Analysis of Complex Networks

2.5 Conclusion

We have discussed the question of how to formulate network theory – the theory of how a set of N nodes can get linked through a set of L links – in terms of a statistical mechanics approach. In particular we were interested in clarifying if and how it would be possible to relate microscopic linking rules between individual nodes to the bulk properties of networks, such as degree distributions or clustering. Further we search for the possibility to formulate a meaningful “thermodynamics” of networks. We focused our attention on stationary, nongrowing networks.

We addressed the question from three independent directions. First, starting from considerations of network entropies we discussed a wide class of network relinking models that maximize generalized entropies, such that the observed degree distributions follow certain functional forms. In particular we were interested in the frequently encountered q-exponential degree distributions. Second, we introduced an example of a network Hamiltonian that allowed us to introduce a “temperature” to the system. We demonstrated explicitly how the introduction of an energy dependence of states allows one to consistently use standard concepts of thermodynamics. With this framework we were able to look at thermodynamical quantities and extract a “thermodynamical” entropy, which we compared to the microscopic entropy definition. We found that networks can undergo a phase transition of degree distributions, from starlike networks ...