The game-theoretical modelling of a dynamically evolving network: Revisiting the target sequence 111

Figure 1.  Transition graph of the sequence 111 showing the minimal set in every possible move at each of the six vertices. Vertex 1 represents the sequence 211, i.e. individual $ v_{1} $ is the one in deficit and moves clockwise to sequence 101 with probability $ \frac{p_1}{3} $ and anticlockwise to 110 with probability $ \frac{1-p_1}{3} $, it will stay in the same state with probability $ \frac{2}{3} $. Similarly, vertex 2 represents sequence 101, vertex 3 represents sequence 112, vertex 4 represents sequence 011, vertex 5 represents sequence 121 and vertex 6 represents sequence 110.

Figure 2.  Transition graph of the sequence 111 showing the minimal set in every possible move at each of the six vertices. Vertex $ v_i $ will move clockwise to vertex $ v_{i+1} $ with probability $ p_i $. Vertex $ v_i $ will move anticlockwise to vertex $ v_{i-1} $ with probability $ 1-p_i $.

Figure 3.  Transition graph of the sequence 111 showing the first solution of the cyclic set, solution 4: Straight arrows $ \searrow $: when $ p_i = 1 $, individual $ v_i $ will move clockwise. Straight arrows $ \nwarrow $:when $ p_i = 0 $, individual $ v_i $ will move anti-clockwise. Curved arrows: for $ 0<p_i<1 $ the movement of individual $ v_i $ to the next state could happen in either direction, clockwise with probability $ p_i $ or anti-clockwise with probability $ 1-p_i $.