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

Cannings Chris; Broom Mark

doi:10.3934/jdg.2020003

Article Contents

2020, Volume 7, Issue 1: 37-64. Doi: 10.3934/jdg.2020003

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

Cannings Chris^1, and
Broom Mark^2, ,

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
^* Corresponding author: Mark Broom

Received: November 30, 2018

Revised: August 31, 2019

Published: January 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 05C57, 91A43; Secondary: 05C81.

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Figure 1. 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

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Figure 2. 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

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Figure 3. The Petersen triangle with associated sets. Vertices are joined if the set-intersection is empty

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Figure 4. 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

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Figure 5. 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

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Figure 6. 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)

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Figure 7. Two intersecting 14-cycles with 9 common vertices

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Table 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

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Table 2. 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

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Table 3. 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

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