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Eigenvector centrality of nodes in multiplex networks
Luis Sola, Miguel Romance, Regino Criado, Julio Flores, Alejandro Garcia del Amo, Stefano Boccaletti
TL;DR
Eigenvector centrality must be adapted to multiplex networks because node importance can propagate through multiple interactions and layers with unequal roles. The paper introduces several scalar and vectorial eigenvector-like centralities, establishes their existence and uniqueness under reasonable conditions, and finds that they yield qualitatively different rankings whose correlations depend on multiplex structure and inter-layer influence.
Problem
The paper addresses how to measure node centrality while accounting for interactions across multiple layers that may have different importance.
Method
The paper introduces several eigenvector centralities for multiplex networks, including uniform, independent-layer, local heterogeneous, and global heterogeneous measures, using influence among layers.
Results
The proposed centrality measures exist and are unique under reasonable conditions, and experiments show substantially different results and qualitatively different rankings on the same multiplex networks.
Takeaways & Limitations
The appropriate centrality measure should be selected carefully for each multiplex network because different measures arise from different heuristics and their correlations depend on network structure.
Abstract
from arXiv · showhide
We extend the concept of eigenvector centrality to multiplex networks, and introduce several alternative parameters that quantify the importance of nodes in a multi-layered networked system, including the definition of vectorial-type centralities. In addition, we rigorously show that, under reasonable conditions, such centrality measures exist and are unique. Computer experiments and simulations demonstrate that the proposed measures provide substantially different results when applied to the same multiplex structure, and highlight the non-trivial relationships between the different measures of centrality introduced.
I. NOTATIONS
The paper represents a multiplex network as multiple directed or undirected, unweighted layers sharing the same nodes. It also introduces a projection network and previews the paper’s progression from definitions to theory and experiments.
- A multiplex network has m layers, each an unweighted directed or undirected network on the same n nodes.
- The projection network associated with the multiplex network is defined from the layer adjacencies.
- The paper introduces centrality measures, proves their existence and consistency under reasonable conditions, and evaluates them through computer experiments and simulations.
II. MATHEMATICAL MODELS FOR EIGENVECTOR CENTRALITY IN CONNECTED MULTIPLEX NETWORKS
The paper develops eigenvector-like centralities that account for interactions across multiplex layers and for potentially unequal layer influence. These include uniform, independent-layer, local heterogeneous, and global heterogeneous formulations, with existence and uniqueness established under suitable conditions.
- Multiplex centrality must account for interactions across layers whose importance may differ and whose effects are not necessarily additive.
- Classical eigenvector centrality can be computed separately for each layer and for the projection network when the relevant principal eigenvectors exist.
- Independent layer centrality collects the separately computed layer eigenvector centralities into a matrix whose columns are normalized.
- Uniform centrality treats all layers as equally important and defines a positive, normalized eigenvector-like centrality when it exists.
- An influence matrix W encodes how layer Gj influences layer Gi, supporting local heterogeneous centrality that incorporates cross-layer influence for each layer.
- Global heterogeneous centrality allows a node’s centrality in one layer to depend on neighbors from all other layers and is represented by a positive, normalized eigenvector.
- The proposed framework interprets the multiplex adjacency operator through tensor products and extends Perron-Frobenius-type reasoning to multiplex networks.
III. EXISTENCE AND CONSISTENCY
The paper establishes existence, uniqueness, and consistency properties for three multiplex centralities under connectivity and positivity conditions, including reductions to familiar centralities in special cases.
- Uniform Centrality: Strong connectivity of the projected graph guarantees existence and uniqueness of Uniform Centrality.The result is stated as Theorem 1.
- Local Heterogeneous Centrality: Strong connectivity of the projected graph together with W > 0 guarantees existence and uniqueness of Local Heterogeneous Centrality.The result is stated as Theorem 2.
- Global Heterogeneous Centrality: Strong connectivity of the projected graph together with W > 0 guarantees existence and uniqueness of Global Heterogeneous Centrality.The associated expanded graph need not itself be strongly connected, so the proof uses a permutation-based normal form and an irreducible nonnegative block.
- Consistency: In monoplex networks, all three multiplex centralities coincide with the usual eigenvector centrality of the sole layer.This establishes consistency with the classical definition in the one-layer case.
- Consistency: For identical layers, Uniform Centrality and each Local Heterogeneous Centrality column coincide with the layer eigenvector centrality.Global Heterogeneous Centrality instead follows the principal eigenvectors of W and the common layer matrix through their Kronecker product.
- Consistency: For starred layers, Global Heterogeneous Centrality becomes a diagonal matrix whose diagonal is the eigenvector centrality of W ◦ A.This also connects the construction to eigenvector centrality on a weighted graph with edge weights w_ij.
IV. COMPARING CENTRALITIES OF A MULTIPLEX NETWORK
The paper compares scalar and vectorial multiplex centralities by aggregating layer-specific information and evaluating the resulting node rankings with rank-correlation measures. It applies this framework to a Renaissance Florentine family network while varying inter-layer influence.
- Ranking comparison: Ranking correlations are more informative than vector-norm discrepancies because centrality measures primarily determine node orderings.The paper therefore compares rankings rather than relying only on distances between centrality vectors.
- Ranking comparison: Spearman and Kendall coefficients quantify agreement between rankings, ranging from -1 to 1, with values near 1 indicating stronger correlation and values near -1 anticorrelation.Values near 0 indicate greater independence according to the paper’s description.
- Aggregation: Vectorial centralities must be aggregated into one scalar per node before direct ranking comparisons with scalar centralities.The paper considers independent layer, local heterogeneous, and global heterogeneous vectorial centralities.
- Aggregation: Convex combinations with λ1 + ··· + λm = 1 aggregate layer-specific centralities, with each λj representing a layer’s relative influence.The numerical analysis uses equal weights when no information favors one layer.
- Aggregation: Global Heterogeneous Centrality uses a different aggregation normalization because its matrix is not column-stochastic; layer influence is defined through the norm of each column.This distinguishes its aggregation from the independent-layer and local heterogeneous cases.
- Florentine-family experiment: The Florentine-family experiment compares four centralities using q-dependent Spearman and Kendall correlations while varying the influence matrix.The multiplex network contains marriage and business ties among sixteen Renaissance Florentine families.
V. NUMERICAL TESTINGS
The paper tests its multiplex centrality measures on synthetic networks generated with a growing, preferential-attachment-inspired model and compares their correlations across influence strengths q.
- Synthetic network construction: The synthetic networks follow a growing random model inspired by Barabási–Albert preferential attachment and assortative network models.The construction uses parameters for network size, active nodes per layer, and the probability of adding new nodes.
- Synthetic network construction: The model starts from a complete seed layer, adds new layers with active nodes, and links active nodes to a coordinator before applying assortative linking.The initial seed layer is removed from the final multiplex network because its complete structure would distort projection-network eigenvector centrality.
- Correlation analysis: The experiments compute correlations between projection eigenvector centrality, uniform centrality, and local and global heterogeneous centralities as q varies from 0 to 1.Symmetric W1(q) and non-symmetric W2(q) influence-matrix families are evaluated, with Spearman and Kendall coefficients plotted.
VI. DISCUSSION AND CONCLUSIONS
The paper introduces multiplex-aware eigenvector centralities and proves their existence and uniqueness under reasonable conditions. Experiments show that the measures can rank the same multiplex network differently, with relationships shaped by network structure and influence strength.
- Contributions: The proposed centralities incorporate multiplex layer structure through a directed graph of influences among layers.The paper analyzes how distinguishing interaction types changes eigenvector-like centrality measures.
- Conclusions: The introduced centrality measures are qualitatively different because they produce different rankings on the same multiplex networks.The authors attribute these differences to the distinct heuristics underlying each measure.
- Conclusions: Correlations among the measures depend strongly on multiplex structure, including the number of layers and the number of nodes per layer.In synthetic examples, the total variation with q between heterogeneous and flat measures grows with the layers-to-nodes-per-layer ratio.
- Conclusions: Differences between heterogeneous and flat centralities are broader at lower q, whereas high q yields behavior similar to projected monoplex networks.The relationship between centrality measures and q is non-linear, and heterogeneous centralities detect q as a measure of multiplexity.
- Conclusions: Influence symmetry is not critical for correlations in the random networks studied, but symmetric and non-symmetric results differ significantly for the small Florentine-family network.The conclusion identifies network size and structure as relevant context for interpreting the role of influence symmetry.
- Contributions: Under reasonable conditions, the proposed multiplex centrality measures exist and are unique.These properties are established in the paper’s theorems 1, 2, and 3.