Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products

In graph theoretical models of the spread of disease through populations, the spread of opinion through social networks, and the spread of faults through distributed computer networks, vertices are in two states, either black or white, and these states are dynamically updated at discrete time steps according to the rules of the particular conversion process used in the model. This paper considers the irreversible k-threshold and majority conversion processes. In an irreversible k-threshold (resp., majority) conversion process, a vertex is permanently colored black in a certain time period if at least k (resp., at least half) of its neighbors were black in the previous time period. A k-conversion set (resp., dynamic monopoly) is a set of vertices which, if initially colored black, will result in all vertices eventually being colored black under a k-threshold (resp., majority) conversion process. We answer several open problems by presenting bounds and some exact values of the minimum number of vertices in k-conversion sets and dynamic monopolies of complete multipartite graphs, as well as of Cartesian and tensor products of two graphs.

💡 Research Summary

The paper investigates two irreversible spreading processes on graphs: the k‑threshold process and the majority (or “half‑neighbors”) process. In both models each vertex is either black (infected, faulty, or holding a certain opinion) or white (healthy, correct, or neutral). At discrete time steps a white vertex becomes permanently black if at least k of its neighbors are black (k‑threshold) or if at least half of its neighbors are black (majority). The central combinatorial objects are the smallest initial sets that guarantee eventual total blackening: k‑conversion sets (denoted min k(G)) for the k‑threshold model and dynamic monopolies, also called dynamos (denoted min D(G)), for the majority model.

The authors first treat complete multipartite graphs K_{p₁,…,p_m}. They define X as the set of all vertices whose degree is less than k; because each partite set is either wholly inside X or wholly outside, X must be contained in any k‑conversion set. Consequently min k(G)=max{|X|,k} when the total number of vertices n exceeds k, and min k(G)=n when n≤k. This unifies earlier case‑by‑case results by Dreyer and Roberts. For the majority model they prove that the minimum size of a dynamo is ⌈(n−p₁)/2⌉, where p₁ is the size of the largest partite class. The lower bound follows from the fact that every vertex in the largest class has n−p₁ neighbors, while the upper bound is achieved by initially coloring ⌈(n−p₁)/2⌉ vertices outside that class; the large class becomes black in the first step and the remainder in the second.

Next the paper moves to graph products. For the Cartesian product G□H they establish a simple multiplicative upper bound for k‑conversion sets: min k(G□H) ≤ min k(G)·min k(H). The construction takes the Cartesian product of minimum k‑conversion sets of the factors, and the proof shows that each “layer” (copies of G or H) eventually becomes fully black because the corresponding factor’s set already does so. For dynamos they obtain a more involved bound: min D(G□H) ≤ min D(G)·|V(H)| + min D(H)·|V(G)| − min D(G)·min D(H). The idea is to place a copy of a minimum dynamo of G in every H‑layer and a copy of a minimum dynamo of H in every G‑layer, subtracting the double‑counted vertices where both copies intersect. By inductively tracking the coloring of layers (using the time needed for each factor), they show that every vertex eventually receives at least half black neighbors and thus turns black.

The authors also discuss tighter bounds for graphs without isolated vertices. Lemma 1 shows that if D is a minimal dynamo, then its complement V(G) \ D is also a dynamo and colors the whole graph in a single step. Consequently, any graph without isolated vertices satisfies min D(G) ≤ |V(G)|/2 (Corollary 1). This observation improves the general Cartesian‑product bound in many cases.

Overall, the paper provides exact formulas for min k and min D on complete multipartite graphs, and constructive upper bounds for these parameters on Cartesian products (and, by symmetry, on tensor products, though the tensor‑product results are only mentioned briefly). The results extend known special‑case formulas, unify them under a common framework, and give practical tools for estimating the minimal “seed” size needed to guarantee total spread in various network topologies. Applications include epidemiology (vaccination strategies), opinion dynamics (viral marketing), and fault tolerance in distributed systems (minimal set of initially corrupted nodes that can compromise the whole system). The paper concludes with suggestions for future work such as exact thresholds for other graph products, probabilistic extensions, and algorithmic aspects of finding minimum conversion sets.