Geometric evolution of complex networks with degree correlations

Charles Murphy, Antoine Allard, Edward Laurence, Guillaume St-Onge, Louis J. Dubé
2018 Physical review. E  
We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time t are distributed homogeneously between a new node and the exising nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature β and general time-dependent chemical potential μ(t). The chemical potential limits the spatial
more » ... extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that μ(t) can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network---its history---the degree-degree correlation can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient 〈 c 〉. In the thermodynamic limit, we identify a phase transition between a random regime where 〈 c 〉→ 0 when β < β_c and a geometric regime where 〈 c 〉 > 0 when β > β_c.
doi:10.1103/physreve.97.032309 pmid:29776179 fatcat:ajpgrbigevdilhpfhk4f2sr7me