## Abstract

In this study, we mathematically demonstrate that heterogeneous networks accelerate the social learning process, using a mean-field approximation of networks. Network heterogeneity, characterized by the variance in the number of links per vertex, is effectively measured by the mean degree of nearest neighbors, denoted as \(\langle k_{nn}\rangle \). This mean degree of nearest neighbors plays a crucial role in network dynamics, often being more significant than the average number of links (mean degree). Social learning, conceptualized as the imitation of superior strategies from neighbors within a social network, is influenced by this network feature. We find that a larger mean degree of nearest neighbors \(\langle k_{nn}\rangle \) correlates with a faster spread of advantageous strategies. Scale-free networks, which exhibit the highest \(\langle k_{nn}\rangle \), are most effective in enhancing social learning, in contrast to regular networks, which are the least effective due to their lower \(\langle k_{nn}\rangle \). Furthermore, we establish the conditions under which a general strategy *A* proliferates over time in a network. Applying these findings to coordination games, we identify the conditions for the spread of Pareto optimal strategies. Specifically, we determine that the initial probability of players adopting a Pareto optimal strategy must exceed a certain threshold for it to spread across the network. Our analysis reveals that a higher mean degree \(\langle k \rangle \) leads to a lower threshold initial probability. We provide an intuitive explanation for why networks with a large mean degree of nearest neighbors, such as scale-free networks, facilitate widespread strategy adoption. These findings are derived mathematically using mean-field approximations of networks and are further supported by numerical experiments.

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## Derivation of \(E[\triangle p_U]\) in Sect. 3

### Derivation of \(E[\triangle p_U]\) in Sect. 3

### 1.1 Change in the Probability of Strategy *U*

We examine the dynamics of the mean-field probability \(p_U\), representing the likelihood of a player choosing strategy *U*. The analysis begins with scenarios where a *V*-player switches to strategy *U*, thereby increasing \(p_U\) by 1/*N*. In the mean-field model, a randomly chosen player has a degree of \(\langle k\rangle \).

Consider a *V*-player surrounded by \(k_U\) *U*-players and \(k_V\) *V*-players (\(\langle k\rangle =k_U+k_V\)). The probability of this *V*-player adopting strategy *U* in the next time step is:

The likelihood of encountering a *V*-player with \(k_U\) *U*-neighbors and \(k_V\) *V*-neighbors is:

Thus, the probability that \(p_U\) increases by 1/*N* in the next time step, under mean-field approximation, is:

Conversely, consider a *U*-player switching to strategy *V*, decreasing \(p_U\) by 1/*N*. The probability for this change, given a *U*-player with \(k_U\) *U*-neighbors and \(k_V\) *V*-neighbors, is:

The likelihood of this scenario is:

Therefore, the probability that \(p_U\) decreases by 1/*N* in the mean-field approximation is:

Combining these probabilities, the expected change in \(p_U\) in continuous time, under mean-field approximation, is:

### 1.2 Change in Conditional Probability

To understand the dynamics of the mean-field probability \(p_U\), we need to examine the behavior of the conditional probability \(q_{U|U}\) in the mean-field approximation.

Firstly, consider a scenario where a *V*-player, chosen randomly (with probability \(p_V\)), switches to strategy *U*. Let \(k_U\) and \(k_V\) denote the number of *U*-players and *V*-players, respectively, neighboring this *V*-player (\(k_U+k_V=\langle k\rangle \)). The probability of this *V*-player adopting strategy *U* is:

In this case, the conditional probability \(q_{U|U}\) increases by approximately \(\frac{k_U}{p_U \langle k\rangle N}\), as \(p_U\) changes by an order of *O*(*w*).

The likelihood of encountering this configuration is:

The probability of a *V*-player having \(k_U\) *U*-neighbors and \(k_V\) *V*-neighbors is:

Secondly, consider a *U*-player (chosen with probability \(p_U\)) switching to strategy *V*. The probability of this change, given \(k_U\) *U*-neighbors and \(k_V\) *V*-neighbors, is:

In this scenario, \(q_{U|U}\) decreases by approximately \(\frac{k_U}{p_U \langle k\rangle N}\).

The likelihood of this configuration is:

The probability of a *U*-player having \(k_U\) *U*-neighbors and \(k_V\) *V*-neighbors is:

Combining these scenarios, the expected change in \(q_{U|U}\) in continuous time, under mean-field approximation, is:

### 1.3 Speed of Spread of Better Strategy *U*

To determine the speed at which the better strategy *U* spreads across the network, we perform detailed calculations and expansions of Eqs. (A.7) and (A.14). In the continuous time limit and under the mean-field approximation, we derive the expected changes in \(p_U\) (the probability of choosing strategy *U*) and \(q_{U|U}\) (the conditional probability):

While \({\dot{p}}_U\) changes at the order of *O*(*w*), \({\dot{q}}_{U|U}\) does so at the order of \(O(w^0)\). Given the smallness of *w*, Eq. (A.16) converges much faster than Eq. (A.15). Therefore, we can assume that \({\dot{q}}_{U|U}=0\) is always true, leading to:

Using the mean-field relation \(p_{UV}=q_{U|V}p_V=q_{V|U}p_U\) (as detailed in Sect. 2.7) and Eq. (A.17), we derive:

Consequently, the expected change in the probability of adopting the better strategy *U* per unit time step, under the mean-field approximation, is:

This equation represents the speed at which strategy *U* spreads through the network.

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### Cite this article

Konno, T. Scale-Free Networks Enhance the Spread of Better Strategy.
*Dyn Games Appl* (2024). https://doi.org/10.1007/s13235-024-00571-w

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DOI: https://doi.org/10.1007/s13235-024-00571-w

### Keywords

- Evolutionary games
- Imitation and learning
- Network heterogeneity
- Scale-free networks
- Coordination games
- Cooperation