We analyse the so-called small-world network model (originally devised by Strogatz and
Watts), treating it, among other things, as a case study of non-linear coupled difference
or differential equations. We derive a system of evolution equations containing more of
the previously neglected (possibly relevant) non-linear terms. As an exact solution of this
entangled system of equations is out of question we develop a (as we think, promising)
method of enclosing the “exact” solutions for the expected quantities by upper and lower
bounds, which represent solutions of a slightly simpler system of differential equation.