1 2 Mode and tempo of cultural evolution in video games

Ivan Dmitriy Ortiz Sánchez

BIOMEDICAL BIOMEDICAL ENGINEERING20

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THESIS

S S

´ BACHELOR Mode and tempo of cultural evolution in video games

Ivan Dmitriy Ortiz Sánchez

Bachelor’s Thesis UPF 2020/2021

Thesis Supervisors:

Dr. Sergi Valverde Castillo, (Evolution of Technology Lab, CSIC-UPF)

Dr. Salvador Duran Nebreda, (Evolution of Technology Lab, CSIC-UPF)

Dedicatory

To my family, for their unconditional presence and trust.

To my friends and beloved, for being so far yet so close in such a different year.

Finally, to Juan Ortiz and to Isabel Peñarroya. In Memoriam.

Acknowledgments

I would like to wholeheartedly thank my supervisors, Sergi and Salva, and acknowledge their help, advice and patience during this research. It has been a fascinating and worth- while experience and, without their time and enthusiasm, this study would have not been the same. I am also thankful for all the knowledge about cultural evolution and network theory and the computational methods they have taught me during the research, which have given me a better comprehension of such a beautiful and engaging field of research.

Summary/Abstract

The mechanisms of biological evolution also apply to artificial phenomena such as culture and technology, and the evolution of video games through history has been shaped by the evolution of technology itself. In particular, the so-called speedruns, which consist in completing video games in the least time possible, have become remarkably popular recently. Since the evolution of performance in video games has never been quantita- tively assessed, in the present study, we wonder whether there are universal patterns in the way speedrunning has evolved through video game history. Specifically, we aim to identify relations between performance improvement and the size and structure of the player community. Thus, a reliable dataset with the results of official speedruns has been manipulated and analyzed. First, we describe the dynamics of performance improvement and growth of the community since its origin. Second, we explore the effects of commu- nity structure with a game-player bipartite network framework and an infectious model of strategy and information propagation. Finally, we relate the model to the actual data and establish linkages between the properties of the network and the learning dynamics. Our results show how the growth of the community and the evolution of performance follow exponential descriptions and how the rank-ordered distribution of players accord- ing to their number of playthroughs follows a power law-like behaviour. A first minimal network model to describe the properties of the community is also provided. This study lays the foundation for a quantitative application of biological and evolutionary models to the video game field.

Keywords computational modelling, complex networks, infection models, cultural evolution, learn- ing, video games, speedrunning

Preface

The evolution of video games has been shaped by the evolution of technology itself, and, in the current context, video games are highly influential. The relevance of the role they play in our society, not only for the youth but for people of all ages, especially with the rise of streaming platforms and the so-called speedrunning community, is undeniable.

Cultural patterns are evolutionary, and those mechanisms which define biological evo- lution have been proven to work with cultural phenomena as well, and allow to either understand the past or forecast the future. Given the mathematical tools and the biologi- cal concepts learnt through the Biomedical Engineering degree, an application of cultural evolution methods to the video game field in order to assess how it has changed through time would represent a first insight into the topic from an evolutionary perspective, and a relevant scientific contribution.

Given this context, we are motivated to explore video game performance and the structure of the community of players and to try to identify universal patterns in the way they have evolved which could be explained by means of simple mathematical descriptions, and to provide a basis for further analyses in such an unexplored and relatively novel area.

Index

1 Introduction1

2 Stage I. Evolution of speedrunning and video game performance6 2.1 Methods...... 6 2.1.1 The data set...... 6 2.2 Results...... 6 2.2.1 A DGBD model for cultural evolution in video games...... 6 2.2.2 An exponential decay model for performance improvement..... 10

3 Stage II. A minimal model for the structural growth of the community 12 3.1 Methods...... 12 3.1.1 The model...... 12 3.1.2 Structural analysis of the community...... 15 3.2 Results...... 18 3.2.1 Properties of the community as a network...... 18 3.2.2 Graph visualization...... 23

4 Stage III. Community structure and learning capability of players 25 4.1 Methods...... 25 4.1.1 Learning capability as a node-specific property...... 25 4.2 Results...... 25 4.2.1 Influence of learning capability on the community...... 25 4.2.2 A structural transition in time...... 27

5 Discussion 28

Bibliography 31

Supplementary information 33 S.I Generation of multiple components in simulations...... 33 S.II Learning capability...... 34

List of Figures

1 Growth in the number of speedruns and player productivity...... 7 2 Rank-ordered distribution of players and DGBD fit...... 8 3 Rank-ordered distribution of games and DGBD fit...... 9 4 Example of evolution of performance in a video game...... 10 5 Probability density of learning rates...... 11 6 Rank-ordered distribution of players and games; empirical and simulated. 15 7 Node degree occurrence distribution...... 19 8 Average node degree in projections...... 20 9 Centrality occurrence distribution in bipartite graphs...... 21 10 Centrality occurrence distribution in projections...... 21 11 Global efficiency of communities...... 22 12 Number of edges and modularity in projections...... 22 13 Connectance of communities...... 23 14 Graph visualization: real and simulated...... 23 15 Learning capability and node-specific properties of the community..... 26 16 Graph visualization: community evolution between 2012 and 2015..... 27 S1 Graph visualization: comparison in terms of ρ ...... 33 S2 Mean learning rate and node-specific properties of the community..... 34 S3 Influence of players with zero learning capability...... 35

List of Tables

No tables have been included in this document.

1 Introduction

The evolution of living beings is characterized by certain mechanisms which act in favour of the survival of those organisms who are better adapted to the environment, namely reproduction (and inheritance of genetic traits), mutation and selection, principally. Re- production allows the persistence of living beings through multiple generations, mutation is a source of randomness and, thus, allows to introduce change and innovation. Crossover between individuals with different characteristics also allows to obtain variability. Finally, selection is a natural mechanism by means of which the environment tests living beings’ condition and adaptability and allows to survive only those with the proper character- istics. Evolution does not only allow life to persist in time but also to generate a large variability of species (and even between individuals of the same species) and organisms with high complexity.

Given the potential of these natural mechanisms, the human being has wondered whether they could be applied to artificial processes, systems and dynamics. Examples of these are the so-called evolutionary algorithms or the evolution of human culture. Evolutionary algorithms use the concepts of reproduction, mutation, crossover, selection, migration, etc. to find optimal solutions to a given problem. On the other hand, it is shown that human culture is itself an evolutionary process exhibiting those mechanisms which define Darwinian evolution [1].

Culture can be defined as group-typical behaviour patterns shared by members of a com- munity that relies on socially learned and transmitted information [2,3]. In cultural evolution, apart from culture itself, the concept of cumulative culture is also important, which refers to the fact that cultural traits are based on the legacy from previous gener- ations and the knowledge about that legacy [4]. Hence, cumulative culture means that culture can be spread through generations and grow, but it also depends on the population itself, since cultural traits must be learned properly in order to avoid losses through time. Cumulative culture also implies that the accumulated knowledge overcomes what a single individual would manage to invent on his or her own [5]. The accumulation of culture is also a punctuated process: remarkable innovations might appear after uninterrupted long technologically stable periods [3]. When a large number of innovations appear together or in rapid succession at a certain time or place, it is said that a technological transition has occurred [6,7]. Nevertheless, not many innovations manage to represent turning points in human culture; in human history, only milestones such as the apparition of language or the invention of computers manage to generate such discontinuities.

Network studies have developed multi-layered models in which nodes can either be indi- viduals, communities or even a certain type of cultural trait [3,6,8]. It has been shown that those cultural structures in which nodes are densely connected tend to manifest higher levels of transmission or learning rates than those which rather have high modu- larity. However, modularity allows to increase cultural variability and innovation [8,9]. As a summary, it can be stated that the higher the community size and its variability, and the more communication between the individuals, the easier the overall learning or performance and maintenance of cultural traits [10, 11, 12].

Cultural transmission, however, also has barriers which might constrain the process, or

1 even lead to cultural loss, which can either be structural or behavioural [13]. Structural barriers are directly related to the network itself, by means of affecting the contact be- tween individuals. On the other hand, behavioural constraints depend on the willingness of individuals to share and spread their knowledge. Through generations, there is also a certain degree of inaccuracy in cultural transmission, since information can be misunder- stood or not properly shared.

In cultural evolution, many studies use the so-called neutral models, in the sense that the dynamics have been assumed to rely on a certain stochasticity which can be controlled or defined by certain constant parameters, providing simple approaches to explain quite complex systems [14]. However, there is certain controversy on whether neutral models are conclusive or not, since they are criticized for considering deterministic processes such as adaptation or selection as merely stochastic [14]. Nevertheless, it is also defended that chance and stochasticity are not sufficiently considered in cultural evolution, and that certain collective decisions can statistically behave as if the product of random copying, which would justify the suitability of stochastic models [15].

Minimal models for technological diffusion have been proposed in which the technological level is a variable that improves in a fashion proportional to the size of the community (which increases in a logistic manner) and decreases due to a certain transmission error [6]. Such models have two stable states: low population with low cultural level and high population with high cultural level. This mechanism applies for urban phenomena, for instance: certain urban features increase faster according to the size of the population [16, 17]. These systems show a positive feedback mechanism in which the wise become wiser and those with the least cultural level might even become extinct.

The use of models of this kind has allowed to establish patterns in the evolution of cer- tain cultural traits or to forecast technological progress as well as to provide frameworks for the analysis of such traits [3, 18]. These models have provided mathematical tools in the shape of simple equations based on relations such as exponential or power laws, which allow a better comprehension of the past and to make predictions about the fu- ture. Cultural evolution studies can thus be applied either to archaeology or to current technological dynamics, and one cultural area that could be of particular interest in the current context is the field of video games.

Video games play a very relevant role nowadays in society. Even though they might be thought as a mere source of entertainment, there are at least three different aspects in which their actual importance can be reflected, namely socioeconomic, cognitive and technical.

First, the economic and social impact of video games must be born in mind. As a matter of fact, the global video game market size was valued at 151.06 billion US dollars in 2019 and it is expected to grow at a compound annual growth rate (CAGR) of 12.9% from 2020 to 2027 [19]. The social importance of video games can be observed in the growth of the so-called speedrunning community, considered in this study. The recent growth of the video game community regardless of speedrunning can also be explained as a consequence of the rise of platforms such as Twitch, YouTube and Discord, which allow to stream and

2 spread information about video games.1,2,3 Furthermore, certain video games have also contributed to different economic areas such as cinema, literature or other merchandise thanks to their fame.

Second, video games have been reported to have a beneficial effect on human cognition. A causative relationship has been observed between video game play and augmented spatial resolution [20]. It has also been possible to establish linkages between neural and cognitive aspects regarding attention, cognitive control, cognitive workload, and reward processing [21]. These results have led to the use of video games as therapeutic tools, since the effects on cognition are reflected in brain structure and function. However, no strong scientific evidence about the underlying mechanisms in the brain has been reported when it comes to supporting the clinical application of video games [22].

Finally, a technical factor should be considered: the process of developing a video game involves the integration of a wide range of professionals and fields, such as computer pro- gramming, economic and business management, arts (auditory, visual, narrative, etc.) or marketing.

Hence, the fact that no previous studies on cultural evolution have been conducted in such a relevant field provides the opportunity to perform a first insight into the topic, and to try to establish a first theoretical basis for further research and analysis about the evolution of performance in video games.

Given this scenario, we hypothesize that there might be universal patterns in the way video game performance and the community of players have evolved through video game history. Specifically, we aim to (1) assess the cultural growth of the video game com- munity, (2) analyze its structural properties, and (3) identify possible relations between the rate of improvement and the size and structure of the community of players. Given these three main goals, and the fact that each of them requires the results from the steps followed to achieve the previous one, this study is organized in three different stages which have been treated independently, each one with different methods and strategy, which aim to fulfil each objective.

In order to assess performance in video games, this study is focused on analysing the so-called speedruns, which consist in finishing games in an optimal manner. We consider, then, that the ideal parameter to measure optimality is the time required to complete a game. It must be noticed that speedrunning does not consist in playing as fast as possible per se without a deep understanding of each game but in uncovering and deciphering as much information as possible on how the game is designed and programmed so that players can take advantage of it and find shortcuts, strategies or a priori unexpected tech- niques to reduce the time taken to complete it. This procedure is known as routing.

Since there is a large number of video games as well as a wide range of genres, it should be expected that each of them could be treated in a different manner. However, as previously stated, this study takes the following assumption: optimality in video game

1www.twitch.tv 2www.youtube.com 3www.discord.com

3 performance can be universally assessed measuring the time taken to complete a game. This justifies why speedrunning data is the best source of information about performance.

Furthermore, each video game might have different strategies or ways to be completed. In general, most video games are considered to be completed once the ending credits appear (or right when the last movement before the game finishes is performed). Thus, part of the content of a game can be skipped during a run, that is, a complete playthrough. This is why each game has different categories. The most typical categories are the so-called “Any%”, “100%” and “Low”. The first one aims to complete a game as fast as possible. The second one aims to complete all the content it has to offer. Finally, the Low mode aims to complete the game avoiding as much collectable items as possible. The difference between Any% and Low is that in the first one the number of items gathered does not matter, but they might coincide in certain cases. Hence, Any% is the category in which the lowest scores are achieved. Note that scores between different categories are indepen- dent. A single video game might include other categories than the aforementioned, and they might be due to its individual design, genre or style, for instance. For each category, however, certain rules are set, and they must be respected by all players. Other remark- able categories are the “Glitchless” mode, which does not allow techniques that break a video game’s original rules, or those other categories which rely on minimizing the num- ber of “presses” of a certain button during the run, that is, executing a specific command (e.g. jumping, moving to a specific direction, running, shooting, etc.). Finally, there also exists a special sort of speedruns, which are the so-called Tool-Assisted Speedruns (TAS). Many video games, in order to be completed in the least amount of time, involve certain mechanisms which require too much accuracy for a person. Consequently, even if an in- dividual managed to perform them, it could be considered the result of luck or after an unimaginable number of attempts. Tool-Assisted Speedrunning aims to solve this limita- tion: players do not perform the run but make a computer program execute the run itself. Thus, players analyze and specify each single movement and command frame by frame in order to reach optimal performances. Even though TASs rely on the users’ ability, they do not involve an actual person playing the video game. This is why they can be used to set theoretical perfect scores to video games, but not as actual human runs. Furthermore, non-tool-assisted speedruns can be performed in real time (Real Time Attack) without stops, or in a segmented manner, dealing with the stages of a video game separately and then summing the best scores.

In order to perform this study, a reliable dataset registering scores for a total of 4,962 dif- ferent categories from 693 different video games is manipulated and analyzed. This dataset is provided by a world-wide known speedrunning website (speedrun.com), in which video game players share their results as well as the date of the run and their usernames, among other information.4 As of November 2020, the website had over 500,000 registered users and over 1,500,000 submitted runs in over 20,000 games. The results are the time taken to complete the specific category. Due to the constant updating of scores and the growth of the community, it must be remarked that the data used in this study was collected by November 2020. Hence, later submissions are not considered. Moreover, this study is only focused on real time speedrunning, so no tool-assisted nor segmented speedruns are considered either.

4www.speedrun.com

4 Video games are considered games and, thus, a source of entertainment. Then, it could be thought that the pursuit of completing them as fast as possible breaks the concept of entertainment. However, recall that speedrunning aims to minimize times not by running blindly but by means of exploring in detail the way each video game is programmed and designed to find the optimal path, that is, by identifying possible errors (bugs and glitches) or techniques which require a detailed analysis of the game. Thereby, speedrunning should be considered as some sort of meta-entertainment directed to those insatiable players who remain unsatisfied once they complete a game or desire something more, developing an exercise that transcends the original concept of the video game.

Not surprisingly, between speedrunners, there exists certain competitivity, not so different than the case of sports, in which people compete against each other in the pursuit of the best score or a victory. The oldest information registered about competition in video game performance dates from 1994, in which the website DOOM Honorific Titles was launched, and players could earn titles submitting recordings and compare their performance in the game Doom (1993), and later in Doom II: Hell on Earth (1994) too.5 Doom allowed play- ers to record their playthrough, which is one of the fundamentals for speedrunning. From then on, and as the community grew, competition also increased. Nowadays, as many of the runs are shared and even broadcast using live-streaming platforms such as Twitch, speedrunners can also earn popularity, fans and support from spectators who can even donate money to players. Competitiveness works as a pressure factor that might force players to enhance their performance, which results in constant updates of world records and changes in ranks. As a proof of this, by December 2020, the 7 best scores registered for the Any% mode of the game The Legend of Zelda: Ocarina of Time (which is one of the most played video games, with 2,820 runs submitted at that time) were all submitted in less than a 2 month interval, being the best run the one submitted by the player Am- ateseru on the 4th of that same month. Given this competitive scenario, speedrunning tournaments are also held, as, for instance, Speedrun Weekly, organized by the website speedrun.com itself. There are also fundraising events such as the ones held by Games Done Quick, who organize speedrunning marathons and donate the money collected to charity.6 As a matter of fact, Games Done Quick has raised over 25.7 million US dollars across 25 marathons, showing how influential speedrunning actually is.

5www.cl.cam.ac.uk/~fms27/dht/dht6.html 6www.gamesdonequick.com

5 2 Stage I. Evolution of speedrunning and video game performance

2.1 Methods There are many models designed to study cultural evolution. In the case of speedrunning, models using preferential attachment or duplication, for instance, could be considered, and agents would not only be runners but their runs, and they would be incorporated into the system and allocated to runners in a manner proportional to the runs they already have. In any case, it is necessary to first study how the community evolves, in terms of the number of players, games and speedruns which have been submitted through time, and to analyze how performance in the different video games has evolved as well as their learning rates and the productivity of the players. This first stage, hence, represents a first approach to the growth dynamics of the community.

2.1.1 The data set Previously, an assumption had been introduced: optimality in video game performance can be assessed via speedrunning scores. In order to perform this study, as previously explained, a reliable data set provided by speedrun.com was downloaded and analyzed. As of November 2020, this data set provided information about 85,786 individuals playing 4,962 different categories from 693 games, with 203,009 submissions. This information was only a fraction of the original downloaded data set: during the manipulation of the data, from all runs included, we included only those whose date was reported, because otherwise they would not provide enough information to study cultural evolution. Fur- thermore, as already stated, no tool-assisted nor segmented speedruns were considered.

The data set provided information about each submission, specifying the name of the user who uploaded it (as a username), the score (as a specific time), the date of submission and the game category, which allowed to obtain information about growth in time, per- formance evolution and player productivity (as the frequency of submissions per player). The analysis was performed using Python via the Scientific Python Development Envi- ronment Spyder 4.1.5.

It is important to remark that speedruns on the website are supervised so that they are reliable and to avoid counterfeit.

2.2 Results 2.2.1 A DGBD model for cultural evolution in video games The first step was to determine the growth in the number of players, games and submis- sions through time, and to try to fit the dynamics to a specific mathematical description.

The first observation, which worked as a starting point for this research, is that, according to the empirical data, the number of submissions follows a quite exponential growth, as can be observed in Figure1 a, although there are some bumps in the time series, possibly

6 Figure 1: (a) Growth in the number of speedruns over time. This growth is cumula- tive since runs are not removed, and is parallel to that of the speedrunning community. Empirical data (in blue) shows a quite exponential behaviour. An exponential fit (in orange) was performed with growth rate µr = 0.507 1/year. (b) Growth in the number of players over time and exponential fit (µp = 0.486 1/year). (c) Growth in the number of games over time and exponential fit (µg = 0.270 1/year). (d) Productivity frequency distribution as number of runs per month. The empirical data (in blue) shows a decay in which players submitting runs with higher frequency become rarer. This distribution has been fitted (in orange) to a gamma process with parameters α = 0.051 and β = 0.131. relating to periods of increased popularity or dissemination. Players and games also follow an exponential description (Figure1 b and c, respectively) and were assumed to join the community but not to leave it, since the mere fact of submitting a run reveals awareness and represents an involvement in the community. Hence, the speedrunning community grows by means of the following equation:

µ(t−t0) N(t) = N0e , (1) in which, according to the fit, time zero would be set at 1996, the initial population

7 Figure 2: Rank-ordered distribution of players according to the number of runs submit- ted: (a) simulated; (b) empirical (in blue); fit (in orange) assuming a DGBD (a = 0.480, b = 0.528). Simulated results are congruent to those shown by the empirical data when population growth and productivity are modelled using exponential dynamics.

according to speedrun.com would be N0 = 2 submissions, and the growth rate would be µr = 0.507 1/year. Players and games have growth rates µp = 0.486 1/year and µg = 0.270 1/year, respectively, and both an initial population N0 = 1. Growth rates can be understood as the chance, for instance, that a player recruited another player in time, or recruitments per time.

We define a player’s productivity π as the number of runs submitted per unit time by a unique user. Results show how most users have less than 1 run per month, and that productivity can be captured by an exponential/gamma distribution, as shown in Figure 1d), in which it has been fitted to a gamma process:

βα p(π) = πα−1e−βπ, (2) Γ(α) where:

Z ∞ Γ(α) = πα−1e−πdπ (3) 0 is the Gamma function. This marginal distribution has a shape parameter α and an inverse-scaling parameter β, and has mean µ = αβ and variance σ2 = αβ2. As can be observed, player productivity can be described as a gamma process with parameters α = 0.051 and β = 0.131, except for those few exceptional players who, in the shape of apparently random bursts in the figure, submit runs with remarkably high frequency.

Then, a simulation was conducted in which, in each iteration, each player might recruit another one to the community with chance µp. A certain productivity was assigned to

8 Figure 3: Rank-ordered distribution of games according to their number of runs: em- pirical (in blue); fit (in orange) assuming a DGBD (a = 0.473, b = 1.311). each new runner following the aforementioned distribution. The simulation was stopped when the empirical population size by November 2020 was reached (85,786 players), and players were ranked according to their number of runs. As can be observed in Figure2, the results of the simulation resemble those shown by the empirical data, whose rank-ordered distribution was also computed. In both cases, with both axes in logarithmic scale, the absolute value of the slope of the curve increases as the rank number increases, in contrast with the long asymptotic tail that characterizes power-law distributions. Moreover, the empirical data also shows higher slopes (in absolute value) for those few players with the highest number of runs.

In order to provide an analytical basis for the rank-ordered distributions obtained, they were fitted to a so-called discrete generalized beta distribution (DGBD)[23]. This distri- bution has the following equation:

(N + 1 − r)b f(r) = A , (4) ra where N is the number of elements in the rank, r ∈ N is the position in the rank, A is a normalization constant and parameters a and b are two fitting exponents. The balance between a and b has its own meaning: a can be related to behaviours generating ordered power-laws, whereas b is usually connected to disordered fluctuations in the distribution. In other words, they represent the role of order and disorder shown by the rank. Many systems from different fields have been observed to follow this distribution, such as the frequency with which the codons appear in the genome of E. coli (a = 0.25, b = 0.50), the number of collaborators a movie actor has worked with (a = 0.71, b = 0.61), the popular- ity of programming languages or the occurrence of musical notes in different pieces [23, 24].

The results were fitted to a DGBD with A = 1, N = 85, 786, a = 0.480 and b = 0.528, as can be observed in Figure2 b. In this case, a is quite similar to b, yet b > a, which leads to think that disorder and fluctuation due to noise or external factors play a more important role than the power-law-like behaviour.

9 Figure 4: Score and date of all runs for different categories of the game New Super Mario Bros. Wii:(a) Any%; (b) 100%. Each dot represents an individual run. Those runs which represent new best scores are highlighted in orange. Using the best scores only, a curve fit was performed in the shape of an exponential decay (see dashed curve in red), with learning rates λ = 0.262 1/year (a) and λ = 1.033 1/year (b). Notice that scores change between categories: whereas the 100% mode requires to complete all what the game has to offer, the Any% mode is only focused on reaching the final credits, allowing the player to skip an important part of the content. Hence, 100% takes more time to complete and scores are larger, yet the learning rates have similar order. It is also remarkable how Any% has many more submissions than 100%.

The same procedure was applied to the distribution of games, as can be observed in Figure 3. In such case, the DGBD fit has parameters A = 1, N = 693, a = 0.473 and b = 1.311. b is also larger than a, even more than in the player scenario. It is remarkable how most video games have at least 100 submissions.

2.2.2 An exponential decay model for performance improvement The progress in individual categories was also assessed, evaluating how performance (scores) changed over time, and an overall exponential decay in the learning was ob- served, with significant improvements when the first speedruns are submitted, which tend to stabilize towards a certain score as more runs are uploaded. This is coherent since, unless new hitherto-unknown strategies are discovered, scores tend asymptotically to a theoretical limit and improvements between new runs in time become smaller.

Even though each category for each video game has its own characteristics, all best scores for each of them were fitted in the shape of an exponential curve:

−λ(t−t0) SC (t) = (S0 − Sbest)e + Sbest, (5)

where SC (t) represents the minimal score at time t for a given video game category C, that is, the minimal time taken for the category to be completed at a certain date. S always tends to improve. S0 is the first score registered and Sbest is the current record

10 Figure 5: Probability density of the learning rates. The distribution has been fitted as a Gamma process with parameters α = 0.00590 and β = 0.00581. However, the empirical distribution shows various bursts for relatively large learning rates. It is coherent that most runs do not show any improvement, and that high improvements are relatively difficult to find, since they usually depend on chance or on discovering new strategies.

score (or, at least, the best score registered by the time the data set was downloaded); t0 (in years) is the time at which S0 was achieved and λ ≥ 0 is the so-called learning rate, which tells about how relevant the improvements of the score are in relation with time. An average learning rate hλi = 114.02 1/year, a maximal rate λmax = 8, 904.37 1/year and a minimal rate λmin = 0 were found.

The exponential decay when fitting was assumed to tend asymptotically towards each best score instead of towards zero, since video games have certain animations and unskippable events which make them technically impossible to be completed in no time, that is, there is a minimal compulsory time that will always work as a lower bound.

These curves only consider those runs which represent an improvement with respect to the latest best score. The exponential fit was only applied to those categories with at least two runs submitted providing the best score at their time (and whose date is known); if there had been no improvement at all since S0, which means that the initial score had always been the best, the learning rate is zero (λmin). The smallest nonzero learning rate is λmin,nonzero = 0.067 1/year. The average learning rate for those categories in which there has been at least one improvement is hλinonzero = 126.66 1/year.

In Figure4, examples for 2 different categories for a specific video game are provided.

Improvements were also assessed in a local manner, by means of the determining the learning rate between individual runs, that is, considering only two runs at a time. The frequency distribution of local learning rates value was visualized, as shown in Figure5. The distribution was fitted to a gamma process, as done with productivity, in this case with α = 0.00590 and β = 0.00581. Regardless of the fit, the empirical results show additional peaks in the shape of bursts, revealing recurrent intervals in which players manage to obtain remarkable improvements in a relatively short period of time.

11 3 Stage II. A minimal model for the structural growth of the community

3.1 Methods After studying how the speedrunning community evolves in terms of its growth in play- ers, games and runs, obtaining a rank-ordered distribution for video game players and determining the learning rates through time as a measure of performance, it is not known whether these results are enough to provide information about the actual structure of the community, that is, how players and games are connected with each other. This informa- tion would allow to establish linkages between games and players and to identify possible subcommunities within the whole structure.

Hence, we can apply network methods in order to determine the structure of the com- munity and, furthermore, to try to replicate its structural growth through time. In the following section, we propose a minimal model for the growth of the speedrunning commu- nity in structural terms based on the information obtained in the previous stage. The aim of this model is to compare its outcome with the actual network and to observe whether the previous results per se are enough to predict such structure or not.

3.1.1 The model The model considers three kinds of individuals: players, games and runs. Each player p and game g are identified by a distinctive number, whereas runs r are tuples in the shape (p, g). Both players and games are stored in vectors p and g and added to them each time they are introduced to the community. Runs are stored in an adjacency list R. Regarding the network, then, each player and game is a node and each run an edge connecting nodes. As can be noticed, the graph is bipartite, since there are two types of nodes and edges can connect only nodes of different kind. Hence, there are no self-edges either. The graph is also undirected and unweighted.

Since we knew how many players, games and runs the community had at each point in time, the growth of the network was simulated through iterations setting the number of entities that there should be when each loop finished. Recall that the aim of the model was to study whether or not there are structural properties of the community which can- not be explained by its observable growth trends, and, thus, the growth in the number of players, games and runs should evolve accordingly. Likewise, the rank-ordered distri- butions for players and games according to their number of runs should also be reproduced.

The simulation starts with a single seed run, with the existence of one single player and one single game. Then, iterations are performed adding as many players, games and runs as required according to the actual data and the iteration step size. Each iteration has three stages: (1) the allocation of new players to the community, (2) the allocation of new games and (3) the allocation of new runs. It is possible that, especially during early iterations, no new players nor games are introduced.

Two variants of the model were designed (which could actually be treated as independent

12 models), each one with a different procedure to allocate runs to games and players. In both scenarios, however, it was imposed that every player and game must always have at least one run (recall that a user with no submissions is not a player).

In the first variant, each time a new player or game is introduced to the community, a run is created linking the player to a random game or the new game to a random player following certain probability functions. When both players and games are introduced, the remaining runs for each iteration are created choosing random players and games each time according to those functions.

The second approach considers an infectious model with duplication in which each new player is introduced due to an elder player attracting the new individual to the commu- nity, which implies that the new player does not play a random video game (as happens in the previous scenario) but a game that the elder player has already played, as if the new player became interested in a game that one of his friends, for instance, told him or her about. Hence, allocated nodes tend to duplicate the behaviour and connections of older ones, as already proposed in certain models for protein-protein interactions, and in a manner related to the number of connections each node already has [25, 26, 27]. This approach aims to be more realistic, and random allocations like those from the first scenario could still occur with a certain probability µ, which consider the possibility that a player discovers and plays a new game on his or her own.

Both versions also consider the possibility that the network is not totally connected, with the existence of multiple components, since it is possible that certain communities of players are restricted to specific games and, thus, completely isolated from other regions of the graph. In order to simulate this, it is possible that a new game and a new player are created simultaneously and linked to one another so that such clusters could emerge: they could either become eventually related to other regions of the network due to new associations or links, or, on the contrary, remain isolated. Such allocations occur with probability ρ.

No matter the way games and players are introduced, the simulation is designed so that after each iteration the number of players, games and runs are the ones which have been imposed by the actual data. Nevertheless, the chances for players or games to be chosen are not equiprobable, and they are related to the number of submissions each player and game has, which was determined in the previous stage of the study (see Figures2 and3).

To reproduce the rank-ordered distributions, two fitness functions were defined in order to generate a probability distribution for players and runs when they have to be chosen randomly. These functions give priority to those players and games which have more runs, that is, those which are more recurrent in the adjacency list R. The default score for each player and game when they are introduced is 1, but it can be increased according to these score functions, one for players φp(n) and another for games φg(n), where n ∈ N stands for the respective number of runs. These functions are only modified once a player or a game have been chosen, so that their frequency is rewarded. These functions have been defined via trial and error so that the best outcomes were obtained.

The fitness score for a given player i follows a linear description:

13 φpi (ni) = 1 + πkp(ni − 1), (6)

where ni is the number of runs player i has, π is a probability which determines whether the fitness of a player should increase or not each time the player is chosen, and kp ∈ N is the number of units the score should be increased each time.

Regarding games, we defined the following nonlinear function:

 1 if n < n  i θ1 φgi (ni) = 1 + k1 if nθ1 ≤ ni ≤ nθ2 , (7)  1 + k1 + γk2(ni − nθ2 ) if ni > nθ2

where γ is a probability analogous to π in the previous case, k1 and k2 are the number of units to increase in each case, and nθ1 and nθ2 are thresholds. This function is defined in a fashion that games with a number of runs above the thresholds become highly popular with time, whereas those which do not overcome them do not reach such popularity and remain with few players.

These functions could also be understood in physical terms: the number of runs associ- ated to a player depends on his or her willingness and persistence. The more submissions a player makes, the more chance a new run will be submitted by that person in com- parison to others which have few. In the case of video games, whether a game is played or not depends on its fame and popularity, so the fact that a function with thresholds can be used could be related to a minimal influence required for the video game to succeed.

Recall that scores are updated each time one player or game is chosen. We define the propensity pi of a player or game i as the probability of i to be chosen. pi is thus given by the following equation:

φi pi = P , (8) j φj

where φi represents either φpi or φgi .

Once the simulation finishes, the resulting community always has as many players, games and runs as the actual one, and their ranks describe the same trend, as can be observed in Figure6.

14 Figure 6: (a) Empirical rank-ordered distribution of players according to their number of runs with the DGBD fit. (b) Rank-ordered distribution obtained via simulation with the following parameters: π = 0.9, kp = 2.(c) Empirical rank-ordered distribution of games with the DGBD fit. (d) Simulated rank-ordered distribution with parameters γ = 0.07, k1 = 5, k2 = 2 and nθ1 = nθ2 = 100. This example is from a simulation using the approach without duplication (ρ = 0.01).

3.1.2 Structural analysis of the community After the simulation was executed, we generated graphs for the community using Python’s library NetworkX 2.5, which allows to create many kinds of graphs for given sets of nodes and edges. 50 simulations were performed, 25 for each approach, thus generating 50 differ- ent graphs. Recall that, at the time the data set was downloaded and after the first stage was performed, we obtained information about 203,009 runs for 693 games and 85,786 players. The network, hence, would have a total of 86,479 nodes connected by 203,009 edges. Considering the bipartite nature of the network, these characteristics imply a the- oretical average degree hkpi = 2.37 for players (runs per player) and hkgi = 292.94 for games (runs per game). It must be remarked, however, that repeated runs are considered as a single edge in the network, reducing the actual degree.

As can be noticed, the size of the community is very large and, when it comes to network

15 analysis, there were measures and operations which represented too much computational load, such as obtaining a network projection so that only players were represented, for instance. Even though a projection of the whole network was obtained, this new graph had a number of edges in the order of 107 matching 85,786 nodes, which made further computations such as centrality parameters inviable by conventional means.

Hence, considering the consistency in the exponential growth of players, games and runs, the analysis was reduced to a simplified but parallel picture of the scenario, and we only considered all submissions, members and games identified in the community up until 2013, ignoring all later information. It has to be remarked that simulations required a different parameter tuning for each specific approach. Such tuning was performed via trial an error. For these simulations up to 2013, parameters used in the approach without duplication were: π = 0.25, kp = 2, γ = 0.4, k1 = 2, k2 = 30, nθ1 = 10, nθ2 = 40 and ρ = 0.01; and, in the design with duplication: π = 0.05, kp = 1, γ = 0.001, k1 = 10, k2 = 2, nθ1 = nθ2 = 1, ρ = 0.4 and µ = 0.1.

Even though it was not so large, by 2013, the speedrunning community was already sig- nificantly dense, with 1,606 runs for 205 games and 1,144 players. In this context, there is an average degree hkpi = 1.404 for players and hkgi = 7.834 for games. Node degrees are smaller than in the previous scenario since the total number of runs is significantly smaller, yet the aim of the model was to identify structural properties in the network that could not be predicted with the information already obtained, and such simplification still represented a comparison between the actual structure and the potential of the proposed model. Projections of the real and simulated networks could also be computed removing games and having only players as nodes. Edges would then connect players sharing one or more video games. Then, structural parameters about the networks and their projections could be determined.

When it comes to node-specific properties, we could first identify node degree k, that is, the number of connections each node in the network has. Computing the distribution of degrees through the network would allow to know whether most of the nodes are highly connected or only a few.

Second, centrality, which measures the influence of nodes in a network. In particular, we determined three types of centrality. The first one is eigenvector centrality, which mea- sures how much connected a node is in the network considering the connectivity of its neighbours too, that is, it provides information about its influence in the network. The higher the eigenvector centrality of a node, the more connections it will have with nodes which have high eigenvector centrality themselves. The second one is closeness centrality, which tells about the influence of nodes in a graph in terms of distances. It determines the average farness with respect to all other nodes, and those with the shortest distances are the ones with the highest values. The third one is betweenness centrality, which de- termines how influential each node is when it comes to spreading information through the network. Computing the unweighted shortest paths for all pairs of nodes, it allows to identify nodes which might connect different clusters or modules, behaving as some sort of bridge between them.

Regarding global properties of the network, we computed the following ones:

16 First, modularity Q, which is a measure for the division of a graph in different groups or subcommunities in which nodes are densely connected. It computes the number of edges in a cluster minus the number of edges expected by chance in the cluster, and sums over all clusters. A graph with high modularity possesses many of those clusters or modules and has few connections between nodes from different modules. The software Gephi 0.9.2 offers a tool to compute the modularity of a graph. All other parameters were determined via NetworkX.

Second, the global number of edges l. Even though it could be related to the number of runs, it must be taken into account that repeated runs are only counted once. Further- more, the number of edges generated in projections do not depend on runs but on the relations between players through games. The average node degree hki for projections was also obtained.

Third, global efficiency Eglobal, which is the average inverse of the shortest path lengths in the network. Since it is possible that some clusters are disconnected from others (graphs with multiple components), shortest path lengths and averages could possibly not be com- puted due to infinite distances. Thus, global efficiency would turn infinite distances into zero, allowing computations.

Finally, connectance C0, which represents the ratio between the number of edges in a net- work and the theoretical number of possible connections. Connectance does not consider self-edges nor repeated links, and represents a constraint for the number of different graphs possibly generated, since it is proven that diversity or variability decreases when C0 is too high or too low and that it is maximal when C0 = 1/2 [28]. Connectance is thus defined as:

l C = , (9) 0 n(n − 1) where l is the number of edges in a graph and n the number of nodes. Notice that the factor in the denominator M = n(n − 1) represents the total number of possible unique connections removing self-edges.

Connectance could be measured both for bipartite graphs and projections. However, the definition for each of them is necessarily different. It is not hard to observe that, in the bipartite case, l represents the number of unique runs nr, that is, without counting re- peated ones (games played by the same player more than once). Furthermore, since the edges of the bipartite graphs link players to video games, the number of possible edges is M = np × ng, where np and ng are the number of players and games, respectively. On the other hand, even though projections have nodes of the same kind (players), the number of links l results from the connections between players through games, which depends on the intrinsic structural properties of the network.

Hence, the connectance for the player-game bipartite networks turns out to be:

17 nr C0 = , (10) npng and the one for projections with only players is:

l C0 = . (11) np(np − 1)

Finally, the projections with only players were visualized using Gephi 0.9.2.

3.2 Results 3.2.1 Properties of the community as a network As aforementioned, the algorithm was executed for a total of 50 realizations, 25 for each approach (with duplication and without), generating 50 different graphs. A graph for the actual network was also generated. First, we determined the node degree distribution for each graph, and compared it with the real data. In Figure7 a and c, a comparison of the degree distribution from the actual bipartite graph with that from the simulations (as an average) is shown. Since the degree, that is, the number of runs per player and per game, and its distribution have to do with the rank-ordered distribution of players and games, which was preserved in the simulation, it is logical that both cases follow the same dis- tribution of degree occurrence: only a few players have a large number of runs, whereas the majority has less than 10. Figure7 b and d show the same information but from projections. It can be observed how most of the players in the actual community have a very low degree (they are not much connected with each other), whereas simulations show distributions remarkably different, with high occurrences for relatively large degrees, implying that the model connects players through games more than in reality. The cases with duplication (Figure7 d), however, show better results than those without (Figure 7b): with a duplication model, players tend to play the same games which people who introduce them into the community play, thus making it difficult for players to connect with others who play different games.

We also computed the average node degree as a global property of the projections. The projection of the actual community has average degree hkireal = 31.813. As can be ob- served in Figure8, both approaches tend to show larger values than hkireal, as can be expected from Figure7. This reinforces the observation that, even though simulations follow the growth properties found in the first stage of the study, they establish more connections between players than in real life. If an individual plays a large number of games, it will certainly be connected to many more players than if the individual played a few or only one. Even if a game has a massive amount of popularity, players will still be more likely to be connected to others if they play multiple games. This might explain why when allocations were performed only according to the fitness functions without du-

18 Figure 7: Node degree average occurrence distribution among the 25 networks gener- ated in each approach (in blue: no duplication; in orange: duplication) and the actual community (in green). (a) and (c) show distributions for the respective bipartite graphs, whereas (b) and (d) show the ones for projections with only players. Since all graphs have the same number of nodes, this figure takes the sum of the occurrence of each degree in each of the 25 graphs for each case and divides it by the number of graphs. plication, larger degrees were reached.

We then computed centrality parameters, and their distribution can be observed in Fig- ure9 and Figure 10 regarding the bipartite graphs and their projections, respectively, as an average of the occurrences among each of the 25 simulations conducted with each approach. In the figures, information about both kinds of simulations and about the real graph is included. Regarding eigenvector centrality, it is observed how, both in the case of bipartite graphs and projections, the actual network has low values with more frequency than simulations, which reveals that simulations were not able to consider the fact that only a few nodes have special influence in the network whereas most are not prominent. Even though simulations with the duplication model managed to obtain lower values, they are still too high and their distribution is closer to those without duplication than to the actual one. Closeness centrality and betweenness centrality, on the other hand, show more accurate results. In the case of closeness centrality, especially in simulations with the duplication model (Figure9 e and Figure 10e), distributions are similar to the original, yet many nodes still have too high values. Real data shows a higher number of

19 Figure 8: Average node degree among the 25 networks generated in projections (blue: no duplication; orange: duplication model). The real average degree is depicted with a red dot on the vertical axis of the violin plot. It can be observed how simulations without a duplication model lead to a wider range of values. occurrences in relatively large values of closeness centrality: players could be clustered in dense modules which could be connected by remarkably short paths (a very small number of players behaving as bridges between modules). Regarding betweenness centrality, in Figure9 c and f and Figure 10c and f, it can be observed how, even though simulations without duplication show high values with higher frequency than the original graph and those with duplication, most nodes have their betweenness centrality equal to zero in all scenarios: not many nodes play an important role in connecting modules or substruc- tures in the networks. It should also be considered that graphs can have more than one component, with two or more unconnected structures (which could also explain the high occurrence of nodes with zero eigenvector centrality in the real bipartite network, as ob- served in Figure9 a or d). Those few nodes with high betweenness centrality, on the other hand, lie around the same interval in all cases. Regarding centrality in general, it could be thought that, the higher the connection between players (as observed with node degrees), the more frequent influential players are (and games too in the bipartite case), leading to a higher frequency of high centrality values in simulations with no duplication, in which individuals play different games each time allocated randomly and, thus, players are connected with more players and from different areas in the network.

When it comes to the global efficiency, it can be observed in Figure 11 how in all simu- lations for both kinds of graphs (bipartite and projections) values are remarkably larger than real efficiencies (0.083 for the bipartite case and 0.197 for the projection). Recall that global efficiency is the average inverse of the shortest path lengths in a graph: higher efficiencies imply that distances between nodes are smaller. Hence, in simulations, given the higher connection between individuals playing different games and, in projections, the higher number of connections between players, individuals are globally closer to one another than in the real community. This higher number of connections can be observed in Figure 12a, in which the number of edges in projections is shown. Notice, however, that they can be determined as the sum of all degrees divided by 2, so they have a meaning similar to average node degrees. It can be highlighted again how most of the simulated graphs create too many connections between players in comparison to the actual network (whose number of connections is lreal = 36, 394 edges).

20 Figure 9: Centrality distribution in simulated bipartite graphs (in blue: no duplication; in orange: duplication) in comparison to actual data (in green). (a) and (d): eigenvector centrality; (b) and (e): closeness centrality; (c) and (f ): betweenness centrality. Since all graphs have the same number of nodes, this figure takes the sum of the occurrences for each centrality in each of the 25 graphs and divides it by the number of graphs, as an average.

Figure 10: Centrality distribution in projections (in blue: no duplication; in orange: duplication) in comparison to actual data (in green). (a) and (d): eigenvector centrality; (b) and (e): closeness centrality; (c) and (f ): betweenness centrality. Since all graphs have the same number of nodes, this figure takes the sum of the occurrences for each centrality in each of the 25 graphs and divides it by the number of graphs, as an average.

21 Figure 11: Global efficiency of communities generated by both approaches (in blue: no duplication; in orange: duplication model), and of the actual community (depicted as a red dot on the vertical axis of the violin plots). (a) Bipartite graphs. (b) Projections.

Figure 12: (a) Number of edges and (b) modularity from projections in simulations with and without duplication (in orange and blue, respectively). Values from the actual graph are also depicted as a dot in red on the vertical axis of the violin plots.

Regarding projections’ modularity, shown in Figure 12b, it can be observed how, even though the duplication model should apparently lead to more remarkable subcommuni- ties of players in the network, simulations without duplication show larger values, and also more similar to the actual modularity (Qreal = 0.634). This could be explained with the existence of substructures (isolated or not) with only a few players, too small to be identified as modules. The algorithm used to compute modularity was the tool offered by the software Gephi 0.9.2, with resolution 1.0 (default) for all graphs. The high modular- ity of the original graph could be explained with the existence of many distinguishable subcommunities of members playing specific games and genres.

The last parameter studied was connectance. In Figure 13, all distributions can be ob- served. In the bipartite case, better results are achieved with the duplication model. However, the difference between the real (C0bipartite,real = 0.00558) and simulated values is small. This makes sense since the number of edges in this kind of graphs is directly related to the number of runs, and differences in connectance can only be explained due to the removal of edges because of repeated runs. Hence, the difference is small and sim- ulations with duplication, in which players are more likely to play the same games more

22 Figure 13: Connectance distribution for the bipartite graphs and their projections with only players (in blue: no duplication; in orange: duplication model). The connectance of the actual community is also depicted in red on the vertical axis of the violin plots.

Figure 14: Visualization of the projections of the bipartite graphs showing only players. (a) Actual community. (b) Random sample from the 25 graphs generated without du- plication. (c) Random sample from the 25 graphs generated with the duplication model. Structures are coloured according to the modules identified. than once, show lower connectance. Recall that connectance is the fraction of edges in the graph with respect to the theoretical number of possible edges. In the case of projections, the values of this fraction increase and are more diverse, since they depend on the connec- tions between players though games. Connectance values in this context are remarkably higher than the real one (C0projection,real = 0.0278), due to the higher connection of players through different games in simulations, yet executions with the duplication model do not reach values as high as the others, since players play more specific games. Recall that the fact that an individual plays more than one game is more likely to make him or her be connected to other players than playing a single game with a lot of users.

3.2.2 Graph visualization Projections were visualized using Gephi 0.9.2. in order to observe how players are con- nected with each other, as shown in Figure 14a, which depicts a first representation of how the speedrunning community was structured by 2013. In Figure 14b and c, samples among simulated graphs are shown. It is remarkable how, from a qualitative point of

23 view, better results were achieved with the duplication model. The community, at least by 2013, had many groups of players connected through specific games and completely isolated from the others. With duplication, this property is reproduced, whereas, other- wise, those isolated components are extremely rare. Further details about the generation of such components have been included in Section S.I in the end of the document.

24 4 Stage III. Community structure and learning capa- bility of players

Once information about the growth and the structural properties of the network was obtained, the last step of this study was to connect that information to the learning ca- pability of each individual.

4.1 Methods 4.1.1 Learning capability as a node-specific property In order to connect the learning properties with the structural characteristics of the net- work, we used learning capacity as a particular attribute of each individual. In Figure5 from Section2, the distribution of the different learning rates in video game performance is shown. Recall that those learning rates were the decay parameters of the exponential approximations applied to performance evolution, but computed as the improvement be- tween two runs being the best score at their respective time. Each learning rate is, hence, a measure of performance, and it can be attributed to the player submitting each latest score. We thus define the learning capability λc of an individual as the sum of all learning rates shown by the player considering all his or her submissions. In Figure5 it can be observed that most submissions have learning rate λ = 0, since they did not represent any improvement with respect to the latest best scores. As a result, most players have zero learning capability, according to the way it has been defined.

Once we attached the learning capabilities to players, we could compare this property with the intrinsic node-specific properties of the network studied in the second stage of the study. Since in this new stage no simulations were performed and we only required the original network, the study could be extended to 2015 instead of 2013, since compu- tations were still possible to be executed (recall that simulations generated graphs with a number of edges significantly larger than the original community, and the computational load it represented). Consequently, in this scenario, only submissions up to 2015 were con- sidered, with a total number of 5,389 players submitting 9,332 runs for 348 different games.

Even though computations were performed using Python via Spyder 4.1.5, figures in this section were created with R using RStudio 1.4.1717 and Gephi 0.9.2, since visualizations of the player graphs in the period between 2012 and 2015 were also achieved.

4.2 Results 4.2.1 Influence of learning capability on the community We compared the learning capability of each player with each of the node-specific param- eters previously explained, namely node degree and centralities, in order to observe if the distribution followed any specific pattern. The results are shown in Figure 15. As can be observed, in all histograms there is a dispersion of players which does not show any sort of tendency relating learning capacity with any of the parameters, that is, there is no

25 Figure 15: Histograms showing the relation between the learning capability of players with their individual properties: (a) node degree, (b) eigenvector centrality, (c) closeness centrality, (d) betweenness centrality. Coloured dots represent combinations of parameters within specific ranges shown by players, and, the brighter they are, the more players within those ranges. The color scale was not set uniform for all histograms since learning capability is aimed to be related to each parameter in an independent manner, so that possible patterns in each distribution could be identified, and hence maximal contrast is achieved in the range of player counts for each plot. Players with zero learning capability have not been included. estimation on whether higher or lower node-specific values lead to higher or lower learn- ing rates. Structural parameters follow distributions analogous to those from the previous stage of the study, now updated to 2015. In Figure 15, however, players with zero learning capability are not shown, since we were interested in relations between players who had performed at least one improvement. In Section S.II, further details about the concept of learning capability and the results obtained are discussed, and it is shown how players with no learning capability are dispersed along the horizontal axis.

Even though no intrinsic relations between learning capability and the other parameters are observed, there are patterns in the distributions which can be identified. For instance, regarding node degree (Figure 15a), the distribution shows high density of nodes within a specific region. As a matter of fact, the average learning capability found in the community

26 Figure 16: Visualization of the actual speedrunning community (projection with only players) and its evolution between 2012 and 2015. Structures are coloured according to the modules identified.

was hλci = 0.688 1/year, which lies in the interval with high node density. This interval is consistent in all four distributions. The average node degree was hki = 213.457, slightly higher than the apparent mean in the distribution (around 40 and 60). This could be influenced by the distribution of players with zero learning capability, which could still have high node degree. Eigenvector, closeness and betweenness centralities have means 0.00567, 0.376 and 0.000294, respectively, which are not exactly the values at which nodes are accumulated (except for closeness centrality), probably influenced by players with no learning capability or because, in the case of betweenness centrality, as can be noticed, most nodes are accumulated around zero, but there are still other intervals of values with significantly different order of magnitude and certain density which affect the value of the mean. Therefore, it can be observed how, in all scenarios, players are gathered around specific values of learning capability, degree and centrality, but no direct relation between parameters can be established.

4.2.2 A structural transition in time Finally, a visualization of the projections from 2012 to 2015 was achieved via Gephi 0.9.2, in order to observe how the community grew during that period at least in a qualitative manner, as shown in Figure 16.

It can be observed how the density of the community increased through time, with a remarkable shift by 2014, in which most subcommunities of players which were isolated were suddenly connected with each other, with an observable increase in network connec- tivity. As discussed in the following section, the structural properties of the community nowadays could significantly differ to how they were in 2013 (which is the latest year we considered in the second stage of the study).

27 5 Discussion

With this study, we have provided a first approach for the analysis of video game per- formance and community in the light of cultural evolution, focusing in particular on speedrunning, under the assumption that optimality in video game performance can be measured with the time taken to complete a video game. This analysis has been possible due to the quantitative nature of speedrunning information, and thanks to the speedrun- ning community, who shares, supervises and gathers information into public platforms such as speedrun.com.

We have managed to study the cultural growth of the community through time, and observed how players, games and submissions are introduced to the community in an exponential fashion, as if players and games were recruited like in a scenario of infection propagation. The popularity reached by speedrunning in the recent years and also the rise of streaming platforms lead us to think that its growth will keep increasing during the following years far from saturation at least from a short term perspective. The intro- duction of new members to the community will not only increase its influence but also the probability of finding new discoveries which could improve performance in video game solving. Performance in individual video games evolves through time via an exponential decay-like description, approaching theoretical limits in an asymptotic manner.

Regarding players’ productivity and learning capabilities, they were distributed and fitted to a Gamma process. The rank-ordered distributions of players and video games were also fitted as discrete generalized beta distributions (DGBDs). All of them were observed to follow power law-related behaviours, with most of the individuals having small pro- ductivity and most learning rates being low (or zero), and only a few showing the most remarkable outcomes.

We have also managed to provide a first evolutionary model to reproduce the growth of the community in terms of the aforementioned properties. The model allows to generate artificial communities with structural properties similar to those from the original, even though it fails to predict the clustering of players in specific modules and hence connects them more systematically than in reality, leading to a significant increase in parameters such as node degree, connectance or the frequency of nodes with high centrality, which are not directly related to the parameters assessed in the first stage.

We also linked the role of performance and learning capability to node-specific proper- ties of the networks. We expected to observe patterns in the distributions revealing how players with certain properties or influence in the community tended to show higher learn- ing skills than others, for instance, since, according to many cultural evolution studies, the higher the communication among individuals, more advanced the learning traits that could be expected [10, 11]. However, no actual tendency was observed relating each other, yet individuals are congregated around specific intervals or values for each specific param- eter. The lack of direct relation between parameters leads us to think that performance is possibly not so related to the learning capability, but rather to perhaps the persistence of players with specific games as well as luck or serendipity. It must also be remarked that learning was defined as the exponent of the decay curve described by player performance. Players performing initial runs of a certain game might perform the best improvements,

28 reducing the difference between scores as the game is mastered. Thus, those players with the latest and best scores might show low learning attributes yet they would still have performed the best playthroughs.

Finally, we observed in a qualitative fashion how the player community changed through a specific period (2012-2015), and noticed a remarkable transition in the connectivity of the network between 2013 and 2014, in which, before that time, most players were clus- tered in isolated modules or components. The sudden emergence of a giant component could be due to a significant increase in the number of runs with respect to the number of individuals (recall that the number of submissions grows with a larger rate than games and players) and participation of players in different video games, thus increasing edge density and communication between subcommunities [29].

Therefore, recalling our initial hypothesis that there could be universal patterns in the way performance and the video game community have evolved through video game his- tory, it has been proven that the properties of such cultural evolution can be explained by simple mathematical descriptions, yet a direct connection between performance and net- work structure has not been identified. The goals of this research have also been achieved in the shape of three different stages in which (1) we assessed the cultural growth of the community, (2) we analyzed its structural properties and (3) we related performance with that structure.

This study, nonetheless, presents certain limitations which have constrained the acquisi- tion of further results. On the one hand, due to the density the community has achieved in the recent years and to computational limitations, the study has had to consider only its development up until 2013 in the second stage and 2015 in the third stage (the first stage has information up to November 2020, when the data set was downloaded), possibly ignoring crucial information about events or phase transitions occurred recently. Second, it must be remarked that the data used during the analysis was only a fraction of the complete data set according to our own criteria, as explained in Section2, and that it does not represent the whole speedrunning community. Third, the model proposed in the second stage represents a minimal model for the growth of the speedrunning com- munity; hence, further and more complex implementations could be applied in order to achieve more accurate outcomes. Fourth, in this study we have provided our own def- inition for performance and learning capability using the learning rates from individual improvements. Studying performance using other definitions and parameters could still possibly allow to identify relations with structural properties of the community which in this study have not been uncovered. Finally, recall the assumption that optimality in video game performance is related to the time required for game completion, not taking into account other communities and possible video game characteristics which are not related to speedrunning.

Additionally, and, as further work, it would be interesting to observe how players’ pro- ficiency in video games change according to the number of runs they have submitted. It would still be possible that many players only shared the best results they achieve, not providing enough information; yet, in such analysis, proficiency could be defined as a parameter with inverse relation to the scores players achieve.

29 Furthermore, if the computational potential to determine structural properties of the com- munity through time later than 2015 was given, it would be of particular interest to study how those properties evolved, and whether learning skills change according to the density and modularity of networks, since, probably, in more recent years, the relation between learning capability and the structural properties of the community follows patterns which did not occur in 2015. It could be possible that the higher size of the community could have increased the overall learning skills and knowledge, and that those modules of players clustered around specific games with larger density also show better performance, con- nected through a small number of players with high centrality and possibly more skilled [6,8,9, 16, 17].

As a conclusion, the fact that all information about video game performance is stored on the Internet, and that games can be promoted through many different platforms in order to reach the influence they have today, it can be stated that information in this field is effectively spread and its legacy will be preserved, promoting the maintenance of their traits in terms of cultural evolution [4, 10, 11]. It can be expected that the video game community will keep increasing encouraging new individuals to participate and with the discovery of new strategies and with new trends in the future, as well as new technologies to improve player experience. With this study, we have provided a first framework and insight into video games from the perspective of cultural evolution, which represents an innovative contribution and a first quantitative application of biological and evolutionary models to such a novel, albeit relevant field.

30 Bibliography

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32 Supplementary information

S.I Generation of multiple components in simulations One observation about the generation of isolated communities is that, as explained in Section 3.1.1, in simulations we defined a parameter ρ which determined the probability of a new player and a new game to be introduced simultaneously and attached to one another. This is the only mechanism which allows the emergence of those components. Figure S1 shows a comparison of the real network (a) with simulated samples using the duplication model and different values for ρ, namely 0.1 (b) and 0.4 (c) (the one that was used). It makes sense that, the higher the value of ρ, the more likely it will be to have more communities which remain isolated through time. However, if too high, the duplication model and fitness functions would lose their influence. In simulations without duplication, it is more difficult for those groups to persist through time, since games are allocated randomly only according to fitness functions and not to whether players have already played them; however, increasing ρ results in too many games and players being introduced without following any fitness nor duplication mechanism, and the rank- ordered distributions imposed cannot be achieved. Hence, in the case without duplication, as observed in Figure 14b in Section 3.2.2, either the rank-ordered distributions or the emergence of an acceptable number of components can occur, but not both at the same time. This is why the value of ρ used in this case was 0.01: rank-ordered distributions had higher priority.

Figure S1: Visualization of the projections of the bipartite graphs showing only players with different chance of introducing new players and games simultaneously (ρ). (a) Actual community. (b) Sample generated with the duplication model and ρ = 0.1.(c) Sample generated with the duplication model and ρ = 0.4. Structures are coloured according to the modules identified.

33 S.II Learning capability In this study, we defined the concept of learning capability as a player-specific attribute which determines the skill and performance of players by means of computing the im- provement their contributions represented assuming an exponential decay between each pair of scores which were the best at their respective time. We considered each learning rate λ as the decay rate in each individual improvement, and we then defined the learning capability λc of an individual as the sum of all his or her learning rates. Even though each learning rate is the exponent of a different individual improvement, they always have the same units (1/year), and summing them would reward those players which have more contributions. Other definitions were also considered, such as the average instead of the sum of learning rates, yet this case penalizes players which have more than one im- provement, and a cumulative definition was preferred. In Figure S2, results with learning capability as the mean instead of the sum are shown. Both outcomes follow similar dis- tributions, yet with the sum higher learning values are reached (Figure 15 in Section 4.2.1).

Figure S2: Histograms showing the relation between learning capability as the mean learning rate of each player with each individual property: (a) node degree, (b) eigenvector centrality, (c) closeness centrality, (d) betweenness centrality. Coloured dots represent combinations of parameters within specific ranges shown by players, and, the brighter they are, the more players within those ranges. The color scale was not set uniform for all histograms since learning capability is aimed to be related to each parameter in an independent manner. Players with zero learning capability have not been included.

34 On the other hand, both defining learning capability as a sum or a mean imply that most players have zero learning capability, since only a few runs represented an improvement of previous best scores. In Figure 15 and Figure S2, only players with non-zero learning ca- pability were considered, since the aim of the stage was not to assess player participation but how performance connects to other parameters. However, in order to prove and not to skip the existence and role of those individuals, in Figure S3 a reproduction of Figure S2d is shown, but including players with zero learning capability. Even though this figure considers learning capability as a mean, players with λc = 0 are the same as with the sum. It can be observed how the vast majority of players have zero learning capability, and most of them are gathered together with zero betweenness centrality, which is the value with higher density among players with non-zero learning capability too. Recall that, by 2015, 9,332 runs had been submitted, and many players had more than one run; this figure shows then the large fraction of players who did not manage to obtain a best score at some point in time.

Figure S3: Histogram showing the relation between learning capability as the mean learning rate of each player with betweenness centrality. Coloured dots represent com- binations of parameters within specific ranges shown by players, and, the brighter they are, the more players within those ranges. Players with zero learning capability are also included, which represent the vast majority of players.

35