Abstract
We consider the preferential attachment model with location-based choice introduced by Haslegrave et al. (Random Struct Algorithms 56(3):775–795, 2020) as a model in which condensation phenomena can occur. In this model, each vertex carries an independent and uniformly distributed location. Starting from an initial tree, the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting r candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices’ degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation–noncondensation phase.
Highlights
The study of complex networks is a prevalent area of interest for researchers as many seemingly dissimilar structures observable in the real world can be modelled using a common set of techniques
It has been observed that the empirical degree distribution of many large-scale real world networks follows an approximate power law over a large finite range of degrees
We show that the condensation phase transition as well as the power-law exponent can be derived from the maximum value of f
Summary
The study of complex networks is a prevalent area of interest for researchers as many seemingly dissimilar structures observable in the real world can be modelled using a common set of techniques. The tail of the asymptotic proportion of vertices of degree at least k behaves like k−τ for some power-law exponent τ The critical value αc for the condensation phase transition matches the one for which the power-law exponent is large enough for the degree distribution’s first moment to exist. This behaviour coincides with our understanding of condensation. We determine the concrete degree distribution of a vertex at a given location whose tail behaviour follows a power-law distribution dependent on f from which we derive the final result. We show numerical results and simulations for some interesting and important choices of underlining our understanding and results
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