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Register a new userA preferential attachment model with random initial degrees
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In this paper, a random graph process ${G(t)}_{tgeq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{tgeq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $tau=min{tau_{W}, tau_{P}}$, where $tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{tgeq 1}$ and $tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.
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We propose a random graph model with preferential attachment rule and emph{edge-step functions} that govern the growth rate of the vertex set. We study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function $f$ is a emph{monotone regularly varying function} at infinity, the sequence of graphs associated to it obeys a power-law degree distribution whose exponent is related to the index of regular variation of $f$ at infinity whenever said index is greater than $-1$. When the regularly variation index is less than or equal to $-1$, we show that the proportion of vertices with degree smaller than any given constant goes to $0$ a. s..
We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting $r$ vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant $alpha$. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in $alpha$ below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.
In this paper we investigate geometric properties of graphs generated by a preferential attachment random graph model with edge-steps. More precisely, at each time $tinmathbb{N}$, with probability $p$ a new vertex is added to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove that the global clustering coefficient decays as $t^{-gamma(p)}$ for a positive function $gamma$ of $p$. We also prove that the clique number of these graphs is, up to sub-polynomially small factors, of order~$t^{(1-p)/(2-p)}$.
We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as $nuparrowinfty$, for the counts of the number of nodes with degree $kgeq 1$ at time $ngeq 1$. These limits rigorously verify and extend a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.
We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.
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