Fig 1.
Schematic example of a multiplex network.
The multiplex is made of N = 5 nodes embedded within M = 3 layers, each one containing 3 links. The size of nodes is proportional to centrality measure. The dashed lines represent interlayer connections, while the continuous lined represent the intra-layer connections.
Fig 2.
Centrality distribution in the multiplex network.
The multiplex is made of N = 1000 nodes embedded within M = 3 layers, each one modelled by a different scale-free network. The size of nodes is proportional to centrality measure.
Fig 3.
The Critical Mass Density as a function of the population’s size N and the number of layers M.
Table 1.
Payoff Matrices of the Prisoner’s Dilemma Game.
Fig 4.
Evolution of cooperation considering different interlayer interaction strength.
The evolution of cooperation against the round as a function of interlayer interaction strength. The ‘blue’ plot represents the case of constant interlayer strength: ωαβ = 0.4. The ‘red’ plot represents the case of variable interlayer strength (one dominant layer): ωαβ = 0.3 between layers 1 and 3; ωαβ = 0.6 between the layer 2 and the other layers of the multiplex. We show the evolution of cooperation until 200 rounds as, in correspondence of that value, the convergence has already been reached. It can be observed that the emergence of cooperation is quicker considering a variable interlayer strength (one dominant layer), than the constant case. The dominant layer acts as a behaviour’s polarizer of the nodes in the other layers.
Fig 5.
Emergence of cooperation over time.
The figure illustrates the fraction of cooperative nodes against the rounds or time steps: low homophily (A) and high homophily (B). The figure shows the evolutionary dynamics of the PD game played between the interacting nodes in a multiplex network with M = 3 layers. In both cases N = 1000 nodes. The results are obtained choosing a fixed number of simulations and the colour corresponds to the density: ‘red’ indicates the highest density (that is the maximum number of overlapping points), while ‘blue’ means the lowest density. As can be observed, increasing the homophily value of the multiplex network , we note a faster emergence of cooperation.
Fig 6.
Temporal evolution of cooperation.
The figure highlights the microscopic emergence of cooperation in the evolutionary process. The formation of cooperative groups in the network and also the group size depend on the homophily value. Figs A, B, C—in the low homophily case (σ = 8), the defective behaviour tends to persist more in the population, not favouring the formation of cooperative groups and globally slowing the emergence of cooperation. Yet, the group size will be smaller in this case of low homophily. Figs D, E, F—in the high homophily case (σ = 1), the convergence towards cooperation becomes quicker, and there is a natural formation of larger cooperative groups than in low homophily case. Analysing the corresponding figures of the evolution, we see clearly this difference, both in speed and size, in the formation of cooperative groups.