Table 1 The notations which are used to formulate the problem

Sets and indices
$$i,j$$ Index of source and destination persons $$i,j=1,\dots ,n;$$
$$t$$ Index of discrete time periods $$t=\mathrm{0,1},\dots ,T;$$
$$s$$ Index of scenarios $$s=1,\dots ,S;$$
$${N}_{i}$$ Out-degree of person i (i.e., the persons that $$i$$ has their phone number in her contact list);
$${K}_{i}$$ In-degree of person i (i.e., the persons that have $$i$$’s phone number in their contact list)
Parameters
$${a}_{ijs}$$ Binary parameter representing that the message may be forwarded by the person $$i$$ to the person $$j$$ in scenario $$s;$$
$${\pi }_{s}$$ Probability of scenario $$s;$$
$${x}_{i}^{0}$$ $$=\left\{ {\begin{array}{*{20}{c}} 1 & {{\text{if the person}}\, i \, {\text{received a message at the initial time of the message diffusion,}}} \\ 0 & {{\text{otherwise;}}}\end{array}}\right.$$
$${x}_{is}^{t}$$ $$=\left\{{\begin{array}{*{20}{c}}1 & {{\text{if the person}}\, i \, {\text{received a message at time period}} \, t \, {\text{in scenario}} \, s (t=1,\dots ,T),}\\ 0 &{\text{otherwise;}}\end{array}}\right.$$
$${l}_{ijs}^{t}$$ $$=\left\{{\begin{array}{*{20}{c}} 1 & {{\text{if the person}}\, i \, {\text{forward the message to the person}}\, j \, {\text{at period}} \, t \, {\text{in scenario}} \, s,}\\ 0 & {{\text{otherwise;}}}\end{array}}\right.$$ 