    January  2020, 7(1): 37-64. doi: 10.3934/jdg.2020003

## Game theoretical modelling of a dynamically evolving network Ⅱ: Target sequences of score 1

 1 School of Mathematics and Statistics, The University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK 2 Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK

* Corresponding author: Mark Broom

Received  December 2018 Revised  September 2019 Published  December 2019

In previous work we considered a model of a population where individuals have an optimum level of social interaction, governed by a graph representing social connections between the individuals, who formed or broke those links to achieve their target number of contacts. In the original work an improvement in the number of links was carried out by breaking or joining to a randomly selected individual. In the most recent work, however, these actions were often not random, but chosen strategically, and this led to significant complications. One of these was that in any state, multiple individuals might wish to change their number of links. In this paper we consider a systematic analysis of the structure of the simplest class of non-trivial cases, where in general only a single individual has reason to make a change, and prove some general results. We then consider in detail an example game, and introduce a method of analysis for our chosen class based upon cycles on a graph. We see that whilst we can gain significant insight into the general structure of the state space, the analysis for specific games remains difficult.

Citation: Chris Cannings, Mark Broom. Game theoretical modelling of a dynamically evolving network Ⅱ: Target sequences of score 1. Journal of Dynamics & Games, 2020, 7 (1) : 37-64. doi: 10.3934/jdg.2020003
##### References:
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Szolnoki, Coevolutionarygames - A mini review, BioSystems, 99 (2010), 109-125.   Google Scholar  E. Ruch and I. Gutman, The branching extent of graphs, J. Combin. Inform. Systems Sci., 4 (1979), 285-295. Google Scholar  L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U.S.A., 39 (1979), 1095-1100.  doi: 10.1073/pnas.39.10.1953.  Google Scholar  A. M. Sibbald and R. J. Hooper, Sociability and willingness of individual sheep to move away from their companions in order to graze, Applied Animal Behaviour, 86 (2004), 51-62.  doi: 10.1016/j.applanim.2003.11.010. Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs: Ⅰ pure reproduction, Applied Mathematics, 1 (2010), 137-145.  doi: 10.4236/am.2010.13018. Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅱ. Age capped reproduction, Applied Mathematics, 1 (2010), 251-259.   Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅲ. Game based reproduction, Applied Mathematics, 1 (2010), 335-343.  doi: 10.4236/am.2010.15044. Google Scholar  B. Voelkl and C. Kasper, Social structure of primate interaction networks facilitates the emergence of cooperation, Biology Letters, 5 (2009), 462-464.  doi: 10.1098/rsbl.2009.0204. Google Scholar  B. Voelkl and R. Noë, The influence of social structure on the propagation of social information in artificial primate groups: A graph-based simulation approach, J. Theor. Biol., 252 (2008), 77-86.  doi: 10.1016/j.jtbi.2008.02.002. Google Scholar  J. Wiszniewski, C. Brown and L. M. Möller, Complex patterns of male alliance formation in dolphin social networks, Journal of Mammalogy, 93 (2012), 239-250.  doi: 10.1644/10-MAMM-A-366.1. Google Scholar

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##### References:
  B. Allen and M. A. Nowak, Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.  doi: 10.4171/EMSS/3.  Google Scholar  V. Bala and S. Goyal, A model of non-cooperative network formation, Econometrica, 68 (2000), 1181-1230.  doi: 10.1111/1468-0262.00155.  Google Scholar  M. Broom and C. Cannings, A dynamic network population model with strategic link formation governed by individual preferences, J. Theor. Biol., 335 (2013), 160-168.  doi: 10.1016/j.jtbi.2013.06.024.  Google Scholar  M. Broom and C. Cannings, Graphic deviation, Discrete Mathematics, 338 (2015), 701-711.  doi: 10.1016/j.disc.2014.12.011.  Google Scholar  M. Broom and C. Cannings, Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences, Journal of Dynamic Games, 4 (2017), 285-318.  doi: 10.3934/jdg.2017016.  Google Scholar  C. Capitanio, Sociability and response to video playback in adult male rhesus monkeys (macac mulatta), Primates, 43 (2002), 169-177.   Google Scholar  R. C. Connor, M. R. Helthaus and L. M. Barre, Superalliances of bottlenose dolphins, Nature, 397 (1999), 571-572.  doi: 10.1038/17501. Google Scholar  P. I. M. Dunbar, Neocortex size as a constraint on group size in primates, J. Human Evoluion, 22 (1992), 469-493.  doi: 10.1016/0047-2484(92)90081-J. Google Scholar  B. Dutta and M. O. Jackson, The stability and effeciency of directed communication networks, Rev. Econ. Design, 5 (2015), 251-272.   Google Scholar  C. S. Elton, Animal Ecology, Sidgwick & Jackson, London, 1927. Google Scholar  P. Erdos and P. T. Gallai, Grafok eloirt fokszamu pontokkal, Matematikai Lapo, 11 (1960), 264-274.   Google Scholar  S. L. Hakimi, On the realizability of a set of integers as degrees of the vertices of a graph. Ⅰ, J. Soc. Indust. Appl. Math., 10 (1962), 496-506.  doi: 10.1137/0110037.  Google Scholar  W. Hässelbarth, Die Verzweightheit von Graphen, Comm. in Math. and Computer Chem. (MATCH), 16 (1984), 3-17. Google Scholar  V. Havel, A remark on the existence of finite graphs, Časopis Pěst. Mat., 81 (1956), 477–480. Google Scholar  M. O. Jackson, The stability and efficiency of economic and social networks, Networks and Groups, Springer, (2003). Google Scholar  M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ, 2008. Google Scholar  M. O. Jackson, An overview of social networks and economic applications, Handbook of Social Economics, Elsevier, (2011), 512–585 Google Scholar  M. O. Jackson and A. Wolinsky, A strategic model of social and economic networks, J. Econ. Theory, 71 (1996), 44-74.  doi: 10.1006/jeth.1996.0108.  Google Scholar  R. Merris and T. Roby, The lattice of threshold graphs, J. Inequal. Pure and Appl. Math., 6 (2005), Art. 2, 21 pp. Google Scholar  R. Noë, Biological markets: Partner choice as the driving force behind the evolution of cooperation, Economics in Nature. Social Dilemmas, Mate Choice and Biological Markets, (2001), 93–118. Google Scholar  R. Noë and P. Hammerstein, Biological markets: Supply and demand determine the effect of partner choice in cooperation, mutualism and mating, Behav. Ecol. Sociobio., 35 (1994), 1-11.   Google Scholar  J. M. Pacheco, A. Traulsen and M. A. Nowak, Active linking in evolutionary games, J. Theor. Biol., 243 (2006), 437-443.  doi: 10.1016/j.jtbi.2006.06.027.  Google Scholar  J. M. Pacheco, A. Traulsen and M. A. Nowak, Coevolution of strategy and structure in complex networks with dynamical linking, Phys. Rev. Lett., 97 (2006), 258103. doi: 10.1103/PhysRevLett.97.258103. Google Scholar  J. Pepper, J. Mitani and D. Watts, General gregariousness and specific social preferences among wild chimpanzees, Int. J. Primatol., 20 (1999), 613-632.   Google Scholar  M. Perc and A. Szolnoki, Coevolutionarygames - A mini review, BioSystems, 99 (2010), 109-125.   Google Scholar  E. Ruch and I. Gutman, The branching extent of graphs, J. Combin. Inform. Systems Sci., 4 (1979), 285-295. Google Scholar  L. S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U.S.A., 39 (1979), 1095-1100.  doi: 10.1073/pnas.39.10.1953.  Google Scholar  A. M. Sibbald and R. J. Hooper, Sociability and willingness of individual sheep to move away from their companions in order to graze, Applied Animal Behaviour, 86 (2004), 51-62.  doi: 10.1016/j.applanim.2003.11.010. Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs: Ⅰ pure reproduction, Applied Mathematics, 1 (2010), 137-145.  doi: 10.4236/am.2010.13018. Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅱ. Age capped reproduction, Applied Mathematics, 1 (2010), 251-259.   Google Scholar  R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅲ. Game based reproduction, Applied Mathematics, 1 (2010), 335-343.  doi: 10.4236/am.2010.15044. Google Scholar  B. Voelkl and C. Kasper, Social structure of primate interaction networks facilitates the emergence of cooperation, Biology Letters, 5 (2009), 462-464.  doi: 10.1098/rsbl.2009.0204. Google Scholar  B. Voelkl and R. Noë, The influence of social structure on the propagation of social information in artificial primate groups: A graph-based simulation approach, J. Theor. Biol., 252 (2008), 77-86.  doi: 10.1016/j.jtbi.2008.02.002. Google Scholar  J. Wiszniewski, C. Brown and L. M. Möller, Complex patterns of male alliance formation in dolphin social networks, Journal of Mammalogy, 93 (2012), 239-250.  doi: 10.1644/10-MAMM-A-366.1. Google Scholar The transition graph for the target $111$ with six states. The vertex in deficit in each state is highlighted by a dot, and corresponding possible transitions are shown The two possible configurations for members of the minimal set with target 11111. The uppermost vertex is the one not attaining its target. There are 15 different cases of the configuration on the left, and 30 of the configuration on the right The Petersen triangle with associated sets. Vertices are joined if the set-intersection is empty The transition graph for the graphs within the minimal set for target 11111. Each triangle is labelled internally with the edge fixed in its configurations. Note that each edge has a vertex at its mid-point. Each vertex is labelled $a/ b$ with a unique state number $a$, and the vertex $b$ which is deficient in state $a$. Three vertices 1, 2 and 32 are duplicated (shown as different shapes) to allow a planar representation The choice graph for the graphs within the minimal set for target 11111. Each vertex has an arrow which is the choice that the deficient vertex would make in that state. There is a stable cycle of length 30, specifically (36; 37; 38; 31; 29; 25; 18; 19; 20; 14; 11; 15; 22; 27; 40; 41; 32; 33; 34; 24; 16; 12; 7; 3; 1; 4; 9; 5; 2; 43; 36), which is shown in bold The choice graph for the minimal set for target 11111. Here we have a stable 24-cycle. (1; 4; 9; 5; 2; 6; 11; 14; 20; 21; 22; 27; 40; 41; 32; 23; 16; 17; 18; 13; 7; 3; 1)
Score 1 sequences generated from the graphic sequence 766543221. In each case we identify the set, one of $S_{A}$, $S_{B}$, $S_{J}$ or $S_{N}$ that each element is in, in the corresponding position to the target sequence
 766543221 $f_{1}=-1, f_{2}=-2, f_{3}=-1, f_{4}=0,$ $\lambda=4, K=4, K'=4$ 866543221 $i=1, \lambda*=5$ for $m=5$ up JJJJNBBBB 666543221 $i=1, \lambda*=5$ for $m=5$ up AAAAAAAAA 776543221 $i=2, 3, \lambda*=5$ for $m=5$ up JJJJNBBBB 765543221 $i=2, 3, \lambda*=5$ for $m=5$ up AAAAAAAAA 766643221 $i=4, \lambda*=5$ for $m=5$ up JJJJNBBBB 766443221$^{a}$ $i=4, \lambda*=5$ never AAAAAAAAA 766553221 $i=5, \lambda*=5$ for all but $m=4, 5$ down JJJJJBBBB 766533221 $i=5, \lambda*=5$ never JJJJBBBBB 766544221$^{b}$ $i=6, \lambda*=5$ for $m=5$ up AAAAAAAAA 766542221 $i=6, \lambda*=5$ for $m=5$ up JJJJNBBBB 766543321 $i=7, 8, \lambda*=5$ for $m=5$ up AAAAAAAAA 766543211 $i=7, 8, \lambda*=5$ for $m=5$ up JJJJNBBBB 766543222 $i=9, \lambda*=5$ for $m=5$ up AAAAAAAAA 766543220 $i=9, \lambda*=5$ for $m=5$ up JJJJNBBBB
 766543221 $f_{1}=-1, f_{2}=-2, f_{3}=-1, f_{4}=0,$ $\lambda=4, K=4, K'=4$ 866543221 $i=1, \lambda*=5$ for $m=5$ up JJJJNBBBB 666543221 $i=1, \lambda*=5$ for $m=5$ up AAAAAAAAA 776543221 $i=2, 3, \lambda*=5$ for $m=5$ up JJJJNBBBB 765543221 $i=2, 3, \lambda*=5$ for $m=5$ up AAAAAAAAA 766643221 $i=4, \lambda*=5$ for $m=5$ up JJJJNBBBB 766443221$^{a}$ $i=4, \lambda*=5$ never AAAAAAAAA 766553221 $i=5, \lambda*=5$ for all but $m=4, 5$ down JJJJJBBBB 766533221 $i=5, \lambda*=5$ never JJJJBBBBB 766544221$^{b}$ $i=6, \lambda*=5$ for $m=5$ up AAAAAAAAA 766542221 $i=6, \lambda*=5$ for $m=5$ up JJJJNBBBB 766543321 $i=7, 8, \lambda*=5$ for $m=5$ up AAAAAAAAA 766543211 $i=7, 8, \lambda*=5$ for $m=5$ up JJJJNBBBB 766543222 $i=9, \lambda*=5$ for $m=5$ up AAAAAAAAA 766543220 $i=9, \lambda*=5$ for $m=5$ up JJJJNBBBB
Score 1 sequences generated from the graphic sequence 766444221. In each case we indentify the set, one of $S_{A}$, $S_{B}$, $S_{J}$ or $S_{N}$ that each element is in, in the corresponding position to the target sequence
 766444221 $f_{1}=-1, f_{2}=-2, f_{3}=-1, f_{4}=-2,$ $\lambda=4, K=3, K'=0$ 866444221 $i=1, \lambda*=5$ never JJJAAABBB 666444221 $i=1, \lambda*=5$ never AAAAAAAAA 776444221 $i=2, 3, \lambda*=5$ never JJJAAABBB 765444221 $i=2, 3, \lambda*=5$ never AAAAAAAAA 766544221$^{b}$ $i=4, 5, 6, \lambda*=5$ for $m=5$ up AAAAAAAAA 766443221$^{a}$ $i=4, 5, 6, \lambda*=5$ never AAAAAAAAA 766444321 $i=7, 8, \lambda*=5$ never AAAAAAAAA 766444211 $i=7, 8, \lambda*=5$ never JJJAAABBB 766444222 $i=9, \lambda*=5$ never AAAAAAAAA 766444220 $i=9, \lambda*=5$ never JJJAAABBB
 766444221 $f_{1}=-1, f_{2}=-2, f_{3}=-1, f_{4}=-2,$ $\lambda=4, K=3, K'=0$ 866444221 $i=1, \lambda*=5$ never JJJAAABBB 666444221 $i=1, \lambda*=5$ never AAAAAAAAA 776444221 $i=2, 3, \lambda*=5$ never JJJAAABBB 765444221 $i=2, 3, \lambda*=5$ never AAAAAAAAA 766544221$^{b}$ $i=4, 5, 6, \lambda*=5$ for $m=5$ up AAAAAAAAA 766443221$^{a}$ $i=4, 5, 6, \lambda*=5$ never AAAAAAAAA 766444321 $i=7, 8, \lambda*=5$ never AAAAAAAAA 766444211 $i=7, 8, \lambda*=5$ never JJJAAABBB 766444222 $i=9, \lambda*=5$ never AAAAAAAAA 766444220 $i=9, \lambda*=5$ never JJJAAABBB
Frequencies of stable cycles resulting from 2, 000 simulations starting from randomly generated initial choice graphs
 Cycle-length 6 10 12 14 16 18 20 22 24 26 28 30 Frequency 358 436 735 319 96 29 17 7 3 0 0 0
 Cycle-length 6 10 12 14 16 18 20 22 24 26 28 30 Frequency 358 436 735 319 96 29 17 7 3 0 0 0
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