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In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
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In social network analysis, the co-stardom network represents the collaboration graph of film actors i.e. movie stars. The co-stardom network can be represented by an undirected graph. Nodes correspond to the movie star actors and two nodes are linked if they co-starred (performed) in the same movie. The links are un-directed, and can be weighted or not depending on the goals of study. If the number of times two actors appeared in a movie is needed, links are assigned weights. Initially, the network was found to have a small-world property. Afterwards, it was discovered that more precisely it exhibits a scale-free (power-law) behavior.
Hierarchical network models are iterative algorithms for creating networks which are able to reproduce the unique properties of the scale-free topology and the high clustering of the nodes at the same time. These characteristics are widely observed in nature, from biology to language to some social networks.
Réka Albert is a Romanian-Hungarian scientist. She is a distinguished professor of physics and adjunct professor of biology at Pennsylvania State University and is noted for the Barabási–Albert model and research into scale-free networks and Boolean modeling of biological systems.
In a scale-free network the degree distribution follows a power law function. In some empirical examples this power-law fits the degree distribution well only in the high degree region; in some small degree nodes the empirical degree-distribution deviates from it. See for example the network of scientific citations. This deviation of the observed degree-distribution from the theoretical prediction at the low-degree region is often referred as low-degree saturation. The empirical degree-distribution typically deviates downward from the power-law function fitted on higher order nodes, which means low-degree nodes are less frequent in real data than what is predicted by the Barabási–Albert model.
The initial attractiveness is a possible extension of the Barabási–Albert model. The Barabási–Albert model generates scale-free networks where the degree distribution can be described by a pure power law. However, the degree distribution of most real life networks cannot be described by a power law solely. The most common discrepancies regarding the degree distribution found in real networks are the high degree cut-off and the low degree saturation. The inclusion of initial attractiveness in the Barabási–Albert model addresses the low-degree saturation phenomenon.
In network science, the configuration model is a method for generating random networks from a given degree sequence. It is widely used as a reference model for real-life social networks, because it allows the modeler to incorporate arbitrary degree distributions.
References
- ↑ Erdős, Paul; Rényi, Alfréd (1960). "On the evolution of random graphs" (PDF). Publication of the Mathematical Institute of the Hungarian Academy of Sciences. 5: 17–61.
- ↑ Meusel, R.; Vigna, S.; Lehmberg, O.; Bizer, C. (2015). "The Graph Structure in the Web - Analyzed on Different Aggregation Levels" (PDF). Journal of Web Science. 1 (1): 33–47. doi:10.1561/106.00000003. hdl: 2434/372411 .
- ↑ Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-law distributions in empirical data". SIAM Rev. 51 (4): 661–703. arXiv: 0706.1062 . Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111. S2CID 9155618.
- ↑ Barabási, Albert-László; Albert, Réka (October 1999). "Emergence of scaling in random networks" (PDF). Science. 286 (5439): 509–512. arXiv: cond-mat/9910332 . Bibcode:1999Sci...286..509B. doi:10.1126/science.286.5439.509. PMID 10521342. S2CID 524106..
- ↑ Brin, Sergey; Page, Lawrence (1998-04-01). "The anatomy of a large-scale hypertextual Web search engine". Computer Networks and ISDN Systems. Proceedings of the Seventh International World Wide Web Conference. 30 (1): 107–117. doi:10.1016/S0169-7552(98)00110-X. ISSN 0169-7552.
- ↑ Glen Jeh and Jennifer Widom. 2003. Scaling personalized web search. In Proceedings of the 12th international conference on World Wide Web (WWW '03). ACM, New York, NY, USA, 271–279. doi : 10.1145/775152.775191
- ↑ Kumar, Ravi; Raghavan, Prabhakar; Rajagopalan, Sridhar; Tomkins, Andrew (1999). "Trawling the Web for emerging cyber-communities". Computer Networks. 31 (11–16): 1481–1493. CiteSeerX 10.1.1.89.4025 . doi:10.1016/S1389-1286(99)00040-7. S2CID 7069190.