Transistors, especially FETs, are lovely little devices. They have very high energy gain — effectively infinite in the FET case. They respond very quickly, and aside from that finite response speed, their response is historyless, which makes the electronic design and test problems considerably more tractable. In particular, circuits that are linear, noninverting, or non-amplifying are not capable of universal computation, and the transistor immediately takes care of all three of those barriers at once. (I’m fairly sure it’s possible to imagine nonlinear, inverting, amplifying circuit elements that are still incapable of universal computation, but I don’t know what they’d look like.)
Transistors do have a few problems, though. One is that their response speed, while good, could still be better; it’s in the picoseconds range for sizes that are practical to fabricate. Another is that the transistors that we know how to fabricate are still somewhat tricky to fabricate, requiring exotic materials, materials not found in nature and of very high purity.
Braess’s Paradox gives us a way to convert some kinds of noninverting nonlinearities into inversion; the well-known construction with resistors and a Zener diode is an example. In general, the bridge topology (used in the Braess circuit) provides a way to materialize a difference in your circuit, which gives you a way to get inversion and amplification, in some sense anyway. A change of 1% in the voltage across one of the “shores” of the bridge can easily result in a change of 10,000% in the voltage across the bridging element.
Nonlinear circuit behavior and behavior with memory is actually fairly universal in real circuit elements; it’s just that normally we consider it a problem to solve, because especially in analog circuits, the distortions introduced by nonlinearity are difficult to characterize and predict. But, for example, capacitor databooks are full of information about the capacitors’ deviations from linearity, which often reach 20% or more despite their best engineering efforts to remove them. What if we could take advantage of such everyday nonlinearities to implement digital logic?