Encyclopedia of Social Network Analysis and Mining

2018 Edition
| Editors: Reda Alhajj, Jon Rokne

Semirings and Matrix Analysis of Networks

  • Monika Cerinšek
  • Vladimir Batagelj
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-7131-2_152



Algebraic structure

A set with one or more operations defined on it and rules that hold for them

Network analysis

A study of networks as representations of relations between discrete objects

Sparse matrix

A matrix with most of entries equal to zero

Large network

A network with several thousands or millions of nodes

Complete graph

Kn A graph in which every pair of nodes is linked


A network can be represented also with a corresponding matrix. Using matrix operations (addition and multiplication) over an appropriate semiring a unified approach to several network analysis problems can be developed. Matrix multiplication is about traveling on network.


Semirings are algebraic structures with two operations that provide the basic conditions for studying matrix addition and multiplication and path...

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The first author was financed in part by the European Union, European Social Fund. The work was partially supported by the EUROCORES Programme EUROGIGA of the European Science Foundation, project GReGAS and by Slovenian Research Agency (ARRS) projects J5-5537, J1-6720, and Z7-7614 (B), and grant P1-0294.


  1. Abdali SK, Saunders BD (1985) Transitive closure and related semiring properties via eliminants. Theor Comput Sci 40:257–274MathSciNetzbMATHCrossRefGoogle Scholar
  2. Baras JS, Theodorakopoulos G (2010) Path problems in networks. Morgan & Claypool, BerkeleyzbMATHGoogle Scholar
  3. Batagelj V (1994) Semirings for social networks analysis. J Math Soc 19(1):53–68MathSciNetzbMATHCrossRefGoogle Scholar
  4. Batagelj V, Cerinšek M (2013) On bibliographic networks. Scientometrics 96(3):845–864CrossRefGoogle Scholar
  5. Batagelj V, Mrvar A (2008) Analysis of kinship relations with Pajek. Soc Sci Comput Rev 26(2):224–246CrossRefGoogle Scholar
  6. Batagelj V, Praprotnik S (2016) An algebraic approach to temporal network analysis based on temporal quantities. Soc Netw Anal Min 6(1):1–22zbMATHCrossRefGoogle Scholar
  7. Burkard RE, Cuninghame-Greene RA, Zimmermann U (eds) (1984) Algebraic and combinatorial methods in operations research, Annals of discrete mathematics, vol 19. North Holland, Amsterdam/New YorkzbMATHGoogle Scholar
  8. Carré B (1979) Graphs and networks. Clarendon, OxfordzbMATHGoogle Scholar
  9. Fletcher JG (1980) A more general algorithm for computing closed semiring costs between vertices of a directed graph. Commun ACM 23(6):350–351CrossRefGoogle Scholar
  10. Glazek K (2002) A guide to the literature on semirings and their applications in mathematics and information sciences. Kluwer Academic Press, DordrechtGoogle Scholar
  11. Golan JS (1999) Semirings and their applications. Springer, DordrechtGoogle Scholar
  12. Gondran M, Minoux M (2008) Graphs, dioids and semirings: new models and algorithms. Springer, New YorkzbMATHGoogle Scholar
  13. Kepner J, Gilbert J (2011) Graph algorithms in the language of linear algebra. SIAM, PhiladelphiazbMATHCrossRefGoogle Scholar
  14. Quirin A, Cordón O, Santamaria J, Vargas-Quesada B, Moya-Anegón F (2008) A new variant of the Pathfinder algorithm to generate large visual science maps in cubic time. Inf Process Manag 44(4):1611–1623CrossRefGoogle Scholar
  15. Schvaneveldt RW, Dearholt DW, Durso FT (1988) Graph theoretic foundations of Pathfinder networks. Comput Math Appl 15(4):337–345MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Abelium d.o.o, Research and DevelopmentLjubljanaSlovenia
  2. 2.Faculty of Mathematics and Physics, Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Department of Theoretical Computer ScienceInstitute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  4. 4.University of Primorska, Andrej Marušič InstituteKoperSlovenia

Section editors and affiliations

  • Suheil Khoury
    • 1
  1. 1.American University of SharjahSharjahUnited Arab Emirates