Network Structures and Poverty Traps

Jackson and Watts [25] and Jackson [26] pointed out that a network g is stochastically stable if its steady state probability \(\mu (g, \varepsilon )\) is bounded below as the error rate \(\varepsilon \) aspires to 0 that is \(\mu (g, \varepsilon ) \rightarrow _\varepsilon a>0\), \(a\in [0,1]\).

All experiments have been extended to the greater number of nodes to verify and estimate the stability of received results. We have reproduced the experiments on graphs with higher numbers of nodes (up to 100), and it turned out that incrementation of the number of nodes does not affect the overall result.

In simulation analysis, we consider two rules to describe the imitation process in populated networks over a period of time. Rule 1: Mean field dynamics by imitation of success. Each agent changes her behavior if at least one neighbor has better payoff. In this case the network remains the structure but the ratio of agents will be changed. There are two ways for the switching rule [23, 35]: (i) Initial distribution of agents is nonuniform. If agent i receives an opportunity to revise her strategy, then she estimates her neighbors as one homogeneous player with aggregated payoff function. This payoff function is equal to mean value of payoffs of players, considered as one homogeneous player. If payoff function of homogeneous player is better then agent i changes her strategy to the strategy of her more popular neighbor. (ii) Initial distribution of agents is uniform. In this case agent i keeps her own strategy. Rule 2: Mean field dynamics by imitation under dissatisfaction. Dynamic process occurs according to the rule in formula (3). In this case, if agent i has a payoff which is less or equal a payoff of her neighbor, then she remains with her strategy; if agent i has the same strategy as her neighbor, then she also keeps her strategy even if the neighbor’s payoff is bigger; if neighbor’s payoff is bigger and corresponds to another/different strategy, then agent i changes her strategy for copying the strategy of the neighbor.