= \int_{-\infty}^{y_i}x \ \;\text{d}F_j^i(x) + {y_i}' \int_{y_i}^{\infty}\;\text{d}F_j^i(x) - y_i -c_i^j/K.$$ If $\Lambda = 0$, agent $i$ is just indifferent between rewiring $j$ and not rewiring. Otherwise, the larger $\Lambda$ value indicates that agent $j$ is more beneficial to rewire.

For each unknown partner $j$, the expected payoff satisfies the cumulative distribution function $F_j^i(x)$ and the rewiring cost is $c_i^j$. We can calculate an index $\Lambda_i^j$ through Eq.9, which fully captures the relevant information about agent $j$: it should be rewired when it has the highest positive index and exceeds the interaction baseline of current neighborhood. It is proven in [@weitzman1979optimal] that this strategy is optimal in terms of expected reward Eq.5.

The overall K-sight rewiring strategy is shown in Algorithm 2. At each rewiring phase, each agent $i$ first calculates the baseline interaction value $y_i$ and second minimum expected value $y_i^{sec}$ in $O_i$ (Line 1-6). Second, for each potential partner $w \in \bar O_i$, its corresponding index $\Lambda_i^w$ is computed following Eq.9 (Line 7-9). Finally, agent $i$ makes the rewiring decision accordingly — to rewire target agent $\tau$ with the maximum value of $\Lambda_{max}$ at cost $c_i^{\tau}$ if $\Lambda_{max} \ge 0$ or not to rewire otherwise (Line 10-15). Agent $i$ 