How to solve a resistor nest efficiently
Well in the particular diagram you have shown there appears only one wire coming in. So there will be no current flow in any of the resistors and all the nodes will be at the same voltage.

Edit: looking at the full nonsense circuit on XKCD there is a wire coming into the top of the subcircuit you pictured which you cropped off. If you connected a voltage source to the top and bottom of the circuit you could solve it by the means below.

But in general you can solve a circuit with lots of resistors by the following procedure.


Choose one node to be your 0V reference.
Assign a variable for the voltage of each node other than the reference node.
Assign a variable and direction for the current through each component (including your voltage source). Don't worry if the direction you assign is backwards, that just means you will get a negative answer for it's value.
Write and equation for each node setting the current into the node equal to the current out of the node.
Write an equation for each component (including your voltage source). The voltage source equation will relate the voltages on two nodes directly. The resistor equations will relate the voltages on two nodes to the current through the resistor.
Solve (or get a computer to solve for you) the resulting simultanious equations.