Preferential attachment with location-based choice: Degree distribution in the noncondensation phase

Abstract

We consider the preferential attachment model with location-based choice introduced by Haslegrave, Jordan and Yarrow as a model in which condensation phenomena can occur [Haslegrave et al. 2020]. In this model every vertex carries an independent and uniformly drawn 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.

Figures (3)

• Figure 1: Plots of a simulated tree for $\Xi = (0,1,0)$ after $500$ vertices have been added. On the left, a realization for $\alpha>\alpha_c$ and on the right, a realization with $\alpha<\alpha_c$. In both cases, the start configuration consists of a root vertex and a single child, both with uniform drawn location. In the plot, the size of a vertex corresponds to its degree. We use colour saturation to indicate how close to the maximum value of $f$ a vertex's location is.
• Figure 2: Simulation of the local degree distribution for the three examples of this section. We have inserted picture (d), which coincides to standard preferential attachment, for comparison. The red surface shows the simulation results while the blue curves depicts the analytical result of Lemma \ref{['LemLocalDegree']} for each $k$. Each plot is generated for $\Psi(x)\in(0,1)$ and $k\in[10,25]$ and $\alpha = 0$.
• Figure 3: Simulations of the power law exponent of the degree distribution for each example for $\alpha$ between the corresponding $\alpha_c$ and $1/2$. The lines show the analytical result of Theorem \ref{['ThmPowerLaw']}.

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof