Eliminating Illusion in Directed Networks

Abstract: We study illusion elimination problems on directed social networks where each vertex is colored either red or blue. A vertex is under \textit{majority illusion} if it has more red out-neighbors than blue out-neighbors when there are more blue vertices than red ones in the network. In a more general phenomenon of $p$-illusion, at least $p$ fraction of the out-neighbors (as opposed to $1/2$ for majority) of a vertex is red. In the directed illusion elimination problem, we recolor minimum number of vertices so that no vertex is under $p$-illusion, for $p\in (0,1)$. Unfortunately, the problem is NP-hard for $p =1/2$ even when the network is a grid. Moreover, the problem is NP-hard and W[2]-hard when parameterized by the number of recolorings for each $p \in (0,1)$ even on bipartite DAGs. Thus, we can neither get a polynomial time algorithm on DAGs, unless P=NP, nor we can get a FPT algorithm even by combining solution size and directed graph parameters that measure distance from acyclicity, unless FPT=W[2]. We show that the problem can be solved in polynomial time in structured, sparse networks such as outerplanar networks, outward grids, trees, and cycles. Finally, we show tractable algorithms parameterized by treewidth of the underlying undirected graph, and by the number of vertices under illusion.

• The paper demonstrates that eliminating majority and p-illusions in directed networks is NP-complete, even under structural constraints like acyclicity and bipartiteness.
• It introduces tractable cases by designing polynomial-time algorithms for trees, cycles, and outerplanar graphs, and establishes fixed-parameter tractability under certain parameters.
• The study provides actionable insights for algorithm design in opinion dynamics and highlights challenges that motivate further exploration into non-classical parameterizations.

Eliminating Illusion in Directed Networks: Computational Complexity and Algorithms

Problem Formulation

The paper "Eliminating Illusion in Directed Networks" (2604.02395) addresses the algorithmic and complexity-theoretic aspects of mitigating the majority illusion and its generalization, $p$-illusion, in directed graphs. In this setting, vertices represent agents in social networks, colored red or blue depending on their state or opinion. The majority illusion occurs when a node perceives the local majority of its out-neighbors to differ from the actual global majority, due to structural biases in the directed network. The $p$-illusion further generalizes this by considering fractional thresholds for the out-neighbor color composition.

The central problems analyzed are:

• Difr: Given a directed graph $G$, an initial coloring, and integer $k$, determine whether at most $k$ vertices can be recolored so that no vertex is under majority illusion (i.e., more red than blue out-neighbors when blue is the global majority).
• $p$-Difr: Similar, but no vertex is allowed to be under $p$-illusion for a user-specified $p \in (0,1)$.

Both problems seek recolorings that restore an "illusion-free" state, minimizing the number of changes.

Main Results: Complexity Landscape

The paper provides a rigorous classification of the computational complexity of these manipulation problems under various structural constraints. The findings sharply contradict any intuition that acyclicity or bipartiteness might enable tractable solutions.

Hardness Results:

• Difr is NP-complete on directed grids. This holds even when the underlying undirected graph is a simple grid. The reduction from Planar Monotone Rectilinear 3SAT is constructive and establishes that spatially local interaction constraints do not eliminate combinatorial intractability.
• $p$-Difr is NP-complete and W[2]-hard parameterized by $k$ for all rational $p$0, even on bipartite DAGs. The reduction from Hitting Set demonstrates that neither bipartiteness nor acyclicity suffices for FPT tractability under standard parameterizations.
• Parameter-Based Hardness: The NP-hardness persists for bounded maximum deficiency, and the problem remains intractable even when standard digraph width parameters (feedback arc set, feedback vertex set, directed treewidth, etc.) are bounded.

These results collectively show that the natural algorithmic approaches (dynamic programming on acyclic structures, parameterized algorithms by solution size, or structural width parameters) are ineffective for exact solutions on general directed networks.

Efficient Algorithms: Tractable Cases

Despite the general intractability, the paper identifies and characterizes non-trivial graph classes and parameterizations admitting polynomial-time or FPT algorithms.

Polynomial-Time Cases:

• Trees and Cycles: The $p$1-Difr problem is polynomial-time solvable on both directed trees (arborescences) and underlying undirected cycles. A dynamic programming algorithm achieves $p$2 time for trees.
• Outward Grids and Outerplanar Graphs: On outward-oriented grids and $p$3-outerplanar graphs, efficient algorithms exist leveraging bounded treewidth and sparsity. In particular, the minimum recoloring task reduces to minimum vertex cover on bipartite graphs induced by local out-neighbor relationships, solvable in $p$4 time for $p$5 grids.

Parameterized and Approximation Algorithms:

• Fixed-Parameter Tractability: For graphs with treewidth $p$6 and bounded maximum deficiency $p$7, $p$8-Difr is FPT with runtime $p$9.
• FPT by Affected Set Size: An integer linear programming formulation yields $G$0 time, where $G$1 is the number of vertices initially under $G$2-illusion.
• PTAS: A Baker-style layering yields a polynomial-time approximation scheme on planar graphs, achieving a $G$3-approximation in $G$4 time.
• FPT via ILP Treedepth: The problem is also FPT parameterized by the treedepth of the primal or dual constraint matrices in the ILP encoding.

As a technical highlight, the paper formalizes a reduction of optimal recoloring to minimum hitting set on the neighborhoods of affected vertices, allowing direct transfer of classical parameterized complexity results and facilitating the ILP encoding.

Implications and Theoretical Insights

A key contribution of the work is the explicit demonstration that mitigation of local perception illusions in directed social networks is fundamentally harder than in undirected analogues—even under seemingly benign structural constraints. The presence of non-symmetric influence relations destroys many algorithmic shortcuts available in undirected models, as evidenced by the gap between the strong NP-completeness/W[2]-hardness results on (bipartite/acyclic) digraphs and prior polynomial-time or FPT results in undirected settings.

The findings have practical significance for the design of interventions in opinion dynamics: efficiently identifying minimal edits that neutralize perception biases is computationally prohibitive in general, unless network structure is exceptionally sparse or regular. In particular, algorithmic countermeasures to majority illusion or high-threshold $G$5-illusion in realistic directed social graphs are unlikely to be precisely optimal and will require approximation or heuristic approaches.

Theoretically, the work motivates further inquiry into non-classical parameterizations (e.g., by the number of influenced vertices, specific neighborhood profiles, or treedepth parameters arising from ILP encodings). The positive results on sparse and low-treewidth structures provide a foundation for the development of hybrid algorithms and conditional lower bounds.

Future Research Directions

Several open problems and directions are highlighted by the analysis:

• Kernelization and Tight FPT: Are there polynomial kernels for $G$6-Difr under joint treewidth and solution size parameterizations?
• Approximation Ratios in General Digraphs: What are the best achievable approximation ratios with or without explicit degree/width restrictions?
• Extended Structural Classes: Beyond outerplanar graphs and grids, which other subclasses of planar or near-planar digraphs admit efficient illusion-elimination schemes?
• Generalization to Multivalued States: Extension of the $G$7-illusion model to opinion vectors or hypergraph-based influence, beyond binary colorings.

Conclusion

The paper comprehensively elucidates the computational boundaries of the illusion elimination problem in directed networks. The work characterizes complexity transitions across graph classes, constructs explicit reductions demonstrating hardness, designs specialized algorithms for tractable instances, and formalizes new parameterized and approximation approaches. The rigorous negative results underscore the need for careful structural or parameter-based choices in algorithm design for network interventions, and the identified positive cases offer actionable strategies for certain practical settings. The theoretical framework and algorithmic tools developed here are poised to inform both further research in computational social choice and practical applications in controlling information diffusion and belief formation in complex networks.