely dominate the network even when transitivity is low, as Behavior Behind also acquires a substantial proportion of adopters. When transitivity is high, the difference in adopters is even smaller. For instance, when transitivity is higher than 0.6, the difference between the two is very small. Connecting early adopters diminishes the advantage of the head-start.

Again the results differ for high-threshold behaviors. Connecting early adopters allows diffusion to occur even for behaviors of very high thresholds. Recall that when early adopters were scattered, diffusion did not occur when the threshold was too high (i.e., four or five). This is not the case when early adopters are connected. Even when the threshold is very high, the red and green lines can be non-zero, especially when transitivity is high. The combination of connections among early adopters and high network transitivity promotes diffusion even for very high thresholds. Although both behaviors diffused, we see that neither behavior dominates the network, and each behavior owns a small portion of adopters. Such an outcome is typical of settings involving organized movements, where the initial connected organizers try to persuade others to join their collective cause. These organized efforts can further benefit from high transitivity networks, such as the church networks involved in the civil rights movement (Morris 1984). In these high-cost collective action scenarios, some diffusion may occur, but it is difficult to persuade many bystanders.

5.3. Additional experiments and robustness checks

The present study experimented with transitivity, adoption threshold, and connectedness among early adopters. However, one may wonder if other factors related to network structures may affect diffusion outcomes. Thus, I conducted additional experiments with network density and network evolution. Network density refers to the number of ties in a network. The more ties there are, the more people are connected, and it is easier for anyone to be connected to an adopter (Kim and Bearman 1997; Watts 1999). Network evolution refers to how the ties in a network form and dissolve, often according to similarity in behavior (Baldassarri and Bearman 2007; McPherson et al. 2001). I present the results in the online supplementary, but briefly summarize them here.

Regarding network density, the general mechanisms in the main analyses remain the same regarding the three focal factors. Still, density has the effect of connecting more people, thus easing diffusion. The effect is strongest in high-threshold behaviors, which require many sources of contact. Given certain combinations of density, transitivity, and connectedness among early adopters, the head-start advantage can be negligible.

Network evolution does not appear to impact the diffusion outcomes strongly, and the general mechanisms for the three factors are similar to the main analyses. However, network evolution dramatically increases the transitivity of the network. Thus, if another diffusion event occurs, the outcome would be very different due to the change in transitivity.

I also conducted two robustness checks, which are available in the online supplementary: I experimented with (1) adding a small random probability in adoption, and (2) changing the adoption threshold from a number-based to a proportion-based. Neither of these checks altered the mechanisms in the main analysis.

6. DISCUSSION

Why did Facebook overpower its opponents? Can we provide answers to whether multiple militias lead to frozen conflict? Can we predict the vote share of political candidates? This study used a network perspective to explain such scenarios of competitive diffusion.

The outcome of competitive diffusion is likely to differ from single-behavior diffusion. The results showed that two-behavior diffusion could lead to one behavior dominating the network, two behaviors splitting the network, two behaviors each securing a small portion of the network, or absence of diffusion. Dependencies render the process much more complicated and highlight the importance of a model that incorporates the countervailing forces of competing behaviors. Lessons from one-behavior diffusion may not necessarily apply to competitive diffusion.

One example of the difference is the effect of network transitivity. Centola and Macy (2007) argued that transitivity would have an inverted-U shape effect on diffusion. In cases of low-threshold behaviors, low transitivity benefits diffusion; in cases of high-threshold behaviors, moderate to high levels of transitivity benefit diffusion. However, the results indicate when behaviors compete for diffusion, the effect of transitivity depends on the adoption threshold and whether early adopters are connected. Transitivity may be beneficial for both behaviors, beneficial for one but detrimental for the other, or there may be no effect. The differing effects of network structure on diffusion suggest that one-behavior models may not apply to competing behaviors.

In cases of competitive diffusion, small differences in initial adoption may lead to significant subsequent differences. In the initial setup, the difference in behavioral adoption rates was a mere 2%, but Behavior Ahead attracted a much larger proportion of adopters in most scenarios than Behavior Behind. These results support the view that diffusion processes are path-dependent (Mahoney 2000); a small head-start increases diffusion in the network, which can snowball to outstrip the opponent completely. For example, in cases of technology adoption such as rivals to the QWERTY keyboard (David 1985), early entrance to the market can populate the network, ultimately wiping out competitors.

Evidence of the head-start advantage is nothing new (David 1985; Petersen et al. 2011; Salganik et al. 2006). However, the present results indicate that the extent of that advantage depends on network conditions. The results invite further questions about early market advantage—for example, when can the first firm to market expect to enjoy a sufficient advantage? In a political context, when does early support for the first “alternative candidate” (such as a populist) suffice to prevent imitators from stealing votes?

Such questions are also important for relatively disadvantaged groups such as minority movements or latecomers to the market. The results indicate two pathways. First, if network transitivity is high, a disadvantaged group can still gain a foothold, especially for low-threshold behaviors. Network transitivity creates social circles with many overlapping ties that are difficult to penetrate. Such circles are much easier to create in the case of low-threshold behavior, as the disadvantaged behavior can quickly populate the social circle. In contrast, although low-transitivity networks may extend the reach of competing behaviors, the network structure also magnifies the head-start advantage and demolishes the disadvantaged. In this case, “small world” dynamics do not always benefit diffusion (Watts 1999; Watts and Strogatz 1998), as such dynamics enhance the diffusion of one behavior but obliterate the other.

For cases of high-threshold behavior, both behaviors benefit from transitivity, although the behavior with an initial head-start will diffuse more rapidly. Still, the disadvantaged behavior can establish a support base. For example, the literature on polarized societies indicates that segmented societies—where networks are highly transitive—tend to intensify polarization (Buskens et al. 2008; Esteban and Ray 2008; Montalvo and Reynal-Querol 2005). The present results add to this view by showing that network transitivity not only creates social cleavages, but also widens the difference between advantaged and disadvantaged groups. For instance, in civil conflict situations where both sides seek to recruit support from densely knit villages, it is rarely the case that both sides are equally powerful. Instead, the disadvantaged group grows slowly while the stronger opponent grows at a faster rate. The disadvantaged group may then choose to hold their base and resort to alternative tactics such as guerilla combat (Berman 1990; Wood 2003).

As a second condition for reducing head-start advantage, early adopters can connect to one another. Centola (2013) showed that such connections can generate the critical mass required for diffusion in high-threshold behaviors. The present results show that connectedness among early adopters not only enhances diffusion for both groups but also prevents the disadvantaged group from being overwhelmed. Early coalitions can help to ensure that a behavior spreads and prevents the advantaged group from infiltrating converted clusters. In high transitivity scenarios such as smaller villages, the disadvantage may be almost negligible, further confirming the importance of early coalition for disadvantaged groups.

For both groups, coalitions among early adopters facilitate diffusion in scenarios involving higher thresholds, especially when transitivity is also high. Given the path-dependent nature of diffusion, connectedness among early adopters is again a necessary condition for high threshold scenarios. In the results, when early adopters were scattered, a threshold of four or five prevented diffusion, however high the transitivity. However, the diffusion of such behavior becomes possible when early adopters are connected because of the snowballing nature of early clusters, especially in highly transitive networks.

The critical mass literature pointed to the importance of coalitions among early adopters for initiating high-cost movements (Centola 2013; Marwell and Oliver 1993). This study extends their perspective by showing that the take-off does not necessarily lead to full-blown diffusion that converts all bystanders. Both behaviors are more likely to account for only a small fraction of the network, with many more bystanders than adopters. The presence of overlapping ties prevents the initial cluster of early adopters from breaking the boundaries of their social circle to reach other circles, thus limiting diffusion. On that basis, the primary focus for both groups should be to recruit bystanders rather than seeking to defeat their competitors. For example, there is evidence that, in many African countries, Christians and Muslims focus on converting remaining non-believers rather than trying to convert each other and even cooperate on public issues (Frederiks 2010). The intersection of network structures and adoption threshold affects the incentive for competing groups to clash, with implications for the conditions of civil conflict (Kalyvas 2006).

The results also have implications for studies of digital networks. Digital networks tend to create highly transitive clusters. These “filter bubbles,” where users are exposed to similar ideas (Flaxman et al. 2016; Pariser 2011), enable different groups to emerge without allowing one to become dominant. The present results add to this discussion by suggesting that network transitivity allows large groups to grow at a faster rate, which may help explain the emergence of large-scale movements (Bennett and Segerberg 2012; Castells 2015). On the other hand, the inability of large groups to infiltrate smaller ones also explains the emergence of many small segmented networks online (Hsiao 2017), including extremist groups (Wojcieszak 2010).

7. IMPLICATIONS FOR FUTURE RESEARCH

The findings indicate avenues for future research. Crucially, by assuming the same adoption threshold and connectedness of early adopters for both behaviors, the only difference was the proportion of early adopters, and the two behaviors were otherwise substitutable. Ceteris paribus, the advantaged behavior always outperforms the disadvantaged behavior. However, this may be unduly simplistic, and future work should explore how to incorporate different characteristics for different behaviors. For instance, the intrinsic qualities of different behaviors may render one more appealing than the other, as in the case of a technology with better features than its competitor. In such scenarios, if network structure minimizes the difference between the head-start behavior and the lagging behavior, the lagging behavior even diffuse better if it has superior intrinsic qualities. It would also be interesting to investigate whether a better quality “late entry” behavior can prevail, for instance where Google was the latecomer but outperformed Yahoo. In the online supplementary, I outline how to model many behavioral differences, including quality of behavior, resource of organizations, or inertia of behavior. The interested reader is encouraged to delve into the developments of such examinations.

I also assumed that all actors are homogenous, but in reality, the distribution of qualities such as resources, interests, or influential power may vary (Kim and Bearman 1997; Marwell, Oliver and Prahl 1988; Siegel 2009). Actor heterogeneity may also interact with network structure to affect diffusion, as in the case of fashion diffusion in classrooms, where popular students with higher social status have more influential power. In the online supplementary I also discuss how to incorporate such factors in the model formally.

The model described here applies to competing behaviors that are mutually exclusive, but there may be cases of “bi-directional diffusion” that involve either a spectrum of possible behaviors (Baldassarri and Bearman 2007), hybrids, or innovative outcomes (Burt 2001). In such cases, it would be valuable to experiment with network structures and the connectedness of early adopters. Again in the online supplementary I outline how to incorporate such possibilities in the current model.

Finally, while this paper followed the framework of threshold models when modeling social influence (Centola and Macy 2007; Granovetter 1978; Schelling 1978; Valente 1996), the concept of adoption by threshold is only a special case of social influence processes (see Kitts and Shi 2018; Valente 1996). Experimenting with a wide range of social influence mechanisms would expand the scope of the present study.

The question of how network structures affect competitive contagion raises further intriguing issues, and studies on competitive contagion are needed to analyze countervailing webs of influence and the conditions under which social divisions may or may not occur.

Supplementary Material

Appendix

Footnotes

1

I use the term behavior in relation to a wide range of adoption phenomena, such as using technology, wearing a particular fashion item, joining a collective action, and practicing a religion. The key notion is that the behavior is a dichotomous choice: one can either adopt or not adopt.

2

There are many ways to consider the connectedness of initial adopters. In this article, I connect them as a connected component where all initial adopters are connected as one group through social relationships. Also see section on experiment design.

3

This is equivalent to “rewiring” the network.

4

These examples are broadly illustrative rather than strictly true for each case. For instance, although transitivity is generally low in social media networks, it may be high for the social media pages of certain groups where users connect more (e.g., support pages for patients with the same disease).

5

This refers to the probability of deleting ties from the original network and replacing them with new random ties.

6

Actors may adopt different behaviors or none at different iterations.

7

I assume the same threshold and setup of early adopters for both behaviors, rendering the behaviors substitutable.

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