More algorithmic results for problems of spread of influence in edge-weighted graphs with and without incentives

TL;DR

This work investigates algorithmic questions for influence spread in edge-weighted graphs with and without incentives. It establishes a gap-preserving hardness result for optimal target sets on complete graphs, showing strong inapproximability, and provides polynomial-time algorithms for several incentive- and degenerate-threshold variants, including $OTVW$ and $OTVW(degenerate)$. The paper also connects degenerate thresholds to degeneracy-order techniques and extends the analysis to broader directed and bi-directed graph settings, proving NP-completeness in several complete-graph and tournament cases. Together, these results delineate the boundaries between tractable and intractable instances across edge-weighted, incentive-based, and degenerate-threshold influence spread models, with implications for efficient design of incentives in social networks.

Abstract

Many phenomena in real world social networks are interpreted as spread of influence between activated and non-activated network elements. These phenomena are formulated by combinatorial graphs, where vertices represent the elements and edges represent social ties between elements. A main problem is to study important subsets of elements (target sets or dynamic monopolies) such that their activation spreads to the entire network. In edge-weighted networks the influence between two adjacent vertices depends on the weight of their edge. In models with incentives, the main problem is to minimize total amount of incentives (called optimal target vectors) which can be offered to vertices such that some vertices are activated and their activation spreads to the whole network. Algorithmic study of target sets and vectors is a hot research field. We prove an inapproximability result for optimal target sets in edge weighted networks even for complete graphs. Some other hardness and polynomial time results are presented for optimal target vectors and degenerate threshold assignments in edge-weighted networks.