Dynamic monopolies in directed graphs: the spread of unilateral influence in social networks

Let $G$ be a directed graph such that the in-degree of any vertex $G$ is at least one. Let also ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset $M$ of vertices of $G$ is called a dynamic monopoly for $(G,\tau)$ if the vertex set of $G$ can be partitioned into $D_0\cup… \cup D_t$ such that $D_0=M$ and for any $i\geq 1$ and any $v\in D_i$, the number of edges from $D_0\cup… \cup D_{i-1}$ to $v$ is at least $\tau(v)$. One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment in which for any vertex $v$, $\tau(v)=\lceil (deg^{in}(v)+1)/2 \rceil$, where $deg^{in}(v)$ stands for the in-degree of $v$. By a strict majority dynamic monopoly of a graph $G$ we mean any dynamic monopoly of $G$ with strict majority threshold assignment for the vertices of $G$. In this paper we first discuss some basic upper and lower bounds for the size of dynamic monopolies with general threshold assignments and then obtain some hardness complexity results concerning the smallest size of dynamic monopolies in directed graphs. Next we show that any directed graph on $n$ vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with $n/2$ vertices. We show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the best known result. The final note of the paper deals with the possibility of the improvement of the latter $n/2$ bound.

💡 Research Summary

The paper studies the problem of dynamic monopolies (also called target sets) in directed graphs, where influence spreads only along the direction of arcs. Formally, a directed graph G = (V, E) is given together with a threshold function τ: V → ℕ. A set M ⊆ V is a dynamic monopoly for (G, τ) if the vertex set can be partitioned into D₀ ∪ D₁ ∪ … ∪ D_t such that D₀ = M and for every i ≥ 1 and every vertex v ∈ D_i the number of arcs from the already activated vertices D₀ ∪ … ∪ D_{i‑1} to v is at least τ(v). This models unilateral influence propagation in social networks (e.g., Twitter follow relations).

The authors first derive elementary upper and lower bounds for the minimum size of a dynamic monopoly under arbitrary thresholds. The lower bound follows from a simple counting argument: the total number of required incoming arcs is Σ_v τ(v), and no vertex can contribute more than the maximum indegree Δ, yielding a bound of ⌈Σ_v τ(v)/Δ⌉. For the upper bound they present a greedy algorithm that repeatedly selects a vertex whose activation satisfies the largest number of still‑inactive vertices; this yields an O(log n)‑approximation, analogous to set‑cover results.

Complexity results are then established. The decision problem “does there exist a dynamic monopoly of size at most k?” is shown to be NP‑complete even when τ is part of the input. Moreover, the problem is W