Dynamics of helping behavior and networks in a small world

To investigate an effect of social interaction on the bystanders’ intervention in emergency situations a rescue model was introduced which includes the effects of the victim’s acquaintance with bystanders and those among bystanders from a network perspective. This model reproduces the experimental result that the helping rate (success rate in our model) tends to decrease although the number of bystanders $k$ increases. And the interaction among homogeneous bystanders results in the emergence of hubs in a helping network. For more realistic consideration it is assumed that the agents are located on a one-dimensional lattice (ring), then the randomness $p∊[0,1]$ is introduced: the $kp$ random bystanders are randomly chosen from a whole population and the $k-kp$ near bystanders are chosen in the nearest order to the victim. We find that there appears another peak of the network density in the vicinity of $k=9$ and $p=0.3$ due to the cooperative and competitive interaction between the near and random bystanders.

Figure 1

The diagram for the explanation of the existence of two transition points $k_1$ and $k_2$, where $k_1=\frac{c-\alpha }{\beta }+1$ and $k_2=\frac{1-c}{\beta }+1$ from the mean-field approximation Eq. (9). Here $\alpha =0.1$, $\beta =0.01$, and $c=0.25$ are used.

Figure 2

The numerical results of $a_k\left(t\right)$ for $k=9$ (lower gray line), $16$ (upper gray line), $17$ (lower black line), and $18$ (upper black line), respectively. $a_9\left(t\right)$ fluctuates around some value. $a_{16}\left(t\right)$ and $a_{17}\left(t\right)$ repeat the slow saturations punctuated by the following abrupt declines. $a_{18}\left(t\right)$ increases monotonically but very slowly and finally approaches $1$ as expected in Eq. (11). Here $N=100$, $\alpha =0.1$, $\beta =0.01$, and $c=0.25$ are used.

Figure 3

The numerical results of the network densities and success rates for $N=100$ (circles), $N=500$ (squares) and $N=1000$ (plus signs), respectively. Each data point of $s_k$ is obtained by averaging over last $5\times {10}^6$ time steps of entire $1.5\times {10}^7$ $(5\times {10}^7)$ time steps for $N=100$ (for $N=500,1000$). And for $a_k$ the time span is the same as $s_k$ but each point is averaged over every ${10}^3$ time steps. For $k\ge k_1$ $a_k\left(t\right)$ increases monotonically as expected in Eq. (11), however, the points are obtained for finite time steps. Here $\alpha =0.1$, $\beta =0.01$, and $c=0.25$ are used.

Figure 4

The numerical results of the helping networks in the one-dimensional world for various $k$. The helping network for $k=11$ is drawn in a circular style (a) and redrawn in (b). (c) and (d) are for the case with $k=31$ and (e) and (f) are for the case with $k=55$, respectively. The hubs emerge from the homogeneous nonadaptive population. Here $N=100$, $\alpha =0.1$, $\beta =0.01$, and $c=0.25$ are used and the networks have been produced with the PAJEK software.

Figure 5

(Color online) The numerical results of the network density and the success rate for $1\le k\le 30$ and $0\le p\le 1$ when $\beta =0$. Here $N=100$, $\alpha =0.1$, and $c=0.25$ are used.

Figure 6

(Color online) The numerical results of the network density and the success rate for $1\le k\le 30$ and $0\le p\le 1$ when $\beta =0.01$. It is found that there appears another peak of $a_{k,p}$ and $s_{k,p}$ in the vicinity of $k=9$ and $p=0.3$. Here $N=100$, $\alpha =0.1$, and $c=0.25$ are used.

Figure 7

The numerical results for verifying the importance of clustered near bystanders. (a) Time evolutions of two network densities introduced in Eqs. (14, 15); $a^{\left(n\right)}\left(t\right)$ for near bystanders (black line) and $a^{\left(r\right)}\left(t\right)$ for random ones (gray line). (b) Time evolutions of the degrees of willingness; $x_{vn}\left(t\right)$ for near bystanders (black line) and $x_{vr}\left(t\right)$ for random ones (gray line). For (a) and (b) each point is averaged over $4000$ time steps. (c) The verification of Eq. (16) with $k=9$ by numerical simulations; $x_{vn}$ (crosses), $x_{vr}$ (plus signs), $a_{k,p}^{MF}$ (squares), and $a_{k,p}$ (circles). Each point is averaged over $50$ realizations after saturated. Since there is no random bystanders for $p\le 1∕k$ and there is no near ones for $p=1$, in these ranges of $p$ either $x_{vr}$ or $x_{vn}$ cannot be defined. Here $N=100$, $\alpha =0.1$, $\beta =0.01$, and $c=0.25$ are used.

Figure 8

(Color online) The numerical results of the landscape of network density for $1\le k\le 20$ (horizontal axis) and $0\le p\le 1$ (vertical axis), changing from the sharp peak to a plateau according to $\alpha$ and $\beta$: (a) $\alpha =0.05$, $\beta =0.005$, (b) $\alpha =0.05$, $\beta =0.02$, (c) $\alpha =0.2$, $\beta =0.005$, and (d) $\alpha =0.2$, $\beta =0.02$. Each point is averaged over ten realizations. Here $N=100$ and $c=0.25$ are used.