Adaptive networks of trading agents

Multiagent models have been used in many contexts to study generic collective behavior. Similarly, complex networks have become very popular because of the diversity of growth rules giving rise to scale-free behavior. Here we study adaptive networks where the agents trade “wealth” when they are linked together while links can appear and disappear according to the wealth of the corresponding agents; thus the agents influence the network dynamics and vice versa. Our framework generalizes a multiagent model of Bouchaud and Mézard [Physica A 282, 536 (2000)], and leads to a steady state with fluctuating connectivities. The system spontaneously self-organizes into a critical state where the wealth distribution has a fat tail and the network is scale free; in addition, network heterogeneities lead to enhanced wealth condensation.

Figure 1

The distribution of wealth has a power-law tail in generic networks; furthermore the corresponding exponent is not sensitive to the network structure as shown here for different kinds of networks (Erdös-Rényi, scale free or with fixed connectivity). The network size is $N=1000$, the coupling $J=0.005$ and the fitted slopes equal $1+\mu =1.447\left(2\right)$ to 1.465(2). Inset: The same, but after imposing a lower cutoff $W_i>0.01$ on agent’s wealth. Here, the essentially common slope is $1+\mu =1.691\left(1\right)$.

Figure 2

The inverse participation ratio is an order parameter for wealth condensation. One goes from a homogeneous phase at large $J$ to a condensed phase at low $J$ where a finite number of agents hold a finite fraction of the total wealth (we have set $\sigma =1$). Shown are data for adaptive networks with $\beta =0.020$ (squares) and $\beta =0.20$ (circles) when $N=1000$. The analogous data for quenched random networks with $⟨q⟩=4$ are displayed using triangles. The lines are here to guide the eye. Note that the transition point is insensitive to the type of network: Quenched or adaptive.

Figure 3

Adaptive networks: Wealth and degree distributions for $N=1000$, $J=0.005$, and $\beta$ ranging from 0.020 to 0.120. The slopes are $1+\mu =1.644\left(2\right)$ and $\gamma =2.105\left(5\right)$, respectively. The lines are to guide the eye. The scale-free shape of both distributions is evident. We have plotted the degree distribution using the rescaling $q\to q∕⟨q⟩$.

Figure 4

Adaptive networks: Wealth and degree distributions for $N=1000$, $\beta =0.020$, and $J$ ranging in small steps from 0.001 to 0.010. The lines are to guide the eye.

Figure 5

Adaptive networks: Wealth and degree distributions for $J=0.005$ at $N=1000$ (dashed line; $\beta =0.20$) and 10 000 (solid line; $\beta =0.02$). For wealth the slopes are $1+\mu =1.697\left(1\right)$ and $1+\mu =1.749\left(9\right)$ at $N=1000$ and 10 000, respectively. The slope for the tail of the degree distribution is $\gamma =2.069\left(13\right)$. The figure illustrates that the exponents depend very weakly on the network size as expected in a thermodynamic limit.

Figure 6

The number $L$ of links in the network fluctuates substantially in the steady state. Here $N=1000$, $J=0.005$, and $\beta =0.020$, with a binning of size 100.

Figure 7

Adaptive networks: Wealth inverse participation ratio (dotted line) and the total number of links (divided by 2000; solid line) versus computer time. Here $N=1000$, $J=0.005$, and $\beta =0.020$.