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Register a new userFluctuations in a general preferential attachment model via Steins method
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We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, this model can show power law, but also stretched exponential behaviour. Using Steins method we provide rates of convergence for the total variation distance. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour.
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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.
We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between the model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation phenomenon, in which unusually fit vertices may obtain abnormally high degree: it arises from a growth-extinction dichotomy within the branching part of the dual. We show further that the condensation is extensive. As the graph grows, unusually fit vertices become, each only for a limited time, neighbouring to a non-vanishing proportion of the current graph.
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 $r>2$.
Distributed Ledger Technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, DAG-type Distributed Ledgers that are based on directed acyclic graphs (DAGs) were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type Distributed Ledger. We propose a simple model for DAG-type Distributed Ledgers that are obtained from a recursive Young-age Preferential Attachment scheme, i.e. new connections are made preferably to transactions that have not been in the system for very long. We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the `young-age preference that tunes how quickly old transactions become unattractive.
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~$t^{(1-p)/(2-p)}$.
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