Mapping to a Bose gas

Figure 3. A schematic illustration of the mapping between the scale-free model with fitness and a Bose gas. (a) In the network each node is characterized by a randomly selected fitness, eta_i, shown by the different colours. The fitness describes the node's ability to compete for links with other nodes ­ the fittest are more likely to acquire more links as the network grows. We assign the energy epsilon_i to each node with fitness eta_i using eta_i = exp(­ß epsilon_i) to obtain a Bose gas with random energy levels. In the mapping, the fittest nodes (high eta_i) result in the lowest energy levels (small epsilon_i). A link from node i to node j in the network corresponds to a particle in level epsilon_j in the Bose gas. The network evolves over time by adding a new node (eta₆) that connects to two other nodes (dashed lines). In the Bose gas this corresponds to the addition of a new unoccupied energy level (epsilon₆, dashed), and the deposition of two new particles in epsilon₁ and epsilon₅, the energy levels to which eta₆ connects. As the network grows, the number of energy levels and particles increase linearly in time. The calculations show that, depending on the shape of the distribution from which the energy levels (fitnesses) are selected, two distinct phases can develop. (b) In the "fit-get-rich" phase there is no clear winner. The particle density decreases as the energy level increases. (c) In contrast, when Bose­Einstein condensation takes place, the fittest node attracts a significant fraction of all links. This node appears as a highly populated, lowest energy level while higher energies remain only sparsely populated.

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