We investigate the behavior of two dimensional resistor networks, with finite sizes and different kinds (rectangular, hexagonal, and triangular) of lattice geometry. We construct the network by having a network-element repeat itself $L_x$ times in $x$-direction and $L_y$ times in the $y$-direction. We study the relationship between the effective resistance ($R_\mathrm{eff}$) of the network on dimensions $L_x$ and $L_y$. The behavior is simple and intuitive for a network with rectangular geometry, however, it becomes non-trivial for other geometries which are solved numerically. We find that $R_\mathrm{eff}$ depends on the ratio $L_x/L_y$ in all the three studied networks. We also check the consistency of our numerical results experimentally for small network sizes.

中文翻译：

---

二维电阻网络的有效电阻

我们研究二维电阻网络的行为，具有有限的尺寸和不同种类（矩形、六边形和三角形）的晶格几何形状。我们通过让网络元素在 $x$-方向上重复 $L_x$ 次并在 $y$-方向上重复 $L_y$ 次来构建网络。我们研究了网络在 $L_x$ 和 $L_y$ 维度上的有效阻力 ($R_\mathrm{eff}$) 之间的关系。对于具有矩形几何形状的网络，该行为是简单而直观的，但是，对于其他以数值方式求解的几何形状，它变得非常重要。我们发现 $R_\mathrm{eff}$ 取决于所有三个研究网络中的 $L_x/L_y$ 比率。我们还通过实验检查了小网络规模的数值结果的一致性。