I am reading Lemma~1.(a) in the paper "A tutorial on modeling and analysis of dynamic social networks. Part II" where it says $w_{ij}(k) \in \left\{ 0 \right\} \cup \left[ {\begin{array}{*{20}{c}} \delta &1 \end{array}} \right] $ where $\delta >0$
what does this mean?
The reason is that the authors want to ensure the graph weights for those edges that connect two nodes to be lower bounded, regardless of , while at the same time allowing $w_{ij}(k)=0$ for the case where nodes , are not connected. So, both two cases are covered here. A case in which this does not hold is $w_{ij}$()=1/ (nodes slowly "disconnect" over time). However, using $w_{ij}$()∈[,1] does not allow to define actual disconnection between nodes as $w_{ij}$()=0 consistently. The only way out is to write $w_{ij}$()∈{0}∪[,1].
In addition, it isn't that is arbitrary small. You cannot chose it to be arbitrarily small. You just assume it exists. It might be =0.8 or =$10^{−200}$. But it is fixed by the assumption. The theorem holds for some (might need to be sufficiently small for the theorem to hold), but not necessarily for arbitrarily small .