Fair Centrality Maximization

• The paper introduces fair centrality maximization by modeling fairness via Group Shapley values and multiobjective submodular optimization to balance network influence.
• It leverages scalable greedy and LP-based sampling methods that achieve provable approximations despite strong computational hardness results.
• Empirical evaluations demonstrate that methods like LP-Greedy improve min-group centrality by 5–10% while requiring significantly fewer computations than traditional approaches.

Fair centrality maximization encompasses algorithmic paradigms and optimization frameworks that select network interventions (such as seed nodes or edges) to improve node centralities equitably across demographic or attribute groups, or to ensure robustness to unknown preexisting conditions. The defining feature is the pursuit of objective functions that explicitly model fairness among groups or across uncertainty sets, as opposed to classical centrality maximization, which may amplify structural disparities. Recent advances leverage the Group Shapley value from cooperative game theory and multiobjective submodular maximization to formulate and solve such problems, with theoretical hardness results and scalable algorithms emerging in both the influence maximization and shortest-path domains (Becker et al., 2020, Spaeh et al., 14 May 2025).

1. Conceptual Foundation and Problem Statements

Fair centrality maximization formalizes interventions that yield centrality gains distributed fairly with respect to uncertainty or group structure. The primary settings are:

• Unknown pre-existing seeds: Given a directed graph $G=(V,E)$, select a seed set $S$ of at most $k$ nodes to maximize influence, measured fairly in the presence of an unknown initial set of seeds $T\subseteq V\setminus S$. This models real-world scenarios where the status of some influencers or informed nodes is not fully known (Becker et al., 2020).
• Group-centered fairness: Given a partition of nodes into groups (e.g., by attribute or “color”), select interventions (such as new in-edges to a target node) that maximize the worst-case centrality improvement across all groups. This reflects a desire to equalize algorithmic benefits or access to central positions (Spaeh et al., 14 May 2025).

These formulations depart from single-function maximization by embedding fairness objectives—either via expectation (Group Shapley) or minimax (multiobjective)—directly into the optimization problem.

2. Group Shapley Value and Influence-based Centrality

The Group Shapley value generalizes the classical Shapley value to measure a coalition’s collective marginal contribution over all possible disjoint coalitions in a cooperative game. For influence-based centrality in a network:

Given $G=(V,E)$ under the Independent Cascade (IC) or Triggering model, define the influence spread function $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$. The pair $(V,\sigma)$ defines a cooperative game: the payoff to coalition $S$ is $\sigma(S)$, known to be monotone and submodular.

The Group Shapley value $\varphi(S)$ for $S$0 is

$S$1

An equivalent characterization: in a random permutation of $S$2 plus a "supernode" representing $S$3, $S$4 is the expected marginal gain of $S$5 over predecessor seed sets.

The corresponding Max-Shapley-Group problem seeks $S$6 with $S$7 maximizing $S$8—which reflects $S$9’s expected marginal influence “fairly,” i.e., over all possible (random) disjoint pre-existing seed sets (Becker et al., 2020).

3. Hardness and Algorithmic Results: Influence-based Fairness

Fair centrality maximization under the Max-Shapley-Group framework faces strong computational barriers. Assuming the Gap Exponential Time Hypothesis, no polynomial-time algorithm can approximate the optimum within $k$0, implying that for large $k$1 a polynomial factor is the best asymptotic worst-case ratio achievable (Becker et al., 2020).

Despite this, a polynomial-time greedy algorithm achieves a $k$2-approximation for any $k$3 when $k$4 is small. The algorithm uses the following insights:

• Reverse-Reachable (RR) set characterization: Let $k$5 be a random RR set. The Group Shapley value can be written as

$k$6

• Empirical surrogate estimation: With $k$7 sampled RR-sets,

$k$8

uniformly approximates $k$9 up to $T\subseteq V\setminus S$0 for all $T\subseteq V\setminus S$1.

• Reduction to Harmonic-Max-Hitting-Set: Maximizing $T\subseteq V\setminus S$2 is equivalent (up to a factor $T\subseteq V\setminus S$3) to maximizing the (monotone submodular) sum

$T\subseteq V\setminus S$4

which is optimized via the standard greedy approach.

For constant $T\subseteq V\setminus S$5, this yields a practical and scalable method; for larger $T\subseteq V\setminus S$6, the guarantee deteriorates as $T\subseteq V\setminus S$7. Whether this barrier can be overcome by structural graph properties or alternate group values (e.g., Group-Banzhaf) is an open question (Becker et al., 2020).

4. Multiobjective Submodular Maximization and Group Fairness

Alternative models address fair centrality via multiobjective submodular maximization. Given a directed network $T\subseteq V\setminus S$8, designated target node $T\subseteq V\setminus S$9, and partitioning of $G=(V,E)$0 into $G=(V,E)$1 groups (colors), the task is to add at most $G=(V,E)$2 new in-edges for $G=(V,E)$3 to maximize the minimum group centrality. Each group $G=(V,E)$4 is associated with the function

$G=(V,E)$5

the average reciprocal distance over group $G=(V,E)$6.

The problem: $G=(V,E)$7 where $G=(V,E)$8 is the set of possible new edges.

Each $G=(V,E)$9 is nonnegative, monotone, and submodular. The joint minimax objective embeds group-fairness directly: interventions are evaluated by their worst-case effect across all groups, contrasting sharply with maximizing aggregate or average centrality (Spaeh et al., 14 May 2025).

5. Scalable LP-based Sampling-Greedy Algorithms

Classical multiobjective submodular maximization approaches often rely on multilinear extension evaluations, which are computationally prohibitive. The LP-based sampling-greedy algorithm of Spaeh & Miyauchi circumvents this by working directly with discrete marginals (Spaeh et al., 14 May 2025):

• In each round, given the current selection $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$0, solve the linear program:

$$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$1

• Sample $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$2 according to $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$3 and add it to $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$4.
• Repeat for $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$5 rounds and run $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$6 repetitions for high confidence; return the best set obtained.

No multilinear extensions or continuous relaxations are involved, enhancing scalability.

For $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$7 (with $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$8), the algorithm outputs $$\sigma(S) = \mathbb{E}[\lvert\text{Activated nodes at termination under seed set }S\rvert]$$9 of size $(V,\sigma)$0 such that $(V,\sigma)$1 with high probability. Each iteration is efficiently solvable using a polynomial number of multiplicative weights updates (MWU) and lazy marginal evaluations (Spaeh et al., 14 May 2025).

6. Empirical Evaluation and Practical Implications

Experimental results in (Spaeh et al., 14 May 2025) assess LP-Greedy and several baselines on Amazon co-purchasing networks, measuring the fair harmonic centrality (minimum across all groups) at various budget levels. Key comparative points:

• Efficiency: LP-Greedy achieves higher min-group centrality (often by 5–10% over bi-criteria or round-robin heuristics), with 2–5$(V,\sigma)$2 fewer marginal-gain evaluations compared to previous MWU-based algorithms.
• Scalability: On graphs with up to $(V,\sigma)$310,000 nodes, LP-Greedy runs in seconds to tens of seconds, outperforming methods that invoke multilinear extensions.
• Fairness dynamics: As intervention budget $(V,\sigma)$4 increases, worst-group centrality improves smoothly; notably, the method delivers uniform centrality improvements across color groups.
• Representative result: On Arts, Crafts & Sewing graph ($(V,\sigma)$5, $(V,\sigma)$6), min-group centrality at $(V,\sigma)$7 was 0.38 (LP-Greedy), compared to 0.31–0.33 for leading baselines; marginal evaluations to $(V,\sigma)$8 were 30,000 (LP-Greedy) versus 150,000 (Udwani MWU).

This demonstrates that theoretical rigor in fairness-oriented objectives does not entail prohibitive computational overhead, and may yield superior fairness-performance trade-offs in practice (Spaeh et al., 14 May 2025).

7. Open Directions and Theoretical Challenges

Major avenues for future research in fair centrality maximization include:

• Algorithmic gaps: Breaking the $(V,\sigma)$9-scaling barrier in the influence-based Max-Shapley-Group approximation, possibly via alternative group values or exploiting specific graph structures, remains unresolved.
• Bi-criteria and relaxations: Investigation into near-optimal solutions under relaxed constraints—either larger intervention sets or relaxations of the fairness utility—may yield practical approximations even under the established computational hardness.
• Empirical systematics: Systematic evaluation on real and synthetic networks, especially under highly skewed group memberships or pre-existing seed conditions, is needed to assess robustness across application domains.
• Generalization to other centrality measures: Extending the fairness paradigm from influence-based and harmonic centralities to additional network centrality concepts may amplify the reach of these methods.

These directions highlight the deep interplay between combinatorial optimization, submodular analysis, and social network fairness in the ongoing evolution of fair centrality maximization (Becker et al., 2020, Spaeh et al., 14 May 2025).