This central chapter builds on Chapter 2 to give sufficient conditions for an interacting particle system to be uniquely defined by a Poisson point process called a graphical representation. The generator construction is also discussed. The basic construction result is applied to prove that interacting particle systems with sufficiently weak interactions are ergodic in the sense that they have a unique invariant law that is the long-time limit started from any initial state. The theory is demonstrated on a stochastic Ising model, leading to a proof that the magnetization is zero at sufficiently high temperatures.