Direction-Sensitive Centrality Measures

• Direction-sensitive centrality measures are techniques that quantify node influence in networks with directional and weighted interactions, emphasizing mediation, broadcasting, and propagation.
• Various methods like quasi-centrality, L1 prominence, Laplacian influence, bow-tie centrality, viral centrality, and spectral approaches extend traditional metrics to capture asymmetric relationships.
• These measures enhance network analysis in fields such as trade, communication, and biological regulation by revealing hidden influence and resilience under node or edge perturbations.

Direction-sensitive centrality measures quantify the importance or influence of nodes and edges in complex networks while explicitly accounting for the directionality and weight of interactions. Unlike traditional centrality concepts designed for undirected graphs, direction-sensitive measures reveal roles such as mediation, broadcasting, reception, and propagation pathways within asymmetric and weighted systems. These measures are essential for analyzing systems where interactions are fundamentally non-reciprocal, such as trade, communication, biological regulation, and citation networks.

1. Conceptual Foundations and Rationale

Direction-sensitive centrality arises from the need to capture aspects of network importance that depend upon directional flows or asymmetric relationships. Traditional centrality scores—such as degree, betweenness, closeness, eigenvector centrality, and undirected PageRank—either disregard direction or only apply trivially to directed settings. This can obscure “shock source” nodes in supply chains, super-spreaders in epidemic graphs, or pivotal brokers in directed knowledge or social networks (He, 2022).

The primary methodological challenge is to extend centrality to respect (i) edge directionality, (ii) link weights, and (iii) the effect of node/edge removal or perturbation on global topological or dynamical properties, including the non-local propagation of changes.

2. Key Families of Direction-Sensitive Centrality Measures

The following table summarizes principal methods by their core principles and distinguishing features:

Measure Type	Brief Description	Notable Attributes
Quasi-centrality	Change in persistent homology upon node deletion	Topological, non-local, handles directed graphs
L1 Prominence	L1-depth-median analogues for directed graphs	Geodesic-sum based, tunable locality
Laplacian Influence	Stationary density of random walker with teleportation	Absorption/interpolation between roots and periphery
Bow-Tie Centrality	Corrected eigenvector-style, cycle-robust	Incorporates node properties and cycles
Viral Centrality	Analytical solution rooted in directed cascade models (ICM)	Explicitly process-driven, accurate for non-cyclic spread
Spectral (HITS, SVD)	Coupled SVD for hub/authority roles	Multi-component, captures directional modes

Quasi-centrality (He, 2022)

Quasi-centrality evaluates, for each node, the difference in 0-dimensional persistent homology (over the Dowker sink complex) between the original network and the node-deleted subnetwork, augmented with the minimal effective distance from the node. This incorporates global propagation and bridging roles, outperforming simple in/out degree and PageRank/Katz/HITS in detecting nodes whose removal causes non-trivial reconfiguration of network topology in directed, weighted settings.

L1 Prominence Measures (Kang et al., 22 Aug 2024)

L1 prestige and L1 centrality generalize the data-depth concept to strongly connected, directed graphs by minimizing weighted sums of shortest-path lengths under a symmetry correction. These measures distinguish “receiving” and “giving” prominence and permit a tunable locality parameter, enabling multiscale structural analysis. They are robust to edge/vertex weighting and accommodate both local and global views.

Laplacian-based Influence (Masuda et al., 2010)

This approach computes a stationary density for a continuous-time random walker with teleportation on the link-reversed network, yielding a centrality interpolating between dominance by root nodes and uniformity. The method uniquely incorporates dynamics, interpolates with a parameter (teleport rate), and is robust to arbitrary node/edge weights and graph structure.

Bow-Tie Centrality (Glattfelder, 2019)

The bow-tie centrality corrects for cycle-induced inflation by analytic rescaling at the matrix level, ensuring balanced representation of nodes in “core,” “in,” and “out” sections of bow-tie topologies. This measure integrates intrinsic node properties, handles weights and directions, and sits between naive path-counting (access centrality) and simple-path indices. The matrix formulation enables efficient computation.

Viral Centrality (Fink et al., 2023)

Specifically tailored for the independent cascade model (ICM) on weighted, directed graphs, Viral Centrality computes the expected reach from a node by closed-form summation of path contributions—accurate when cycles are weak (small spectral radius). This method approximates the true influence in spreading processes, outperforming weighted degree, PageRank, and Katz for realistic transmission networks.

Spectral Hub/Authority Decomposition (Sciarra et al., 2018)

Spectral centralities in directed networks emerge as low-rank approximations to the adjacency matrix via singular value decomposition. The hub (left singular) and authority (right singular) components correspond to directional outflow and inflow roles. Multi-component (truncated SVD) analyses enable discovery of distinct functional modes not captured by single-scalar scores, providing richer representations for strongly directional networks.

3. Mathematical Formulation and Computation

Quasi-centrality computes for node $x$: $$C(x) = \sum_{b \in P_0(f(\gamma(G), x))} l(b) - \sum_{b \in P_0(\gamma(G))} l(b) + \mu(x)$$ where $P_0$ denotes the set of 0-dimensional persistence barcodes, $l(b)$ their length, and $\mu(x)$ the minimal effective sink-distance (He, 2022).

L1 Prestige, for vertex $v_k$: $$\mathrm{Pres}(v_k) = 1 - c(G) \cdot \max_{j \neq k} \left\{ \frac{ \sum_{i=1}^n \eta_i (d(v_i, v_k) - d(v_i, v_j)) }{ \eta\cdot d(v_k, v_j) } \right\}^+$$ with $c(G) = \min_{i \neq j} d(v_i,v_j)/d(v_j,v_i)$, and $d$ the (directed) geodesic distance matrix (Kang et al., 22 Aug 2024).

Laplacian Influence solves $(L + q I) v = (q/N)\mathbf{1}$, where $L$ is the continuous-time Laplacian of the link-reversed network and $q$ the teleport rate (Masuda et al., 2010).

Bow-Tie Centrality is given by: $\zeta = \overline{W} v = W W^* v$ where $W$ is the adjacency matrix, $v$ the node property vector, and $W^*$ is built from the analysis of $(I - W)^{-1}$ and subsequent rescaling/augmentation for cycle correction (Glattfelder, 2019).

Viral Centrality for node $j$: $$\mathrm{VC}_j = \left[ (I - \mathcal{P})^{-1} \mathcal{P} \mathbf{1} \right]_j$$ where $\mathcal{P}_{ij}$ is the edge transmission probability in the ICM (Fink et al., 2023).

HITS/Authority-Hub:

$$A a = \sigma_1 h,\quad A^T h = \sigma_1 a,\quad \|h\| = \|a\| = 1$$

The main singular vectors $(h,a)$ capture, respectively, hub and authority scores; higher-rank SVD extensions admit richer directional role decomposition (Sciarra et al., 2018).

Computation typically reduces to linear algebraic routines: Cholesky or sparse solvers for Laplacian-based and bow-tie measures, union-find algorithms for persistent homology in quasi-centrality, and Dijkstra or Floyd-Warshall for L1 and shortest-path-based measures.

4. Algorithmic Properties and Limitations

Algorithmic complexities depend on the measure: L1 prominence and quasi-centrality incur $O(n^3)$ runtimes at scale due to all-pairs shortest-path or homology calculations (He, 2022, Kang et al., 22 Aug 2024). Bow-tie and Laplacian-influence measures require inversion or solution of sparse linear systems, which is tractable for large graphs with modern numerical methods. Viral Centrality may operate in closed form when $\rho(\mathcal{P}) < 1$; otherwise, iterative or truncated summation suffices (Fink et al., 2023).

Quasi-centrality and TDA-based measures are computationally limited beyond $n \simeq 80$ due to memory and processing bottlenecks (He, 2022). L1 prominence measures achieve multiscale analysis at significant, though manageable, cost for networks up to several hundred nodes (Kang et al., 22 Aug 2024). Approximations and parallelization are active research directions.

5. Interpretative and Practical Impact

Direction-sensitive centrality measures have demonstrated utility in a range of empirical contexts:

• Trade networks: Quasi-centrality uncovers systemically critical supply nodes missed by classic in/out-centrality (He, 2022).
• Urban mobility: L1 prestige and centrality distinguish structural hierarchy in city-scale flow networks, identifying commuter vs. commercial hubs and periphery “outliers” (Kang et al., 22 Aug 2024).
• Spreading processes: Viral Centrality quantitatively matches ICM outcomes for epidemic and information diffusion, reliably ranking spreaders and outperforming degree/PageRank in both simulated and real-world networks (Fink et al., 2023).
• Economic networks: Bow-tie centrality gives cycle-corrected evaluation of relevance integrating both network topology and intrinsic node values, supporting value-flow analytics that are unattainable with HITS or PageRank (Glattfelder, 2019).
• Multi-modal functional decomposition: Multi-component HITS/SVD-based methods reveal subcommunity hub/authority roles that single-component spectral scores conflate (Sciarra et al., 2018).

6. Comparative Analysis and Theoretical Distinctions

Several salient contrasts emerge across the measures:

• Quasi-centrality and L1 prominence directly quantify effects of node/edge removal on global connectivity, bridging local and non-local criteria.
• Spectral (HITS/SVD, Laplacian influence, bow-tie) methods are grounded in matrix decompositions, capturing recurrent flows and systemic eigenmodes.
• Viral Centrality is uniquely process-level, with accuracy limitations set by the prevalence of cycles ($\lambda_{\max}\ll1$ requirement).
• L1 prominence and bow-tie approaches integrate weight asymmetry and adjustable scale or context—features missing from classical eigenvector, PageRank, or degree-based metrics.
• TDA-based and absorption-probability measures provide robust handling of strongly asymmetric, weighted, and non-reciprocal topologies.

A summary:

Measure	Incorporates Edge Direction	Handles Cycles Robustly	Distinguishes In/Out/Broadcast Roles	Handles Edge Weights	Captures Global Propagation
Quasi-centrality	Yes	Yes	Yes	Yes	Yes
L1 Prominence	Yes	N/A (Shortest Paths)	Yes	Yes	Yes (scale-tunable)
Laplacian Influence	Yes	Yes	Partially	Yes	Yes
Bow-Tie Centrality	Yes	Yes	Yes	Yes	Yes
Viral Centrality	Yes	Only if $\lambda_{\max} \ll 1$	Partially	Yes	Yes (spread dynamics)
Spectral (HITS, SVD)	Yes	Yes	Yes	Yes	Yes

7. Open Problems and Future Directions

The computational tractability of topological centralities in large directed networks remains a significant challenge (He, 2022). Theoretical stability and robustness, particularly for measures based on homological or absorption frameworks, are not fully characterized, especially under small edge or node perturbations. Extensions to group centrality via higher-dimensional persistence or multiparameter TDA are undeveloped but may yield richer models of group mediation and control.

Further, no unique “gold standard” exists for direction-sensitive centrality. Different measures are suited for different networks and application demands: propagation versus structure, local versus global, weighted versus unweighted, or process-specific versus process-agnostic. Ongoing research seeks to clarify theoretical relationships, computational efficiency, and domain-specific effectiveness among these diverse approaches.