The idea of the model is this: on any given day, person $i$ looks at the world around him or her, and sees some previous day's version of everyone else. This information is $s_j(t - \tau_{ij})$.

The amount that person $j$ influences person $i$ is given by the influence matrix, $J_{ij}$, and after putting all the information together, we see that person $i$'s mean impression of the world's style is

$$ m_i(t) = \frac{1}{N} \sum_j J_{ij} \cdot s_j(t - \tau_{ij}) $$

Given the problem setup, we can quickly check whether this impression matches their own current style:

if $m_i(t) \cdot s_i(t) > 0$, then person $i$ matches those around them

if $m_i(t) \cdot s_i(t) < 0$, then person $i$ looks different than those around them

A hipster who notices that their style matches that of the world around them will risk giving up all their hipster cred if they don't change quickly; a conformist will have the opposite reaction. Because $\epsilon_i$ = $+1$ for a hipster and $-1$ for a conformist, we can encode this observation in a single value which tells us what which way the person will lean that day:

$$ x_i(t) = -\epsilon_i m_i(t) s_i(t) $$

Simple! If $x_i(t) > 0$, then person $i$ will more likely switch their style that day, and if $x_i(t) < 0$, person $i$ will more likely maintain the same style as the previous day. So we have a formula for how to update each person's style based on their preferences, their influences, and the world around them.