Collective helping and bystander effects in coevolving helping networks

We study collective helping behavior and bystander effects in a coevolving helping network model. A node and a link of the network represents an agent who renders or receives help and a friendly relation between agents, respectively. A helping trial of an agent depends on relations with other involved agents and its result (success or failure) updates the relation between the helper and the recipient. We study the network link dynamics and its steady states analytically and numerically. The full phase diagram is presented with various kinds of active and inactive phases and the nature of phase transitions are explored. We find various interesting bystander effects, consistent with the field study results, of which the underlying mechanism is proposed.

Figure 1

The phase diagram for the case with $b=0$. The parameter space $\left(a,c\right)$ is divided into four regions by two transition lines of $c=1$ and $c=a$.

Figure 2

(Color online) The steady-state link density ${\rho }_{\infty }$ along the $a=0.5$ line for $k=2$, 4, and 6.

Figure 3

(Color online) The scaling collapse of numerical data for the fluctuations (a) ${\chi }_{\infty }^{\left(2\right)}$ and (b) ${\chi }_{\infty }$ at $k=2$ and $a=0.5$ in the active phase $\left({\epsilon }_2>0\right)$.

Figure 4

The phase diagram for the case with $a=0$ and $k=2$.

Figure 5

The phase diagram for the case with $k=2$ and $0<b<\frac{1}{2}$. The parameter space $\left(a,c\right)$ is divided by four transition lines such as $c=1\pm b$, $c=a-b$, and $c=\left(1+2a-b\right)/3$. The dashed lines are 7 nonanalytic lines for the transition rates.

Figure 6

The phase diagram for the case with $k=3$ and $0<b<\frac{1}{6}$.

Figure 7

(Color online) The various bystander effects: (a) NBE, (b) CBE, and (c) RBE when $b=0.05$ and $a=0$, $b$, and $3b$. (d) Critical potential cost lines $c_{\text{R}}\left(a\right)$ and $c_{\text{N}}\left(a\right)$ in the $\left(a,c\right)$ phase diagram at fixed $b=0.05$. We find the NBE for $c>c_{\text{N}}\left(a\right)$ and the RBE for $c<c_{\text{R}}$. In between these two boundaries, the CBE is found. When identifying the BE’s we consider only the active phase, where only one stationary link density exists.