david demper
NETWORK STRUCTURE,
NORMS, AND THE PROMOTION
OF SOCIAL INNOVATION.
MASTER THESIS, MSC. BEHAVIOURIAL ECONOMICS & GAME THEORY, UNIVERSITY OF AMSTERDAM. AUTHOR: DAVID DEMPER
SUPERVISOR: PROF. DR. MATTHIJS VAN VEELEN JULY 2016
This document is written by David Demper, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
TABLE OF CONTENTS
TABLE OF CONTENTS ... 2
INTRODUCTION ... 4
LITERATURE REVIEW ... 6
Norms ... 6
Norm dynamics / shift... 8
Norms and networks. ... 9
Characteristics of real social networks ... 9
Network generation algorithms... 10
viral spread versus the spread of norms ... 11
the stochastic evolutionary game theory approach ... 12
THE CURRENT RESEARCH ... 21
Output variables ... 23 initial stability ... 23 Network topologies ... 24 METHODS ... 25 The model ... 25 timing ... 26 simulations ... 26 FINDINGS ... 26 circle ... 27
initial stability & foothold ... 27
tipping time ... 27
end stability ... 27
lattice ... 27
tipping time ... 28 end stability ... 28 scale-freee ... 29 initial stability ... 29 initial foothold ... 30 tipping time ... 31 end stability ... 31
CONCLUSION AND DISCUSSION; BACK TO REAL WORLD APPLICATION ... 33
INTRODUCTION
A significant proportion of the behavior and believes people exhibit is at least partly
determined by the behavior and believes of the people surrounding them. Groups of people may hold certain believes, stick to certain norms and conventions, use the same technologies and have opinions not because of their intrinsic value, but because others around them do, and it makes sense to do the same. This need not be a rational consideration people may simply internalize the behavior or opinions of people around them, or it may be (socially) costly to deviate from a behavior that is considered ‘normal’ within the set of people an individual may encounter. If everyone drives on the right side of the road, it makes no sense to drive on the left side. Likewise, it might not be beneficial for an individual to start using a superior
technology, or adopt a superior believe or opinion, before some proportion of the people around them do.
How do these norms and conventions arise, and what dynamics govern their change over time? Several models have been proposed that try to capture this. An established norm, used by a large proportion of some population, can be regarded as an equilibrium of a game with several possible equilibria (Young, 2015). Individuals use a certain convention or norm, because they expect a person they encounter to expect the same. The question then becomes: what governs these expectations? Schelling (1960), being one of the first to study why individuals end up using one convention instead of another, emphasized the importance of focal points: some strategies may be more prominent or conspicuous than others, leading individuals to ‘play’ this strategy. Later authors such as young (1993) have emphasized that over time, expectations may converge due to positive feedback effects: if one encounters a lot of people playing one
strategy, one comes to expect that people will play this strategy, and hence, one will play that strategy too. This highlights the importance of social network structure: expectations and strategy depend on the individuals interacting with a certain individual. Once some norm has become established in a population, it is clear that can be very hard to break, since the expectations of interacting individuals need to change simultaneously.
In its most basic form, which norm to follow is an issue of coordination, which can be modeled as a repeated coordination game played between individuals connected to each other in some network structure (Montanari & Saberi, 2010; young, 2011). Once some behavior, some technology or believe has become the norm, a population has (unknowingly) coordinated on some equilibrium, that tends to be self-enforcing, since it does not pay for individuals to deviate. Punishment of deviants is generally not needed, since it is inherently costly to deviate. (young, 2015). If everyone is using Microsoft word documents, it makes no sense to use
another file format that no one can open, even though the format is superior. The equilibrium may not be the best possible one. It may have been the best one available when the norm arose, or it may have been inferior from the start. Either way, circumstances tend to change, and improved options become available. The abandonment of these inferior equilibria, and the coordination on beneficial social equilibria is what Montanari & Saberi (2010) call ‘social
innovation’. This thesis will evaluate which structural characteristics of social networks may promote or inhibit such social innovation.
Following Montanari & Saberi, (2010) and young (2011) I will assume that agents are not perfectly rational, but instead play a best response to their neighbors with some probability <1, and play a suboptimal strategy instead. The probability of this rational or irrational behavior depends on the expected payoff of the 2 optional strategies, following a log-linear learning rule (see equation 1). This reflects the fact that agents do not have perfect information, or may be more inclined to experiment when differences in payoff are small. I primarily address the spread of norms in Barabási-Albert (BA) networks with scale-free degree distribution, since these offer a realistic representation of the degree distribution that are found in real social networks (Barabási & Albert, 1999), and have been studied very little with regard to the spread of social innovation. It turns out that while initial clusters of social innovation arise rapidly in these networks, it may take a longer time for the innovations to conquer the complete population.
LITERATURE REVIEW
The spread of social innovation on networks has been studied from several perspectives. A relatively big branch of research has evaluated the spread of information and infectious diseases on networks. Further research has tried to evaluate whether the same kinds of network structures that promote the spread of information, or the spread of a contagious diseases, promote the spread of norms. The essential characteristic of the spread of social innovation that is different from ‘standard’ contagion, is that one needs several sources, or some proportion of neighbors to become ‘infected’ (Centola, 2007) with a new norm, whereas one needs only needs one infected individual to become infected with a disease he is carrying, or to obtain a piece of information he holds. In these contagion-based models, only the number separate encounters matter, not the fraction of neighbors at a given point in time. Another branch of literature has taken the tools provided by stochastic evolutionary game theory to evaluate the spread of norms on networks, starting with the early explorations of local interaction and coordination by Ellison, G. (1993). Before going in to the difference in both approaches and their findings, we will first explore some general characteristics of the process of norm change.
NORMS
Before discussing the characteristics of a norm change, and the spread of a beneficial norm through a network, It is worthwhile to describe what exactly we mean by a norm. Norms, as I define them, have several properties:
For the person in question, the ‘payoff’ of following the ‘norm’ increases, the higher the fraction of people interacting with the individual who follow the norm. It is rational for an individual to conform to the norm, as long as the people around him conform too. The set of of ‘interacting people’ may differ in size and shape, depending on the norm. When discussing driving on one side of the other road or the other, everyone ‘interacts’ with a very large amount of strangers, from many different locations. When talking about some ritual in a remote village, or the behavior within a remote cult, the set of ‘interacting’ individuals is much
smaller. For some norms, only a small group of individuals ‘close’ to the individual in question matters. When deciding whether to use Whatsapp or Telegram, whether to use skype or google hangouts, it matters who of your friends and family which you wish to contact use one of the technologies.
A logical result of this property is that norms tend to take over densely interconnected populations fairly completely, since people interact with each other, and individual payoffs increase when coordination is successful. Especially when there is no inherent benefit in one behavior over the other, relatively isolated groups may coordinate on different equilibria between groups, but within the connected group, adhere to the same norm. It may come as no surprise that the countries on the European mainland all drive on the right side of the road, but in England, they drive on the left. In fact, the norm on which side to drive on differed from region to region on the European mainland too, until increased (and longer-distance) travel extended the coordination on one equilibrium (Hamer, 1986). Centola (2014) performed an experiment where participants were paid to coordinate with others on the name of a person in a picture. When participants randomly interacted with another person every round, a global convention regarding which name to use would arise fairly rapidly. When participants
interacted only with their closest few neighbors, local coordination would arise rapidly, which inhibited the formation of global coordination. Young (2015) goes as far as to describe this as a general tendency of norms: there tends to be global diversity, and local conformity. More accurate would be to state that this depends on network structure, the timeframe which is evaluated , and the underlying payoffs of the coordination game.
Norms typically evolve without top-down direction; they arise in a process of experimentation, trial and error (young, 2015). Although authorities and media can exert great influence, they will be kept outside of the scope of the this paper. They may also not greatly change the dynamics, but may simply change the (perceived) payoffs of the local coordination game individuals find themselves in.
Besides the coordinating aspect of norms, social norms may also serve as important signaling devices, which people tend to use to indicate that they belong to some group or class. Mackie
introduced in harems to promote fidelity, came to be signals of class and fidelity. Once a very large proportion of some social system starts performing the signaling behavior, the signaling value decreases, and the behavior gets the characteristics of a norm: it can be very costly to belong to the first people to deviate. Centola, Willer, & Macy (2005) describe that behaviors and opinions that very few individuals actually believe in / agree with can take over a
population, when people enforce the norm to signify that the belong to a group, and incorrectly assume that everyone else in the group follows these norms. An interesting empirical
demonstration is for instance the classic study by Schank (1932), of a religious community, who found that members privately rejected strict norms on drinking and gambling behavior, but endorsed them in public because they expect the fellow members of the group to believe in these norms.
NORM DYNAMICS / SHIFT
Dynamics of norm shifts possess some essential features that an agent based model should be able to reflect in order to capture reality. First of all, Norms tend to be static for long periods of time, because they are inherently self-enforcing (young, 2015; Mackie, 1996; ). Deviating from the norm does not pay off, before others around you do the same. Secondly, when a norm is replaced by another one, the equilibrium tends to tip, and the new norm replaces the old norm in a relatively very short time frame (young, 2001; Mackie, 1996).
The fact that norms tend to be stable, undergo a period of much activity when they ‘tip’ to another equilibrium, has much in common with other critical mass phenomena, where positive feedback loops are present (See Schelling, 1978; Gladwell, 2006). The reason such a period of rapid change occurs at a certain time can depend on several factors. The first factor is chance. Imagine a repeated game, where every individual has to pick whether or not to conform to the norm in every period. With large probability, individuals play a best response to their neighbors. With some small probability, an individual does not play a best response in some period, due to experimenting, imperfect information, forward looking behavior or other factors. If a large enough group of people interacting with each other adopts this new norm in the same time
period, it may become sustainable for this group of people to keep following this new norm. It may take a very long time for such a foothold to form, but if it does, and network structure allows, it can spread fast across the population, leading to the observed tipping behavior. Of course, this process may be aided by external factors. External factors may decrease the payoff to the old norm relative to some alternative, or the network structure may change in such a way that norms that previously did not interact, start interacting with each other. If the norm used to be to believe in god, the progression of science may have changed the payoffs to believing or not believing. This, in turn, increases the chance that an initial threshold of non-believers may become sustainable, after which a wave of secularization may occur. On the other hand, roads, the internet and other technologies may alter (increase) the connectivity within networks where norms interact.
NORMS AND NETWORKS.
Since the literature on the spread of information and diseases in networks is very extensive, and quite intuitive, some authors have taken it as a starting point for studying the spread of norms (eg centola, 2007) Before discussing the findings of the papers attempting to make this step, it makes sense to look at some basic characteristics of social networks. These
characteristics have been formalized in some network generation algorithms, that we will briefly discuss below, before evaluating which of these structures promote the spread of information, and may or may not promote the spread of norms.
CHARACTERISTICS OF REAL SOCIAL NETWORKS
First of all, social networks tend to exhibit very small average shortest path lengths, also called a low degree of separation (Milgram, 1967; Watts & Strogatz, 1998). That is to say, every node in the network is connected to every other node in the network via very few intermediary nodes. This property is often described as the small world phenomenon (Milgram, 1967; Watts & Strogatz, 1998). Secondly, social networks tend to feature a large amount of transitive relations: if person A knows person B and C, person B and C tend to know each other too
(Granovetter, 1973), This property is also called clustering (Watts & Strogatz, 1998). Thirdly, the degree of nodes (the number of connections to other nodes one node possesses) tends to be heterogeneous, that is; some nodes have more connections to other nodes. The degree distribution of many social networks tends to be scale-free: some individuals have very much social connections, whilst the majority of people only interacts with few other individuals (Barabási & Albert, 1999).
NETWORK GENERATION ALGORITHMS
There are several standard network generation methods which achieve some of the 3 properties mentioned above:
1. Small world networks (Watts & Strogatz, 1998). Small world networks are constructed by randomly rewiring some edges in a ring lattice or regular lattice. This achieves the ‘small world’ property: on the one hand, the average degree of separation between nodes becomes small with some random rewiring, but most interactions are still local. This means that the second property mentioned above can be realized: many transitive relations.
2. Barabási-Albert (BA) scale-free networks (Barabási & Albert, 1999). BA networks are constructed by a mechanism called preferential attachment: new nodes have a probability of attaching an edge to an existing node with a probability relative to de degree of that existing node. Starting with 𝑚𝑚0 initially connected nodes, nodes are
added and connected to the existing network. Every newly attached node connects with 𝑚𝑚 new edges to the network. This algorithm provides network with scale-free degree distributions and very short average longest path length, but lacks the property of transitive relations. The average degree depends on the number of links added per added node. These networks reflect property 1 and 3, but lack property 2.
The spread and emergence of information, norms, conventions, culture, cooperative behavior etc. has been studied quite extensively in the Barabási-Albert model and the Watts & Strogatz Model (next to the less realistic randomly connected networks, regular lattices and well-mixed
populations) (see eg Centola, & Macy, 2007; Santos, Pacheco, & Lenaerts, 2006; Perc, et al., 2013; Centola, 2010; Watts, 2002; Klemm, Eguíluz, Toral, & San Miguel, 2003; Centola, & Baronchelli, 2015). Some less used network generation methods try to capture all 3 features of real social networks above, such as Holme and Kim (2002) who provide a network generation algorithm that extends the Barabási-Albert model to include transitive relations.
VIRAL SPREAD VERSUS THE SPREAD OF NORMS
In the age of social networking websites, ‘viral’ spread may be the first thing we think of when we consider how new behaviors, opinions and norms may spread through a population. However, viral models assume that a certain proportion of individuals become ‘infected’ as soon as they come in contact with the new norm / behavior, an assumption that may not hold for most norms. Consider driving on the right side of the road. If we encounter someone who drives on the left side of the road in a right-driving country, no reasonable person encountering this individual will adopt the norm of the deviant. We will signal with lights, we will honk our horn, and maybe alert the police. Only if we were to encounter a large proportion of people driving on the left for an extended period of time, we would consider revising our strategy, which might greatly change the dynamics.
3 Topological features of networks have been shown to greatly increase the speed of viral spread (Montanari & Saberi, 2010). 1) high (average) degree of nodes; much interconnectivity. 2) ‘Long links’ or ‘weak ties’ connecting distant parts of the population (see Granovetter, 1973). 3) heterogeneity of degree, and especially the presence of hubs, nodes who have a very high degree. Transitivity and dense local connectivity is less important for viral spread, since nodes near infected nodes will probably become infected anyway. Therefore, randomly connected networks and well-mixed populations are very efficient for viral spread.
These results only to some degree extend to the spread of norms. The difference in dynamics stems from 2 determinants of expected waiting time before some social innovation takes over a population: time till an initial foothold is formed, and the speed with which the norm spread to the complete population. The first aspect captures the difference in how a social innovation
comes about. For viral spread, only one initially infected individual is needed, from here on, other infections will follow, given that it is sufficiently contagious and the host (or the behavior / idea itself) does not die before others are infected. This is not the case for the spread of a social innovation in the sense discussed here: when one individually (irrationally) adopts some social innovation, it may not be rational for his network neighbors to adopt it too, since the majority of his neighbors are still using the old norm. In other words, ‘mutants’ adopting some new norm need to arise in large enough clusters, so it becomes rational for the people
surrounding the mutants to adopt the new norm too. The second aspect influencing the expected waiting time till some social innovation takes over the population is the speed with which a new norm spreads through the population once it has established this initial foothold. The second aspect of expected waiting time to take over the population has been researched in extensions of contagion dynamics. Centola (2007) defined ‘complex contagion’ as contagion for which several, or a fraction of neighbors need be infected before you become infected too. After assuming an initially infected cluster of individuals, he measured the expected waiting time for a new, beneficial norm to take over the population, as a function of the fraction of randomly rewired ties in a small world model (0% rewired = regular lattice, 100% rewired is = random network). He found that whereas simple contagion profits greatly even from low levels of random (long) connection, complex contagion only profits from a small interval of
reconnection fraction, centered around approximately 10% long ties. Low fractions of random long connections do not speed up cascades, whereas high fractions inhibit them altogether. In order to capture the first aspect of expected waiting time, however, complex contagion is not enough, and stochastic evolutionary game theoretical approaches have been the most fruitful. Before discussing to the results, the section below will illustrate how Evolutionary game theory has been applied to research the spread of social norms.
THE STOCHASTIC EVOLUTIONARY GAME THEORY APPROACH
The coordination game has been the basis for evolutionary game theoretical research into the evolution of norms and conventions. It reflects the basic feature of norms that it tends to pay to
follow them as long as others around you do, and if you play an alternative strategy, it pays if the people around you start doing the same. Given that you meet someone using norm A, you would obtain a higher payoff using norm A too. Consider the symmetric 2-player game in Table 1. If 𝑎𝑎 > 𝑐𝑐, and 𝑑𝑑 > 𝑏𝑏, (𝐴𝐴, 𝐴𝐴) and (𝐵𝐵, 𝐵𝐵) are both Nash equilibria, and the game is a coordination game. In the most simple coordination games, the values b and c are zero. If 𝑝𝑝𝑎𝑎𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎 > 𝑝𝑝𝑎𝑎𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑, strategy 𝐴𝐴 is the payoff dominant strategy. Furthermore, (A,A) is said to be the risk dominant equilibrium (Harsanyi & Selten, 1988). If b and c are nonzero, the payoff dominant strategy and the risk dominant strategy need not be the same. In this case, the payoff dominant equilibrium is still the equilibrium that yields the highest payoffs. The risk dominant strategy is the strategy that gives the highest expected payoff given that you attach equal probability to your opponent playing 𝐴𝐴 or 𝐵𝐵, that is, If 𝑎𝑎 – 𝑐𝑐 > 𝑑𝑑 – 𝑏𝑏, the equilibrium (𝐴𝐴, 𝐴𝐴) is the risk dominant equilibrium (Harsanyi & Selten , 1988).
Table 1
A B
A a b
B c d
UPDATE MECHANISMS.
Several ways in which agents may update their strategy have been proposed within evolutionary game theory. These update mechanisms are based on different assumptions regarding the rationality and information agents possess. Young (2001) identifies 4 basic ways in which agents may adapt to their environment.
1. Natural selection. With this update regime, players who obtain a higher payoff are at a reproductive advantage relative to players who obtain a lower payoff. Over time, therefore, players who play a strategy that yields a higher payoff tend to be at a reproductive advantage, which makes this strategy grow, given that network structure
allows for this. There are several ways in which this natural selection process can be modeled. One of the possibilities is a dead-birth process, where one individual is randomly chosen to die. The neighboring individuals then compete for the empty spot, and have a chance to fill the empty spot with one of their offspring with a probability proportional to the payoff they get from the strategy they are playing (see Nowak, 2006) It is important to consider that the payoffs of the game do not reflect the agents’ preferences in this case. The only thing the payoffs reflect are reproductive success rates (young, 2001). This update regime does not suppose rationality of agents, it only supposes that the strategies that increase the relative fitness of an individual are more likely to replicate.
2. Imitation. With this update regime, individuals can usually copy the strategy played by one of their neighbors, or keep their own strategy. The choice of strategy depends on the relative payoffs of strategies: whilst the payoff to the own strategy is inversely related to the chance off imitating someone else’s strategy, a neighboring strategy with high payoff increases the chance of this strategy being imitated. Simpler models are also possible, such as models where an individual chosen to update simply copies the
neighbor who obtains the higher payoff, or picks a strategy of one of the neighbors at random (Nowak, 2006).
3. Reinforcement learning. In this model, individuals only look at the history of their own payoffs. Historical payoffs to playing one strategy are compared with historical payoffs of playing another strategy, and the historically higher paying strategy is chosen with greater probability. The ‘history’ considered may be an x number of previous rounds, or memories may ‘fade away’, with more recent payoffs more important for the
consideration which strategy to use than older payoffs. Although it seems fair that reinforcement learning plays an important role in many behaviors, it assumes very limited rationality of agents as an update strategy: individuals simply experiment and play the strategy that used to work, regardless of what others around them are currently doing.
4. Best reply. If an individual plays the strategy that maximizes his expected payoffs given the strategies he expects others to play, he is playing a best reply. In the simplest version of best reply, the strategies of neighbors are observable for the individual, and do not change after the observation. Hence, the reply is optimal per se, since he knows which strategies others will play. In more complicated models, individuals do not know which strategies neighbors will play, but take the historical strategies played by
neighbors into account, form an expectation based on these, and adjust their own strategy to this expectation. Another way to capture imperfect information or
tendencies to experiment that individuals may have, is to let them play a best response with some probability, but assign a small positive probability to alternative strategies too. This approach has been used by many authors discussing the evolution of norms. Some authors, such as (Ellison, 1993) use a fixed probability that an individual will make a random choice instead of a best reply. A more common approach is to use a log-linear learning model, where the chance of playing a suboptimal strategy is not fixed, but depends on the ‘cost’ of playing this strategy. This model will be discussed in more detail later.
When discussing the evolution of norms, none of the abovementioned strategies offers a perfect reflection of reality. It seems more likely that the way in which people actually learn what behavior to use is a combination of rational consideration of the environment, past own experience via reinforcement learning, and imitation of successful others, depending on the situation. Of course, even such a blend of update rules with added irrationality would still be an incomplete reflection of reality. People may differ in their preferences, status may play a role, etcetera. However, considerable insights have been be gained by using these simplified
representations. Almost all of the papers on the evolution of norms use some form of the best reply update mechanism. It captures the idea that people actively adapt to what they expect others to do, which is an essential feature of norms (e.g. schelling, 1968, young, 2015). When studying the spread of norms trough social networks, it makes no sense to assume that agents only interact with one other individual. The way this problem is usually solved, is by assuming that an individual can pick one strategy per round, and plays the two-person game
with all of his neighbors once per round. The payoff of these interactions is summed. If an individual is chosen to update under a best-reply regime, he or she will pick the strategy that will optimize his (expected) payoff to the strategies of others he is interacting with. Usually, agents are assumed to not always play the best reply: this reflects the fact that they are not completely rational, like to experiment, or do not have perfect information (see Young, 2011; Montanari & Saberi, 2010; Ellison, 1993).
Whilst some papers have assumed initial situations where both strategy A and B are present in equal quantities, most work has been conducted in studying how a population of players in some equilibrium may be invaded by another (beneficial) strategy. The papers starting with equal fractions of A and B players have primarily dealt with the question whether the risk dominant, or payoff dominant strategy tends to take over populations. The answer to this question seems to be that populations playing 2x2 coordination games with symmetric strict Nash equilibria tend to converge to playing the risk dominant strategy, given a combination of experimentation (or errors) and myopic behavior. This seems to hold holds for populations playing one strategy which are invaded (Blume, 1995; Kandori, Mailath & Rob, 1993; Young, 1993; Ellison 1993). Off course, depending on the amount of ‘errors’ people make, network structure and size, it may take a nearly infinite time to reach this risk dominant equilibrium. When individuals make errors, defining when a system is in equilibrium is not as
straightforward as with completely rational agents. With completely rational agents, an
equilibrium is reached when every individual in the system is playing one strategy. In this case, if a ‘mutant’ playing the alternative strategy enters the population, his strategy will be selected away, since the new strategy does not manage to coordinate with anyone. Hence, this situation is evolutionary stable (of course, if the mutant strategy is extremely beneficial, and the
individuals around the mutant update their strategy to that of the mutant before the mutant plays his best response, it may not be evolutionary stable. However, for the sake explanation, we will not consider this case here). When individuals are allowed to make mistakes /
experiment with other strategies, there will never be a stable state where the whole population plays the same strategy. Hence, one cannot define equilibrium as a state in which everyone plays some strategy, since this state is very unlikely to occur. Since norms tend to ‘tip’ abruptly
from almost everyone playing one strategy to almost everyone playing the other (beneficial) strategy, the expected time it takes for this tipping point to occur may be of just as much value as the time it takes to reach the actual new equilibrium. Some authors, such as Ellison (1993) have therefore used the time till some fraction of players has adopted the beneficial norm as the variable of interest, not the time till some form of new equilibrium has been reached. Others use the concept of stochastically stable equilibrium (Foster & Young, 1990), which is defined as a state in which “a distribution [of strategies] is restored repeatedly when the
evolutionary process is constantly buffeted by small random shocks” (young, 1993). When using 2-player coordination games, the stochastically stable state is the state in which individuals play the risk-dominant strategy, except for some fraction the individuals who make
mistakes/experiment (young, 1993).
Since the risk dominant strategy tends to take over populations, it makes sense to simplify the game in Table 1 to the one in Table 2, where the off-diagonal payoffs are zero. Following Ellison (1993), young (2011) and Montanari & Saberi, (2010), I will use this simple coordination game, where (A, A) is the risk and payoff dominant strategy, and alpha is the per-period, benefit of mutually adopting strategy A. In the terminology of Young (2011) and Montanari & Saberi, (2010), α is the size of the social innovation, the benefit of mutually adopting the new norm.
Table 2
A B
A 1 + α 0
B 0 1
FINDINGS EVOLUTIONARY GAME THEORY
Evolutionary game theory has revealed that the network structures that tend to promote viral spread, often inhibit the spread of norms (Montanari & Saberi, 2010). Where viral spread is
promoted by interconnectivity (individuals interact with much others), global, random
connections and high degree hubs, norms tend to spread better on locally connected networks, with limited interconnectivity, and no high-degree hubs. This seems to be due to two essential differences between norms and infection. Firstly, an initial foothold needs to be enabled, a location where coordination on a different norm than the rest of the population can be sustained (young, 2011). Secondly, a norm can only spread fast to other areas under the
conditions that it becomes rational for people in these area to adopt the new norm, which may not be the case if the people in this area only interact with separate individuals playing the new norm, not with a crowd big enough to change their behavior.
This basic dynamic can be illustrated by Mackie’s (1996) study on the ending of foot binding and infibulation. These mutilations were very costly norms: families of boys expected ‘decent’ girls to be mutilated. Families of girls expected families of boys to expect them to be mutilated, in order to be seen as a marriage candidate. Even though a majority of parents may have been opposed to the practice (Mackie provides evidence that they were), the mutual expectations kept the practice in place. After many attempts, what seems to have worked, is target fairly isolated marriage communities: groups of families that intermarry a lot. by having a proportion of these families vow not to mutilate their daughters, and forbid their sons to marry mutilated girls, the local payoff to mutilation changed. It became a viable strategy for this group to follow the new norm of rejecting mutilation, given some proportion of the marriage market the found themselves in kept following the new norm. It is clear to see, that such an initial foothold of a beneficial norm would be very hard to established if marriage markets were not local but global: the amount of people to initially adopt the new norm would have to be incredibly large, for it to become a beneficial strategy for an individual. After the initial stable foothold has been initiated, of course, there need to be connections to the rest of the population for the norm to spread.
The outcomes discussed below all assume an initial population of B players, playing the game in Table 2. Individuals play the game with their network neighbors; that is, nodes in the network play the game with the nodes they are connected to. Edges between nodes are undirected, and have equal weights of 1. Agents update with using best-reply strategy with noise. In Ellison’s
(1993) model, agents play a best response with probability 1 − 𝜀𝜀. With probability 𝜀𝜀, they play a random strategy, resulting in a probability 1 −12𝜀𝜀 of playing the optimal strategy, and
probability 12𝜀𝜀 of playing the suboptimal strategy. this is called the uniform error model. Young (2011) and Montanari & Saberi, (2010) have used a log linear model to model best reply with some error, using function 1. In this function, the probability for node i, of playing strategy A equals:
𝑃𝑃𝑃𝑃𝑖𝑖(𝐴𝐴) = 𝑒𝑒
𝛽𝛽𝑈𝑈𝑖𝑖(𝐴𝐴,𝑥𝑥−𝑖𝑖𝑡𝑡 )
𝑒𝑒𝛽𝛽𝑈𝑈𝑖𝑖(𝐴𝐴,𝑥𝑥−𝑖𝑖𝑡𝑡 )+ 𝑒𝑒𝛽𝛽𝑈𝑈𝑖𝑖(𝐵𝐵,𝑥𝑥−𝑖𝑖𝑡𝑡 )
1
Where 𝑈𝑈𝑖𝑖(𝐴𝐴, 𝑥𝑥−𝑖𝑖𝑡𝑡 ) is player i’s utility obtained by playing strategy A in period t, considering his
opponents’ strategies in that period 𝑥𝑥−𝑖𝑖𝑡𝑡 . 𝛽𝛽, which is always ≥ 0, measures the rationality or
responsiveness of agents. For 𝛽𝛽 = 0, 𝑃𝑃𝑃𝑃𝑖𝑖(𝐴𝐴) = 0.5. If 𝛽𝛽 becomes large, agents almost always
play best response. The log-linear error model incorporates the idea that error rates are not fixed, but depend on the relative payoff of the options. In Figure 1 the two error models are compared. It needs be noted that both update models do not allow to separate initial mutation rate from noise during updating, which simplifies matters, but can also be seen as a limitation.
Figure 1. Pri(A), as a function of the number of neighbors of player i playing strategy A. In this graph, i has 8 neighbors (Moore
neighborhood). In this graph, social innovation α is taken to be zero, so the payoff to (A, A) in the coordination games equals the
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 pri (A ) neighbors of i playing a
payoff to (B, B), and agents are indifferent between choosing A and B when half their neighbors play one of the strategies. For the error rate update regime, 𝜀𝜀 is taken to be 0.2. For the log linear update regime, 𝛽𝛽 is taken to be 0.5
Ellison (1993) was one of the first to illustrate the importance of local interactions for the evolution of norms. In well mixed populations, the risk dominant equilibria is eventually selected. However, the time it takes to reach this equilibrium grows exponentially? in the number of individuals in the population, leading to extremely long waiting times when the population is not very small or 𝜀𝜀 very high. With these extremely long waiting times, evolutionary forces may not hold any relevance for realistic situations. Ellison posits that if waiting times are short, the norm is due to evolution. If waiting times are (extremely) long, the norm is more likely to be determined by whichever situation is the starting position. Local interactions make waiting times independent of population size, and significantly reduce them. Ellison evaluates several network topologies with local matching: neighbors interacting on a ring (1 dimensional), in a regular lattice (2 dimensional) and on 4 dimensional lattice structures. Furthermore, he evaluated the influence of the number of neighbors individuals have.
Whilst holding α constant at 1, and evaluating the waiting time till 70% of individuals adopt strategy A in a population of initial B players, he found that the 𝜀𝜀 greatly influences results. That is, bigger neighborhoods (more interaction) lead to much larger waiting times for low values of 𝜀𝜀, but waiting times stabilize for higher 𝜀𝜀. The same holds for overlap in neighborhoods. If one interacts with 8 neighbors on a one-dimensional structure, overlap in the neighborhoods of players is very large. In more-dimensional structures, this overlap is smaller. Ellison found that for low 𝜀𝜀, waiting times for more-dimensional structures are greatly increased. For higher values of 𝜀𝜀, more dimensional structures with less overlap seem to decrease waiting times. While Ellison (1993) was interested in actual waiting times until a fraction of the population plays adopt the new strategy, the majority of authors have only considered whether the
expected waiting time to reach a high level of new-norm penetration in some network topology is bounded above for a given advance α, and a given degree of noise. ‘Bounded above’ implies that if the size of the network goes to infinity, the expected waiting time remains bounded above, that is, does not go to infinity too. (see e.g. Montanari & Saberi, 2010; young, 2011; Ellison, Fudenberg, & Imhof, 2016; Kreindler, & Young, 2012) ). Although this approach does
reveal very general, mathematical sound conclusions, it obscures a lot of characteristics of the tipping process in these networks.
One of the main contributions to this strand of research has been provided by Young. In Young (2011). He identifies 3 criteria needed for fast convergence. Firstly, the level of noise should not be too high or too low. Too much noise, and coordination on one equilibria cannot be
sustained. Too little noise, and it takes very long for an initial foothold to form. Secondly, the network structure. Thirdly, the payoff gain alpha. In terms of network structure that enable bounded above waiting times, young identifies the presence of autonomous enclaves as the main condition for bounded above waiting times. That is, the network contains, or can be divided in (small) subsets, where nodes have a considerable proportion of connections to other nodes within the subset relative to nodes outside of the subset. Depending on this proportion and the advance α, these subsets can be autonomous: given that all players outside of the subset play B, and players in the subset play A, it is stochastically stable for individuals in the subset to keep on playing A. This means that a stable initial foothold has formed, from which strategy A can spread.
THE CURRENT RESEARCH
As mentioned above, the evolutionary game theoretical approach has primarily dealt with the question whether waiting times are bounded above for specific topologies, which obscures a lot of the underlying dynamics. Agents are mostly assumed to play a near perfect best response, so results only hold in the limit of rational agents where 𝛽𝛽 → ∞ or 𝜖𝜖 → 0 (Montanari and Saberi, 2010). This may be a useful assumption for mathematically sound derivations, but again, it may obscure the underlying dynamics in more realistic cases, where agents are not completely rational, do not observe the situation perfectly, or like to experiment. Also, this focus has likely prevented research of scale-free networks, since spread may not take place under the
abovementioned circumstances. In order to capture the influence of network topology on the spread of social innovations more completely, I will evaluate several aspects of the tipping dynamics. I will focus primarily on networks with scale-free degree distributions, since very little game-theoretical research has evaluated their behavior.
Output variables
Firstly, we will evaluate the initial ‘stability’ of a network, that is, the initial fraction of A-players needed to achieve a foothold with some specified probability. Secondly, we will evaluate the time it takes till an initial foothold is formed. This ‘initial foothold’ is the topic of interest in much of the existing literature. Since local footholds are needed for waiting times not to scale together with network size, the main question becomes under what condition local footholds can develop. Thirdly, the time from initial foothold till a target fraction of A-players in the population is reached. This variable captures the spread of the social innovation after the initial foothold has been formed, which may be an essential component of total waiting time that is often not evaluated. Fourthly, the end stability of the network is evaluated, that is, the fraction of A players in the stochastic equilibrium that is reached after the system has ‘tipped’ to the situation where A has become the norm.
INITIAL STABILITY
Initial stability is defined as the fraction of initial A-players needed for the system to develop a foothold with some probability. That is, more stable network topologies require more initial A-players for a foothold to be obtained. In order to control for differing degrees of rationality for a specific fraction of A-players in different network topologies, measurements were taken using completely rational agents, combined with an initial number of A-players. The results can be found in Figure 6.
INITIAL FOOTHOLD
In this paper, we do not use the concept of stochastic stability to evaluate mathematically whether some initial group of A-players is stochastically stable, an approach taken by Young (2011). Instead, we define the waiting time till an initial foothold has been formed as the
expected number of periods it takes for the first node to play A for 10 subsequent periods. That is, player i updates 10 times, and picks strategy A during all these updates. The first period of these 10 subsequent periods in which this player plays A, is taken to be the time till a foothold has been formed. The number 10 is not too low to detect nodes randomly playing A due to
irrationality, and not too high to not detect stable footholds where nodes play B due to irrationality. The results can be found in Figure 7.
TARGET FRACTION
To be meaningful, the target fraction of A-players needs be close to, but lower than the final level of A-players in equilibrium. If the target fraction is set too high, it may never be reached. If it is set to low, it does not reflect the time till the population has ‘tipped’ to playing the new norm. I use 0.7 as the target fraction, and then evaluate the average fraction of A players in the equilibrium that follows (see below). After this, the time from initial stable foothold till final equilibrium can be computed.
END STABILITY / FINAL EQUILIBRIUM
This final, stochastically stable equilibrium is captured by evaluating the average final fraction of A-players. In order to make sure we measure equilibrium values, we take a waiting time into account after the target fraction has been reached. One way to do this would be to wait a number of periods after the target fraction has been reached. In order to make simulations run more efficient, we let the waiting time depend on the time it takes the system to go from initial foothold to target fraction. More specifically, we wait 1x this time, before we start measuring the final equilibrium. This final average fraction of A-players is computed by taking the average over 20 periods. Results can be found in Figure 9.
NETWORK TOPOLOGIES
The variables mentioned above will be evaluated in several topologies. Firstly, the basic results in regular lattices and on the circle will be evaluated. These will be taken as a starting point for further evaluation. Secondly, the influence of heterogeneity of degree will be evaluated, by studying the spread of norms trough several forms of Barabási-Albert scale free networks. Small world networks were briefly evaluated, but long ties did not greatly influence dynamics in the networks of limited size we considered, other than preventing cascades for high levels of rationality. A simplified version of the Holme and Kim (2001) model was also explored.
METHODS
THE MODEL
Consider a graph with undirected edges. Every edge has an equal weight of 1. Individuals, which are represented by the nodes of the graph, can play strategy 𝐴𝐴 or 𝐵𝐵 (see Table 2). Strategy B is the status quo behavior initially played by everyone, and 𝐴𝐴 is an innovative behavior with potential benefit α, if coordination on this new equilibrium is achieved. The state of the process at time 𝑡𝑡 can be defined as the vector 𝑥𝑥𝑡𝑡 ∈ {𝐴𝐴, 𝐵𝐵}𝑛𝑛. Let player 𝑖𝑖’s strategy at time 𝑡𝑡 be 𝑥𝑥
𝑖𝑖𝑡𝑡. In
every period, player 𝑖𝑖 plays the strategy of his choice in that period against all of his neighbors. That is, he cannot play A to some of his neighbors, and B to other neighbors. Player 𝑖𝑖’s payoff in a period is the sum of the payoffs obtained from the individual games played in that period. So, if we take 𝐾𝐾𝑖𝑖 to be the set of 𝑖𝑖’s neighbors, we have:
𝑈𝑈𝑖𝑖�𝑥𝑥𝑖𝑖, 𝑥𝑥−𝑖𝑖� = � 𝑈𝑈𝑖𝑖(𝑥𝑥𝑖𝑖, 𝑗𝑗∈Ki
𝑥𝑥𝑗𝑗) 2
In this payoff function (2), nodes with a higher degree have more incentive to conform to the norm that exists around them. Consider a population of B-players. If a node with degree 1 deviates and plays 𝐴𝐴, he forgoes a payoff of 1. If a node with degree 𝑘𝑘 deviates, he forgoes a payoff of 𝑘𝑘. Although it seems fair to assume that more connected individuals have more to lose when failing to coordinate with their network neighbors, it would also be fair to assume that regardless of the number of the connections an individual has, it is just as costly to fail to coordinate with some fraction of neighbors. This would lead to the following payoff function, where payoffs are scaled to account for the number of neighbors an individual has. If we take 𝑘𝑘𝑖𝑖 to be the degree of node i we have:
𝑈𝑈𝑖𝑖�𝑥𝑥𝑖𝑖, 𝑥𝑥−𝑖𝑖� =
∑𝑗𝑗∈𝐾𝐾𝑖𝑖𝑈𝑈𝑖𝑖(𝑥𝑥𝑖𝑖,𝑥𝑥𝑗𝑗)
𝑘𝑘𝑖𝑖 3
The difference between scaled and non-scaled payoff has an effect on which 𝛽𝛽 leads to the same level of rationality (see equation 1). When payoffs are summed as in equation 4, 𝛽𝛽 should be smaller to produce the same 𝑃𝑃𝑃𝑃𝑖𝑖(𝐴𝐴) . Consider a regular lattice with Moore neighborhoods,
function in comparison with the non-scaled payoff function, in order to obtain the same behavior. The distinction between scaled and non scaled payoffs only matters for log-linear update regimes, not for constant-error rate regimes, since there, absolute payoffs do not matter, only if 𝑈𝑈𝑖𝑖�𝐴𝐴 , 𝑥𝑥−𝑖𝑖�>𝑈𝑈𝑖𝑖�𝐵𝐵 , 𝑥𝑥−𝑖𝑖�.
In order to easily compare network topologies, we primarily used the scaled payoff function. By using this function, rationality is constant, regardless of the numbers or neighbors an individual has in the network. Qualitative differences arise in networks where nodes have hetrogeneous degree. Here, both update rules were evaluated.
TIMING
Every individual node updates once per period or ‘tick’. The order in which nodes update within this tick is random. The information of nodes is updated every time one node updates, that is, information is updated 𝑛𝑛 times per tick. If we speak about a state at time 𝑡𝑡, this moment can fall in between ticks, i.e. when some fraction of nodes has already updated during that period, and some fraction still needs to update. If we speak about a state at a specified number of periods or ticks, this is the state of the system at the end of this tick, where every node has updated.
SIMULATIONS
Simulations were run using Netlogo agent-based modeling software (Wilensky, 1999). This well-tested and commonly used software package offers relatively easy coding, and visual displays. The main drawback of this tool in combination with the hardware is the fact that we were somewhat limited in the size of networks we could evaluate. For the results presented below, we used networks with 𝑛𝑛 = 400, with 50 repetitions per observation, and a maximum of 3000 periods per run. A small number of runs were performed in large networks to evaluate whether qualitative results differed, which did not happen.
CIRCLE
INITIAL STABILITY & FOOTHOLD
The initial stability of the circle topology, with each node having 2 neighbors, is very low. That is, if one node ‘mutates’ and starts playing strategy 𝐴𝐴, the best response of its neighboring nodes is start to playing strategy 𝐴𝐴 too, as long as 𝛼𝛼 > 0 (see Figure 2) Whether this actually happens, depends on the timing of the nodes updating, and possible non-best-response
updates. Given rational updates, the chance of one mutant A player causing a stable foothold is 2/3 (see Figure 6).
Figure 2. Node 2 starts playing A. node 1 and 3 can obtain 1+α
when playing A, or 1 when playing B. When node 2 updates again, before node 1 or 3 adopts strategy A, the system falls back to 0 A-players.
TIPPING TIME
The time from initial foothold to taking over the population depends primarily on numbers of footholds forming in the meantime due to non-best replies. Since the initial foothold can only spread in 2 directions, one node at a time, the tipping time after one foothold with no further footholds can be quite long for large 𝑛𝑛 and high levels of rationality, but tends to be very small if non-best replies allow quick formation of several footholds see Figure 8.
END STABILITY
End stability of this network is fairly low, given low levels of α. Since nodes have only one 2 neighbors, if one node mutates and start playing B, it may ‘convince’ a neighboring node to do the same (see Figure 9).
LATTICE
2 3
INITIAL STABILITY & FOOTHOLD
The initial fraction of A-players in a lattice can be fairly high, before a stable foothold is formed. this is the case because several neighboring individuals need to ‘mutate’ and start playing A, before a foothold can be formed (see Figure 3 and Figure 6). This causes the lattice with smallest neighborhoods (𝑘𝑘𝑖𝑖 = 4) already to have the highest initial stability of the evaluated
topologies. The bigger the neighborhood of interaction, the more nodes around one node need to play strategy A before strategy A is rationally adopted. Hence, regular lattices are fairly stable, especially with larger neighborhoods.
TIPPING TIME
Tipping time is the smallest of network types evaluated, given that α does not become too small. Initial stable footholds need some irrationality to spread in the neuman-4 lattice, since A-players cannot convert B A-players rationally as long as the cluster of A-A-players is a square. In this case, B-players interact with at most 1 A-players, so given that 𝛼𝛼 < 2, B players need to play a non-best response to start playing A. However, few of these irrational steps are needed, and the new norm can spread in 2 dimensions instead of one, as is the case in the circle. This leads to the fastest tipping times of the networks we evaluated with 𝛼𝛼 = 1 and β ≤ 10 (see Figure 8). For 𝛼𝛼 = 0.1, and high levels of rationality however, tipping times go up dramatically because less B-players at the border of the square cluster of A players play the suboptimal response required for further spread.
END STABILITY 10 14 13 9 5 6 12 16 15 11 7 8 2 1 3 4
Figure 3. regular lattice with Neumann neighborhoods of radius 1. Yellow=strategy A, rest plays B. If node 7 and 10 start playing A (independently of each other), and node 6 and 11 update rationally before 7 and 10 10 fall back to playing B, a stable foothold is created.
Together with scale-free network with 𝑚𝑚 = 2, the regular lattice has the highest end stability (see Figure 9). This is probably due to the fact that it is more costly to deviate when one has many neighbors playing another strategy. If one has 3 neighbors playing A, and 1 playing B, it is relatively more costly to play B than the case where one has only 2 neighbors, one playing A and one playing B. The difference in end stability is most pronounced for low levels of 𝛼𝛼, since in this case, it is not too costly to adopt strategy B if a neighbor does so too. If 𝛼𝛼 is large, regardless of the network, almost no one will start playing B in the final equilibrium, let alone that other nodes follow suit.
SCALE-FREEE
We evaluated 2 versions of the BA Scale-free network with 𝑚𝑚0 = 2: One with 𝑚𝑚 = 1, the other
with 𝑚𝑚 = 2. In the generation of the former, every new node attataches with 1 edge to the existing network, leading to an average degree of 2 as 𝑛𝑛 → ∞, and a minimum degree of 1. In the latter, every new node attaches with 2 edges to the network, leading to an average degree of 4 as 𝑛𝑛 → ∞, and a minium degree of 2. The main 2 differences between the two networks are the following. Firstly, 𝑚𝑚 = 1 contains ‘end nodes’ where 𝑚𝑚 = 2 does not (see Figure 4 and Figure 5). Secondly, in 𝑚𝑚 = 2 connections tend to be more global: one node can attach to one hub of the network, and another hub far away. These connections to several hubs are unlikely when 𝑚𝑚 = 1 networks, since new nodes added only have one edge to preferentially attach to the existing network. Because of these differences, the difference between 𝑚𝑚 = 1 and 𝑚𝑚 = 2 is much more pronounced than between for instance 𝑚𝑚 = 2 and 𝑚𝑚 = 3.
INITIAL STABILITY
Initial stability in scale-free networks is low, due to the fact that initial footholds are easily formed. The initial fraction of A-players for a given level of (average) rationality depends on the payoff regime. If payoffs are scaled, the low-degree nodes obtain a significantly different payoff when playing best reply or non-best reply, hence, footholds tend to be stable. When payoffs are not scaled, low degree nodes are much more irrational than high-degree nodes, since if payoff of neighbors is summed, the difference in payoff between optimal and suboptimal
strategy is much more pronounced for high degree nodes. Therefore, under non-scaled payoffs, a much higher fraction of initial A-players can be observed. For the same average rationality, low-degree nodes will behave relatively irrational, which causes a high initial fraction of A-players.
INITIAL FOOTHOLD
Initial footholds are very easily formed in m=1 scale free networks, since only one node with degree 1 or 2 needs to adopt strategy A to create the possibility of a stable foothold via best reply of neighboring nodes (see Figure 4, Figure 6 and Figure 7). In scale-free networks with m=2, much less of these places where one initial A-player can trigger a foothold (see Figure 5) exist, and hence, the same level of initial A-players takes a longer waiting time to trigger a foothold (see Figure 6 and Figure 7).
If payoffs are not scaled, nodes with degree one or two are very irrational, and switch strategies a lot. In this case, the stable footholds tend to form in higher-degree nodes, and variation
persists in low degree nodes.
Figure 4. Scale free network with M = 1. Yellow=strategy A, rest plays B. If node 2 updates before 1 updates, a stable foothold is formed, given not too much irrational behavior. If another foothold is formed in node 3 and 4, node 5 can
sustainably start playing A too, given 𝛼𝛼 > 0.
1 4 5 2 3 2 3 4 1
Figure 5. Scale free network with M = 2. Yellow=strategy A, rest plays B. When node 3 updates before node 2, a stable
foothold is created, given 𝛼𝛼 > 0. If node 2 and 3 are playing A,
1 irrationally plays A and 4 updates after this, strategy A can
TIPPING TIME
Although an initial foothold is formed very easily in (especially m=1) scale-free networks, the time for such a foothold to spread to the rest of the population can be very long (see Figure 8). Non-best replies, and large numbers of footholds are needed for a network to ‘tip’ to playing 𝐴𝐴. The reason for this is that stable footholds can only form in low-degree nodes, unless a hub and a significant proportion of nodes around this hub mutate and start playing A at the same time, which is unlikely to happen and be sustainable. The usual pattern of tipping in these networks is that first, a large number of footholds form in low degree nodes. Some higher degree nodes will now be connected to several footholds, and can sustainably start playing 𝐴𝐴, after which higher degree hubs may follow. For this tipping, non-best replies are still needed, Especially when M = 1. In this case, hubs tend to be connected to some stable footholds, and several nodes who are only connected to the hub. In order for the hub to tip, either the nodes on these spokes, or the hub needs to play an irrational strategy. The tipping is usually a step-wise process, because it takes time to sustainably convert hubs, but when these hubs are converted, they advance the spread very quickly to the rest of the network.
END STABILITY
The end stability of scale free networks with m=2 is fairly high, especially when compared with scale-free networks with m=1 (see Figure 9). This seems to be due to the fact that there are many nodes in the m=1 network that only connect to one hub. If this hub plays the suboptimal strategy B, the best response for all of these nodes is to play B too. although this strategy is not sustainable, it sometimes leads to bursts of B-players. More generally, the best response of end nodes with 𝑘𝑘 = 1 is always to follow the strategy of its neighbor, so mutations of these
neighbors are reinforced. In M2 networks, the lowest-degree nodes often connect to different parts of the world, so that individual B-players have a hard time convincing their neighbors to follow this strategy. Non-scaled payoffs lead to considerably lower end stability, since low-degree nodes keep experimenting more.
0 0,2 0,4 0,6 0,8 1 0 10 20 30 pr oba bi lit y f oo tho ld initial a-players circle lattice sf m=1 sf m=2
Figure 6: Initial stability: probability of a foothold arising as a function of initial A-players. n=400). High chance of foothold = low stability. Rational updates combined with an initial fraction of A-players was used to isolate stability. The same pattern can be observed with log-linear updates, but here, the varying rationality is a confounding factor. The initial A-players are part of the stable cluster in all cases, except for 3 out of 1500 runs in the scale free lattice with m=1. 0 100 200 300 400 500 600 700 800 900 1000 4 6 8 10 tick s t ill fo ot ho ld 𝛽𝛽 circle lattice sf m = 1 sf m=2
Figure 7. Expected ticks till initial foothold as a function of
rationality 𝛽𝛽. 𝛼𝛼=1, n=400, scaled log-linear update.
0 100 200 300 400 500 600 700 800 900 1000 0 5 10 tip pi ng t im e 𝛽𝛽 circle lattice sf m=1 sf m=2
Figure 8. Expected waiting tiime from first initial foothold to
fraction of A-players 0.7, as a function of rationality 𝛽𝛽. 𝛼𝛼=1,
n=400, scaled log-linear update. Since scale free networks
with m=1 need many initial footholds and irrationality for strategy A to spread, the expected waiting time goes up fast with increased rationality. For very low levels of 𝛼𝛼 (not shown) waiting times for lattice and scale free with m=2 go up rapidly with rationality too, since non-best responses in favor of A are not only needed to gain an initial foothold, but also for further spread low 𝛼𝛼 means that non-best responses become more costly.
CONCLUSION AND DISCUSSION
This thesis has sought to evaluate not only the possibility of footholds of social innovation arising is specific classes of networks, but also evaluated the time for footholds to spread to the rest of the network, and the initial-and final stability of the networks. Even though especially scale free networks with m=2 feature a lot of random, ‘global’ connections instead of local ones, the low degree nodes enable initial footholds. Hence, not only local connections, but aIso heterogeneity of degree promote social innovation. This finding agrees with young’s (2011) that networks need small autonomous enclaves for footholds to arise. Low degree nodes connected to other low-degree nodes are autonomous in that they are primarily connected to each other, and are not to a large amount of nodes at different locations in the network. Not only can footholds easily form in low-degree nodes, these nodes are also easily converted to the norm that a higher-degree hub is playing, especially in the hub and spoke structures found in scale free networks with m=1. What often happens is that initial infections slowly climb up to ever higher degree nodes, after which these high degree nodes rapidly infect other nodes.
On the other hand, high degree nodes may slow down the spread of social innovation, when low-degree nodes situated around the hub cannot convince each other to start playing the new norm directly, but only via hubs. This finding, however, might be primarily due to the lack of clustering in BA networks, and not necessarily due to the high degree hubs themselves. Exploring simulations in scale free networks with high clustering (constructed with a simplified
0,7 0,75 0,8 0,85 0,9 0,95 1 3 4 5 6 7 8 end s ta bi lit y 𝛽𝛽 circle lattice sf m=1 sf m=2
Figure 9. Final fraction of A-players as a function of
rationality 𝛽𝛽. 𝛼𝛼=0.1, n=400, scaled log-linear update. Not
all lines are defined for all 𝛽𝛽, since tipping may not have
occurred within the timespan of our simulations (upper bound) or because there is too much noise, and an end fraction above 0.7 was never reached (lower bound).
version of the Holme and Kim (2001) network) revealed that this clustering (which leads to edges between spokes) may indeed increase spread under some circumstances. Further research should evaluate this role of clustering in scale-free networks more thoroughly. As has been revealed by previous research in other network topologies, the dynamics of norm shifts greatly differs from the dynamics of viral spread. One of the main implications of the current research, is provided by the fact that footholds only arise in low-degree nodes. When trying to achieve social innovation, it may be more fruitful to target several less-connected individuals, rather than social ‘hubs’ with many connections, since the social innovation may become sustainable faster amongst less-connected individual, and spread from there.
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