Understanding mobility in a social petri dish

Abstract

Despite the recent availability of large data sets on human movements, a full understanding of the rules governing motion within social systems is still missing, due to incomplete information on the socio-economic factors and to often limited spatio-temporal resolutions. Here we study an entire society of individuals, the players of an online-game, with complete information on their movements in a network-shaped universe and on their social and economic interactions. Such a “socio-economic laboratory” allows to unveil the intricate interplay of spatial constraints, social and economic factors, and patterns of mobility. We find that the motion of individuals is not only constrained by physical distances, but also strongly shaped by the presence of socio-economic areas. These regions can be recovered perfectly by community detection methods solely based on the measured human dynamics. Moreover, we uncover that long-term memory in the time-order of visited locations is the essential ingredient for modeling the trajectories.

Understanding the statistical patterns of human mobility, predicting trajectories and uncovering the mechanisms behind human movements¹ is a considerable challenge with important practical applications to traffic management²,³, planning of urban spaces⁴,⁵, epidemics⁶,⁷,⁸,⁹, information spreading¹⁰,¹¹, and geo-marketing¹²,¹³. In the last years, advanced digital technologies have provided huge amounts of data on human activities, allowing to extract information on human movements. For instance, observations of banknote circulation¹⁴,¹⁵, mobile phone records¹⁶, online location-based social networks¹⁷,¹⁸, GPS location data of vehicles¹⁹, or radio frequency identification traces¹,⁵,²⁰, have all been used as proxies for human movements. These studies have provided valuable insights into several aspects of human mobility, uncovering distinct features of human travel behaviour such as scaling laws¹⁴,²¹, predictability of trajectories²², and impact of motion on disease spreading⁷,⁸,⁹,²³. However, from a comparative analysis of the different works it emerges clearly that a “unified theory” of human mobility is still outstanding, since results, even on some very basic features of the motion, often appear to be contrasting¹. One example is the measured distribution of human trip lengths in various types of transportation: some studies agree that mobility is generally characterized by fat-tailed distributions of trip lengths¹⁴,²¹, while others report exponential or binomial forms¹,⁵,¹⁹. The discrepancies arise due to the different mobility data sets used, where mobility is indirectly inferred from some specific human activity in a particular context. For instance, mobile phone records typically provide location information only when a person uses the phone²¹, while radio frequency identification traces like the ones of Oyster cards in the London subway⁵ only log movements based on public transportation systems. Analyses of these data sets can then result in a possibly biased view of the underlying mobility processes. Furthermore, most of the analyzed data sets have poor information on how socio-economic factors influence human mobility patterns. More generally, the lack of an all-encompassing record set with positional raw data, including complete information on the socio-economic context and on the behaviour of all members of a human society, has so far limited the possibilities for a comprehensive exploration of human mobility.

Here, we address the issue of mobility from a novel point of view by analyzing, with unprecedented precision, the movements of a large number of individuals, the players of a self-developed massive multiplayer online game (MMOG). Such online platforms provide a fascinating new way of observing hundreds of thousands of interacting individuals who are simultaneously engaged in social and economic activities. The potential of online worlds as largescale “socio-economic laboratories” has been demonstrated in a number of previous studies²⁵,²⁶,²⁷,²⁸. For the MMOG at hand²⁹, we have access to practically all actions³⁰, including movements, accumulated over several years. This MMOG can therefore be considered as a “socio-economic petri dish” to study mobility in a completely controlled way. We can in fact observe the long-time evolution of a social system at the scale of an entire human society, having a perfect knowledge of all the spatio-temporal and socio-economic details. In contrast to traditional studies in social science which are typically biased by well-known “interviewer-effects”, in MMOGs the socio-economic measurements are objective and unobtrusive, since subjects are not consciously aware of being observed.

Using positional data of the players in the game universe, in combination with other socio-economic information from the game, we uncover various fundamental features of mobility, and we provide a complete description of the mechanisms causing the observed anomalous diffusion. Two are the main results of our work. First, we find the emergence of different spatial scales, due to the strong tendency of the players to limit their economic activities to some specific areas over long time periods and to avoid crossing the borders between different areas. Making use of this observation, we propose an efficient method to identify socio-economic regions by means of community detection algorithms based solely on the measured movement dynamics. Our second result unveils the driving mechanism behind the movement patterns of players: Locations are visited in a specific order, leading to strong long-term memory effects which are essential to understand and reproduce the observed trajectories. Finally, we provide large-scale evidence that neglecting either of these spatial or temporal constraints may obstruct the possibility of understanding the processes behind human mobility.

Pardus is a massive multiplayer online game running since 2004, with a worldwide player base of more than 350,000 individuals. It is an open-ended game whose players live in a virtual, futuristic universe and interact with each other in a multitude of ways. The topology of the universe can be represented as a network with 400 nodes, called sectors, embedded in a two-dimensional space, the so-called universe map shown in Fig. 1. Each sector is like a city where players can have social relations (establish new friendships, make enemies and wage wars), and entertain economic activities (trade and production of commodities). Typically, sectors adjacent on the universe map, as well as a few far-apart sectors, are interconnected by links which allow players to move from sector to sector. At any point in time, each sector is usually attended by a large number of players. The network is sparse and, similarly to other spatial networks, is not a small world. It has a characteristic path length L = 11.89 and a diameter d_max = 27, which means that, on average, players have to move through a non-negligible number of sectors to traverse the universe. See Supplementary Section S3 and Supplementary Table I for a detailed characterization of the universe network structure.

Figure 1. The universe map of the massive multiplayer online game Pardus.

The universe of Pardus can be represented as a network²⁴ with N = 400 nodes, called sectors (playing the role of cities), and K = 1160 links. Sectors are organized into 20 different regions, called clusters, shown in the figure as different colour-shaded areas. There is no explicit set of goals in the game. Players are free to interact in a number of ways to e.g. increase their virtual wealth or status. Players move between sectors to interact with other players, e.g. to trade, attack, wage war, or to explore the virtual world.

The sectors have been originally organized by the developers of the game into 20 different clusters, which are perceived by the players as different political or socio-economic regions such as countries. For example, a player who is member of a political faction in the game is provided some game-relevant protection in all clusters which are controlled by the faction, and has the opportunity of social promotion when accomplishing certain tasks within these clusters. Each cluster is shown in Fig. 1 with a different background colour. All clusters contain about 20 sectors each, with the exception of the central cluster, consisting of just one sector, and its surrounding three clusters having only 6–7 sectors. Sectors belonging to the same cluster are geographically close on the map, meaning that the distance between any two sectors in the same cluster is small, with an average distance around 3. Players typically have a “home cluster” where they focus their socio-economic activities over long time periods. Occasionally, they also move to sectors belonging to other clusters in order to explore the universe, to relocate their home (migrate), or during extreme game events such as wars.

In Pardus, players are free to pursue whichever role they like to take. Many of them focus on expanding their social relations or political influence, some play the role of “scientists” exploring the universe, while others choose their main goal in trade and optimizing the amount of virtual money earned²⁵. The large variety of complex socio-economic behaviours emerging in this online society, results in high heterogeneity in the mobility patterns, such as observed in real human motion. However, differently from other empirical studies on human movements, mobility in Pardus can be investigated in a controlled way, since complete information on actions of players is available²⁵,²⁶. In this article we consider a data set consisting of movements in the network universe of all players who were active over a period of 1,000 days, as well as of socio-economic information about their environment. This opens the possibility of investigating motion in relation to other social and economic factors. Note that we do not have to address the common issues of relying on incomplete data, on data that are only a proxy of mobility, or on data that are aggregates of different types of transportation⁹. See Supplementary Section S2 for more details on the data set.

Results

Basic features of the motion

The position of each player in the universe, namely the ID number of the sector where the player is currently situated, is logged once a day. In this way the motion of each player becomes a time series of 1,000 sector positions. A jump occurs when a player's sector position changes from one day to the following. The associated length d of a jump is measured in terms of graph distance, an integer value between 1 and d_max = 27. The probability distribution of jump distances, computed for all players over the whole observation period, is reported in Figure 2 (a). For d ≤ 15, the distribution is well-fitted by an exponential:

with a characteristic jump length λ ≈ 3. The existence of a typical travel distance, as also recently found in other mobility data⁵,¹⁹, is related to the use of a single transportation mode in Pardus³¹. This allows to disentangle the intrinsic heterogeneity of the players from the effects due to the presence of different means of transportation⁹, which might be the cause of the scale-free distributions found in mobile phone or other mobility data sets¹⁴,¹⁶. It has in fact been suggested that power laws in distance distributions of movement data may emerge from the coexistence of different scales¹,³².

Figure 2. Distribution of jump distances and of waiting times.

To each player a time series consisting of the sector positions over 1000 days is associated. A jump is said to occur when the sector position in the time series changes from one day to the following. The length d of a jump is measured in terms of graph distance and can take an integer value between 1 and d_max = 27, the diameter of the network. (a) The probability distribution of jump distances is reported in a semi-log plot. For d ≤ 15, the distribution follows an exponential with a characteristic length λ ≈ 3. Players can also remain in the same sector for more days, without moving to other sectors. We define as waiting time Δt the number of consecutive days a player spends in only one sector. (b) We show the probability distribution of waiting times Δt in a log-log plot, which is well fitted by a power-law P(Δt) ∼ Δt^−β, with β ≈ 2.2.

In some cases, players stay in the same sector for a number of consecutive days. For instance, 11 of the 1458 considered players, although being active in the game, never jump within the entire observation period. On average, a player does not change sector in approximately 75% of the days. To better characterize the motion, we computed the waiting times Δt (measured in terms of number of days) between all pairs of consecutive jumps, over all players. The distribution of these waiting times, shown in Fig. 2 (b) follows a power-law distribution:

with an exponent β ≈ 2.2, in agreement with other recent measurements on human dynamics³³. In addition, we found that the average waiting times of individual players are distributed as a power-law (see Supplementary Fig. 2). This implies a strong heterogeneity in the motion of different players, which is related to the heterogeneity in their general activity (see Supplementary Section S1 and Supplementary Fig. 1).

Mobility reveals socio-economic clusters

Mobility patterns are influenced by the presence of the socio-economic regions in the network, highlighted in colours in Fig. 1. The typical situation is illustrated in Fig. 3 (a), with jumps within the same cluster being preferred to jumps between sectors in different clusters. In order to quantify this effect, we report in Fig. 3 (b), blue circles, the observed number of jumps of length d within the same cluster, divided by the total number of jumps of length d. This ratio is a decreasing function of the distance d, and reaches zero at d = 12, since no sectors at such distance do belong to the same cluster. As a null model we report the fraction of sector pairs at distance d which belong to the same cluster, see red squares in the same figure. The significant discrepancy between the two curves indicates that players indeed tend to avoid crossing the borders between clusters. For example, a jump of length d = 8 from one sector to another sector in the same cluster is expected only in 3% of the cases, while it is observed in about 20% of the cases. Now, the propensity of a player to spend long time periods within the same cluster might be simply related to the topology of the network, as in the case of random walkers whose motions are constrained on graphs with strong community structures³⁴. Nodes belonging to the same cluster are in fact either directly connected or are at short distance from one another. This proximity is reflected in the block-diagonal structure of the adjacency matrix A and of the distance matrix D, respectively shown in Fig. 4 (a) and (b). We have therefore checked whether the presence of the socio-economic clusters originally introduced by the developers of the game can be derived solely from the structure of the network. For this reason we adopted standard community detection methods based on the adjacency and on the distance matrix³⁵,³⁶. The results, reported respectively in Fig. 4 (d) and (e), show that detected communities deviate significantly from the clusters, implying that in our online world the socio-economic regions cannot be recovered merely from topological features. In comparison we considered the player transition count matrix M, shown in Fig. 4 (c), which displays a similar block-diagonal structure as A and D, but with the qualitative difference that it contains dynamic information on the system. Figure 4 (f) shows that community detection methods applied to the transition count matrix M reveal almost perfectly all the socio-economic areas of the universe. This finding demonstrates that mobility patterns contain fundamental information on the socio-economic constraints present in a social system. Therefore, a community detection algorithm applied to raw mobility information, as the one proposed here, is able to extract the underlying socio-economic features, which are instead invisible to methods based solely on topology. For a detailed treatment of adopted community detection methods and measures see Supplementary Section S4, Supplementary Table II and Supplementary Figs. 4 and 5.

Figure 3. Influence of socio-economic clusters on mobility.

(a) Sketch of jump patterns from a sector i to sectors within the same cluster, j and l, and to sectors in a different cluster, j′, l′. Although sectors j′ and l′ have the same graph distance from sector i as sectors j and l respectively, transitions across cluster border have smaller probabilities. (b) Quantitative evidence of the tendency of players to avoid crossing borders. Red squares show the null model, i.e. the fraction of all pairs of sectors at a given distance d being in the same cluster. Blue circles show the fraction of measured jumps leading into the same cluster, per distance. Coincidence of the two curves would indicate that clusters have no effect on mobility. Clearly this is not the case – there is a strong tendency of players to avoid crossing the borders between clusters.

Figure 4. Extracting communities from network topology and from mobility patterns.

(a) The adjacency matrix A of the universe network, (b) the matrix D of shortest path distances, and (c) the matrix M of transition counts of player jumps. Each of the three matrices contains 400 × 400 entries, whose values are colour-coded. Sector IDs are ordered by cluster, resulting in the block-diagonal form of the three matrices. We have used modularity-optimization algorithms to extract community structures from the information encoded in the three matrices. Different node colours represent the different communities found, while the 20 different colour-shaded areas indicate the predefined socio-economic clusters as in Fig. 1. The displayed Fowlkes and Mallows index quantifies the overlap of the detected communities with the predefined clusters. The closer is to 1, the better the match, see Supplementary Section S4. (d) Although information contained in the adjacency matrix A allows to find 18 communities, a number close to the real number of clusters, the communities extracted do not correspond to the underlying colour-shades areas (). (e) Extracting communities from the distance matrix D only results in 6 different groups (). (f) The 23 communities detected using the transition count matrix M reproduce almost perfectly the real socio-economic clusters (), with only a few mismatched nodes detected as additional clusters. For more measures quantifying the match of communities, see Supplementary Table II.

A long-term memory model

In order to characterize the diffusion of players over the network, we have computed the mean square displacement (MSD) of their positions, σ²(t), as a function of time. Results reported in Fig. 5 (a) indicate that, for long times, the MSD increases as a power-law:

with an exponent υ ≈ 0.26. This anomalous subdiffusive behaviour is not a simple effect of the topology of the Pardus universe. In fact, as shown in Fig. 5 (b), gray stars, the simulation of plain random walks on the same network produces a standard diffusion with an exponent υ ≈ 1 up to t ≈ 100 days, and then a rapid saturation effect which is not present in the case of the human players.

Figure 5. Diffusion scaling in empirical data and simulated models.

(a) The mean square displacement (MSD) of the positions of players follows a power relation σ²(t) ∼ t^υ with a subdiffusive exponent υ ≈ 0.26. The inset shows the average probability for a player to return after τ jumps to a sector previously visited. The curve follows a power law with an exponent of α ≈ 1.3 and an exponential cutoff. We report, for comparison, (b) the MSD for various models of mobility. For random walkers and in the case of a Markov model with transition probability π_ij = m_ij/Σ_j m_ij we observe an initial diffusion with an exponent υ ≈ 1 and then a rapid saturation of σ²(t), due to the finite size of the network. A preferential return model also shows saturation and does not fit the empirical observed scaling exponent υ. Conversely, a model with long-time memory (Time Order Memory) reproduces the exponent almost perfectly. Such a model makes use of the empirically observed while the Markov model and the preferential return model over-emphasize preferences to locations visited long ago and do not recreate the empirical curve well. Curves are shifted vertically for visual clarity.

Insights from the previous section suggest that the anomalous diffusion behaviour might be related to the tendency of players to avoid crossing borders. We have therefore considered a Markov model in which each walker moves from a current node i to a node j with a transition probability π_ij = m_ij/Σ_l m_il, where m_ij is the number of jumps between sector i and sector j, as expressed by the transition count matrix M of Fig. 4 (c). The probabilities π_ij are the entries of the transition probability matrix Π, which contains all the information on the day-to-day movement of real players, such as the preference to move within clusters, the length distribution of jumps, as well as the tendency to remain in the same sector. Despite this detailed amount of information used (the matrix Π has 160,000 elements), the Markov model fails to reproduce the asymptotic behaviour of the MSD, see magenta diamonds in Fig. 5 (b). Since the model considers only the position of the individual at its current time to determine its position at the following time, deviations from empirical data appear presumably due to the presence of higher-order memory effects³⁷. For this reason we have considered the recently proposed preferential return model²¹ which incorporates a strong memory feature. The model is based on a reinforcement mechanism which takes into account the propensity of individuals to return to locations they visited frequently before. This mechanism is able to reproduce the observed tendency of individuals to spend most of their time in a small number of locations, a tendency which is also prevalent in the mobility behaviour of Pardus players (see Supplementary Fig. 3). However, the implementation of the preferential return model on the Pardus universe network is not able to capture the scaling patterns of the MSD, as shown in Fig. 5 (b). The reason is that in the model the probability for an individual to move to a given location does not depend on the current location, nor on the order of previously visited locations. Instead, we observe that in reality individuals tend to return with higher probability to sectors they have visited recently and with lower probability to sectors visited a long time before. Consequently a sector that has been visited many times but with the most recent visit dating back one year has a lower probability to be visited again than a sector that has been visited just a few times but with the last visit dating back only one week.

To highlight this mechanism we measured the return time distribution in the jump-time series (see Methods). In particular, we extracted the probability for an individual to return again (for the first time) to the currently occupied sector after τ jumps. As shown in the inset of Fig. 5 (a), we found that the return time distribution reads

with an exponent α ≈ 1.3. We used this information for constructing a model which takes into account the higher re-visiting probability of recently explored locations. In this way we can capture the long-term scaling properties of movements. Exactly these asymptotic properties are fundamentally relevant for issues of epidemics spreading or traffic management.

This “Time Order Memory” (TOM) model incorporates a power-law distribution of first return times, together with a power-law distribution of waiting times and an exponential distribution of jump distances, as those observed empirically in Fig. 2. We show below that these ingredients are sufficient to reproduce the subdiffusive behaviour reported in Fig. 5 (a). The model works as follows: an individual stands still in a given sector for a number of days drawn from the waiting time distribution, Eq. (2). Then, the individual jumps. There are two possibilities: (i) with a probability v she returns to an already visited sector, (ii) with the probability 1 – v she jumps to a so far unexplored sector. In case (i), one of the previously visited sectors is chosen according to Eq. (4). In the exploration case (ii), the individual draws a distance d from the distance distribution, Eq. (1), and jumps to a randomly selected, unexplored sector at that distance. The model has four parameters. The parameters λ, β and α of equations (1), (2) and (4) respectively, are fixed by the data. Further, averaging over all jumps and players, the probability of returning to an already visited location is v ≈ 0.83. Similarly to the measured data, the MSD of the TOM model, black squares in Fig. 5 (b), exhibits no saturation effects and displays an exponent υ_TOM = 0.23 ± 0.02 (the error is calculated over an ensemble of realizations) in agreement with the exponent observed for the players.

Discussion

The flat slope of υ ≈ 0.26 and the lack of saturation of the MSD of the players over the whole observation period exposes the significant level of subdiffusivity in the motions of individuals, consistent with previous findings²¹,³⁸,³⁹,⁴⁰,⁴¹. However, the mere tendency of individuals to return to already visited locations is not sufficient to capture these subdiffusive properties of the MSD, but it is fundamental to consider a mechanism that takes into account the temporal order of visited locations, as achieved by the TOM model. Moreover, the TOM model is realistic in the sense that, in contrast to Markov models, it takes into account the tendency of individuals to develop a preference for visiting certain locations. At the same time it allows for the possibility that a previously preferred location becomes not frequented anymore. This view provides an alternative to recently suggested reinforcement mechanisms in preferential return models²¹. The possibility for individuals to “change home” is relevant when the model should be able to account for migration, which is an important feature in the long-time mobility behaviour of humans.

Finally, we discuss to which extent the findings from our “social petri dish” are valid also for human populations unrelated to the game. Previous analyses of human social behaviour in Pardus²⁵,²⁶ have shown agreement with well-known sociological theories and with properties on comparable behavioural data. Examining the preference of players to move within socio-economic regions is of obvious importance for clearing up the role of political or socio-economic borders on the movement and migration of humans, where the presence of borders has a strong influence on mobility¹⁵,⁴²,⁴³,⁴⁴. Online societies as the one of Pardus have the evident potential to serve as “socio-economic laboratories”, where the complete knowledge of activities, social relations, and positions of all individuals can significantly advance our understanding of large-scale human behaviour, in particular of mobility.

Methods

Data set

We focus on one of the three Pardus universes, Artemis. For this universe, we extract player mobility data from day 200 to day 1200 of its existence. We discard the first 200 days because social networks between players of Pardus have shown aging effects in the beginning of the universe, i.e. there seems to exist a transient phase in the development of the society, possibly affecting mobility, which we would like to avoid considering²⁵. To make sure we only consider active players, we select all who exist in the game between the days 200 and 1200, yielding 1458 players active over a time-period of 1000 days. The sector IDs of these players, i.e. their positions on the universe network's nodes, are logged every day at 05:35 GMT. Players typically log in once a day and perform all their limited movements of the day within a few minutes, see Supplementary Section S1. The legal department of the Medical University of Vienna has attested the innocuousness of the used anonymized data.

Transition count matrix and transition probability matrix

The entry m_ij of the transition count matrix M is equal to the number of times a player's position was on sector i and then, on the following day, on sector j. This number is cumulated for all players. The entry π_ij of the transition probability matrix Π corresponds to the probability that a player moves to a sector j given that on the previous day the player's location was sector i. It reads: , where m_ij is the number of observed player movements from sector i to sector j, and the sum over l is over all sectors of the universe. The matrix Π is a stochastic matrix, i.e. it has the property that the entries of each row sum to one.

MSD and diffusion

The MSD is defined as σ² (t) = 〈(r (T + t) − r (T))²〉, where r (T) and r (T + t) are the sectors a player occupies at times T and T + t respectively, and where by (r (T + t) − r (T)) we denote the distance between the two sectors. The average is performed over all windows of size t, with their left boundaries going from T = 0 to T = 1000-t, and over all the 1458 players in the data set. If σ² has the form σ²(t) ∼ t^υ with an exponent υ < 1, the diffusion process is subdiffusive, in the case υ > 1 it is super-diffusive. An exponent of υ = 1 corresponds to classical brownian motion³⁸,³⁹.

Jump-time and first return time distribution

We transform the time-series of daily sector IDs occupied by the players from real-time to jump-time, in order to be able to compare time-series of different length and to focus on the movements between sectors. An example of this conversion is provided: a time series [5, 5, 5, 32, 32, 104, 5, 5, 104, 104, 104, 32, 337, 337, 32…] becomes in jump-time [5, 32, 104, 5, 104, 32, 337, 32, …]. We denote jump-time by the greek letter τ, that is, at jump-time τ a player has performed exactly τ jumps. We use τ in the computation of the first return time distribution. In the hypothetical time series of sectors [5, 32, 104, 5, 104, 32, 337, 32] a first return to a sector lying τ = 1 jumps back happens 2 times (104, 5, 104 and 32, 337, 32), for τ = 2 this happens once (5, 32, 104, 5), for τ = 3 also c, , , where the sum over all is equal to 1.

Author Contributions

All the authors have equally contributed to the design of the study, to the analysis and interpretation of the results and to the preparation of the manuscript.

Supplementary Material

Supplementary Information

Supplementary Material