Copyright © 2007 Elsevier B.V. All rights reserved.
Preferential deletion in dynamic models of web-like networks
Received 29 September 2005.
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Abstract
In this paper a discrete-time dynamic random graph process is studied that interleaves the birth of nodes and edges with the death of nodes. In this model, at each time step either a new node is added or an existing node is deleted. A node is added with probability p together with an edge incident on it. The node at the other end of this new edge is chosen based on a linear preferential attachment rule. A node (and all the edges incident on it) is deleted with probability q=1−p. The node to be deleted is chosen based on a probability distribution that favors small-degree nodes, in view of recent empirical findings. We analyze the degree distribution of this model and find that the expected fraction of nodes with degree k in the graph generated by this process decreases asymptotically as k−1−(2p/2p−1).
Keywords: Web-like networks; Interconnection networks; Dynamic random graph modeling; Preferential node deletion; Scale-free; Power-law degree distribution







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2.5 s, close to the previously reported time delay between consecutive ELF transient signals in the Schumann resonance range [Füllekrug, M., 1995. Schumann resonances in magnetic filed components. Journal of Atmospheric and Terresterial Physics 57, 479–484]. We propose that this behavior is similar to the collective dynamics of a network of weakly coupled limit-cycle oscillators [Strogatz, S.H., 2000, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica, D, 1–20]. Thunderstorm cells embedded within a mesoscale convective system (MCS) constitute such a network, and their lightning frequency is best described in terms of phase-locking of a globally coupled array [Kourtchatov, S.Y., Yu, V.V., Likhanskii, V.V., Napartovitch, A.P., Arecchi, F.T., Lapucci, A., 1995 Theory of phase locking of globally coupled laser arrays. Phys. Rev. A 52, 4089–4094]. Comparison of basic parameters of the lightning networks with predictions of random-graph models reveals that the networks cannot be described by the classical random-graph model [Erdos, P., Renyi, A., 1960. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci., 5, 17–61], but are more compatible with generalized random-graphs with a prescribed degree distribution [Newman, M.E.J., Strogatz, S.H., Watts, D.J., 2001. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118] that exhibit a high clustering coefficient and small average path lengths. Such networks are capable of supporting fast response, synchronization and coherent oscillations [Lago-Fernandez, L.F., Huerta, R., Corbacho, F., Siguenza, J.A., 2000. Fast response and temporal coherent oscillations in small-world networks. Physical Review Letters 84, 2758–2761]. Several physical mechanisms are suggested to explain the observed phenomenon.




