IIL NUOVO CIMENTO
Virtual social science
Stefan Thurner
Section for the Science of Complex Systems, Medical University of Vienna - Vienna, AustriaSanta Fe Institute - Santa Fe, USAIIASA - Laxenburg, AustriaComplexity Science Hub Vienna - Vienna, Austria
Summary. — Can we describe social systems quantitatively and predictively, whenwe know all the actions, interactions, and states of individuals? We interpret humansocieties as co-evolutionary systems of individuals and their interactions. Based onunique data of a society of computer game players, where all actions and interactionsbetween all players are known, we show that this might indeed be possible. Withinthis framework we address a number of sociological classics, including formation ofsocial networks, strength of relations, group formation, hierarchical organization,aggression management, gender differences, mobility, and wealth-inequality. Wediscover behavioral and organizational patterns of the homo sapiens and its societythat were not visible with traditional methodology from the social sciences.
1. – Introduction
In the first years of the nineteenth century Auguste Comte suggested to copy whathad been done in physics at this point in time: to establish a natural and experimentalscience of social systems. He suggested to call this hopeless endeavor sociophysics . No-onefollowed him. He died poor and alone, without any acknowledgement of this vision of his.Some remember him today as the inventor of the term sociology . Of course, his vision wasbound to fail. There was no way of understanding what the rules of interactions betweenthe components of societies were. Even had he known the interpersonal interaction laws,he would have failed. He would not yet have had a way to aggregate information of manyinteractors to levels that could have been interpreted in any meaningful way. Statisticalmechanics was not invented yet, nor was the computer; Gauss was barely born. Had heovercome all these difficulties, he would have most likely failed because he did not havea way to know what non-linearity can do to the predictability of dynamical systems. Hehad to fail.Now, more than 200 years later, we try it again: to transform the social science aswe know it today, into an experimental physical science: fully quantitative, predictive,and experimentally testable. A science that is based on the elementary interactions c (cid:13) Societ`a Italiana di Fisica a r X i v : . [ phy s i c s . s o c - ph ] N ov STEFAN THURNER between humans, and between them and their environment. Immediately criticism fromthe traditional representatives of the social sciences and humanities arises: (i) humanshave a free will, their actions and interactions can not be predicted; (ii) societies have waytoo complicated interactions, which will never be quantifiable, and (iii) data on actionsand interactions on a society-wide level does not exist.In the 21st century we do not have to accept these arguments. None of them states afundamental reason, why a new attempt would have to fail. In response to the free willargument (i), we can argue that atoms have even more “free will” than humans do, andthat on a quantum mechanics level the situation is even worse: not only could an atomdecay spontaneously, not even its position and momentum can be known simultaneously.How should we ever be able to predict properties of matter that is composed of suchvolatile things? But we can. A counter argument to statement (ii), that human interac-tions are too complicated to be quantified, is found in the very existence and availabilityof exactly this kind of data, together with the computer power to process it. We mightbe able to record a large fraction of all interactions soon, given that we are not stoppingthe current trends in digitation. The same holds true for argument (iii), that there is alack of data on social actions and interactions. The world has changed.This does not mean that those who try to realize the vision of Comte again will besuccessful. In this lecture I will show that in fact, we are getting to the point, wherewe will be able to record every single interaction in a social system, meaning that wehave records about all interactions, at all times, and between all individuals. I willshow a special example where we have exactly this situation: a dataset of a humansociety of players “living” in a massive multiplayer online game, where all informationabout actions, interactions and properties of all avatars is available. The game is called
Pardus , and was developed, programmed, and maintained by my former students andcollaborators Michael Szell and Werner Payer. In this game, which attracted almost half amillion players since 2004, avatars act out an open-ended “second life”, over large periodsof time—often over years. All actions and interactions, all properties and characteristicsof all players are recorded, at every point in time.We have studied this human society in the past years. In this lecture notes, I summa-rize some of the work that was done together with Michael Szell, Peter Klimek, BenediktFuchs, Renaud Lambiotte, Vito Latora, Roberta Sinatra, Didier Sornette, Olesya Mry-glod, Yurko Holovatch, Bernat Corominas-Murtra, Maximilian Sadilek, and others. Theoriginal works are found in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. .1. What is social science? . – Social science is the science of social interactions andtheir implications for society. Traditionally, social science has neither been very quanti-tative or predictive, nor does it produce experimentally testable predictions. It is largelyqualitative and descriptive. Until recently, there has been a tremendous shortage of datathat are both, time-resolved (longitudinal) and multidimensional. As stated, the situa-tion is changing fast with the tendency of homo sapiens to leave electronic fingerprintsin practically all dimensions of life. The century-old data problem of the social sciencesis rapidly disappearing. Another fundamental problem in the social sciences is the lackof reproducibility and repeatability. On many occasions, an event takes place once inhistory, and no repeats are possible.As in biology, social processes are hard to understand mathematically because theyare evolutionary, path-dependent, out-of-equilibrium, and context-dependent. They arehigh-dimensional and involve interactions on multiple levels and scales. The methodolog-ical tools used by traditional social scientists, which rarely extend much beyond linear
IRTUAL SOCIAL SCIENCE regression models, and basic Gaussian statistics, are not powerful enough to address theseissues appropriately. However, two important innovations have been developed withinthe social sciences that play a crucial role in the theory of complex systems: Multilayer interaction networks.
In social systems, interactions between individualsand institutions, happen simultaneously on more or less the same strength scale on a mul-titude of superimposed interaction networks. Social scientists, in particular sociologists,have recognized the importance of social networks, starting in the 1970s [16, 17].
Game theory.
Another contribution is game theory, a concept that allows us to deter-mine the outcome of rational interactions between agents trying to optimize their payoffor utility [18]. Each agent is aware of the fact that the other agent is rational and thathe/she also knows that the other agent is rational. Before computers arrived on thescene, game theory was one of the few methods of dealing with complex systems (inequilibrium). Game theory can easily be transferred to dynamical situations, and it wasbelieved for a long time that iterative game-theoretic interactions were a way of under-standing the origin of cooperation in societies. This view is now severely challenged bythe discovery of so-called zero-determinant games [19]. Game theory was first developedand used in economics, but later penetrated other fields of the social, behavioural, andlife sciences. .1.1. Social systems are continuously restructuring networks. Social systems can bethought of as time-varying multilayer networks. Nodes are individuals or institutions;links are interactions of different type. Interactions change over time. The types of linkcan be friendship, family ties, processes of good exchange, payments, trust, communica-tion, enmity, and so on. Every type of link is represented by a separate network layer,see e.g. Fig. 1. Individuals interact through a superposition of these different interac-tion types (multilayer network), which happen simultaneously, and are often of the sameorder of magnitude in “strength”. Often, networks at one level interact with networks atother levels. Networks that characterize social systems show a rich spectrum of growthpatterns and a high level of plasticity. This plasticity of networks arises from restruc-turing processes through link creation, re-linking, and link removal. Understanding anddescribing the underlying restructuring dynamics can be challenging. However, there area few typical and recurring dynamical patterns that allow us to make scientific progress.Individuals are represented by “states”, a set of characteristics that describe theirwealth, gender, education level, political opinion, age, and so on. Some of these stateschange dynamically over time. States typically have an influence on the linking dynamicsof their corresponding node. If that is the case, a tight connection exists between networkstructure and node states. The joint dynamics of network re-structuring and changes ofstates by individuals is a classic example of co-evolution . .2. Social systems are complex systems . – If social systems are complex systems—what are complex systems? We follow the definition of complex systems given in [20]:
Complex systems are co-evolving multilayer networks.
This statement summarizes tenfacts about complex systems:1. Complex systems are composed of multiple elements, labelled by latin indices, i .2. These elements interact with each other through one or more interaction type, la-belled by greek indices, α . Interactions between elements are specific, not everyoneinteracts with all the others. To keep track of which elements interact we use net-works. Interactions are represented as links, the interacting elements are nodes. STEFAN THURNER
Fig. 1. – Schematic representation of a multilayernetwork. Here nodes are characterized by a two-dimensional state vector. The first component isgiven by colors (red, blue), the second by shapes (cir-cles, squares). Nodes interact through three typesof interaction that are represented by (full, broken,and dotted) lines. The system is complex if statessimultaneously change as a function of the interac-tion network, and at the same time, if interactionschange as a function of the states, see Eqs. (1). It isa co-evolving system, where states and interactionsupdate each other, similar to what an algorithm isdoing.
Every interaction type can be seen as one network layer in a multilayer network,see Fig. 1. A multilayer network is a collection of networks linking the same setof nodes. If these networks evolve independently, multilayer networks are super-positions of networks. However, there are often interactions between interactionlayers.3. Interactions change over time. We use the following notation to keep track ofinteractions in the system. The strength of an interaction of type α between twoelements i and j at time t is denoted by, M αij ( t ) interaction strength . Interactions can be physical, financial, emotional, economical, hostile, or symbolic,to just mention a few. Most interactions are mediated by an exchange process ofsome sort between nodes. In that sense, interaction strength is often related tothe quantity of “things” or units exchanged (money for financial interactions, loveletters for emotional interactions, bullets for hostile interactions, bottles of winefor positive social interactions, and so on). Interactions can be deterministic orstochastic.4. Elements are characterized by states. States can be scalar; if an element has variousindependent states, it will be described by a state vector, or a state tensor. Statesalso evolve over time. We denote the state vectors by, σ i ( t ) state vector . States can be the education level, gender, political believe, wealth, aggression levelof a person, or the capitalization, and risk aversion levels of a bank. State changescan be deterministic or stochastic. Changes can be the result of the endogenousdynamics in the multilayer network, or of external driving.5. States and interactions are often not independent but evolve together by mutuallyinfluencing each other; states and interactions co-evolve . The way in which statesand interactions are coupled can be deterministic or stochastic.6. The dynamics of co-evolving multilayer networks is usually non-linear.
IRTUAL SOCIAL SCIENCE
7. Complex systems are context-dependent. Multilayer networks provide that context.To be more precise, for any dynamic process happening on a given network layer,the other layers represent the “context” in the sense that they provide the onlyother ways in which elements in the initial layer can be influenced. Multilayernetworks sometimes allow complex systems to be interpreted as “closed systems”.They can be externally driven. Then they are dissipative and non-Hamiltonian.8. Complex systems are algorithmic , they behave rather like an algorithm than asystem that is described by a set of differential equations. The algorithmic natureis a direct consequence of the discrete interactions between interaction networksand states.9. Complex systems are path-dependent, and therefore often non-ergodic. Given thatthe network dynamics (the dynamics of links) is sufficiently slow, the networks inthe various layers can be seen as a “memory” that stores the recent past of thesystem.10. Complex systems often have memory. Information about the past can be storedin nodes if they have explicit memory, or in the network structure in the variouslayers.In the following, we assume that a co-evolving multilayer network structure is the fun-damental dynamical backbone of social systems. A snapshot of a co-evolving multilayernetwork is shown in Fig. 1. Nodes are given by a state vector with two components,colour (blue, red) and shape (circles and boxes). Nodes interact through three types ofinteraction (full, broken, and dotted lines). The system is a complex system if stateschange as a function (deterministic or stochastic) of the interaction network and, simul-taneously, interactions (the networks) change as a function of the states. The shownmultilayer network could represent a social network of individuals with certain time-dependent properties. Think of wealth represented by shape (rich=circle, poor=square),and gender (blue=male, red=female), and the different different link-types represent com-munication, trade, and friendship. While the state of wealth can change as a functionof the trading links, gender will not change because of trading links, but might becauseof friendship links. Changes in wealth will have a positive influence on the creation offuture trading links, and maybe a negative effect on friendship links. .2.1. What is co-evolution?. Interactions can change the states of elements. Theinteraction partners of a node in a (multilayer) network can be seen as the local “envi-ronment” of that node. The environment often determines the future state of the node.In social systems, interactions do change over time. For example, people establish newfriendships or economic relations, countries terminate diplomatic relations. The stateof nodes determines (fully or in part) the future state of the links, whether it exists inthe future or not, and if it exists, the strength and the direction that it will have. Theessence of co-evolution is expressed in the statement: The state of the network (topologyand weights) determines the future states of the nodes. The state of the nodes determinesthe future states of the links of the network.
STEFAN THURNER
Formally, co-evolving multilayer networks can be written as ddt σ i ( t ) ∼ F (cid:0) M αij ( t ) , σ j ( t ) (cid:1) and ddt M αij ( t ) ∼ G (cid:16) M βij ( t ) , σ j ( t ) (cid:17) . (1)Here, the derivatives indicate “change within the next time step”, and are not continuousderivatives. The first equation means that the states of element i change as a function, F , that depends on the present states of σ i , and the present multilayer network states, M αij ( t ). The function (or functional), F , can be deterministic or stochastic and containsall summations over all greek indices and over j . The first equation depicts the analyticalnature of physics that characterized the past 300 years of science. Once one specifies F ,and the initial conditions, say, σ i ( t = 0), the solution of the equation provides us withthe trajectories of the elements of the system. In physics the interaction matrix, M αij ( t ),could represent the four forces. Usually it only involves a single interaction type α , thatis static, fully connected, and interaction strength only depends on the relative distancebetween i and j . Typically, systems that can be described with the first equation aloneare not complex systems, however complicated they may be.The second equation specifies how the interactions evolve over time as a function G that depends on the same inputs, states and interaction networks. G can be a de-terministic or stochastic function or functional. Now, interactions evolve in time. Inphysics this is very rarely the case. The combination of both equations makes the systemco-evolving and complex. Co-evolving systems of this type are, in general, no longer ana-lytically solvable. One cannot solve these systems using the rationale of physics becausethe environment—or the boundary conditions—specified by M , change as the systemevolves. Equations (1) are not useful until the functions G and F are well specified. Thescience of complex systems often tries to identify these functions for a concrete systemat hand. Often this is done in an algorithmic way, meaning that F and G can be givenas “update rules”.More and more data sets containing full information about an entire system are be-coming available, meaning that all state changes and all interactions between the elementsare recorded. It is becoming technically and computationally possible to monitor cell-phone communications on a national scale [21], to track all airplanes in motion, or totrack all legal financial transactions on the planet. Longitudinal data about states andinteractions can be used to visualize Eqs. (1); all the necessary components are in thedata at any point in time: the interaction networks, M αij , the states of the elements, σ i ,and all the changes ddt σ i and ddt M αij . Even though Eqs. (1) might not be analyticallysolvable, it is becoming possible for more and more situations to “watch” them.The structure of Eqs. (1) is not the most general. One generalization is to endow mul-tilayer networks with a second greek index, M αβij , that captures cross-layer interactionsbetween elements [22]. It is conceivable that elements and interactions are embeddedin space and time; indices labelling the elements and interactions could carry such ad-ditional information, i ( x, t, ... ) or { ij } αβ ( x, t, ... ). One can introduce memory to theelements and interactions. We will make use of these generalizations in the following,where we use complete information on all the actions and interactions of ten thousands ofplayers engaged in a massive multiplayer online game, with their millions of interactionsand state-changes. IRTUAL SOCIAL SCIENCE
2. – A virtual society2 .1.
The universe: the Pardus game . – Our habitat is
Pardus
Pardus players, about 16,000 active players on a given day, the game is online sinceSep 2004, and is free of charge. MMOGs are characterized by a substantial number ofusers playing together in the same virtual environment connected by an internet browser[23, 24]. In
Pardus every player owns one account with one single character or avatar.This avatar is a pilot owning a spacecraft with a certain cargo capacity, moving aroundin a virtual universe to produce and trade commodities and products, socialize, engagein social activities, wage wars, engage in administration, and much more. An importantcomponent of
Pardus is that the actions and interactions of players are strongly drivenby social factors such as friendship, cooperation, and competition. There is no explicitgoal in
Pardus , nor are there forbidden moves (with a few exceptions concerning decentbehavior).
Pardus is a virtual world with a gameplay based on socializing and role-playing, with massive interactions between players’ avatars, and with interactions witha non-player environment. Players simultaneously engage in three forms of “life”:
Economic life.
They produce, distribute and trade goods and services, and makeprofit from economic activity. They spend money on goods and services, such as newspace ships, equipment, and consumables. Status symbols play a big role in the purchaseand consumption of goods.
Social life.
Players communicate and share information toorganize in social structures and collective actions, be it social, political, legal, hostile, oreconomical. Often actions are driven by the wish to accumulate social status through oneof many forms of earning recognition.
Exploratory life.
Players explore their universe.They produced maps of the universe and it resources, classified its game-specific fauna.Some even have engaged in research on the “physical nature” of the game.While playing, players form groups of different sizes and structures. These include po-litical parties, special interest groups, criminal gangs, cartels, banks, courts, self-defencegroups, or armies. They also organize in clubs that are called alliances .Daily database backups are recorded, and are available for more than a decade, start-ing from Sept 2005. These backups contain longitudinal information of the actions andproperties (states) of the avatars, as well as all the interactions between them, and be-tween them and the environment. .1.1. The census of avatars. Age and nationality.
In a poll taken in 2005, 5% reportedan age below 15 years, 18% between 15–19, 34% between 20–24, 23% between 25–29, and20% are older than 29 years. The distribution of player nationalities was estimated asfollows: US 40%, UK 14%, Canada 5%, Austria 4%, Germany 4%, Australia 4%, other29%.
Lifetimes of characters.
Next to automatic deletion after 120 inactive days, everyplayer can delete her account at any time. Of all characters, about 8% have a lifetime of0 days, i.e. they delete themselves on the same day they signed up. About 13 .
4% of allcharacters become inactive after their first day. The probability that players play morethan 120 days is about 0 .
7. More than 31% of all deletions are self-induced.
Gender composition.
When signing up for the first time, players have to choosebetween a male and female character. The decision is irreversible. Depending on gender,a male or female avatar is displayed in various places and occasions in the game. About90% of all characters are male.
STEFAN THURNER
Fig. 2. – (left) Map of a universe. Squaresare sectors consisting of 15 ×
15 fields. Col-ors indicate clusters. Sectors are connectedby wormholes (streets, not shown). (right)Space chart. 7 × .1.2. The structure of the universe. Space in Pardus is two-dimensional. The universeis divided into 400 sectors , Fig. 2 (top), each sector consisting of about 15 × fields , thesmallest units of space. They form a square grid on which ship movement is possible byclicking on the desired destination field within the space chart . This chart is a 7 × wormholes , which play the role of roads. Acollection of about nearby 20 sectors is called a cluster . Typical spatial movements andthe activities of avatars are usually confined to one cluster for several weeks or longer. Action Points – the unit of time.
Many game actions carried out by a player (trade,travel etc.) cost a certain amount of so-called
Action Points (APs). These points cannever exceed a maximum of 6,100 APs per avatar. For avatars owning less APs thantheir maximum, every six minutes 24 APs are automatically regenerated, i.e. 5,760 APsper day. Once a player’s character is out of APs, she has to wait to be able to play on.As a result, the typical player logs in once per day to spend all APs on several activitieswithin a few minutes. This makes APs the game’s unit of time; it is the most valuablefactor. Players that use APs most efficiently can experience the fastest progress or earnthe highest profits. Social activities, such as communication, do not reduce APs. Highlyinvolved players spend a lot of real time on socializing and on planning their futuremoves. .1.3. Trade and economy. The Pardus currency unit is the so-called credit . It isnot convertible to real currencies. Every player starts life with 5,000 credits. Sincemost objects, such as ships, ship equipment, buildings, are traded in credits, it is offundamental interest to earn money. There exist a number of possibilities to do this,usually through participation in the economy. The richest players own hundreds ofmillions of credits. There exist over 30 commodity types. Some of these are renewableand exist in the environment. These “raw materials” can be mined, for example, gas fromnebulas or ore from asteroids. Most commodities, however, are processed from more basicones in player-owned firms. For example, a brewery manufactures expensive liquor outof cheap energy, water, food, and chemical supplies. Every player has the possibility tofound a small number of such firms. Production chains follow a fixed production tree,and coordination of several players is needed to establish a sucessful industry. Mostend-products at the top of the production tree are usable commodities. For example,manufactured drugs may be consumed to create APs, or droid modules can be installedfor powerful building defences against hostile attacks. Needs generated by the societyare the driving force behind the development of industries. Besides player-owned firmsthere are game-owned sites that trade and consume commodities. Prices are exclusively
IRTUAL SOCIAL SCIENCE (a) Pajek (b)
Pajek
Fig. 3. – (a) Accumulated communications over 445 days between 78 randomly selected individ-uals in the early universe, (cid:80) τ +445 t = τ M comm ij ( t ). Light gray, gray, and black correspond to 1–10,11–100, and 101–1000 PMs sent, respectively. (b) Friend (green, solid), M friend ij (445), and en-emy (red, dashed), M enemy ij (445), relations on day 445 between the same individuals. One prettyhated guy is visible. From [1]. determined by local supply and demand: when commodities are abundant, prices arelow, if they are rare, prices rise. Players face an economic life that is known from thereal world. It involves collaboration, competition, cartellization, fraud, and so on. .1.4. Communication. There are three communication channels in the game: The Chat: players can simultaneously communicate with many others. Chat entries scroll upand disappear; they are good for temporary talks. The
Forum: messages, called posts ,consist of several lines and stay for a long time. Posts are organized within threads ,which bundle into topics. The
Private message (PM): a system similar to email, wheremessages can be sent to any other player. The PM content is only seen by sender andreceiver. PMs always have exactly one recipient. These communication channels can beused independently from game-mechanic states, such as ship’s location, wealth, etc. Inthe following we focus on PMs, and call a PM between two players a “communicationevent”. .1.5. Friends and enemies. For a small amount of APs, players can mark others astheir friend or enemy . This can be done for any reason. Marked characters are addedto the markers personal friends or enemies lists . Every player also has a personal friendof and enemy of list, displaying all players who have marked them as friend or enemy,respectively. When marked or unmarked as friend or enemy the player is informed. Onecan mark others as either friend or enemy, not both. Lists are private, meaning that noone except the marking and marked players have information about their ties. In thisrespect the Pardus system does not introduce a bias toward accumulation of friends,and represents a more realistic social situation, where social ties are not immediatelypublicly accessible, but have to be found out by communication, or observation of others.The friends and enemies lists also serve game-mechanic purposes: friends/enemies areincluded/excluded for certain actions. For example, enemies of building owners are notable to use services in the buildings. Friend and enemy markings need not reflect affectiverelations, they rather indicate a degree of cooperativeness. Friend and enemy relationsas well as communication events (PM) are temporal networks, Fig. 3. We denote themby M friend ij ( t ), M enemy ij ( t ), and M comm ij ( t ), respectively. .1.6. Performance measures of players—“states”. Players i are characterized by anumber of time dependent states, σ i ( t ), that may change as a result of interactions with STEFAN THURNER N (a) PMsFriendsEnemies L (b) 051015 k (c)00.20.4 C (d) 050100150 C / C r (e) 00.20.4 E l o c (f)2468 g (g) 0.811.21.4 g / g r (h) 00.20.4 E g l ob (i)0 100 200 300 40000.51 Days ρ (j) 0 100 200 300 400 − r C ( k ) (k) 0 100 200 300 400 − − r und i r (l) Fig. 4. – Network properties over time: (a) number of nodes N , (b) number of (directed) links L ,(c) average degree ¯ k , (d) clustering coefficient C , (e) clustering coefficient C divided by clusteringcoefficient of corresponding random graph C r , (f) local efficiency E loc , (g) average geodesic ¯ g ,(h) average geodesic ¯ g divided by average geodesic of corresponding random graph ¯ g r , (i) globalefficiency E glob , (j) reciprocity ρ , (k) assortative mixing coefficient r C ( k ) , (l) assortative mixingcoefficient r undir . Arrows mark the beginning of a war at day 422. From [1]. others. These states can be achievement-factors that quantify various skills of players.The efficiency in harvesting natural resources is quantified by the farming skill , σ farm i ( t ).Other performance measures are combat skill , σ comb i ( t ), that quantifies fighting skills,and the experience points , σ XP i ( t ), that keep a record of fighting and other activities.Players may become members of political factions (parties), which sometimes engagein large-scale conflicts (wars). Faction rank , σ fr i ( t ), is a measure of influence within afaction: above a certain threshold, the faction rank grants the privilege to take part indecisions on war or peace. Finally, players are characterized by a certain wealth level, σ wealth i ( t ), that depends strongly on their economic activities. Some players regard highcombat skill, faction rank, wealth, or XP as their main goals in their Pardus life. .1.7. Alliances. For various purposes players organize in social groups called alliances .Often players share the same interests, or cooperate in pirate groups, exploration teams,self-defense units, etc. Usually groups do not get larger than about 140 members. Notethe proximity to the Dunbar number, 150 [25]. The game provides administration toolsfor officially declared alliances. Alliances have a common cash pool, which they use fortheir goals, like defence or production. Often alliances are used for economic purposes.There existed 161 alliances with an average size of 23 members at day 1200. Being amember of an alliance is a social commitment.
3. – How do people interact?
One fascinating aspect of this game is that at all times, all social networks are avail-able. Nodes i represent avatars. Links are individual social interactions of type α that IRTUAL SOCIAL SCIENCE Fig. 5. – (left) Illustration of what Granovet-ter means by a weak tie. In network lan-guage, weak ties have high link between-ness. (right) Definition of overlap betweentwo nodes i and j . happen from player j to player i at time t , M αij ( t ). This allows us to measure how peopleinteract and organize. We focus on mainly six interaction types: Communication networks.
We consider all PM communications, usually on an aggre-gated (e.g. weekly) timescale. A weighted link pointing from node i to node j exists ifavatar i has sent at least one PM to j within the aggregation period. The weight is thenumber of sent PMs. Figure 3 (a) illustrates a subgraph of the communication network,accumulated over 445 days between 78 randomly selected characters. Friends and enemies.
A link is defined from i to j if character i has marked character j as friend/enemy. Friend/enemy markings exist until they are actively removed bythe players. Friend and enemy networks are unweighted. Since links of friend- andenemy links never coincide, we can see the union of friend- and enemy networks as signed networks. Figure 3 (b) shows a signed friend–enemy network as observed on day 445.Note the cliquishness and reciprocity of friends, and a strong enemy in-hub. Commercial (trading) networks.
Trade networks are extracted by considering twokinds of trading between players: either players meet and exchange credits for com-modities, or they visit commercial outlets of other players and buy/sell commodities orequipment there.
Aggression networks.
Bounties and Attacks are two forms of how aggression can beexpressed in the game.
Attack links are defined as attacks carried out by one playeron another (or on her commercial outlets).
Bounty links represent (weighted) bounties,which are amounts ( credits ) placed on other players. Any player can collect a bounty byattacking the bountied player, or by harming his commercial outlet.Other networks in the game are for example production and mobility networks. Thesedo not directly constitute social interactions and will not be of interest here.Interaction networks change over time. In Fig. 4 we show a number of network mea-sures as they unfold for communication, friendship, and enmity during the first 422 days,clearly a period in transition. Networks are characterized by growing average degrees, andshrinking diameters; networks densify. After about 1,000 days, most measures becomeapproximately stationary (not shown). Real world communication [21] and friendshipnetworks networks are similar to those observed in
Pardus . Not many real world studiesexist on enmity networks—people seem to avoid to list their enemies. .1. Testing a classic sociological hypothesis of social interaction: weak ties . – Giventhese networks, we can immediately test a classic hypothesis in sociology stated by MarkGranovetter in the 1970s. The so-called the weak ties hypothesis makes a statement aboutthe importance of weak links, that connect communities, see Fig. 5. It states that “thedegree of overlap of two individual’s friendship networks varies directly with the strengthof their tie to one another”, [17]. Weak ties (for example casual acquaintances) areassumed to be important because they can link communities, which would otherwise beseparated. While weak ties are local bridges between communities, strong ties (e.g., goodfriendships) are easily replaceable intra-community connections. In network language, STEFAN THURNER −2 −1 γ = −0.54(a) b O ( b ) −2 −1 γ = 0.30(b) w O ( w ) Fig. 6. – Overlap versus (a) betweenness, and versus (b) weightin largest connected component of the communication networkat day 422. Grey markers show individual overlap values of thelinks. Black markers denote logarithmically binned averages,green lines are least squares fits. From [1]. weak ties are links with a high link-betweenness . Link-betweenness of link l ij is(2) b ij = (cid:88) m ∈N (cid:88) n ∈N \{ m } ρ mn ( l ij ) ρ mn , where ρ mn is the number of all paths between m and n , and ρ mn ( l ij ), is the number ofpaths that contain the link between i and j .For testing the hypothesis, we have to clarify how to measure “strength”. We de-fine interaction strength between two individuals as the number of exchanged messages w = M comm ij ( t ), over a given aggregation period. The hypothesis predicts an increasingfunction of overlap, O ( w ) , versus weight w . The overlap between two nodes i and j is(3) O ij = n ij ( k i −
1) + ( k j − − n ij , where n ij is the number of common neighbors of the nodes. The expected relationship isclearly realized for communication networks, see Fig. 6 (b), where we find an approximatecube root law(4) O ( w ) = w . ∼ √ w . By the weak ties hypothesis, the overlap O ( b ), as a function of betweenness, shoulddecrease. Figure 6 (a) confirms this prediction, and suggests an inverse square root law(5) O ( b ) = b − . ∼ √ b . These results are in agreement with real world communication networks as obtainedfrom mobile phone call data [21], and are robust across game universes and variousaccumulation times. The weak ties hypothesis was been tested with real small-scalesocial networks [26]. The weak ties hypothesis is fully confirmed in the
Pardus society.
IRTUAL SOCIAL SCIENCE Fig. 7. – Interactions mediated by exchange of particles. (a) Electromagnetic interaction betweena proton and an electron is mediated by the exchange of virtual photons [27]. (b) Two playersinteract by exchanging messages, M comm ij ( t ), and goods, M trade ij ( t ). (c) Map of the universe,where nodes are sectors (cities), lines are connections (wormholes or streets). Colors representdifferent regions (countries). From [10]. .1.1. How strong do people interact?—Kepler’s law. We can eliminate the overlapfrom the equations obtained from Fig. 6, b = O and w = O , and get(6) w = (1 /b ) , which immediately reminds us at Kepler’s third law of the motion of planets. Thisrelation is interesting in the sense that it relates interaction strength, which is a localquantity between individuals, with the betweenness of a link, which is a global, society-wide quantity: the strength of a (positively connoted) individual relation seems to berelated to the structure of the entire network. It remains to be verified if this relation isgenerally true for real societies. .2. Forces between avatars—Newton’s law for social interactions? . – In physics thereare four fundamental forces, the electromagnetic the weak force, the strong force, andgravitation. The origin of the forces has been clarified in the 20th century. The currentview is that they result from the exchange of virtual gauge bosons between interactingparticles, see e.g. [27]. Electromagnetism results from the exchange of photons, see Fig.7 (a), the weak and strong force comes from the exchange of W- and Z-bosons, andgluons, respectively. In classical physics a force can be expressed as a negative gradientof a potential V ( x )(7) m a = m d dt x = −∇ V ( x ) . If a central force is present, meaning that only the distance r between two bodies matters,the potential becomes a function of r , V ( x ) = V ( r ), and we get(8) ma = − ddr (cid:2) V ( r ) + V ( r ) (cid:3) , where V ( r ) is an effective-potential. Similar to physics, many human interactions arealso based on exchange. Exchanged objects can be messages, goods, money, presents,promises, aggression, bullets, and so on. In Fig. 7 (b) we schematically show the tra-jectories of two individuals, who exchange messages and a gift; as a result their relative STEFAN THURNER
Fig. 8. – Interaction-specific potentials for messages,trade, and attacks. Solid lines are least-squares fits to aharmonic potential in Eq. (9). V is a result of the back-ground motion of non-interacting pairs of players. Theinset is a blow-up for small distances. The potential forattack shows a minimum at r attack ∼
3. From [10]. distance reduces over time. Up to now it was not possible to determine if exchangeevents generate effective attractive or repulsive forces that influence relative motion.This is due to the lack of simultaneous information on exchange events and the trajec-tories of the involved individuals. This situation is about to change. Data from mobilephone networks, email networks, and online social networks show that the probabil-ity for interaction events decays with distance as an approximate power law, P ∼ r − γ [28, 29, 30, 31, 32, 33, 34, 35, 36], with exponents ranging from γ = 0 .
83 [35] to γ = 2 . r ij ( t ), between players i and j , their relative velocities v ij ( t ), and accelerations a ij ( t ). Atthe same time, we have the exchange densities given by the multilayer network, M αij ( t ).We can now address the question of how social interactions between humans influencetheir relative motion. We obtain the interaction potentials by integrating Eq. (8) forcases where particular interactions are predominant. We get(9) V α ( r ) = κ α r − b α r . For details, see [11]. The resulting potentials for the three interaction types, commu-nication, trade and attack are shown in Fig. 8. They follow a harmonic and a linearpotential, where κ β is the respective “force constant”. For communication this resultis consistent with real-world observations [35]. For trade and attacks, players need toreduce their distance to zero so that an interaction is possible. We see that attractiveforces are due to the exchange of messages and trade, whereas repulsive and attractiveforces arise from hostile actions (“hit and run” strategy). To confirm this finding in thereal world, mobile phone data could be used to perform a similar study.
4. – How do people organize?4 .1.
Dynamics of the “atoms of society”: triadic closure . – Some sociologists considertriangles (or triads), as the relation between three individuals, as the elementary buildingblock of societies. The fractions of different types of triads within a society provideinformation about its structure, stability, and efficiency. Granovetter stated in 1973:“The triad which is most unlikely to occur, [. . . ] is that in which A and B are strongly
IRTUAL SOCIAL SCIENCE Fig. 9. – The 16 types of triads and their ids. linked, A has a strong tie to some friend C, but the tie between C and B is absent.”Again he is talking about friendly interactions. This statement means that one shouldexpect over-representation of closed triads and a suppression of open ones. Does thismean that closed triangles should be under-represented in enmity networks? We will see.For directed networks, there exist 16 types of triads, see Fig. 9. In dynamical terms thismeans that, over time, open triangles should tend to close. For example, in Fig. 10 wewould expect to see triad 6 close to form triad 13. Since we know networks over time,we can count transitions from open to closed triads. The transition counts between alltriad types are collected in Fig. 10 (middle). .1.1. Testing triadic closure—the triad-significance profile. A way to visualize theover/under-representation of closed triads is to use the Z -score. It measures the over/under-representation of specific triads with respect to the number of triads that one wouldexpect in a random graph with the same number of nodes and links. The resulting num-bers are collected in the triad significance profile shown in Fig. 10 (right). For friendlyinteractions, i.e. friendship and communication networks, open triads have negative (nor-malized) Z scores (under-represented), closed ones are positive (over-represented). Wesee explicit evidence for triadic closure for friendship and communication networks inFig. 10. For negatively connoted ties, we find triad types 1–6 over-represented, and7–13 under-represented in enemy networks. Note the exceptions: triad id 4 is not clearlyoverrepresented, ids 9 and 11 are not clearly under-represented. If one wants to modelsocial dynamics, triadic closure must be taken into account. We will see in the next K D ( D = 50) a b c 1 2 3 4 5 6 7 8 910111213abc12345678910111213 0510152025 Fig. 10. – (left) Illustration of triadic closure as the tendency to close open triads. (middle)Transition counts from one type of triad to another. Red circles mark the rates between 6 → →
6. Clearly, there is a tendency to close open triads. (right) Triad significanceprofile for the three network types at day 445. From [1]. STEFAN THURNER triadic closure random link1 - rrtt+1 ii ij j jk kk
Fig. 11. – Simple triadic closure model. From time t to t + 1 a new link is created. With probability r it closes atriangle, with 1 − r it does not. From [7]. section that triadic closure alone is sufficient to explain a number of statistical facts innetworks associated with positive interactions. Triadic closure turns out to be a drivingforce for social network formation, well beyond Granovetter’s initial ideas. .2. Taking triadic closure seriously—understanding social multilayer network struc-ture . – We now show that triadic closure is able to explain three important statisticalfacts in three positive networks (communication, trade, and friendship): the degree dis-tribution is an approximate power law with exponent q , the node attachment kernel isan approximate power law with exponent γ , and the clustering coefficient as a functionof node degree is an approximate power law with exponent β .We use a simple model [7] that is shown in Fig. 11. The network is initialized with N nodes, each having one link to a randomly chosen node. The dynamics is completelyspecified by the iteration of the following steps,1. Pick a node i at random. If i has less than two links, create a link between i andany randomly chosen node, and continue with step (iii). If i has two or more links,choose one of its neighbors at random, say, node j , and continue with step (ii).2. With probability r (triadic closure parameter), create a link between j and anotherrandomly chosen neighbor of i , say k . With probability 1 − r , create a link between j and a node randomly chosen from the entire network, see Fig. 11.3. With probability p (node-turnover parameter) remove a randomly chosen nodefrom the network along with all its links, and introduce a new node linking to m randomly chosen nodes. Then continue with time-step t + 1.For p > L ( t ), and the network measures Π( k ), P ( k ), and c ( k ) fluctuate around stationary levels. The model is a variation of the modelproposed in [37], which appears as the special case for r = 1, in the above protocol.For similar models, see also [38, 39, 40, 41]. The model is completely specified by fourparameters, N , r , p , and m , all of which can be read off from the actual game data, i.e. M αij ( t ). In this sense, the model does not have a single free parameter. All parameters,including the node- and link-generation rates are obtainable from the game for the threeinteraction types. The model can now be simulated; the emerging networks are ana-lyzed with respect to the degree distribution, the attachment kernel, and the clusteringcoefficients. .2.1. Characteristic exponents. Simulation results for the values of the characteristicexponents γ, q , and β depend on the parameters p and r , see Fig 12. Given p and r (asmeasured in the data), we can read off the corresponding values of the scaling exponentsfrom Fig 12. These can be compared to the direct measurements from the data, i.e. the IRTUAL SOCIAL SCIENCE γ q β Fig. 12. – Dependence of scaling exponents γ , q , and β on model parameters p and r . (a) γ increases in p or r , and is confined to 0 < γ <
1. (b) q is large for small p and large r ; itapproaches 1 for large p . (c) β is close to zero for small r , and approaches β = −
1, for largevalues of p and r . N = 10 , m = 0, averages over 500 realizations for each ( p, r ). From [7]. model can be validated. Figure 13 shows the attachment kernel Π α ( k α ) (the probabilityfor a new node to attach to an existing link with degree k α ), the degree distribution P α ( k α ), and the clustering coefficients c α ( k α ), for the three sub-networks M α in theempirical multilevel network data. They are compared to the respective distributions ofthe calibrated model. Data and model agree nicely.These results suggest that triadic closure may play an even more fundamental role insocial multilayer network formation than previously anticipated [42, 17]. Given that all model parameters can be measured in the data, it is remarkable that the three important −2 −1 Π α ( k α ) friends ( α = 1) datamodel γ = 0.88(4) γ mod = 0.77(2)10 −5 −4 −3 −2 −1 k P α ( k α ) q = 1.16(1)q mod = 1.116(2)10 −2 −1 k α c α ( k α ) β = 0.66(3) β mod = 0.69(3) 1 2 345678910 2030405060708090100 messages ( α = 2) datamodel γ = 0.84(1) γ mod = 0.76(2) q = 1.24(1)q mod = 1.148(3)10 k α β = 0.59(3) β mod = 0.78(3) trade ( α = 3) datamodel γ = 0.83(1) γ mod = 0.80(1) q = 1.073(1)q mod = 1.102(1)10 k α β = 0.63(3) β mod = 0.60(3) Fig. 13. – Scaling exponents of the multilayer network as explained by the triadic closure model.Friendship ( α = 1, left column), communication ( α = 2, middle column), and trade ( α = 3,right column). (top row) Attachment kernels scale sub-linearly with the degree in each case.Data and model are barely distinguishable. (middle row) Degree distributions for α = 1 , , q -exponential. (bottom row) Clustering coefficients as a function of degree for dataand model. From [7]. STEFAN THURNER −4 −3 −2 −1 P ( ≥ k ) Positive ties
Friendship (a) −1.0Negative ties
Enmity (d)10 −4 −3 −2 −1 P ( ≥ k ) Communication (b) −1.7
Attack (e)10 −4 −3 −2 −1 P ( ≥ k ) k Trade (c) 10 −1.5k Bounty (f) k in k out Fig. 14. – (left) Degree distributions for positive inter-actions (communication, trade, friendship) follow an ap-proximate Poisson distribution, while negative interac-tions (enmity, attacks, bounties) show fat tailed distri-butions that could be power laws (right). From [2]. scaling laws are simultaneously explained by this radically simple model. The exponentsfound in the model compare well to those of real-world networks. Sub-linear preferentialattachment was reported in scientific collaboration networks and the actor co-starringnetwork, Π( k ) ∼ k . and ∼ k . , respectively [43]. Degree distributions of many socialnetworks often fall between exponential and power law distributions [44, 45, 21, 1, 46],and scaling of the average clustering coefficients as a function of degree, is observed inscientific collaboration and actor networks, c ( k ) ∼ k − . and ∼ k − . , respectively(when the same fitting as in Fig. 13 is applied). Mobile phone and communicationnetworks give ∼ k − [47]. .3. Degree distributions for negative ties are power laws—positive are not . – Forthe (cumulative) in- and out-degree distributions, we find approximate power laws foraggressive behavior: attacking (out-degree for attacks), being declared an enemy (in-degree for enmity), and punishing/being punished (out- and in- degree for bounty).Power laws are absent for positive (friendship, communication, trade) and passive links(being attacked), see Fig. 14. This suggests different linking/rewiring processes forpositive and negative ties. Moreover, we find that positive links are highly reciprocal(directed links go in both directions, M ij = M ji ), while negative links are not [2]. Lowreciprocation in enemy networks may partially be explained by deliberate refusal ofreciprocation to demonstrate aversion by total lack of response [1]. For attack networks,it may originate from the asymmetry in the strength of the players (a strong playeris more likely to attack someone weaker). We also find that positively connoted linksshow higher clustering coefficients than negatively connoted ones [2]. High values ofclustering are expected for positive interactions because they signal a cohesive structureand seems to benefit performance [48]. The significantly lower clustering for negativeinteraction types suggests that triadic closure [42] is irrelevant for negative interactions.Their formation might be the result of a “balance” of signed motifs—which is at the coreof social balance theory [49]. .4. Social Balance . – Social balance focuses on signed triads, where the sign is theproduct of the signs of its three links. In the following we assign +(-)1 to a positive(negative) link, e.g., friendship links have +1, enemy links -1. Social balance theory,
IRTUAL SOCIAL SCIENCE Strong formulation of balanceWeak formulation of balance B + + +- - -- --+ ++ B UU B UB B26,329 4,428 39,519 8,03210,608 30,145 28,545 9,00971 -112 47 -5
Fig. 15. – Signed triads, balanced (B) orunbalanced (U), according to the strong orweak formulation of structural balance. Wesee the number of each type of triad N ∆ inthe friendship-enmity bilayer, the expectednumber of triads in a null model with sign-randomization, N rand∆ , and the correspond-ing Z-score (standard deviation from thenull model). + + + and + − − are over-represented, + + − are under-representedwith extraordinary significance. From [2]. in its strong form [50], claims that positive triads are “balanced” and negative triadsare “unbalanced”, see Fig. 15. Unbalanced triads are sources of social stress and tendto be avoided. They are therefore under-represented. There is a “weak” formulationof structural balance [51] that assumes that triads with exactly two positive links areunder-represented in real networks, while the three other triads should be abundant. Inthe weak formulation only situations like, “the friend of my friend is my enemy” areunstable, whereas in the strong form of structural balance, “the enemy of my enemy ismy enemy” is also unstable, see Fig. 15.To test social balance, we focus on the bilayer network of friendship and enmityinteractions. The number of the different signed triads is N ∆ . They are compared tothe expected number of such triads in a null model, N rand∆ , where we re-shuffle the signsof links. In Fig. 15 the Z-score shows that + + + and + − − triads are heavily over-represented, while + + − triads are under-represented. Triads of type − − − are under-represented to a lesser degree than the three other types, favoring the weak formulationof structural balance. .4.1. Origin of social balance. A dynamical analysis reveals that changes in the net-work are driven by the creation of new positive and negative links, and not by switchingsigns of existing links. To illustrate this, we define a wedge as a signed, open triad withtwo links and one link missing (hole). There are three possible wedge types: ++, + − , −− . We measure day-to-day transitions from wedges to other triadic structures. Foralmost all cases ( > . − close preferentially (11 times more likely) with a negative link.We collect empirical transition rates in a transition matrix A STC , which we use in asimple dynamical model for
Signed Triadic Closure (STC). A STC is simply applied onthe state vector that contains all signed wedges at time t . At time t + 1, these wedgesare either closed or left unchanged. This model reproduces the empirical observationsreasonably well, see Fig. 16 (right). .5. Avatars organize in multiples of four . – Humans dominate their environment bythe way they organize in groups. Societies consist of hierarchically layered, nested groups STEFAN THURNER
Days R a t i o o f t r i ad s Data0 200 40000.250.50.751 DaysNull model0 200 400 DaysSTC model0 200 400 −−−++−+−−+++
Fig. 16. – Ratio of signed triads over timeas (left) seen in the data, (center) expectedfrom the random null model, and (right)simulation of signed triadic closure. Mea-sured ratios in the data deviate from thosein the null model, except for − − − tri-ads. The model explains the observed ra-tios much better. From [2]. of various quality, size, and structure, such as support cliques, sympathy groups, bands,cognitive groups, tribes, linguistic groups, and so on, [53, 54, 25]. Combining data onhuman group formation patterns, a discrete hierarchy of group sizes with a preferredscaling ratio close to 3 was identified [55]. It was later confirmed for hunter-gatherergroups [56] and mammal societies [57]. Do we see such a hierarchical organization in the
Pardus society? In particular, do we see the Dunbar numbers? .5.1. Dunbar numbers. In a nutshell, the concept of the Dunbar numbers is thatsocieties are approximately organized in multiples of three. In its simplest form, thismeans that groups of size 3, 3 ×
3, 3 × ×
3, ..., 3 n , and so on, should be over-represented.In Fig. 17 we see two indications for that hierarchical scaling. The so-called Horton plotshows the average size of groups per order. These orders have the following, somewhatsubjective, meaning. Horton order h = 1, is the trivial group consisting of one person,the “ego”. Layer 2 ( h = 2) contains closest friends of the ego, defined by both a friendshipmarking and at least one communication event within the last 30 days. Layer 3 ( h = 3)includes more casual relations, in particular all players that ego has marked as a friend,or by whom ego was marked as friend. Layer 4 ( h = 4) contains the alliance membersof the ego. Layer 5 ( h = 5) is obtained by applying a community detection algorithm(Louvain algorithm) [58, 59] to the communication network of the players. We tested thatlayer 5 is an organisational layer in its own right, whose communities are predominantlysubsets of the factions ( h = 6), and supersets of the alliances ( h = 4). Layer 6 ( h = 6)contains the three factions (political parties), and layer 7 ( h = 7) is the entire society.The size of the groups behind these orders scales like a power law with an exponentof roughly 4, indicating the hierarchical scaling, see Fig. 17 (a). The distribution ofgroup sizes after a smoothing procedure (with a Gaussian kernel) is shown in Fig. 17(b). It is immediately visible that groups of approximate sizes 1, 4, 16, 30, 250, 1500are over-represented. This is in line with an hierarchical organization of about four, 4 n ,1 , , , , , Pardus , where “life” is detached from most real-world
Fig. 17. – Group size scaling. (a) Hor-ton plot: average size of groups per order.(b) Estimated probability density of groupsizes s in Pardus, obtained with a Gaussiankernel estimation ( σ = 0 .
14 acting on thelogarithm of group sizes, ln( s )). From [11]. IRTUAL SOCIAL SCIENCE Fig. 18. – Segment of action sequencesof three players. Some actions of play-ers 146 and 701 are directed toward player199. This results in a sequence of received-actions for 199, R = {· · · ATTCT · · · } .(bottom line) Combined sequence of ac-tions (originated from–and directed to)player 199, C . Red letters mark actionsfrom others to player 199. constraints, suggests that this particular form of hierarchical organisation of societies isdeeply rooted in human psychology. .6. The Behavioral Code . – To describe a human, one way of doing it is to list thetemporal sequence of her actions. In real life there is a huge number of such actions, likecooking coffee, brushing teeth, washing cars, and so on. In the computer game there aremuch less actions that players can act out. But we have these action sequences ready foranalysis. We limit ourselves to eight different actions that every player can execute atany time, and which are observable as changes in M αij ( t ). These are communication (C),trade (T), setting a friendship link (F), removing an enemy link (forgiving) (X), attack(A), placing a bounty on another player (punishment) (B), removing a friendship link(D), and setting an enemy link (E). While C, T, F and X are positive (good) actions,A, B, D and E are hostile or negative (bad). We classify communication as positivebecause only a negligible fraction of communication takes place between enemies [1]. Weignore other possible actions like movement, production, working, sleeping, and so on.Segments of action sequences of three players are shown in Fig. 18.We consider three types of sequence. The first is the (time-ordered) stream of N consecutive actions A i = { a n | n = 1 , · · · , N } , which player i performs during his “life”in the game. The second is the stream of actions that player i receives from all theother players, i.e. all the actions which are directed towards player i . Received-actionsequences we denote by R i = { r n | n = 1 , · · · , L } . The third sequence is the time-orderedcombination of player i ’s actions and received-actions, which is a chronological sequenceof the elements of A i and R i in their order of occurrence. The combined sequence wedenote by C i ; its length is L + N , see Fig. 18. The n th element of one of these seriesis denoted by A i ( n ), R i ( n ), or C i ( n ). We do not consider the actual time between twoconsecutive actions, which can range from seconds to weeks. We work in “action-time”. .6.1. Two ways of seeing the same data. Since individual actions are directed, theinformation-content of both, the temporal multilayer data, M αij ( t ), and the behavioralcode ( A , R , C ) are identical, except that in the latter we use action time. The situationis similar to the Heisenberg- and Schr¨odinger picture in quantum mechanics. In the multilayer picture the focus is on the topological linking structure, temporal informationis hard to visualize (“Heisenberg picture”). In the Behavioral code picture the temporalinformation is clear, linking structure is harder to visualize (“Schr¨odinger picture”). .6.2. Behavioral code and predicting behavior. By p ( Y | Z ) we denote the probabilitythat an action of type Y follows action of type Z in the behavioral sequence of a player. Y and Z stand for any of the eight actions, executed or received (received is indicatedby a subscript r ). In Fig. 19 the transition probability matrix, p ( Y | Z ), is shown. The y STEFAN THURNER
Fig. 19. – Transition probabilities p ( Y | Z )for actions (or received actions) Y at time t +1, given that a specific action Z was exe-cuted (or received) in the previous timestep t . Received-actions are indicated by sub-script r . Normalization is such that rowsadd up to one. Large values in the di-agonal show that human actions are oftenrepetitive. Large values for C → C r and C r → C reveal that communication is ananti-persistent activity—it is more likely toreceive a message after sending one, andvice versa, than to send consecutive mes-sages. From [3]. axis indicates the action (or received-action) happening at a time t , the probabilities forthe actions (or received-actions) that immediately follow are given in the correspondinghorizontal place. We find that the probability to perform good actions is significantlyhigher if in the previous time-step a positive action has been received. Similarly, it ismore likely that a player is the target of a positive action, if at the previous timestephe executed a positive action. Conversely, it is highly unlikely that after a good action,executed or received, a player acts negatively, or is the target of a negative action.Instead, if a player acts negatively, it is very likely that he will perform another negativeaction in the following timestep. Finally, if a negative action is received, it is likely thatanother negative action will be received in the following timestep. The high statisticalsignificance of the cases P ( −|− ) and P ( − r |− r ) hints at a high persistence of negativeactions in the players’ behavior. For details see [3].An important finding is obtained by considering pairs of received actions, followedby performed actions. This approach allows us to quantify the influence of receivedactions on the performed actions. For these pairs we measure a conditional probabilityof 0 .
02 of performing a negative action right after receiving a positive action. This valueis significantly lower when compared to the probability of 0 .
10, which is obtained fromrandomly reshuffled sequences. Similarly, we measure a probability of 0 .
27 of performinga negative action, right after receiving a negative action. These results agree with arecent study, where the emotional content of posts in online fora was analyzed [60]. Thehomo sapiens seems to become drastically more aggressive, almost by a factor of ten,immediately after being treated badly. .6.3. Worldlines of players. We can interpret action sequences as random walks. Byassigning +1 to positive actions C, T, F or X, and − A i , into a binary sequence, A bin i . The cumulative sum of the binary C A C A A E C C E A C C C A A A+ − + − − − + + − − + + + − − − Fig. 20. – Illustration of a worldline W good − bad i as a binary random walk in“good-bad” action space. Positive actions(C, T, F or X) produce an upward move,negative ones (A, B, D and E) go down-ward. Good people have rising worldlines. IRTUAL SOCIAL SCIENCE Fig. 21. – Worldlines of good-bad actionrandom walks (top) of the 1,758 most activeplayers. (bottom) action-reaction world-lines of the same players. Red lines showfemale avatars. sequence is the “worldline” or a random walk for player i , W good − bad i ( t ) = (cid:80) tn =1 A bin i ( n ),see Fig. 20. Similarly, we define binary sequences from the combined sequence C i ,where we assign +1 to an executed action, and − C bin i ; its cumulative sum, W act − rec i ( t ) = (cid:80) tn =1 C bin i ( n ) is the “action-receive” worldline.Worldlines are shown in Fig. 21 for good-bad action sequences (top), and action-reaction(bottom). Figure 21 (a) also shows that the lifetime of players with many negative actionsis often short. The average lifetime for players with a slope A < , ± ,
856 actions,compared to players with a slope
A > , ± ,
559 actions. The average lifetimeof the whole sample of (very active) players is 3 , ± ,
484 actions.To characterize worldlines we define the slope A of the line connecting the origin ofthe worldline with its end point. It is an approximate measure for “altruism”. A = 1( − W good − bad indicates that the player performed only positive(negative) actions. The histogram of the slopes is shown in Fig. 22 (a) for good-badsequences. For the action–received-action worldline the slope is a measure of how wella person is integrated in her social environment. If A = 1, the person only acts andreceives no input, she is “isolated” but dominant. If A = −
1, the person is driven by theactions of others and never acts nor reacts. The histogram is shown in Fig. 22 (c).As a second measure we use the mean square displacement of worldlines to quantifythe persistence of action sequences(10) S ( τ ) = (cid:104) (∆ W ( τ ) − (cid:104) ∆ W ( τ ) (cid:105) ) (cid:105) t ∼ τ α , where ∆ W ( τ ) = W ( t + τ ) − W ( t ), and (cid:104) . (cid:105) t is the average over t . The asymptotic exponent α quantifies the “persistence” of a worldline. α = 1 / α > / α < /
2, anti-persistence. Persistence means that the probabilityof making an up (down) move at time t + 1 is larger (less) than p = 1 /
2, given the moveat time t was up. The histogram of exponents α for the good-bad random walk is shownin Fig. 22 (b), for the action–received-action world line in (d). In both cases persistentbehavior is obvious. STEFAN THURNER
Fig. 22. – Distribution of worldline slopes, A , for good-bad action (a), and action-received actionworldlines (c). Distribution of scaling exponents α for good-bad action (b), and action-receivedaction worldlines (d). .6.4. Zipf’s law in the human behavioral code. The ensemble of sequences of allactions A i of all players i allows us to analyse the frequencies of the occurring n -strings,see [61, 62]. An n -string is a subsequence of n adjacent actions in an action sequence.Given our 8-letter action alphabet, (A, B, C, D, E, F, T, X), there are 8 n different n -strings, or “words” i , that occur with probability P ( n ) i . We partition the action sequencesinto “words” of length n . Fig. 23 (b) shows the rank distribution of word occurrencesfor different lengths n . The distribution shows an approximate Zipf law [61] (slope of κ = − κ ∼ − .
5. TheShannon n -tuple redundancy, see e.g. [63, 64], for sequences composed of 8 letters is(11) R ( n ) = 1 + 13 n n (cid:88) i =1 P ( n ) i log P ( n ) i . For the equi-distribution, P i = 8 − n , we have R ( n ) = 0, and in the other extreme of onlyone single letter in the sequence, R ( n ) = 1. Figure 23 (a) shows R ( n ) as a function of n . R ( n ) increases with n , which indicates strong structure in the sequences, since Shannonentropy is not an extensive quantity for action sequences [65]. .7. Network–network interactions . – Social interaction networks of are not indepen-dent. How do they influence each other? An answer to this question would contributemuch to a deeper understanding of how societies work. We try to take simple first steps inthis direction by interpreting several measures that quantify inter-dependencies betweenpairs of networks. We follow two approaches.In the first, we focus on the link-overlap between networks and calculate the Jaccardcoefficient, J αβ , between two interaction layers α and β . It measures the tendency that −6 −4 −2 rank o cc u r an c e k=−1 R ( n ) n=2n=3n=4n=5n=6(a) (b) Fig. 23. – (b) Rank ordered probability dis-tribution of 1 to 6 letter “words” in the 8-letter action alphabet. The slope of κ = − n -tuple redundancy increases as a function ofword length n , a sign for structures in thesequences. From [3]. IRTUAL SOCIAL SCIENCE L i n k o v e r l ap D eg r ee / R an k c o rr e l a t i on C:F T:C E:A C:A E:B A:B C:E T:F F:A T:A C:B T:E F:B F:E T:B00.040.080.120.160.2 Link overlapDegree correlationDegree rank correlation 00.20.40.60.81
Fig. 24. – Measures to quantify network-network interactions. Node degree correla-tion (red), overlap (blue), and rank correla-tion (green). For an interpretation, see thetext. From [2]. links are simultaneously present in both layers, and is defined as the size of the intersec-tion of the link sets, α and β , divided by the size of their union, J αβ = | α ∩ β | / | α ∪ β | .The link overlap, O αβ = (cid:80) ij M αij M βij , measures the overlap between networks α and β . In the second approach, we compute Pearson correlation coefficients, ρ ( k α , k β ) = E (cid:2) ( k α − ¯ k α )( k β − ¯ k β ) (cid:3) / ( σ k α σ k β ), between node degrees in the different layers. Theymeasure to which extent degrees of avatars in one layer correlate with degrees of thesame avatar in another. If ρ ( k α , k β ) ∼
1, players who have many (few) links in layer α have many (few) links in layer β . Note that correlation coefficients might be influencedby different network sizes or different average degrees. To account for this possibility,we additionally compute correlations ρ (rk( k α ) , rk( k β )) between ranks of node degrees.Overlap and correlation provide complementary views on the organization of social struc-tures. All three measures are shown in Fig. 24 for all possible combinations of networklayers. It suggests the following set of conclusions: Communication–Friendship.
The pronounced overlap implies that friends tend to talk with each other , which is of course not unexpected. Strong correlation means that playerswho communicate with many (few) others tend to have many (few) friends, see also [1].
Trade–Communication.
The high overlap shows that trade partners have a tendencyto communicate with each other, while high correlations indicate a tendency of commu-nicators also being traders.
Enmity–Attack.
The high overlap shows that enemies tend to attack each other, orthat attacks are likely to lead to enemy markings. The high correlations imply thataggressors (or victims) tend to be involved in many enemy relations.
Communication–Attack.
The relatively high overlap shows that there is a tendency forcommunication taking place between those players that attack each other. A relativelyhigh correlation implies that players who communicate with many (few) others tend toattack or be attacked by many (few) players. Aggression is not anonymous, but is mostlyaccompanied by communication.
Enmity–Bounty and Attack–Bounty.
The situation is similar to the Enmity–Attackcase.
Communication–Enmity.
The situation is similar to the Communication–Attack case.
Trade–Friendship.
Similar to Trade–Communication, however, with a smaller overlap.It is more difficult for traders to become friends than to just communicate.
Friendship–Attack.
The low overlap shows that attacks tend to not take place betweenfriends, or that fighting players do not tend to become friends. The relatively highcorrelations mean that players with many (few) friends do attack or are attacked bymany (few) others.
Trade–Attack.
Is similar to the Friendship–Attack case.
Communication–Bounty.
Similar to Communication–Attack and Communication– STEFAN THURNER
Fig. 25. – Players choose a male or fe-male gender when joining the game. Someavatars that players can choose from. Allpossible avatars come in two genders.
Enmity, however, with much smaller overlap and degree correlations.
Trade–Enmity.
For this and all other interactions, overlap vanishes. Players whotrade with each other almost never become enemies and vice versa.
Friendship–Bounty.
Is similar to the Communication–Bounty case.
Friendship–Enmity.
The degree (rank) correlation is substantial, suggesting thatplayers who are socially active tend to establish both, positive and negative links. How-ever, vanishing overlap indicates the absence of ambivalent relations. Friends can not beenemies.
Trade–Bounty.
This interaction shows the smallest values for all measures. Therelatively small correlation suggests that players who are experienced in trade have atendency to not act out negative sentiments by spending money on bounties.The values of the two correlation measures must be interpreted with some care [2].Low values of ρ ( k α , k β ) indicate that hubs in one network are not necessarily hubs inanother (see e.g. the Trade–Enmity case). This suggests that avatars play very differentroles in different network layers. For example they can be central for flows of informationbut peripheral for flows of goods [66].
5. – Gender differences
When signing up for the first time, players chose to be a male or female avatar, Fig.25. We have no information about the biological sex of players. Selecting a genderdifferent from the biological is called gender swapping , which is common in online games[67]. A survey on 8,694 players in
Everquest found 15.5% gender-swappers, 17% of themales and 10% of the females [68]. Similar values are reported for
Second life (10%swapping of all, 16% of males, 2.7% of females) [69], or for
World of Warcraft (23% ofmales, 3% of females) [70].We observe that on average females are less risk-taking but wealthier [6]. Femalesaccumulate significantly more wealth (4 sigma level) than males. At the same time,male players experience significantly more deaths (2 sigma level), due to more risk-taking and/or aggressive behavior. This points to a much larger engagement of females
IRTUAL SOCIAL SCIENCE Fig. 26. – Network differences for male and female players on day 856 (male minus female).(a) average degree k , (b) clustering coefficient C , (c) average neighbour degree k nn . Femaleshave higher average degrees in communication and trade networks, as well as a higher clusteringcoefficient in trades, but considerably lower k nn for communication. (d) reciprocity betweenmale-male and female-female links. Females are much more reciprocal in friendships and trades.Errorbars are standard deviations of the network measures obtained from 8 male control groups,each of the same size as the female sub-group. in economic, rather than destructive activities. Concerning overall activity, experiencepoints, kills and collected bounties, female and male players perform comparably ( H can not be rejected).Females show homophily, males are heterophiles [6]. Homophily is the tendency of in-dividuals to associate and link with similar others [71]. A straightforward way to measurehomophily is to compare the numbers of directed links between all gender-combinations(MM, MF, FM, FF) in all network layers to the corresponding numbers from surro-gate data, where the gender of nodes is randomized (re-shuffled) but the topology ofthe network is left intact. To measure statistical significance of differences betweenthe various combinations of genders, we compare each real network to 1,000 reshuf-fled surrogate networks. Female-to-female trading and communication are the mostsignificantly over-represented link types, with a Z-score of approximately 4 sigmas [6].Male-to-female trades ( Z = 2 .
7) and communication ( Z = 2 .
7) are also strongly over-represented, whereas the opposite, female-to-male trades and communication is muchless substantial ( Z = 1 . .
6, respectively). Male-to-male trades and communicationare under-represented ( Z = − . .1. Gender differences in networking . – How do male and female players create andmanage their local networks? Are they different? .1.1. Gender differences in network topology. Figure 26 shows gender differencesin four network properties as observed on day 856, illustrating that males and femalesstructure their local network layers in very different ways. We compute the averagedegree k , the clustering coefficient C , and the average nearest neighbour degree k nn . Wethen subtract the value for the female group from the average of 8 male control groups ofequal size. This difference is shown in the panels. Errorbars are the standard deviationsof the means of the control groups. We find the following differences: STEFAN THURNER −4 −3 −2 −1 Positive ties
Friendship (g) Negative ties
Enmity (j)10 −4 −3 −2 −1 Communication (h)
Attack (k)10 −4 −3 −2 −1 k outTrade (i) 10 k outBounty (l) Fig. 27. – (left panel) degree distributions for six social interaction layers for males (red) andfemales (blue). (right panel) differences (male minus female) out-degree, plotted against thedegree. Gender differences become apparent. Female players appear to be super-likers, andsuper-haters. Females have more communication partners for almost all degrees. For aggression-management, males prefer attacks, females prefer bounties—at all degrees. From [6].
Females have more communication partners.
As seen in Fig. 26 (a), on averagefemales have about 5 more communication and 2-3 more trading partners than males.
Females organize in clusters.
Female trading networks show a clustering coefficientthat is much (about 25%) higher than the one of males, Fig. 26 (b). This means thatfemales tend to trade with people who trade among themselves. Also the clustering offemale friendship networks is significantly higher than those of males, showing a prefer-ence for stability in local networks [17]. Surprisingly, also for attacks females are morelikely than men to attack people who are already in conflict with each other.
Males prefer well-connected communication partners.
From Fig. 26 (c) we learnthat the communication partners of males have more communication partners than thecommunication partners of females. The same tendency is seen for male enmity networks,meaning that the typical enemy of a male player has more enemies than the typical enemyof a female. In relative terms, both effects are in the 10% range [6].
Females reciprocate friendships.
Females invest more effort in reciprocating positivelinks. Figure 26 (d) shows that females reciprocate more friendship and trading linksthan males. For most negative links, there are no substantial gender differences.On average, females have 5 more communication partners, females build more trian-gles, males link to well-connected communicators, and females reciprocate more. Thishas practical consequences for social life: while females focus on stable structures (highclustering), males seem to optimize communication speed by linking to well-connectedcommunication partners. Male networks are less stable, but information spreads faster.In the left panel of Fig. 27 we compare degree distributions of the different interactionlayers, for females (red) and male (blue). In the right panel we see the degree-differencesas a function of the degree. These plots illustrate that women are super-communicators(dominating, whenever many friends are involved), and females are the super-enemies(whenever the degree in enmity networks becomes very large). Males play out aggressionthough attacks, while females prefer bounties to manage their negative feelings towardsothers—at all degrees.
IRTUAL SOCIAL SCIENCE −2 −1 P ( tt r ≥ t ) (a) λ MF = −0.0060 λ FM = −0.0078 MFFM F r i end s (b) λ MM = −0.0075 MMFF −1 t [days](c) λ MF = −0.0055 λ FM = −0.00650 100 200 300 E ne m i e s t [days](d) λ MM = −0.0068 Fig. 28. – Time-to-respond distributions fordifferent gender combinations. (a) Timefor females to reciprocate a friendship linkfrom a male initiator (MF), and vice versa(FM). On long time scales ( >
30 days),males are much faster to reciprocate femalefriendship initiatives than the other wayround. (b) Equal sex reciprocation MM,and FF. Decay times are somewhere be-tween the MF and FM case. (c) Time-to-respond for enemy links. Males are consid-erably slower to reciprocate within the first180 days if the initiator was a female thanthe other way. (d) Equal sex reciprocationfor enemy links. .1.2. Gender differences in temporal behavior. Imagine one player marks anotheras a friend. How long does it take to respond to this action? We can measure thetime-to-respond to any action that is directed from one player to another. Males appearto respond fast (slow) to female friendship (enmity) initiatives. We measure the time-to-reciprocate (ttr) it takes for individuals to reciprocate actions of a given type. InFig. 28 (a) and (b) we show the cumulative distributions for the time-to-reciprocate forfriendship and enmity links, for the four possible gender permutations: MM, FF, MF,FM. The first letter denotes the gender of the initiator, the second of the reciprocator.The decay rate λ for MF friendship reciprocation is λ MF ∼ − . λ FM ∼ − . λ MM ∼ − . λ FM ∼ − . λ MF ∼ − . fast to female friendship initiatives—females slow to male ones. Females respond fast to female friendship initiatives—males slow tomales ones. And finally, males respond fast to enemy activities from males—but respond slow to female aggression
6. – Mobility—how avatars move in their universe
Pardus is a world with a well-defined space. Players can move on a 2-dimensionalsurface, see Fig. 7 (c). Since we know the location of all avatars at any point in time, wecan study mobility patterns, and compare them to those of humans on the 2-dimensionalsurface of Earth. We locate players in one of the N = 400 nodes, the so-called sectors(cities), that are linked by K = 1160 wormholes (roads). Sectors are arranged into20 different clusters , which are perceived by the players as different political or socio-economic regions, similar to countries. Each cluster is shown with a different backgroundcolor in Fig. 7 (c). Players usually have a “home cluster”, where they focus their socio-economic activities over long time periods. Occasionally, they move to sectors belongingto other clusters to explore the universe, to relocate their home (migrate), or in response STEFAN THURNER −6 −4 −2 d P ( d ) λ = 3(a) 10 ∆ t P ( ∆ t ) β = 2.2(b) Fig. 29. – Distribution of jump distances d (a), and waiting times ∆ t (b). A jump oc-curs whenever the sector position changesfrom one day to the following. The distri-bution of jump distances has a characteris-tic length of λ ∼
3. The waiting time ∆ t is the number of consecutive days a playerspends in the same sector. The distribu-tion is an approximate power law with anexponent of β ∼ .
2. From [4]. to extreme events, such as wars. .1. Jump- and waiting time distributions . – In Fig. 29 we show the distributions ofjump (travel) distance and the waiting times between movements (jumps) from players’trajectories over 1,000 days. In Fig. 29 we show the distributions of jump (travel)distance and the waiting times between movements (jumps) from players’ trajectoriesover 1,000 days. The length d (integer) of a jump is measured in terms of networkdistance, and ranges from 1 to d max = 27, the diameter of the network. The distributionof jump distances for all players over the observation period, is seen in Fig. 29 (a). For d ≤
15, the distribution is approximately exponential(12) P ( d ) ∼ e − dλ , with a characteristic jump length of λ ∼
3. The existence of a typical travel distancewas found in real-world mobility data [72, 73]. The distribution of waiting times, ∆ t (indays), between all consecutive jumps is seen in Fig. 29 (b). It follows an approximatepower law(13) P (∆ t ) ∼ ∆ t − β , with β ∼ .
2, in agreement with recent measurements on human mobility [74]. Wefind that mobility patterns are strongly influenced by the presence of clusters, the socio-economic regions [4]. .2. Long-term memory and mobility . – To understand the diffusion of avatars overthe transport network, we show the mean square displacement (MSD) of their positions inFig. 30 (a), σ ( t ) ∼ t ν , with ν ∼ .
26. This indicates anomalous, sub-diffusive behaviour.This is not an effect from the specific topology of the
Pardus universe. To see this, in Fig.30 (b), we show the situation for random walkers on the wormhole network (gray stars).It produces the expected MSD result that is expected from standard diffusion, i.e. withan exponent ν ∼
1, for up to t ∼
100 days, at which the fine size of the universe beginsto play, which causes the saturation. We tested several potential models to explain theanomalous diffusion behaviour, including a simple Markov model that is based on theobserved node–node transition probabilities, and a preferential-return model that takeshigher-order memory effects into account [75, 76]. Both are not able to capture thecorrect scaling pattern of the MSD, in particular ν ∼ .
26, as seen in Fig. 30 (b).The reason for failure is that the probability to move to a certain location does notdepend on the current location, nor on the order of previously visited locations. Instead,
IRTUAL SOCIAL SCIENCE (a) Players ν = 0.26 σ ( t ) −6 −4 −2 α = 1.3 τ P ← ( τ ) (b) Models t σ ( t ) Time Order MemoryRandomMarkovPreferential Return
Fig. 30. – (a) Mean square displacement(MSD) of players’ trajectories follows apower law, σ ( t ) ∼ t ν , with sub-diffusiveexponent ν ∼ .
26. The inset shows theprobability, P ← (cid:45) ( τ ), for a player to returnto a previously visited sector after τ jumps.(b) MSD for various models to explainthe observed scaling exponent ν . Randomwalkers, a Markov model, and a preferentialreturn model can not explain ν . A modelwith long-time memory (Time Order Mem-ory) reproduces the exponent almost per-fectly. Curves are shifted vertically for clar-ity. From [4]. we observe that individuals tend to return to sectors they have visited recently. Tomodel this mechanism we measure the return time distribution in the jump timeseriesfor returning (for the first time) to the currently occupied sector after τ jumps, andfind P ← (cid:45) ( τ ) ∼ τ − α , with α ∼ .
3, see Fig. 30 (a). We use this information to designa “Time Order Memory” (TOM) model that incorporates a power law distribution offirst return times, power law distributed waiting times, and exponentially distributedjump distances. These ingredients are sufficient to reproduce the sub-diffusive behaviourin Fig. 30 (a). The model works as follows: an individual rests in a given sector fora number of days drawn from the waiting time distribution. Then, she jumps. Thereare two possibilities: (i) with probability v she returns to an already visited sector, (ii)with probability 1 − v she jumps to a sector she never visited before. For (i), one of thepreviously visited sectors is chosen with P ← (cid:45) ( τ ). In case (ii), she draws a distance d fromthe distance distribution, and jumps to a randomly selected sector at that distance. Themodel parameters, λ , β , and α , are all fixed by the data. By averaging over all jumps andplayers, the probability of returning to an already visited location is found as v ∼ . ν TOM = 0 . ± .
02, in agreement with the observed exponent. Black squares inFig. 30 (b) indicate that the model works indeed.In summary, we find that mobility in the
Pardus world is not all that different frommobility on Earth. Locations are visited in specific temporal patterns, leading to strongmemory effects that are essential to understand the statistics of observed mobility tra-jectories. Neglecting either spatial or temporal factors make it hard to understand thestatistics of human mobility. This might be true in the real world too. Interestingly, athorough understanding of human mobility is still outstanding, since some results ap-pear to be contradictory [77]. Some report fat-tailed distributions of trip lengths [78, 76],others exponential or binomial distributions [73, 72, 77].
7. – The wealth of virtual nations
Almost universally, wealth is not distributed uniformly within societies. Even thoughwealth data have been collected in various forms for centuries, the origins behind wealth-,and hence, social inequality are not yet fully understood. This is not different in
Pardus .However, there we can figure out what it needs to be wealthy in terms of your position in STEFAN THURNER −3 −2 −1 A w P ( W > w ) α =2.46UK 2005Sweden 2007Pardus 2010exponentialpower law10 −1 T w =3.6e+070 5 × B Cumulative share of people C u m u l a t i v e s ha r e o f w ea l t h mean alliancenon−alliance playersall playersequality Fig. 31. – (a) Cumulative wealth distributions for Sweden, the UK, and for the Pardus societyon day 1200. Note the similarity. (b) Corresponding Lorenz curve of wealth in Pardus. Forevery alliance, a separate Lorenz curve is calculated; the dashed blue line is their average. (c)Gini index over time, g ( t ). A Christmas charity event on day 562 leds to a re-distribution fromthe wealthy to the poor, resulting in a downward jump of the Gini index. The inset shows theexponential recovery to previous levels. Gray areas indicate periods of war. From [12]. the social multilayer network. Can we finally understand the origin of wealth inequality? .1. More on the Pardus economy . – From interaction data, M αij ( t ), and player states, σ X i ( t ), we can reconstruct all economic activities in the Pardus society. In particular theinput-output production matrix of the economy and the variety of goods are pre-defined.Goods are of uniform quality (homogeneous); consumables and equipment can be par-tially substituted by other types of consumable and equipment. Intermediate goods areneeded for production in exact proportions. There are 5 commodities (natural resources),19 intermediate goods, and 5 end-products, i.e. consumables. Although capital require-ments to establish production facilities are low, there are barriers to entering production.Incumbents may threaten or harm potential new entrepreneurs. Game rules set a max-imum number of production facilities per player. Production facilities can not be sold.Investments in production facilities therefore motivate players to stay in the sector. Nolabor is needed for production itself, but transport of raw materials and intermediategoods requires effort and resources. Because of transport costs, facilities effectively onlycompete with similar facilities nearby. This leads to local oligopolies. Owners of produc-tion facilities are completely free to set the price at which they sell their products. Thereexist non-player facilities (belonging to the game) whose prices depend on local supplyand demand. The monetary currency is called credits . There is no credit or bankingsystem, all transactions are payed and cleared immediately. There is no inflation. .2. Wealth . – There are various ways to accumulate wealth: trading, collecting andselling natural resources, producing goods, working for hire (common jobs are courier,hunter, or bounty hunter), receiving donations or other payments, increase of the alliancefunds, robbing, and stealing. The wealth of player i is the sum of the value of his assets,cash v l,i , equipment v e,i , share of alliance funds v af,i , and inventory v inv,i ; details in [12],(14) σ wealth i ( t ) = v l,i ( t ) + v e,i ( t ) + v af,i + v inv,i . Ways to reduce wealth include consumption, paying for maintenance, investing in pro-duction facilities or equipment, discarding goods, becoming victim of theft or robbery,
IRTUAL SOCIAL SCIENCE giving to fellow players, or paying into alliance funds, a decrease of the alliance funds, ormaking adverse trades.The cumulative wealth distribution of Pardus players, in comparison to the UK andSweden, is shown in Fig. 31 (a) New and inactive players were excluded. The bulk ofthe distribution is compatible with an exponential with decay T w = 3 . × credits.The tail can be seen as an approximate power law with exponent α ∼ .
5. The situationis compatible with real world data [79]. .3. Inequality . – Figure 31 (b) shows the Lorenz curve for the Pardus society (blackline). It is the share of the total wealth as a function of the fraction of the people holdingthat share. The closer the Lorenz curve is to the diagonal (black dotted line) the moreegalitarian is the wealth distribution. Uniform wealth distribution corresponds to thediagonal. Associated to the Lorenz curve is the Gini index, g = 1 − A , with A beingthe area (integral) under the curve [80]. We find g = 0 .
65. We show the Lorenz curvesfor all players and for those that are not organized in any alliance (red dotted line).These players generally operate individually, and show a much more pronounced wealthinequality than the entire society, the respective Gini index being 0 .
70. In contrast, theLorenz curve for the various alliances (dashed blue line) indicates that people within thealliances tend to be much more equal in wealth, when compared to the entire society.The Gini index for the alliances is 0 .
50. The main reason for this higher equality is thesmaller fraction of poor players in alliances: while 79% of the total population, and 92%of the richest 10%, are alliance members, only 28% of the poorest 10% are.Figure 31 (c) shows the time evolution of the Gini index. After an initial steep risein the first 150 days, the Gini index g fluctuates between 0 .
68 and 0 .
63, similar to manyWestern countries. A sharp drop of g from 0 .
67 to 0 .
65 occurred on Christmas day 2008.On this day, a charity event took place, where thousands of players donated cash for theless wealthy. The gained level of equality is lost exponentially fast, within a few daysprevious Gini index levels are reached, see inset. This indicates a remarkable stability ofthe shape of the wealth distribution. It is hard to decrease inequality by re-distributingwealth. Is the origin of wealth inequality based in social behavior? .4. Behavioral factors for wealth . – .4.1. Influence of activity on wealth. We see a trivial strong linear relation between theaverage wealth of a player and her total activity. The corresponding Pearson correlationcoefficient is ρ = 0 .
535 ( p -value < − ). We conduct a partial correlation analysis andfind several significant behavioral factors that explain wealth [12]. We find that the morea player trades compared to his other actions, the higher is his wealth-gain. This isnot surprising since trade is the main source of income in the game. We also find thatthe more of a player’s actions are attacks, the lower is his wealth-gain. This suggeststhat revenue from attacks through robbery and bounty hunting does hardly bear theassociated costs, e.g. for repairing damage caused by fights. There might be secondarydamaging effects of aggressive behavior, such as reduced willingness of others to socialize(and trade) in the future. Another explanation might be that attacks are sometimescarried out without economic interests, but just for creating terror. .4.2. Influence of achievement-factors on wealth. Wealth and other achievement fac-tors, such as skills, XPs, and faction rank, are strongly correlated with total activity. Forthis reason, we define the wealth-gain of player i as η i ( t ) = w i ( t ) /a i ( t ), where a i ( t ) is thecumulative activity of a player approximated by the total amount of APs he has “spent”. STEFAN THURNER combat skill f a r m i ng sk ill a) 20 40 60 80101520253035 experience c o m ba t sk ill b) 10 l og ( η ) f a c t i on r an k c) 10 Fig. 32. – Two-dimensionalbinned averages of the wealth-gain as a function of achievement-factors. Colors represent thelogarithm of the average wealth-gain, log ( η ), over all playersthat fall into that bin. Bluecorresponds to low, red to highvalues, empty bins are white. AXP and faction rank, B XP andcombat skill, C combat skill andfarming skill. From [12]. η i ( t ) can be seen as the efficiency of gaining wealth. To further exclude these spuriouscorrelations, partial correlation coefficients are calculated. Age and faction rank are asignificant factors ( p -value below 1%). Players that are not in any faction, i.e. on aver-age less social, have the smallest possible value of faction rank, and are generally poorer.There is a significant fraction of rich people with low combat skill. Otherwise, we findno correlation between combat skill and wealth-gain. Farming skill has a consistentlypositive and mostly significant correlation with wealth.In Fig. 32 we show the wealth-gain as a function of a combination of different per-formance factors. High farming skills and intermediary combat skills correlate with highwealth-gains that are shown as by colors (high wealth-gain is red). Below a certain levelof experience points, no high wealth-gain seems possible, and a high faction rank togetherwith high experience point scores are a good predictor for wealth. For details, see [12]. .4.3. Wealth depends on how social you are. Alliance members on average are some-what richer than non-alliance members, both in absolute terms and in wealth-gain. Mem-bers also have better skills and a higher faction rank. As seen in Fig. 33 the size of analliance has little influence on wealth and other factors, except for players that are in noalliances or in alliances with only two members. These players are consistently poorerthan players in groups with three and more members. Members of the largest alliancesshow performance indicators below average (dashed line). It does not matter if you aremember of big or small groups; it matters if you are a member of at least one group. .5. Wealth and position in the multilayer network . – We use the trade, commu-nication, friendship, and enemy interactions layers, M αij ( t ), to determine the in- andout-degree, k in i and k out i , the nearest-neighbor degree, k nn i , and the clustering coefficient, c i , for every player i . To show the dependence of wealth-gain on various combinationsof these network parameters, we plot two-dimensional binned averages of wealth-gainversus pairs of network measures in Fig. 34. The color is the logarithm of wealth-gain,log ( η ), from blue (lowest) to red (highest). We find the following results: Trade . The trade layer has the strongest impact on wealth. Trade in-degree hasa significant, positive partial correlation with wealth. The in-degree is defined as tradewith a player’s production facilities and is therefore a proxy for his production. Figure 34(a) confirms the positive connection between trade in-degree and wealth, while showingmuch less influence from trade out-degree. Concerning the undirected degree of the tradenetwork versus the nearest-neighbor degree (b), we see that the richest are found to havean intermediate nearest-neighbor degree of about k tradenn ∼ −
70, well below their
IRTUAL SOCIAL SCIENCE undirected degree. This means that they are selling to people that are less connectedin the trade network than they are themselves. From Fig. 34 (c) we gather that highwealth-gain is made with a combination of high degree and a relatively low clusteringcoefficient, C trade ∼ .
1. This means that rich players avoid cyclical structures in theirtrading networks, which allows them to act as “brokers” between players that do notdirectly trade with each other. The partial correlation between wealth and the tradeclustering coefficient is negative.
Communication . Communication in-degree has a significantly positive partial correla-tion. High communication in-degree means good access to information, which is expectedto be profitable. Since communication links are reciprocal, and in- and out-degree arehighly correlated, there might be a spurious effect of the communication in-degree. Thecommunication nearest-neighbor degree has a negative and mostly significant partial cor-relation. This might indicate that it is advantageous to converse with players who areless informed than oneself.
Friendship . In Fig. 34 (d) the situation for the in- and out-degrees for the friendshiplayer is shown. Players with high wealth-gain are those that are liked by more playersthan they like themselves, k friendin > k friendout . Poor players have marked others as friendsmore often on average than they are marked. The role of these asymmetries might be afruitful direction for future study. Enmity . We see that people with above-average wealth-gain are rarely marked as anenemy by others, but do mark others as enemies, (e). Players who have been marked asenemy by many others are generally poor. In agreement with this finding, the enmityin-degree has a significant negative partial correlation with wealth, while the enmity out-degree has a weak significant positive correlation with wealth [12]. This suggests thatplayers with high wealth-gain actively invest in a good reputation. Finally, players withabove average wealth-gain have a high nearest-neighbor degree (f). Players with highenmity (in)degree are “public enemies” [1]. A high k enemynn indicates that one is the enemyof public enemies and that one has few private enemies.We learn that being wealthy in Pardus is not mere luck—it is highly structural. Itdepends on the actions that you do and toward whom you direct them, or in differentwords, in which sectors of the social multilayer network you are located. a v e r age w ea l t h < w > A a v e r age age B a v . w ea l t h − ga i n < η > C a v . c o m ba t sk ill D a v . f a r m i ng sk ill E a v . f a c t i on r an k F Fig. 33. – (a) Wealth, (b) age, (c)wealth-gain, (d) combat skill, (e)farming skill, (f) faction rank asa function of alliance size. Thefirst bin contains players that arein no alliance, the second bin hasplayers in alliances of size two.Members of the smallest alliancesshow low wealth and achievementscores. Also the largest groupsshow lower levels. The line isthe average over all players in al-liances with at least three mem-bers. From [12]. STEFAN THURNER
Fig. 34. – Wealth-gain (log ( η ))as a function of network proper-ties, blue (lowest) to red (high-est), empty bins are white.(a) trade in- and out-degree,(b) trade undirected degree andnearest-neighbor degree, (c) tradeundirected degree and clusteringcoefficient, (d) friend in- and out-degree, (e) enmity in- and out-degree, (f) enmity undirected de-gree, and nearest-neighbor de-gree. From [12].
8. – Towards a new social science?
What have we learned about the homo sapiens? Methodically we created a unique labsituation. For the first time, we are confronted with almost complete information aboutan entire human society with little-to-no interference with their observers. Howevervirtual
Pardus may be, it is a society where decisions are made by humans, not bybacteria, rats, or algorithms. Sociological predictions and hypotheses, sometimes centuryold, and can now be tested on big data, without the danger of privacy violations, andsometimes, questions can be decided: in summary, we find • The strength of human interactions is measurable. Local tie strength is related(maybe not causally) to the betweenness of links, which is a global property. Weconfirm the weak ties hypothesis [1]; but we can go much further by measuring thestrength of ties as a function of the interaction density between humans [10]. • Humans are triangle-closers. We confirm triadic closure [1]. The new discovery isthat the mechanism of triadic closure is so dominant, that it might be sufficient todescribe the basic statistics of human interactions [7] and group formation [14]. • Humans organize in stable signed triangles. We confirm previously conjecturedsocial balance to unprecedented precision levels. We find evidence for the weakform of social balance [2, 15] . • We find that humans tend to organize in group sizes that are (roughly) multiplesof four [11], a recently conjectured social organizational principle that might bearsignificance for human group formation [55]. • Males and females organize their local social networks in very different ways: fe-males tend to focus on higher clustering and thus stable networks, males focus onmore “fuzzy” networks that allow for fast communication across large parts of thesociety, but which are less stable. • Good and bad actions lead to different organization of interaction networks. Pos-itively connoted interactions show Poissonian degree distributions and high reci-procity, negative ones show fat tails, and are hardly reciprocal [2].
IRTUAL SOCIAL SCIENCE • Females and males handle aggression differently. Males tend to act out aggressiondirectly through attacks, females tend to delegate aggressive acts to others byplacing bounties on others’ heads [6]. • Humans become vastly more aggressive when confronted with hostile behavior to-wards them. The likelihood of negative responses after receiving unfriendly treat-ment increases about 10-fold [3]. • Short-term efforts for wealth redistribution are short-lived. Wealth is found to be afunction of your position in the social multilayer network. Wealth is a consequenceof your behavior that determines that position. In that sense wealth seems to be astructural phenomenon [12]. • We discover new social laws in temporal behavior, such as the characteristic expo-nential decay in response times for reciprocating actions [13].To what extent can we trust these findings? Is
Pardus a good “model” for real soci-eties? We don’t know yet, but we provided evidence that there are striking similarities inthe structures of communication and friendship networks [1], mobility patterns [4], socialbalance [2], gender specific patterns [6], and wealth distributions [12], when compared totheir real-world analogs.Would we finally agree that social dynamics will become a quantitative-predictive ex-perimental science in the near future—that sociology eventually becomes a sub-disciplineof physics, as Comte envisioned? I think we demonstrated in this lectures, that this isa possibility. The limit to understanding societies will neither be data size nor com-putational power—we are almost there. It is not hard to imagine that the data, as weanalyzed it here, can be replaced by real data, including mobility, friendship interactions,financial transactions, medical data, trading, shopping, surfing, and so on. I believe thelimits to understanding societies, as soon as we aim for an understanding at a deeperlevel, is the co-evolutionary complexity of social systems that might impose bounds topredictions; and of course, as always in science, the ultimate limit is the quality of ourquestions.Works summarized here were supported by EC FP7 INSITE, COST MP0801, andthe Austrian Science Fund FWF under P23378-G16.
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