- Tomohiko Konno, Assistant Professor (June, 2015)
Why Does Cooperative Behavior Emerge?
I have been involved in studies on the features of architecture and topology (how they are connected) in complex networks (huge and complex networks found in actual societies or organisms) using network models. From the architecture of the Internet to chemical reactions within cells, the fields to which network science can be applied are numerous. One surprising example is the verification of features of terrorist organizations. Researchers from diverse fields are engaged in studies on networks, so we can say it is an interdisciplinary research field. The research outcome introduced below came from inspiration from a discussion in a seminar on mathematical biology.
Human beings and other organisms occasionally behave altruistically at a cost to themselves in order to benefit others in a group. Whether performing cooperative behaviors at a cost to one’s self is a universal phenomenon has been a major question in evolutionary biology, and there have been discussions to the effect that networks may generate cooperative behaviors.
Each element in a network is called a “node,” and connections between these nodes are called “links.” If we compare this to human relationships, a human being corresponds to a node, and an acquaintance or friendship to a link. My focus in this research was to search for network conditions that promote cooperative behavior between nodes.
Different Features Depending on the Network
In this research, we have dealt with networks with different properties, i.e., regular networks, random networks, and scale-free networks (Figure 1). The circles scattered in each model diagram are the nodes, and the lines connecting them are the links. In regular networks, nodes are connected with regularity. In the network shown in the diagram, each node has four links with the adjacent nodes making a stable network.
On the other hand, random networks have a property where whether two nodes are connected is determined at a fixed probability, or literally at random. A network where every combination of two nodes is connected randomly is a random network. There will be no extreme differences in the number of links between each node because every node has the same probability of establishing links. Therefore, this network is often compared to highways connecting cities. When you look at a highway network, you can see that each city is connected with about the same number of highways, and there is no single interchange that is connected to a hundred roads.
The third example is scale-free networks where, in contrast to the previous two, there is an extreme difference in the number of links between each node. We see this type of network a lot in actual society, including on the Internet. Most of the nodes have a limited number of links, but some nodes have a large number, making them central points in the network. We call this kind of node with a large number of links a “hub,” like Google is in terms of the Internet.
Figure 1: Model diagram of the three networks (Courtesy of Assistant Professor Tomohiko Konno)
Prisoner’s Dilemma
In order to analyze cooperative behavior in these networks, we are going to make each node play games. The basis of this analysis will be the structure of benefits and costs called “the prisoner’s dilemma,” which is often used in game theory. This is a situational model where if two players have both chosen a rational option for themselves, then they both will get unfavorable results as a whole.
This time, we have created a model where if you choose to defect, then your benefit increases, regardless of whether others choose to cooperate (Figure 2). When two people play this game, it ends in a result where both players take uncooperative actions. However, if we place players in a network as nodes and make the adjacent players play games, then there are cases that provoke cooperative behavior.
Figure 2: Payoff matrix used in this analysis. Rational choices are not reflected in the benefit of the whole. (Courtesy of Assistant Professor Tomohiko Konno)
Networks Likely to Provoke Cooperative Behavior—Scaled by the Number of Links on the Adjacent Node
Past research on the prisoner’s dilemma games played in networks tells us that when the benefit-cost ratio is greater than the average number of links of the node, then cooperative behaviors are provoked in the network. However, under these conditions, we have no means of analyzing the difference of how far the architecture of the network can influence cooperative behaviors. For example, in the random network and the scale-free network shown in Figure 1, the average number of links for both networks is three. This means that both should have the same level of likelihood of cooperative behavior. Therefore, I have focused my attention on the number of links of the adjacent node (Figure 3) in the network.
Figure 3: The difference between the links of one node and the links of the adjacent node. The number of links of the adjacent node is signified by the number of links of the white node when viewed from the orange node. (Courtesy of Assistant Professor Tomohiko Konno)
In general, the average number of links of an adjacent node is larger than the average number of links of the other node. Additionally, there is a feature where the average number of links of an adjacent node differs depending on the architecture of the network. The random network and the scale-free network in Figure 1 both have an average of three links. However, the average number of links of adjacent nodes is four in the random network and infinity in the scale-free network. In other words, this difference can be used as the condition to determine how cooperative behavior is provoked according to different network architectures.
As a result of the analysis, when players placed in the nodes of the network play games only with adjacent players according to the payoff matrix shown above, we have found that if the benefit-cost ratio was greater than the average number of links of adjacent nodes, cooperative behavior was provoked. In short, we have found that the likelihood of cooperative behavior in terms of network types is greater in the order of regular networks, random networks, and scale-free networks.
As mentioned in the beginning, since there is a wide range of fields in which network science can be applied, we suppose that discussion with other researchers is necessary to find out how we can use this in other fields.
Interview and Composition: Noel Kikuchi
In cooperation with: Waseda University Graduate School of Political Science J-School
