Individual position diversity in dependence socioeconomic networks increases economic output

The availability of big data recorded from massively multiplayer online role-playing games (MMORPGs) allows us to gain a deeper understanding of the potential connection between individuals' network positions and their economic outputs. We use a statistical filtering method to construct dependence networks from weighted friendship networks of individuals. We investigate the 30 distinct motif positions in the 13 directed triadic motifs which represent microscopic dependences among individuals. Based on the structural similarity of motif positions, we further classify individuals into different groups. The node position diversity of individuals is found to be positively correlated with their economic outputs. We also find that the economic outputs of leaf nodes are significantly lower than that of the other nodes in the same motif. Our findings shed light on understanding the influence of network structure on economic activities and outputs in socioeconomic system.


I. INTRODUCTION
Considerable studies have offered us a deep understanding of the influence of network structures on the dynamics of complex systems, such as the spreading of diseases and information [1] and emerging of collaborations [2]. However, the connection between network position and economic output is less studied. It is reported that specific network structures may enhance economic outputs [3][4][5][6][7]. Eagle et al argue that the diversity of individual relationships within a community strongly correlates with economic development of communities [5]. Furthermore, Bettencourt et al and Ortman et al find that the diversity of relationships is positively correlated with the productivity of individuals and communities [6,7].
Network motifs are building blocks of complex networks [8][9][10]. It is found that network motifs of social networks may reflect the driving forces for forming social structures [11,12].
Similar to the friendship networks of US students [13], Xie et al. studied triadic motifs in dependence networks of virtual societies and found that low level individuals have preference of forming links to high level individuals [4]. Their findings in virtual world are consistent with empirical findings in real society that "collaboration is easier when both partners share the same social status, and the probability of partnership formation decreases significantly as the status gap between the partners increases" [3].
Understanding the structure and function of social networks are of great importance to investigate economic activities and outputs in social systems [11,[14][15][16][17][18][19]. In real world, social networks are not large and samples are biased [13,20,21], which hinders the empirical investigation of relationship between network structures and economic activities in social system. Some other empirical studies have shown that individuals' or firms' network positions are closely related to success measured in terms of economic production [22][23][24][25]. In the era of big data, information technology provides us alternative methods to collect data of social relationships and economic activities, for example, massively multiplayer online roleplaying games (MMORPGs) [26][27][28][29][30], which enable us to study complex social and economic behaviours of human in online social systems [31][32][33]. Bainbridge et al also emphasized the scientific potential of virtual world for future research [31]. Empirical studies show that social behaviours in MMORPGs are representative of human behaviours in many aspects in real society [12,[34][35][36][37][38][39].
In this paper, we construct dependence networks based on weighted friendship networks of individuals and identify 30 distinct motif positions in 13 directed triadic motifs which represent local dependence among individuals. Using the k-means algorithm, we further classify individuals into k clusters based on the motif position profiles. Our results indicate that the motif position of individuals do have great influence on their economic outputs.

A. Data description
We use a huge database recorded from 124 servers to investigate the potential connection between network structure and economic output for individuals within the virtual world of a popular Massively Multiplayer Online Role-Playing Game (MMORPG) in China. There are two opposing camps or societies in a virtual world residing in a server, thus giving us 248 virtual societies. There exist great differences about the numbers of avatars among different virtual societies. The populations of virtual worlds vary from thousands up to fifty thousands. The distribution of the number of avatars in each virtual world is drawn in Fig.   1G. In each society, three professions have different skills. The advantage of warrior's skill is the power of attack, the advantage of mage's skill is the power of defense, and the advantage of priest's skill is the ability to cure illness. An avatar can be a warrior, a priest, or a mage.
To improve their skills, the avatars cooperate with friends to accomplish tasks. The more friends, the higher efficiency. Two individuals i and m are allowed to establish social ties to satisfy their desire of making friends and enhance their utility of collaborations. The strength of social ties is measured by the intimacy I i,m , which increases according to the collaborative activities of i and m if they belong to the same society; otherwise, I i,m remains zero if i and m belong to two different societies. Hence the friendship networks of the two camps are essentially separated. As a measure of closeness to each friendship, the values of intimacy are recorded every day. When two individuals in the same society form a team and collaborate to accomplish a task, their intimacy increases. The evolving intimacy allows us to track the evolution of the cooperation behavior in the socioeconomic networks. Each individual can maintain a friendship list, denote as F i for individual i. The social tie is symmetric: if i ∈ F m , then m ∈ F i . In addition to the friendship network, the game data contains other socioeconomic networks, such as the face to face trading networks between initiators and receivers, the vendor trading networks between vendors and costumers, the mail networks between senders and receivers, the mentor networks between students and mentors, the kill networks between killers and victims. In this paper, our focus is the friendship network.
More details of the database can be found in our earlier works about the triadic motifs in dependence networks [4] and skill complementarity in collaboration networks [40].

B. Economic outputs of individuals
We measure the economic output by converting the virtual money and items into a standardized currency for each individual. There are two virtual currencies, Xingbi and Jinbi . The Xingbi and Jinbi can be exchanged in the built-in platform in each virtual world.
The virtual system has an approximately stable exchange rate between Xingbi and the Chinese currency Renminbi . Jinbi is produced by the economic activities of the individuals when they form a team and collaborate to accomplish the tasks. The currency Jinbi and virtual items, such as weapons, clothes, and medicines, are awarded to individuals when monsters are killed and tasks are accomplished.
We convert the produced items and Jinbi to Xingbi to obtain the real economic output for each individual on each day. On average, the normal life span of virtual societies is close to 5 months [40]. Therefore, we calculate the output of each individual for a fixed period of 145 days for all virtual societies, denoted as y i = ln t=145 t=1 y i,t .

C. Construction of dependence network
For a given friendship network, we construct a dependence network by removing the insignificant edges based on statistical validation [41]. First, we define the relative intimacy of individual m in reference to all friends of individual i as w i,m = I i,m / N m=1 I i,m . Obviously, w i,m = w m,i . Following the statistical validation [41], a directed tie i → m is significant at the level of α if where k i is the degree of individual i. If α m,i < α, the directed tie m → i is significant.
For each society, we set the significant level α and remove the insignificant links, resulting  Hence, we first define the z-sore of occurrence frequency for position j: where all individuals in the dependence network of a given virtual society [8,10]. The structural similarity between individual i and m are thus defined as the correlation coefficients between Z i and Z m , such that, where E[x] is the mathematical expectation of x. value. Then we can get the optimal number of clusters k. The correlation distance defined as one minus the sample correlation between points (treated as Z i ) is used to capture the closeness between individuals in k-means algorithm, such as, The k-means algorithm returns the cluster indices of each individual and the k cluster centroid locations in a k-by-30 matrix. Each centroid is the component-wise mean of the points in that cluster. We denote the k-th cluster centroid locations as P k = (P k,1 , P k,2 , ..., P k,30 ), where P k,j = i∈C k p i,j /c k and j ∈ {1, 2, ..., 30}. c k is the number of individuals in cluster C k . We have 30 j=1 P k,j = 1, and P k is the position ratio profile of cluster k.  The triadic motifs are regarded as building blocks of complex networks [9], suggesting that the 30 positions in triadic motifs may contain important structural information of nodes in complex networks. Here we utilize the occurrence frequency of the 30 different positions in dependence networks to represent the network structure profile for each individual. Based on these profiles, we can assess the structural similarity between individuals, which allows us further to group individuals. In dependence networks, it is observed that some motifs, for example, 3 → 4 → 5 , 9 → 10 ← 9 , 16 → 17 ⇋ 18 , appears more frequency than other motifs [4].
By ordering the individuals according to the rule that nodes with large similarity are close to each other, we plot the structural similarity in Fig. 3 (D). The color bar stands for the value of similarities s i,m . One intriguing observation is that there is a block-diagonal structure, strongly indicating the function of grouping individuals for position ratio profiles.
This inspires us to further classify the individuals into clusters by k-means algorithm (see classifying individuals based on their position ratio profile in section II). In Fig. 3(A) and  Fig. 1 (B). icant profiles is to calculate the difference of position ratio profiles between the dependence networks and the reference randomized dependence networks [9]. But this method is too computationally intensive for our 248 dependence networks and some of the networks' sizes are larger than fifty thousands. highly significant. Our results indicate that the individuals who appear in more triadic motif positions have higher economic outputs. It is easy to find that more active players have more friends (more in-degree and out-degree) and appear in more triadic motif positions. Here we show the plots of the relation between the economic output y i and the individuals' in-degree k in , out-degree k out in Fig. 3 (G, H) respectively. This agrees with Fuchs' result [44] that the output is correlated to both in-and out-degree. To avoid the impact of the in-and outdegree, we investigate the relation between the economic output y i and the position diversity d i of individuals with fixed k in , k out = 1, 3, 5 in Fig. 3 (J, K, L). And we can get the same conclusion that the individual position diversity increases economic output of individuals with fixed k in or k out . What is more, we also find that the economic output is susceptible to asymmetries between individuals' in-and out-degree in our dependence socioeconomic network. The individuals with high in-degree have higher economic outputs than with high out-degree. Because it is difficult to get the records and measure the activity of individuals accurately, we assume that the individuals with the same in-and out-degree have almost the same activity.
The k-means algorithm also gives cluster centroid locations denoted as P k = (P k,1 , P k,2 , ..., P k,30 ) for each class C k . Similar to Eq. 5, we define the cluster position diversity as D k = − 30 j=1 P k,j ln P k,j . In Fig. 4(A-F), the clusters are sorted by the cluster position diversity D k in ascending order. Considering the impact of different significant level α on the relation between the economic output and position diversity, we analysis the dependence network from all the virtual society with different significant level α = 0.01, 0.05, 0.1.
In Fig. 4, the three columns from left to right correspond to the dependence networks with significant level α = 0.01, 0.05, 0.1 respectively. Fig. 4 (A, B, C) show the plots of the average economic output Y of classes and Fig. 4 (D, E, F) illustrate the plots of the value of P k for different significant level α = 0.01, 0.05, 0.1 respectively. The clusters are sorted by the cluster position diversity D k in ascending order, so one can see that the economic output Y increases with the cluster position diversity D k . We can find that lots of clusters are comprised of individuals with P k,3 , P k,9 , P k,16 equalling to one. Although these individuals have the same individual position diversity d i = 0, their economic outputs could be different.
For each dependence network with a given α, we calculate the Davies-Bouldin index values to evaluate the optimal number of clusters and get the optimal number of clusters k. The distributions of the optimal number k are shown in Fig. 4 (G, H, I). The average optimal number k is between 50 and 60.
To compare the relation coefficients between the economic output and the individuals' in-degree k in , out-degree k out , position diversity d i , we denote the coefficients as ρ k in , ρ kout and ρ d i respectively and draw the distribution of the three kinds of coefficients for different significant level α. When the significant levels α is smaller, there are greater difference between the three kinds of coefficients. At the same time, the asymmetries between individuals' in-and out-degree in our dependence socioeconomic network is obvious with the significant level α = 0.01.   4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 23 24 25 26 27 28 29   Much evidence shows that the behaviour of individuals in virtual society is representative in many aspects of human behaviour in physical society. This is rational because individuals' decision process is determined by players that are real human being. The invisible hand operates not only in modern societies but also in ancient societies, not only in real societies but also in virtual societies. Yee et al. argued that online environments such as MMORPGs could potentially be unique research platforms for the social sciences and clinical therapy, but it is crucial to firstly establish that social behavior and norms in virtual environments are comparable to those in the physical world [45]. To investigate the relation between friendship (or socioeconomic) networks in virtual and real worlds, Grabowski and Kruszewska conducted a survey among the players of an online game and construct the off-line network [39]. They showed that the structure of the friendship network in virtual world was very similar to the structure of different social networks in real world.
We believe that our interdisciplinary work represents significant scientific evidence for understanding the behaviour of social systems. It involves topics that range from social science, network science, and economics to human dynamics. It also enriches our understanding on the formation of socio-economic networks and proposes a new method to classify nodes of complex networks for understanding of people's economic behaviour from the big data of massive players. Our work sheds new light on the scientific research utility of virtual worlds for studying human behaviour in complex socio-economic systems.