Effects of substrate network topologies on competition dynamics

We study a competition dynamics, based on the minority game, endowed with various substrate network structures. We observe the effects of the network topologies by investigating the volatility of the system and the structure of follower networks. The topology of substrate structures significantly influences the system efficiency represented by the volatility and such substrate networks are shown to amplify the herding effect and cause inefficiency in most cases. The follower networks emerging from the leadership structure show a power-law incoming degree distribution. This study shows the emergence of scale-free structures of leadership in the minority game and the effects of the interaction among players on the networked version of the game.

Figure 1

Typical time series $A\left(t\right)$ with $N=1001$ and $m=2$ for (a) ER, (b) 1D regular lattice, (c) BA, and (d) GKK substrate networks.

Figure 2

Volatility ${\sigma }^2∕N$ distribution according to the value $\alpha =2^m∕N$, for (a) ER, (b) 1D regular lattice, (c) BA, (d) GKK, (e) bar BBS, and (f) loco BBS substrate networks. The numbers of players are $N=1001$ for (a)–(d), $N=4461$ for (e), and $N=7410$ for (f). For each value of memory $m$, the results of 500 simulations for (a)–(d) are recorded as a histogram, whereas (e) and (f) are from 100 realizations. Average values of the volatility are drawn at the bottom of each graph.

Figure 3

(Color online) Volatility ${\sigma }^2∕N$ according to the value $\alpha =2^m∕N$ for WS small-world network with (a) small, and (b) large values of rewiring probability $p$. The number of players is $N=1001$, and each player is connected to her neighbors two or fewer lattice spacings away. For each value of memory $m$, the results of 500 simulations are averaged. The solid lines are guides to the eyes, and the horizontal line $({\sigma }^2∕N=0.25)$ indicates the coin-toss case. Each (red) line corresponds to a limiting case of $p$, i.e., the 1D regular lattice in (a) and the ER random graph in (b), respectively.

Figure 4

The interrelationship among the average scores of players, their degrees in the substrate network, and the average numbers of outgoing link changes per game. Each data point represents the values corresponding to each player averaged over 500 games, each of which is composed of $20000$ steps. $(m=2)$ The substrate network is the BA model with 1001 nodes.

Figure 5

Predictability $H$ according to the value $\alpha =2^m∕N$, for each substrate network whose size $N=1001$. Each point and error bar are drawn values averaged over 100 games, each of which is composed of $20000$ steps.

Figure 6

(Color online) Snapshot of typical structure of follower network, where the ER random graph is used as substrate in this case. Thin gray lines stand for the substrate network and thick black arrows for the follower network.

Figure 7

Incoming degree distribution of the follower networks for $N=1001$ whose substrate networks are (a) the ER random graph with connecting probability $p=0.1$ [18], (b) the BA model with $m_0=m=4$ [23], and (c) the GKK model with $\gamma =2$ and $\mathrm{average}\mathrm{number}\mathrm{of}\mathrm{degrees}=10$ [24]. Empty squares $(◻)$ correspond to the degrees of substrate networks. The degree distribution of each follower network is drawn from 500 realizations of MG.

Figure 8

Incoming degree distribution of the follower networks for whose substrate networks are (a) the e-print archive coauthorship network with $49983$ nodes [44], (b) the bar BBS network with 4461 nodes [45], and (c) the loco BBS network with 7410 nodes [45]. Empty squares $(◻)$ correspond to the degrees of substrate networks. The degree distribution of each follower network is drawn from 100 realizations of MG.