Groupwise Harmonic Centrality

• Groupwise Harmonic Centrality is a measure that quantifies a group’s collective reach by summing the inverse shortest-path distances from nodes outside the group.
• It addresses computational challenges with proven NP-hardness, submodularity, and tailored approximation algorithms such as greedy and local-search methods.
• The metric extends to stochastic influence models, enabling efficient scalability and parallelized implementations for real-world network analysis.

Groupwise Harmonic Centrality generalizes classic node-centric centrality by quantifying the collective reach of a group of nodes in a network, where the importance of nodes outside the group is inversely proportional to their shortest-path distances from the group. This measure is foundational in group centrality optimization, influence maximization, and diffusion analysis in static and probabilistic networks. Rigorous theoretical and practical advances have established its computational complexity, algorithmic approximability, and generalization to stochastic influence models (Angriman et al., 2020, Chen et al., 2018).

1. Formal Definition and Variants

Given a (possibly directed) graph $G = (V, E, \ell)$ with positive edge-lengths $\ell: E \to \mathbb{R}_{>0}$, let $S\subseteq V$ be a nonempty set of seed nodes. For each $v \in V \setminus S$, define the group-to-node distance as

$d(S, v) = \min_{u\in S} d(u, v),$

where $d(u, v)$ denotes the shortest-path length from $u$ to $v$ (set to $\infty$ if $v$ is unreachable from $\ell: E \to \mathbb{R}_{>0}$0). The groupwise harmonic centrality is

$\ell: E \to \mathbb{R}_{>0}$1

The reciprocal $\ell: E \to \mathbb{R}_{>0}$2 is interpreted as $\ell: E \to \mathbb{R}_{>0}$3 when $\ell: E \to \mathbb{R}_{>0}$4.

Properties include submodularity (diminishing marginal gains), normalization ($\ell: E \to \mathbb{R}_{>0}$5), and monotonicity in the deterministic (static) graph case (Chen et al., 2018). The $\ell: E \to \mathbb{R}_{>0}$6-sphere extension counts the number of nodes within radius $\ell: E \to \mathbb{R}_{>0}$7, interpolating between immediately adjacent sets and full reachability.

Influence-based group harmonic centrality is defined in stochastic or cascade models (such as independent cascade or linear threshold). For an influence instance $\ell: E \to \mathbb{R}_{>0}$8 and $\ell: E \to \mathbb{R}_{>0}$9, let $S\subseteq V$0 denote a random cascading activation sequence generated according to $S\subseteq V$1. For each $S\subseteq V$2, let $S\subseteq V$3 be the first time $S\subseteq V$4 is activated, or $S\subseteq V$5. Define the influence-based centrality as

$S\subseteq V$6

2. Computational Complexity and Inapproximability

The maximization problem — finding $S\subseteq V$7 with $S\subseteq V$8 maximizing $S\subseteq V$9 — is NP-hard even on unweighted, undirected or directed graphs. This is shown via reductions from Maximum Coverage (directed) and Dominating Set (undirected) (Angriman et al., 2020).

Approximation hardness results are as follows:

Graph Type	Inapproximability Bound
Directed	$v \in V \setminus S$0
Undirected	$v \in V \setminus S$1

No polynomial-time algorithm achieves a better approximation unless $v \in V \setminus S$2 (directed: reduction via set-cover bipartite digraphs giving exact correspondence; undirected: via gap-amplified dominating set reductions).

The function is submodular but not always monotone—adding nodes may reduce total centrality due to overlap in coverage, affecting some algorithmic guarantees (Angriman et al., 2020, Chen et al., 2018).

3. Approximation Algorithms

Greedy Algorithm

Despite non-monotonicity, a standard greedy approach is used for maximizing $v \in V \setminus S$3.

Algorithmic Steps:

1. Initialize $v \in V \setminus S$4.
2. While $v \in V \setminus S$5, compute for each $v \in V \setminus S$6 the marginal gain $v \in V \setminus S$7.
3. Select $v \in V \setminus S$8 and set $v \in V \setminus S$9.
4. Return $d(S, v) = \min_{u\in S} d(u, v),$0.

Let $d(S, v) = \min_{u\in S} d(u, v),$1 where $d(S, v) = \min_{u\in S} d(u, v),$2 and $d(S, v) = \min_{u\in S} d(u, v),$3 are respective minimum and maximum edge lengths. The guarantees are:

Graph Type	Guarantee (relative to OPT)
Directed	$d(S, v) = \min_{u\in S} d(u, v),$4
Undirected	$d(S, v) = \min_{u\in S} d(u, v),$5

Scaling from unweighted to weighted graphs reduces the guarantee by $d(S, v) = \min_{u\in S} d(u, v),$6. The analysis exploits surrogates and telescoping inequalities in the undirected case to obtain the relevant bounds (Angriman et al., 2020).

Local-Search Algorithm (Undirected)

For undirected graphs, local search (single-swap) is effective:

• Start from an initial $d(S, v) = \min_{u\in S} d(u, v),$7-set $d(S, v) = \min_{u\in S} d(u, v),$8 (e.g., greedy output).
• While there exist $d(S, v) = \min_{u\in S} d(u, v),$9 and $d(u, v)$0 such that $d(u, v)$1, perform the swap.

This yields a $d(u, v)$2-approximation, supported by submodularity and matroid exchange properties (Angriman et al., 2020).

Influence-Based Extension

For influence models, the reverse-reachable (RR) set framework efficiently approximates influence-based group centrality. For each sampled RR set $d(u, v)$3 from node $d(u, v)$4, the marginal contribution is $d(u, v)$5, and estimates are accumulated over many such sets for unbiased estimators. Relative-error guarantees can be made with appropriate number of samples, adapting the IMM framework (Chen et al., 2018).

4. Shapley-Value Decomposition and Uniqueness

Groupwise harmonic centrality can be interpreted as a cooperative-game value function on $d(u, v)$6. The Shapley value embedding decomposes the group score into individual node scores:

$d(u, v)$7

Equivalently, averaging over random orderings,

$d(u, v)$8

where $d(u, v)$9 is the set of nodes preceding $u$0 in permutation $u$1.

A characterization theorem establishes that influence-based group harmonic centrality is the unique Bayesian (linear in profile), permutation-invariant extension conforming to its deterministic (static-graph) definition. Layered-graph BFS instances form a basis for this uniqueness result (Chen et al., 2018).

5. Engineering and Scalability Considerations

Efficient maximization for large-scale graphs is achieved via several engineering techniques (Angriman et al., 2020):

• Lazy Marginal Evaluation: Maintain a heap of overestimates $u$2 to accelerate marginal computation; recompute exact values only when necessary (following the “lazy greedy” paradigm).
• Pruned SSSP: For candidate $u$3, a BFS or Dijkstra search from $u$4 is terminated at $u$5 when $u$6. Partial sums and upper bounds enable early pruning and avoid traversing the full graph.
• Distance Labels: Precompute and update $u$7 via multi-source BFS/Dijkstra per iteration. For local search, maintain the nearest and second-nearest group members for each $u$8 to speed up swap marginal calculations.
• Parallelization: Both greedy and local-search procedures benefit from multi-threaded computation. Marginals for different candidates, and swap tests in local search, can be evaluated in parallel. Reported scaling to 16–32 cores yields $u$9--$v$0 speedups.

In practice, these optimizations reduce per-iteration work from $v$1 to empirically $v$2–$v$3 (Angriman et al., 2020).

6. Experimental Results and Quality Benchmarks

Comprehensive experiments were conducted on (Angriman et al., 2020):

• Small (n$v$4): KONECT, DIMACS, and set-cover–derived graphs for comparison to ILP optima.
• Medium/Large (n=$v$5–$v$6, m$v$7): Complex networks.
• High-diameter weighted (n$v$8): Road networks.

Findings indicate:

• On small graphs, greedy and local-search are within $v$9 and $\infty$0 of optimum, respectively; random baselines achieve $\infty$1 of OPT.
• On large unweighted graphs, greedy yields a $\infty$2 improvement over random, local search adds $\infty$3 further quality at $\infty$4 higher runtime.
• For high-diameter weighted graphs, greedy achieves $\infty$5 the random baseline, with local-search again gaining $\infty$6 but at $\infty$7–$\infty$8 the runtime.
• Both methods demonstrate excellent parallel scaling.

7. Practical Guidance and Theoretical Significance

• Greedy algorithms are recommended for large-scale problems ($\infty$9), balancing runtime with near-optimal solution quality ($v$0 of optimum), with provable guarantees parameterized by $v$1.
• Local search is favored when highest solution quality is essential and computational resources are less constrained; most beneficial for moderate graphs ($v$2) or small groups ($v$3).
• Pruned search and lazy marginal updates dramatically improve scalability.
• Parallel implementation is highly effective and straightforward for both greedy and local search, justifying use on modern multicore architecture.

Groupwise harmonic centrality encapsulates a rigorous group-based notion of accessibility and influence, with robust theoretical foundations, unique influence-based characterization, and scalable maximization algorithms applicable in real-world network analytics (Angriman et al., 2020, Chen et al., 2018).