probability of this state. In particular, we can define $$E({\mathbf\sigma}) = -\ln \left[ {{P({\mathbf\sigma})}\over{P({\mathbf\sigma}_0)}}\right] .
\label{Edef}$$ In an equilibrium system, this is precisely the energy of each state (in units of $k_B T$), but we can define this “energy” for any probability distribution. As discussed in detail in Appendix \[thermoP\], all of thermodynamics can be derived from the distribution of these energies. Specifically, what matters is how many states have $E({\mathbf\sigma})$ close to a particular value $E$. We can count this number of states, $n(E)$, and define a local (microcanonical) entropy $S(E) = \ln n(E)$. If we can imagine a family of systems in which the number of degrees of freedom $N$ varies, then a thermodynamic limit will exist provided that both the entropy and the energy are proportional to $N$ at large $N$.

In most systems, including the networks that we study here, there are relatively few states that have high probability, and many more states with low probability; mathematically, $n(E)$ is an increasing function. At large $N$, this competition between decreasing probability and increasing numerosity picks out a special value of $E = E^*$, which is the energy of the “typical” states that we actually see; $E^*$ is 