arXiv 0705.41512007-05math.PRmath.CO

具有随机初始度的偏好连接模型

A preferential attachment model with random initial degrees

Maria Deijfen, Henri van den Esker, Remco van der Hofstad, Gerard Hooghiemstra

备注：: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proof
DOI：: 10.1007/s11512-007-0067-4

摘要：

研究了随机图过程${G(t)}_{t\geq 1}$，分析了它的度序列.令$(W_t)_{t\geq 1}$为i.i.d.顺序图形过程被定义为，在每个整数时间$t$，一个新的顶点，与$W_t$边连接到它，被添加到图形。然后，在时间t添加的新边优先连接到较旧的顶点，即，在$G(t-1)$的条件下，给定边连接到顶点i的概率与$d_i(t-1)+δ$成比例，其中$d_i(t-1)$是独立于其他边的顶点$i$在时间$t-1$的度数。主要结果是，该过程的渐近度序列是指数为$τ=\min\{τ_{W}, τ_{P}\}$的幂律，其中$τ_{W}$是初始度$(W_t)_{t\geq 1}$的幂律指数，$τ_{P}$是纯优先连接预测的指数.这一结果扩展了Cooper和Frieze以前的工作，并进行了调查。

英文摘要：

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ 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 $d_i(t-1)+δ$, 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 $τ=\min\{τ_{W}, τ_{P}\}$, where $τ_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $τ_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.

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摘要：

英文摘要：

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