Encyclopedia of Big Data Technologies

2019 Edition
| Editors: Sherif Sakr, Albert Y. Zomaya

Graph Invariants

  • Paolo BoldiEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-3-319-77525-8_77



In this section, we cover some of the basic general definitions related to graphs, with a special emphasis on the most elementary properties of interest for link analysis; for more details and further definitions and results, see, for example, (Bollobás 1998; Diestel 2012).


Graphs are one of the basic representation tools in many fields of pure and applied science, useful to model a variety of different situations and phenomena. Besides the ubiquity of graphs as a representation tool, the emergence of social networking and social media has given a new strong impetus to the field of “social network analysis” and, more generally, to what is called “link analysis.” In a nutshell, link analysis aims at studying the graph structure in order to unveil properties, discover relations, and highlight pattern and trends. This process can be seen as a special type of data mining (Chakrabarti et al. 2006), sometimes referred to...

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  1. Anthonisse JM (1971) The rush in a directed graph. Technical Report BN 9/71, Mathematical Centre, AmsterdamGoogle Scholar
  2. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bavelas A (1948) A mathematical model for group structures. Hum Organ 7:16–30CrossRefGoogle Scholar
  4. Berge C (1958) Théorie des graphes et ses applications. Dunod, PariszbMATHGoogle Scholar
  5. Berge C (1984) Hypergraphs: combinatorics of finite sets, vol 45. Elsevier, North HollandGoogle Scholar
  6. Boldi P, Vigna S (2014) Axioms for centrality. Internet Math 10(3–4):222–262MathSciNetCrossRefGoogle Scholar
  7. Bollobás B (1998) Modern graph theory. Graduate texts in mathematics. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  8. Brin S, Page L (1998) The anatomy of a large-scale hypertextual Web search engine. Comput Netw ISDN Syst 30(1):107–117CrossRefGoogle Scholar
  9. Chakrabarti S, Ester M, Fayyad U, Gehrke J, Han J, Morishita S, Piatetsky-Shapiro G, Wang W (2006) Data mining curriculum: a proposal (version 1.0). In: Intensive working group of ACM SIGKDD curriculum committee, p 140Google Scholar
  10. Craswell N, Hawking D, Upstill T (2003) Predicting fame and fortune: PageRank or indegree? In: In Proceedings of the Australasian document computing symposium, ADCS2003, pp 31–40Google Scholar
  11. Cvetkovic D, Rowlinson P (2004) Spectral graph theory. In: Beineke LW, Wilson RJ, Cameron PJ (eds) Topics in algebraic graph theory, vol 102. Cambridge University Press, Cambridge, p 88zbMATHGoogle Scholar
  12. Diestel R (2012) Graph theory, 4th edn. Graduate texts in mathematics, vol 173. Springer, New YorkGoogle Scholar
  13. Erdős P, Rényi A (1959) On random graphs, I. Publicationes Mathematicae (Debrecen) 6:290–297Google Scholar
  14. Freeman LC (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):35–41CrossRefGoogle Scholar
  15. Gross JL, Tucker TW (1987) Topological graph theory. Series in discrete mathematics and optimization. Wiley, New YorkGoogle Scholar
  16. Harris CD (1954) The market as a factor in the localization of industry in the United States. Ann Assoc Am Geogr 44(4):315–348Google Scholar
  17. Jin R, Liu L, Aggarwal CC (2011) Discovering highly reliable subgraphs in uncertain graphs. In: Proceedings of the 17th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 992–1000Google Scholar
  18. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43MathSciNetzbMATHCrossRefGoogle Scholar
  19. Lovász L (2012) Large networks and graph limits. Colloquium publications, vol 60. American Mathematical Society, ProvidenceGoogle Scholar
  20. Newman M (2003) The structure and function of complex networks. SIAM Rev 45:167–256MathSciNetzbMATHCrossRefGoogle Scholar
  21. Newman MEJ (2010) Networks: an introduction. Oxford University Press, Oxford/New YorkzbMATHCrossRefGoogle Scholar
  22. Vigna S (2016) Spectral ranking. Netw Sci 4(4):433–445CrossRefGoogle Scholar
  23. Watts D, Strogatz S (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–442zbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanoItaly