On the influence of the interaction graph on a finite dynamical system

  • Maximilien GadouleauEmail author


A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images (the so-called rank), the number of periodic points (the so-called periodic rank) and the number of fixed points. In particular, we investigate the minimum, average, and maximum number of images (or periodic points, or fixed points) of FDSs with a prescribed interaction graph and a given alphabet size; thus yielding nine quantities to study. The paper is split into two parts. The first part considers the minimum rank, for which we derive the first meaningful results known so far. In particular, we show that the minimum rank decreases with the alphabet size, thus yielding the definition of an absolute minimum rank. We obtain lower and upper bounds on this absolute minimum rank, and we give classification results for graphs with very low (or highest) rank. The second part is a comprehensive survey of the results obtained on the nine quantities described above. We not only give a review of known results, but we also give a list of relevant open questions.


Finite dynamical systems Boolean networks Interaction graph Fixed points Periodic points Rank 



  1. Aracena J (2008) Maximum number of fixed points in regulatory Boolean networks. Bull Math Biol 70:1398–1409MathSciNetzbMATHGoogle Scholar
  2. Aracena J, Salinas L. Private communicationGoogle Scholar
  3. Aracena J, Demongeot J, Goles E (2004) Fixed points and maximal independent sets in AND–OR networks. Discrete Appl Math 138:277–288MathSciNetzbMATHGoogle Scholar
  4. Aracena J, Richard A, Salinas L (2014) Maximum number of fixed points in AND–OR–NOT networks. J Comput Syst Sci 80(7):1175–1190MathSciNetzbMATHGoogle Scholar
  5. Aracena J, Richard A, Salinas L (2017) Number of fixed points and disjoint cycles in monotone Boolean networks. SIAM J Discrete Math 31:1702–1725MathSciNetzbMATHGoogle Scholar
  6. Atkins R, Rombach P, Skerman F (2017) Guessing Numbers of Odd Cycles. Electron J Comb 24(1):1–20MathSciNetzbMATHGoogle Scholar
  7. Bang-Jensen J, Gutin G (2009) Digraphs: theory, algorithms and applications. Springer, BerlinzbMATHGoogle Scholar
  8. Bridoux F, Castillo-Ramirez A, Gadouleau M (2015) Complete simulation of automata networks. arXiv.1504.00169Google Scholar
  9. Christofides D, Markström K (2011) The guessing number of undirected graphs. Electron J Comb 18(1):1–19MathSciNetzbMATHGoogle Scholar
  10. Comet J-P, Richard A, Noual M, Aracena J, Calzone L, Demongeot J, Kaufman M, Naldi A, Snoussi EH, Thieffry D (2013) On circuit functionality in Boolean networks. Bull Math Biol 75(6):906–919MathSciNetzbMATHGoogle Scholar
  11. Demongeot J, Noual M, Sené S (2012) Combinatorics of Boolean automata circuits dynamics. Discrete Appl Math 160:398–415MathSciNetzbMATHGoogle Scholar
  12. Didier G, Remy E (2012) Relations between gene regulatory networks and cell dynamics in Boolean models. Discrete Appl Math 160:2147–2157MathSciNetzbMATHGoogle Scholar
  13. Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  14. Flajolet P, Odlyzko AM (1989) Random mapping statistics. Research Report, INRIA RR-1114, pp 1–26Google Scholar
  15. Gadouleau M (2017) On the stability and instability of finite dynamical systems with prescribed interaction graphs. arXiv:1709.02171Google Scholar
  16. Gadouleau M (2018a) On the rank and periodic rank of finite dynamical systems. Electron J Comb 25(3):3–48MathSciNetzbMATHGoogle Scholar
  17. Gadouleau M (2018b) On the possible values of the entropy of undirected graphs. J Graph Theory 88:302–311MathSciNetzbMATHGoogle Scholar
  18. Gadouleau M, Richard A (2016) Simple dynamics on graphs. Theor Comput Sci 628:62–77MathSciNetzbMATHGoogle Scholar
  19. Gadouleau M, Riis S (2011) Graph-theoretical constructions for graph entropy and network coding based communications. IEEE Trans Inf Theory 57(10):6703–6717MathSciNetzbMATHGoogle Scholar
  20. Gadouleau M, Richard A, Riis S (2015) Fixed points of Boolean networks, guessing graphs, and coding theory. SIAM J Discrete Math 29(4):2312–2335MathSciNetzbMATHGoogle Scholar
  21. Gadouleau M, Richard A, Fanchon E (2016) Reduction and fixed points of Boolean networks and linear network coding solvability. IEEE Trans Inf Theory 62(5):2504–2519MathSciNetzbMATHGoogle Scholar
  22. Goles E (1985) Dynamics of positive automata networks. Theor Comput Sci 41:19–32MathSciNetzbMATHGoogle Scholar
  23. Goles E, Noual M (2012) Disjunctive networks and update schedules. Adv Appl Math 48(5):646–662MathSciNetzbMATHGoogle Scholar
  24. Goles E, Tchuente M (1983) Iterative behaviour of generalized majority functions. Math Soc Sci 4:197–204MathSciNetzbMATHGoogle Scholar
  25. Noual M, Sené S (2017) Synchronism versus asynchronism in monotonic Boolean automata networks. Nat Comput 17:393–402MathSciNetGoogle Scholar
  26. Paulevé L, Richard A (2010) Topological fixed points in Boolean networks. C R Acad Sci Ser I Math 348:825–828MathSciNetzbMATHGoogle Scholar
  27. Riis S (2006) Utilising public information in network coding. In: Ahlswede R, Bäumer L, Cai N, Aydinian H, Blinovsky V, Deppe C, Mashurian H (eds) General theory of information transfer and combinatorics, vol 4123. Lecture notes in computer science. Springer, Berlin, pp 866–897Google Scholar
  28. Riis S (2007a) Graph entropy, network coding and guessing games. arXiv:0711.4175
  29. Riis S (2007b) Information flows, graphs and their guessing numbers. Electron J Comb 14:1–17MathSciNetzbMATHGoogle Scholar
  30. Robert F (1980) Iterations sur des ensembles finis et automates cellulaires contractants. Linear Algebra Appl 29:393–412MathSciNetzbMATHGoogle Scholar
  31. Scheinerman ER, Ullman DH (1997) Fractional graph theory. Wiley, HobokenzbMATHGoogle Scholar
  32. Shenvi S, Dey BK (2010) A simple necessary and sufficient condition for the double unicast problem. In: Proceedings of ICC 2010Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departement of Computer ScienceDurham UniversityDurhamUK

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