Part IV. Games on networks
IV-2. Different types of networks
1. Goal
Our goal here is to extend the model we have created in the previous chapter to study different types of networks.
2. Motivation. Assessing the significance of network structure
The model we will develop in this chapter will allow us to explore the importance of network structure on evolutionary game dynamics. Consider, for instance, the 2-player 2-strategy single-optimum coordination game of the previous chapter:
| Player 2 | |||
| Player 2 chooses A | Player 2 chooses B | ||
| Player 1 | Player 1 chooses A | 1 , 1 | 0 , 0 |
| Player 1 chooses B | 0 , 0 | 2 , 2 |
In the previous chapter we saw that, in this game, under certain conditions:[1]
- an unstructured population of 100 agents (i.e., complete network) will most likely approach the inefficient state where all agents choose strategy A and spend most of the time around there (see fig. 1),[2] but,
- in stark contrast, if we embed that population on a G(n-of-players = 100, prob-link = 0.02) Erdős–Rényi random network, then agents will most likely approach the state where all agents choose strategy B and spend most of the time close to it (see fig. 2).
What is different in these two networks? For a start, note that the degree (i.e. number of link-neighbors) of players in the unstructured population model is 99 (i.e. complete network), while the expected degree of players in the G(100,0.02) random network is just 0.02 · 99 = 1.98 ≈ 2. Thus, an interesting question is: will agents approach the efficient state in any network where they have an average degree of about 2? or is there something special about the random network?
To explore this question, we should embed the population on other networks with average degree about 2, but with different structure. The following figure shows other types of networks where players have about two link-neighbors on average.
For each of the networks shown above, do you think that we will get similar results as with the random network? The average degree is about the same in all of them, but the network structure is very different in each case.
Let us build a model to explore this question!
3. Description of the model
The only functionality we are going to add to the program implemented in the previous chapter is the possibility of using different network-generating algorithms to create the network. Thus, we refer to the previous chapter to read the basic description of the model. The only information we should add is the following:
Agents are embedded on a network which is created using a network model determined by parameter network-model. This parameter is implemented as a chooser, with 8 possible values:
- “Erdos-Renyi”. The network is created following the G(n-of-players, prob-link) Erdős–Rényi random network model (Erdös and Rényi (1959)).
- “Watts-Strogatz-small-world”. The network is created using the Watts–Strogatz model (Watts and Strogatz (1998)). This model has two parameters:
- avg