Succinct circuit representation of graphs
Your question seems to be asking: how does one represent arbitrary graphs (or even arbitrary path graphs) as a circuit of polynomial size? The answer is, you don't. The number of different path graphs with 2n vertices is (2n)!, far more than the number of different circuits with nc gates (exponential in nc log n). So almost all graphs with this many vertices cannot be represented by a succinct circuit.

Therefore, as you hint, in some sense only graphs that have a high degree of structure can be represented in this way. That's what makes complexity classes like PPAD interesting: despite the structure that we know the input graphs to the EOL problem must have, we don't seem to know how to take advantage of the structure to solve the problem efficiently.

If I'm misunderstanding your question and you're really asking: how does one make a circuit that even meets the input requirements for EOL, for even a very highly structured graph: try the path graph that connects vertex x (considered as a number in binary) to x-1 and x+1, with ends at zero and at 2^n-1. Or if you want something less trivial that seems more difficult to solve EOL for: let E and D be the encryption and decryption functions for a fixed key in your favorite cryptosystem, let the neighbors of x in the graph be E(x) and D(x), and let the ends of the line be 0 and D(0).