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Intervention problems in the Linear Threshold Model: A general formulation and new results
Giacomo Como, Fabio Fagnani, Stephane Durand
TL;DR
The paper studies the minimum-cost threshold intervention needed to ensure global convergence to the all-1 configuration in Linear Threshold Models. The paper reformulates optimal intervention as a combinatorial optimization over agent permutations and analyzes it using graph-theoretic structure. For unweighted graphs with q sink-connected components, the bounds coincide with the least-cost intervention: Cmin(G,θ) = Cmax(G,θ) = ρ.
Problem
The paper studies the minimum-cost threshold intervention needed to ensure global convergence to the all-1 configuration in Linear Threshold Models.
Method
The paper reformulates optimal intervention as a combinatorial optimization over agent permutations and analyzes it using graph-theoretic structure.
Results
For unweighted graphs with q sink-connected components, the bounds coincide with the least-cost intervention: Cmin(G,θ) = Cmax(G,θ) = ρ.
Takeaways & Limitations
The analysis identifies a case where the intervention-cost bounds coincide with the least-cost intervention, with a G-greedy permutation attaining cost ρ.
Takeaways & Limitations
The paper leaves formal complexity proofs and approximation algorithms for the resulting combinatorial optimization problem to future research.
Abstract
from arXiv · showhide
We study an optimal intervention problem for linear threshold models. This is a popular class of dynamical network systems whereby a number of agents, identified with the nodes of a graph, strategically change their binary action (0 or 1) according to a threshold rule. Specifically, an agent adopts action 1 if and only if the fraction of its neighbors in the interaction graph that do so is greater than or equal to a prescribed threshold. Assuming that a planner can modify the agents' thresholds at a cost equal to the aggregate threshold increase, we study the minimum intervention cost needed to ensure global convergence to the all-1 configuration. Our main contribution is the introduction of a new graph-theoretic quantity, called oriented path number, that is the minimum number of disjoint paths needed to cover the graph that can be oriented to form a directed acyclic graph. When thresholds are all equal to 1/2, the optimal cost is shown to coincide with the oriented path number, whereas, in the general case, it turns out to be the main ingredient of a bound on the optimal intervention cost.
1. INTRODUCTION
The paper frames threshold cascades as strategic network dynamics and introduces a minimum-cost intervention problem distinct from conventional seeding. It reformulates this problem through activation orders, enabling graph-structural analysis.
- 1. INTRODUCTION: Network contagion behavior reflects the interaction between individual dynamics and interaction topology, motivating graph-based indices of susceptibility and resilience.
- 1. INTRODUCTION: Linear Threshold Models capture strategic binary-action dynamics in which agents respond to neighbors choosing the same action.The model is interpretable as best-response dynamics in a network game.
- 1. INTRODUCTION: Existing LTM work often studies random networks, while cohesiveness offers a general-graph index that is conceptually simple but difficult to compute.The paper positions its intervention formulation against these limitations in prior work.
- 1. INTRODUCTION: The paper studies minimum-cost threshold interventions that trigger a full cascade, rather than selecting initially activated agents.Intervention cost depends on continuous threshold modifications, making it distinct from classical seeding problems.
- 1. INTRODUCTION: The optimal intervention problem is equivalently reformulated as a combinatorial optimization over permutations representing possible agent activation orders.Each permutation directly determines the minimum intervention cost for that order.
2. THE MODEL
The model represents agents on a weighted directed graph that adopt action 1 when active-neighbor weight reaches their threshold. A planner lowers thresholds through cost functions and seeks interventions guaranteeing convergence to all 1 from every initial state.
- 2. THE MODEL: Agents adopt action 1 when the total weight from previously active neighbors reaches at least their threshold multiplied by weighted degree.The dynamics operate on binary configurations and include an equilibrium notion.
- 2. THE MODEL: The planner’s cost functions are non-decreasing and lower semicontinuous, with C_i(h_i) measuring the cost of lowering agent i’s threshold by h_i.
- 2. THE MODEL: The minimum intervention cost exists because the successful-intervention set is compact under the continuity of the n-step dynamics.
- 2. THE MODEL: The framework includes target-set selection and least-cost influence as special cost-function cases, with the latter known to be NP-complete.Target-set selection uses fixed costs for any nonzero intervention, while least-cost influence uses linear costs.
- 2. THE MODEL: The paper announces a pure combinatorial reformulation for every choice of cost functions, followed by bounds and analytically solvable examples for least-cost influence.
3. A COMBINATORIAL REFORMULATION OF THE PROBLEM
The least-cost influence problem is recast in terms of one-at-a-time activation orders. Each permutation induces the minimum threshold intervention needed at every step, and optimizing over permutations exactly matches the synchronous LTM optimum.
- 3. A COMBINATORIAL REFORMULATION OF THE PROBLEM: The reformulation optimizes over permutations of nodes, each encoding an activation order and an associated intervention cost.This creates an energy interpretation used in the subsequent analysis.
- 3. A COMBINATORIAL REFORMULATION OF THE PROBLEM: For a given permutation, the minimum intervention for node σ_t equals its threshold requirement minus the weight contributed by previously activated neighbors.The intervention vector collects these values, while negative entries are removed through the positive-part operator.
- 3. A COMBINATORIAL REFORMULATION OF THE PROBLEM: The permutation-based optimum coincides with the optimum obtained from synchronous LTM dynamics.
- 3. A COMBINATORIAL REFORMULATION OF THE PROBLEM: Every permutation-generated intervention is successful because agents can be activated sequentially in the prescribed order.Conversely, every successful intervention yields a permutation whose induced intervention is no more costly.
4. GENERAL BOUNDS FOR THE LCIS PROBLEM
The section develops graph-based bounds for the intervention cost by optimizing over activation permutations and introduces G-greedy permutations. The bounds coincide under specific unweighted-graph threshold conditions, with greedy permutations then optimal.
- The problem is reformulated as minimizing a permutation-dependent cost, while general bounds are stated for G-greedy permutations and the graph’s sink components.G-greedy permutations exist when no non-designated connected component is a sink component, and each satisfies C(σ) ≤ C_max(G,θ).
- The optimization can be normalized to unit intervention costs by replacing c with 1 and W with [c]W.The analysis therefore focuses on the resulting combinatorial optimization problem.
- The intervention cost has an energy interpretation: previously activated agents supply w_i(σ), while the remaining requirement (θ_iw_i − w_i(σ))+ is externally provided.Choosing an activation permutation determines the energy each agent receives at activation.
- Under the stated unweighted-graph conditions, C_min(G,θ) = C_max(G,θ) = ρ, and every G-greedy permutation is optimal.The condition is θ_iw_i ≤ 1 for every agent i.
- For a DAG, an activation order consistent with the graph gives zero intervention cost, so C*(G,θ) = 0.Each sink component is a singleton, yielding ρ = 0; the compatible total order satisfies w_k_t(σ) = w_k_t.
5. LCIS OVER UNDIRECTED GRAPHS
For undirected graphs, the paper derives conservation and reversal properties that simplify intervention-cost optimization, then gives exact optimal permutations for several basic topologies.
- 5. LCIS OVER UNDIRECTED GRAPHS: For every undirected weighted graph, the total received energy is permutation-invariant, and the total intervention energy equals total threshold energy minus that constant.These identities provide the conservation law underlying the undirected-graph analysis.
- 5. LCIS OVER UNDIRECTED GRAPHS: Reversing the activation order connects the cost for thresholds θ to the cost for complementary thresholds 1−θ.The connection follows from the relation between received energy under a permutation and its reverse.
- 5.1. Some basic examples.: On a line or ring with uniform thresholds θ≤1/2, the identity permutation is optimal; on a line, the first activation then triggers the remaining cascade without further intervention.For the line, C(id)=w1θ1=min{wiθi}.
- 5.1. Some basic examples.: When line or ring thresholds are heterogeneous, exact solutions are available when all thresholds lie on the same side of 1/2, with complementary-threshold reversal handling the all-above-half case.For all thresholds at most 1/2, permutations satisfying the stated condition are optimal; in particular, the identity is optimal exactly when w1θ1=min{wiθi}.
- 5.1. Some basic examples.: For unweighted trees satisfying θiwi≥1 for every node, minimum and maximum intervention costs coincide, so every G-greedy permutation is optimal.The equality uses W*=n−1 for trees.
- 5.1. Some basic examples.: For complete graphs, every permutation with thresholds in non-decreasing order is optimal, while any permutation violating that order cannot be optimal.Equal adjacent thresholds may be swapped without changing the cost.
6. CONCLUSION
The paper introduces and studies an optimal intervention problem for the linear threshold model. It identifies proving the resulting optimization problem’s complexity and developing approximation algorithms as future work.
- 6. CONCLUSION: The paper introduces and studies a new optimal intervention problem for the linear threshold model.
- 6. CONCLUSION: Future work includes formally proving the resulting combinatorial optimization problem’s complexity and developing approximation algorithms.