Multilayer Betweenness Centrality

• Multilayer betweenness centrality generalizes classic betweenness by incorporating inter-layer transitions and temporal aspects to quantify node brokerage in complex networks.
• It employs tensorial and supra-adjacency frameworks with extensions of the Brandes algorithm to compute shortest-path, random-walk, and Pareto-optimal centrality measures across multiple layers.
• This approach is instrumental in revealing hidden influencers in systems like social, transportation, and biological networks by addressing spurious shortcuts found in aggregated graphs.

Multilayer betweenness centrality generalizes classic concepts of centrality within networks to the setting of multilayer, multiplex, or multidimensional complex networks. In such systems, nodes interact via multiple types of relationships, or over multiple temporal slices, or within multiple ontology domains, each represented as a separate network layer. Betweenness centrality in the multilayer context quantifies the extent to which nodes or node-layer tuples act as brokers or conduits for flow—be it information, physical transport, or other processes—across all available paths and layers. Several rigorous mathematical frameworks have been developed to define, compute, and interpret multilayer betweenness centralities, including shortest-path, random-walk, Pareto-optimal, and occupation-based formalisms. The precise meaning and computation of the metric depend crucially on how the concept of “path,” “distance,” and “inter-layer transition” are defined in the multilayer model.

1. Mathematical Foundations and Key Definitions

The formalization of multilayer betweenness centrality is rooted in the extension of path, walk, and distance concepts to multilayer or multiplex structures. The network is often specified by a rank-4 adjacency tensor $M^{i\alpha}_{j\beta}$, where $i,j \in \{1,\ldots,N\}$ enumerate physical nodes and $\alpha,\beta \in \{1,\ldots,L\}$ index layers. Each nonzero entry gives the (possibly weighted) existence of an edge from $(i,\alpha)$ to $(j,\beta)$. The multilayer setting permits both intra-layer ($\alpha = \beta$) and inter-layer ($\alpha \neq \beta$) edges, the latter typically encoding switching between relationship types, time slices, or ontologies (Domenico et al., 2013).

Betweenness centrality $B_j$ at the node level is usually defined via the aggregation of a more granular state-node betweenness $B^{j\beta}$ over the $L$ replicas of node $i,j \in \{1,\ldots,N\}$0: $i,j \in \{1,\ldots,N\}$1. The combinatorial or probabilistic basis for $i,j \in \{1,\ldots,N\}$2 varies by paradigm, as detailed in later sections.

Several variants coexist:

• Shortest-path multilayer betweenness: Counts the fraction of geodesic (minimum-length) paths—defined on the expanded supra-graph of state nodes—which pass through a given node or node-layer tuple (Domenico et al., 2013).
• Random-walk multilayer betweenness: Evaluates the expected visitation frequency of a node or node-layer by a random walker navigating the multilayer structure, often under absorbing boundary conditions (Solé-Ribalta et al., 2015, Böttcher et al., 2020).
• Pareto-optimal multilayer betweenness: Employs a notion of non-dominated, layerwise-optimal paths, capturing the full combinatorial trade-off across path lengths in different layers (Magnani et al., 2013).
• Temporal/multiaspect betweenness: Generalizes further to networks with temporal or higher-order aspects, where only “realizable” paths—consistent with all structural constraints—are counted (Wehmuth et al., 2020, Zaoli et al., 2020).

2. Tensorial and Supra-adjacency Frameworks

Most rigorous treatments of multilayer betweenness employ a tensorial or supra-matrix representation:

• The adjacency tensor $i,j \in \{1,\ldots,N\}$3 is flattened into an $i,j \in \{1,\ldots,N\}$4 supra-adjacency matrix ordering all state-nodes $i,j \in \{1,\ldots,N\}$5 (Domenico et al., 2013, Solé-Ribalta et al., 2015, Böttcher et al., 2020).
• Path enumeration, shortest-distance, and dependency calculations are then performed analogously to the monoplex case, but at the level of state-nodes, with inter-layer transitions incorporated via appropriate entries in $i,j \in \{1,\ldots,N\}$6 or cost functions.
• The Brandes algorithm, originally developed for efficient calculation of classic betweenness, extends naturally: for each state-node source, shortest paths (BFS or Dijkstra) are computed on the supra-graph, storing predecessor sets and path counts, followed by dependency accumulation and contraction over layers if a node-level metric is desired (Domenico et al., 2013).
• Computational complexity grows as $i,j \in \{1,\ldots,N\}$7 for unweighted networks, as each of the $i,j \in \{1,\ldots,N\}$8 state-nodes is used as a source, and the supra-graph is typically sparse if the underlying layers and their couplings are sparse.

The tensorial framework is especially crucial for disambiguating between path realizations that stay within a layer and those that switch layers, since naive aggregation of layers would either overestimate (by introducing spurious paths) or underestimate (by ignoring cross-layer pathways) true flow potential (Domenico et al., 2013, Wehmuth et al., 2020).

3. Paradigms of Multilayer Betweenness Centrality

a) Shortest-path–based Multilayer Betweenness

In the direct extension of Freeman’s betweenness, multilayer betweenness of a state-node $i,j \in \{1,\ldots,N\}$9 is

$\alpha,\beta \in \{1,\ldots,L\}$0

where $\alpha,\beta \in \{1,\ldots,L\}$1 is the dependency of ordered pair $\alpha,\beta \in \{1,\ldots,L\}$2 on $\alpha,\beta \in \{1,\ldots,L\}$3, defined via the ratio of the number of shortest paths passing through $\alpha,\beta \in \{1,\ldots,L\}$4 to the total number of shortest paths connecting $\alpha,\beta \in \{1,\ldots,L\}$5 and $\alpha,\beta \in \{1,\ldots,L\}$6 (Domenico et al., 2013). Node-level scores $\alpha,\beta \in \{1,\ldots,L\}$7 are obtained by summing over $\alpha,\beta \in \{1,\ldots,L\}$8.

This framework naturally reduces to classic betweenness for $\alpha,\beta \in \{1,\ldots,L\}$9, and emphasizes the multiplicity of available pathways due to layer-switching (or multiplexity).

b) Random-walk–based Multilayer Betweenness

Random-walk betweenness quantifies the expected traversal load rather than strictly minimal paths. The discrete-time, tensorial generalization is given by (Solé-Ribalta et al., 2015): $(i,\alpha)$0 where $(i,\alpha)$1 is the absorbing transition operator (all transitions into $(i,\alpha)$2 in any layer are set to zero), $(i,\alpha)$3 is the fundamental matrix counting expected pre-absorption visits, and $(i,\alpha)$4 are all-ones vectors used for aggregation and averaging over layers and nodes.

Continuous-time variants with occupation-resolvent formalism and even quantum extensions have also been introduced, involving the supra-Laplacian and occupation statistics of walkers with either diffusive or coherent transport (Böttcher et al., 2020). The classical random-walk multilayer betweenness reduces, in the monoplex case, to current-flow (electrical) betweenness.

c) Pareto-optimal and Multi-criteria Betweenness

In systems where edge or path costs are vectors (e.g., hop-counts per layer), shortest paths are replaced by the full Pareto front of walks not strictly dominated in all cost components. Multilayer betweenness $(i,\alpha)$5 is defined as the sum, over all node pairs $(i,\alpha)$6, of the fraction of Pareto-optimal $(i,\alpha)$7 walks that traverse $(i,\alpha)$8 (Magnani et al., 2013): $(i,\alpha)$9 where $(j,\beta)$0 is the Pareto-frontier of walks from $(j,\beta)$1 to $(j,\beta)$2. This formalism captures nuanced multilayer brokerage roles, but the size of Pareto sets can grow exponentially with the number of layers.

d) Temporal, Knowledge-graph, and Multiaspect Generalizations

Further generalizations model “composite vertices” that combine node, layer, and time aspects. Betweenness on such MultiAspect Graphs (MAGs) requires counting only valid (realizable) paths in the full multidimensional structure to avoid spurious shortcuts that arise in naive aggregate graphs (Wehmuth et al., 2020). In temporal multiplexes, generalized distance metrics incorporate convex combinations of topological, inter-layer, and temporal costs, supporting path-finding and betweenness computation sensitive to temporal ordering and switching penalties (Zaoli et al., 2020).

4. Algorithmic Strategies and Computational Complexity

Efficient calculation relies on extensions of Brandes’ algorithm. In the classic supragraph approach (Domenico et al., 2013):

• For each source state-node $(j,\beta)$3, shortest-path distances and counts to all reachable $(j,\beta)$4 are computed.
• Predecessor lists and sigma-path counters are maintained per state-node.
• Dependency accumulation propagates these counts to intermediate nodes, yielding $(j,\beta)$5.

When using Pareto-optimal or multiaspect models, a multi-objective Dijkstra or BFS retains layer-specific cost vectors at each node. The worst-case complexity becomes exponential in the number of layers for multi-criteria variants (Magnani et al., 2013).

For random-walk betweenness, linear algebraic approaches are prevalent: in the tensorial formalism, inversion of the fundamental matrix $(j,\beta)$6 for each absorbing node $(j,\beta)$7 (of size $(j,\beta)$8) is required (Solé-Ribalta et al., 2015). These can be parallelized or, for very large systems, approximated using iterative solvers.

Occupation-resolvent and quantum frameworks entail solving dense or sparse linear systems and may require spectral decomposition per absorbing state, at cubic (or worse) cost in system size (Böttcher et al., 2020). Approximations or block-sparse exploitation are necessary for scalability.

5. Comparison to Mono-layer Betweenness and Aggregated Methods

The multilayer betweenness framework subsumes standard node betweenness as the special case $(j,\beta)$9. However, naive aggregation—collapsing all layers into a single weighted or unweighted graph—can misestimate centrality rankings and functional roles:

• Aggregated graphs can admit spurious “shortcut” paths that violate true layer- or time-consistency (Wehmuth et al., 2020).
• In multilayer formulations, both the overall number of shortest paths and their layer distributions can change substantially, leading to higher or lower centralities for particular nodes depending on their cross-layer connections (Domenico et al., 2013, Magnani et al., 2013).

Empirical studies show that absolute values of multilayer betweenness are typically larger and ranking shifts can be significant for actors who act as “super-brokers” across contexts. In social networks, this reveals previously “hidden” influencers only identifiable when inter-layer pathways are respected (Magnani et al., 2013, Domenico et al., 2013).

The following table synthesizes key distinctions:

Variant	Layer/Aspect Modeling	Path Types	Major Algorithmic Step
Shortest-path supra-graph	Full tensor/supra	State-node geodesic	Brandes on $\alpha = \beta$0 nodes
Random-walk fundamental matrix	Full tensor/supra	All walks	Linear system solves per destination
Pareto-optimal/Multi-criteria	Multi-dimensional	Pareto-optimal walks	Multi-objective BFS/Dijkstra
MultiAspect Graph (MAG)	Composite vertices	Valid multidim. paths	Aggregated Brandes per subdetermination
Temporal multiplex	Static expanded graph	Time-respecting paths	Dijkstra on $\alpha = \beta$1 vertices

6. Impact, Empirical Observations, and Limitations

Multilayer betweenness centrality is critical in identifying brokers whose roles involve traversing multiple relational or temporal contexts. Key empirical findings include:

• In real-world social or transportation multiplexes, multilayer betweenness highlights actors obscured in any single projection or aggregate (Magnani et al., 2013, Domenico et al., 2013).
• Temporal or multidimensional betweenness can diverge strongly from static or per-layer metrics, affecting node rankings and revealing the impact of inter-layer coupling and time-ordering (Zaoli et al., 2020).
• The topological nature of the base network mediates sensitivity: small-world networks are highly unstable to multilayer augmentation, while scale-free networks sustain more robust betweenness rankings (Dörpinghaus et al., 2022).

Limitations include:

• Computational complexity—both in time and space—especially acute for Pareto-based and occupation-resolvent approaches.
• Exponential blow-up of Pareto fronts with layer/criteria number (Magnani et al., 2013).
• In many empirical settings, ranking changes only affect a minority of nodes, suggesting redundancy across social layers or redundancy in infrastructure (Magnani et al., 2013, Domenico et al., 2013).

7. Open Problems and Future Directions

Several challenges and research directions remain:

• Development of efficient or approximate algorithms for Pareto-front maintenance and occupation-based centralities on large multilayer networks (Magnani et al., 2013, Böttcher et al., 2020).
• Incorporation of edge weights, inter-layer switching penalties, and time-respecting constraints, as well as principled approaches for compression or sampling of massive layer sets (Magnani et al., 2013, Zaoli et al., 2020).
• Systematic evaluation of the effect of inter-layer coupling strength and multiplex structure on centrality distributions and network robustness (Solé-Ribalta et al., 2015).
• Release and curation of large, multilayer benchmark datasets for empirical validation.
• Analytical investigation of the relationship between multilayer betweenness, community structure, and other multilayer diagnostics (e.g., motif analysis, multilayer modularity) (Magnani et al., 2013, Domenico et al., 2013).
• Quantification and mitigation of “spurious path” artifacts in practical centrality computation (Wehmuth et al., 2020).

A plausible implication is that as multilayer representations become standard in diverse fields (from social to biological to knowledge networks), robust and accurate multilayer centrality measures will be increasingly central to the quantitative analysis and design of complex systems.