| Abstract |
We analyse the properties of networks formed using a
variant of the preferential attachment algorithm
introduced by Barabási and Albert, where a new node i
connects to the old node j with a probability that is
proportional to its degree k_j. In our model, nodes are
assigned a random position on a ring, and connection
probability is proportional to (k_j)^{\alpha}/(d_{ij})^{\sigma}, where d_{ij} is the distance
between i and j, and \alpha and \sigma are positive parameters. When \gamma=\sigma/\alpha
is fixed and \alpha grows to infinity, an even simpler model
selecting the m nodes having the highest value of k_j/(d_{ij})^{\gamma} is
produced. The resulting family of networks shows
various properties, most interestingly scale-free,
small-world, and hierarchical structure, that are
commonly found in real-world networks.
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