Integer programming approach to static monopolies in graphs

A Polyhedral Approach to Least Cost Influence Maximization in Social Networks

“…Proof. If g i = d i , then h i = σ i d i and inequality (14) coincides with (13a). Similarly, if σ i = 1, we also have g i = h i = d i and inequality ( 14) is reduced to (13a).…”

“…We have j∈N (i) y ji ≥ σ i − 1 in both inequality (13a) and inequality (14). Following Case 1 and Case 2, we have j∈N (i) y ji = σ i − 1 with y ji = 1 and y ij = 0 for some j ∈ N (i) such that |{j : j ∈ N (i)}| = σ i − 1.…”

“…Suppose w = 0 and x i = 0. Inequality (13a) is reduced to j∈N (i) y ji ≥ h i d i , which is dominated by inequality (14) with j∈N (i) y ji ≥ σ i . There exist at least σ i activated neighbors that exert influence toward node i.…”

“…Suppose w = σ i and x i = h i . Inequality (13a) becomes j∈N (i) y ji ≥ 0, which dominates inequality (14) with j∈N (i) y ji ≥ σ i − h i g i as the right hand side here is strictly negative. Similar to Case 1, we must have y ji = 0 and y ij = 1 for all j ∈ N (i) due to the implicit upper bound j∈N (i) y ji ≤ 0 and constraints (13b).…”

“…The minimum subset of nodes required under dynamic activation is also referred to as the dynamic monopoly, where nodes become activated if at least half of the neighbors are activated in the previous time period. Hence, an extra index of time is incorporated into the integer programming formulation to describe step-by-step activation of nodes, see [14,15,16]. We do not consider time dynamics in our problem despite the fact that the influence propagation occurs in discrete steps.…”