Preferential attachment is a central paradigm in the theory of complex networks. In this contribution we consider various generalizations of preferential attachment including for example node removal and edge rewiring. We demonstrate that generalized
preferential attachment networks can undergo a topological phase transition. This transition separates networks having a power-law tail degree distribution from those with an exponential tail. The appearance of the phase transition is closely related to the breakdown of the continuous variable description of the network dynamics.
In this contribution we introduce local attachment as an universal network-joining protocol for peer-to-peer networks, social networks, or other kinds of networks. Based on this protocol nodes in a finite-size network dynamically create power-law con
nectivity distributions. Nodes or peers maintain them in a self-organized statistical way by incorporating local information only. We investigate the structural and macroscopic properties of such local attachment networks by extensive numerical simulations, including correlations and scaling relations between exponents. The emergence of the power-law degree distribution is further investigated by considering preferential attachment with a nonlinear attractiveness function as an approximative model for local attachment. This study suggests the local attachment scheme as a procedure to be included in future peer-to-peer protocols to enable the efficient production of stable network topologies in a continuously changing environment.
