Abstract
When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.
Highlights
In the social and behavioral sciences there is a growing interest in the descriptive power of agent-based models
The constant hidden by the big O notation is relatively small, and for an edge-weighted variant in which integer weights come from the set {1, 2, 3, ..., k} at worst function evaluations cost O(k|E|2) operations, still with constant at most 8. The foundation of these bounds on compute time are new time-to-convergence results we prove for threshold automata in networks: behavior updating is guaranteed to converge in 2|E| + |V | time steps
Conversion optimal value, since every set cover L with ∪i∈LJi = S corresponds to a seed set in our constructed instance of Min-Cost Complete Conversion which converts all of V to Behavior 1
Summary
In the social and behavioral sciences there is a growing interest in the descriptive power of agent-based models. For discrete-time reversible-threshold binary-behavior updating we show that evaluating the long-term effect of an intervention takes O(|E|2) operations where E is the edge set for the network. The constant hidden by the big O notation is relatively small (at most 8 or 12 for the most general cases), and for an edge-weighted variant (which can describe variablestrength ties in a social network) in which integer weights come from the set {1, 2, 3, ..., k} at worst function evaluations cost O(k|E|2) operations, still with constant at most 8 (or 12) The foundation of these bounds on compute time are new time-to-convergence results we prove for threshold automata in networks: behavior updating is guaranteed to converge in 2|E| + |V | time steps (or 2k|E| + |V | time steps for the edge-weighted case). (iii) For background on game play in networks, see the widely-read textbook of Easley and Kleinberg (2010)
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