Ordered increasing k-trees: Introduction and analysis of a preferential
  attachment network model

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Abstract

We introduce a random graph model based on k-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of important parameters for the network model such as the degree, the local clustering coefficient and the number of descendants of the nodes and root-to-node distances. We do not only obtain results for random nodes, but in particular we also get a precise description of the behaviour of parameters for the j-th inserted node in a random k-tree of size n, where j = j(n) might grow with n. The approach presented is not restricted to this specific k-tree model, but can also be applied to other evolving k-tree models.

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Is Open Access

Emergence of scaling in random networks

Reka Albert, Albert-László Barabási (2000)

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scale-free distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.