Source-linked AI summary
Weighted Multiplex Networks
Giulia Menichetti, Daniel Remondini, Pietro Panzarasa, Raúl J. Mondragón, Ginestra Bianconi
TL;DR
The paper asks what full multiplex analysis reveals beyond treating interacting layers as separate networks. It introduces multilink-based weighted measures and an entropy framework, then shows in APS co-authorship and citation multiplexes that weights depend on multiplex structure and that single-layer analysis misses these patterns.
Problem
Single-layer analyses may overlook information encoded in the interacting layers of multiplex networks, leaving the advantage of full multiplex analysis unresolved.
Method
The paper studies APS co-authorship and citation multiplexes using multistrength, inverse multiparticipation ratio, and entropy-based indicators Ψ and Ξ.
Results
Weights are correlated with multilink structure: shared multilinks have larger average weights, and multistrength and inverse multiparticipation behavior differs by multilink type.
Takeaways & Limitations
Weighted multiplex properties reveal information that cannot be recovered by analyzing the constituent layers separately.
Abstract
from arXiv · showhide
One of the most important challenges in network science is to quantify the information encoded in complex network structures. Disentangling randomness from organizational principles is even more demanding when networks have a multiplex nature. Multiplex networks are multilayer systems of $N$ nodes that can be linked in multiple interacting and co-evolving layers. In these networks, relevant information might not be captured if the single layers were analyzed separately. Here we demonstrate that such partial analysis of layers fails to capture significant correlations between weights and topology of complex multiplex networks. To this end, we study two weighted multiplex co-authorship and citation networks involving the authors included in the American Physical Society. We show that in these networks weights are strongly correlated with multiplex structure, and provide empirical evidence in favor of the advantage of studying weighted measures of multiplex networks, such as multistrength and the inverse multiparticipation ratio. Finally, we introduce a theoretical framework based on the entropy of multiplex ensembles to quantify the information stored in multiplex networks that would remain undetected if the single layers were analyzed in isolation.
A. Definition
A weighted multiplex network consists of multiple weighted layers sharing the same node set, with layer-specific links. The formalization assumes integer-valued link weights as multiples of a minimal weight.
- A weighted multiplex network comprises M weighted networks sharing node set V of cardinality N, while each layer has its own link set.
- The multiplex is represented as the ordered collection G = (G1, G2, …, GM), with each layer described by an adjacency matrix.
- Link weights are assumed to take integer values, a condition treated as nonrestrictive when weights represent multiples of a minimal weight.
B. Structural properties of individual layers
Single-layer weighted-network analysis characterizes how total weight and weight heterogeneity vary with node degree. These relationships distinguish linear from super-linear strength growth and quantify whether weights are concentrated across incident links.
- Node strength is the sum of incident link weights, while inverse participation ratio measures how unevenly those weights are distributed.
- The inverse of the participation ratio represents the effective number of links carrying a node’s weight and lies between one and its degree.
- Average strength conditional on degree is modeled as sα(k) and is expected to follow a power law with exponent βα ≥ 1.
- Linear strength growth indicates hub links have average weights like those of less-connected nodes, whereas super-linear growth indicates otherwise.
- Inverse participation ratio is used to characterize the heterogeneity of weights among links incident upon nodes with a given degree.
C. Multilink, multistrength, and inverse multiparticipation ratio
Multilinks encode which layers connect each node pair, enabling multidegree-based measures of weighted multiplex structure. Multistrength sums weights for a specified multilink type, while inverse multiparticipation ratio measures their heterogeneity.
- C. Multilink, multistrength, and inverse multiparticipation ratio: A multilink m records the presence or absence of a link between a node pair in every layer, generalizing link overlap.
- C. Multilink, multistrength, and inverse multiparticipation ratio: Multiadjacency matrices identify node pairs connected by each multilink type, with only 2^M − 1 independent matrices because of normalization.
- C. Multilink, multistrength, and inverse multiparticipation ratio: Multidegree k_m_i counts how many multilinks of type m are incident upon node i.
- C. Multilink, multistrength, and inverse multiparticipation ratio: Multistrength is the sum of weights in a specified layer for multilinks of a chosen type incident upon one node.
- C. Multilink, multistrength, and inverse multiparticipation ratio: Average multistrength and inverse multiparticipation ratio are expected to scale with multidegree using exponents β_m,α ≥ 1 and λ_m,α ≤ 1.
- C. Multilink, multistrength, and inverse multiparticipation ratio: When the number of layers is too large for explicit multilink types, overlap multiplicity groups multilinks by the number of active layers.
II. EMPIRICAL EVIDENCE OF WEIGHTED PROPERTIES OF MULTILINKS
Empirical analyses of APS co-authorship and citation multiplexes find link overlap, correlated layer structure, and weight patterns that depend on multilink type. These differences show why analyzing layers separately misses weighted multiplex information.
- Datasets and multiplex structure: The study analyzes APS bibliographic data containing authors, collaborations, and citations from articles published between 1893 and 2009.
- Datasets and multiplex structure: The two duplex networks are CoCo-PRL/PRE, with PRL and PRE collaboration layers, and CoCi-PRE, with PRE collaboration and citation layers.
- Datasets and multiplex structure: Both multiplexes show significant link overlap and degree correlation, with collaboration hubs tending also to be citation hubs.
- CoCo-PRL/PRE results: In CoCo-PRL/PRE, multilinks present in both collaboration layers have significantly larger average PRL weights than PRL-only multilinks.
- CoCo-PRL/PRE results: In the PRL layer, inverse multiparticipation exponents are λ(1,0),PRL = 0.84 ± 0.03 and λ(1,1),PRL = 0.74 ± 0.05.
- Implication: Analyzing individual layers would miss both the larger weights of shared multilinks and differences in multiparticipation behavior.
- CoCi-PRE results: The CoCi-PRE network shows multilink-dependent multistrength behavior, including citation exponents β(1,1),cit,in = 1.30 ± 0.07 and β(0,1),cit,in = 1.11 ± 0.01.
A. Assessing the informational content of weighted multilinks
The paper uses entropy-based multiplex ensembles and indicators Ψ and Ξ to measure information in weighted multilinks and compare correlated multiplex structure with layer-separated analyses. The CoCi-PRE network shows multilink-dependent weight behavior, while related results indicate that multistrength contains information unavailable from individual layers.
- Entropy-based framework: Entropy-based multiplex ensembles quantify information encoded in weighted multilink properties under specified structural constraints.The framework treats entropy as an informational measure of the constraints imposed on multiplex networks.
- Empirical multilink properties: In CoCi-PRE, collaboration multistrength distributions share exponents across multilink types, but multilinks (1, 1) have larger average weights than multilinks (1, 0).The corresponding collaboration-layer exponents for multilinks (1, 0) are also larger than those for multilinks (1, 1).
- Empirical multilink properties: Citation-layer incoming and outgoing multistrengths vary by multilink type, whereas the average inverse multiparticipation ratio shows no significant cross-type behavioral change.Related correlated ensembles with non-trivial inverse multiparticipation ratios produce similar results.
- Entropy-based framework: Ψ compares a weighted multiplex ensemble with a homogeneous-weight ensemble to measure information carried by weight distributions.The homogeneous reference preserves structural properties while distributing weights uniformly, with each link retaining a minimal weight.
- Entropy-based framework: Ξ measures the additional information encoded by multilink properties in a correlated ensemble relative to an uncorrelated ensemble analyzed layer by layer.The correlated ensemble accounts for multilink properties, whereas the uncorrelated ensemble treats the layers separately.
III. CONCLUSIONS
The paper finds significant cross-layer correlations between weights and multiplex structure in APS co-authorship and citation networks. It introduces multistrength, inverse multiparticipation ratio, and entropy-based indicators Ψ and Ξ to identify information that single-layer analyses miss.
- Findings: Weighted multiplex networks exhibit significant correlations across layers, with weights closely correlated with multiplex structure.The evidence comes from two APS multiplex networks combining co-authorship and citation data.
- Findings: Multistrength and inverse multiparticipation ratio capture weighted multilink properties that cannot be reduced to properties of single layers.These measures characterize how weights depend on the pattern of links connecting node pairs across layers.
- Information measures: The entropy framework uses Ψ to quantify information relative to randomly distributed weights and Ξ to quantify multilink information beyond single-layer properties.The indicators distinguish information in weighted multiplex structure from information contained in separately analyzed layers.
- Conclusion: The analysis shows that partial single-layer analyses cannot capture all non-trivial information in weighted multiplex networks.The conclusion emphasizes shifting attention toward weighted properties of multilinks.
IV. MATERIALS AND METHODS
The study constructs two weighted APS-author multiplex networks and analyzes their topology, multilink overlap, degree correlations, and weight–structure scaling. It also introduces entropy-maximizing multiplex ensembles to model structural constraints.
- Entropy framework: The theoretical framework generalizes entropy-maximizing canonical multiplex ensembles to weighted networks under soft structural constraints.The ensemble probability uses Lagrange multipliers for constraints, with normalization enforced by a partition function.
- Datasets: The analysis uses APS article metadata and citing-article pairs, restricted to PRL and PRE and authorship records with at most 10 authors per paper.The cutoff excludes very large experimental collaborations to reduce bias.
- Weight definitions: Collaborative weights count co-authorship contributions, whereas citation weights count how many times one author cites another.The collaboration network is undirected; the citation network is directed and allows self-loops.
- Multiplex construction: CoCo-PRL/PRE contains PRL and PRE co-authorship layers, while CoCi-PRE combines PRE co-authorship with directed PRE citation links.Both multiplexes use shared author node sets across their two layers.
- Structural analysis: The networks exhibit significant overlap between links across layers, and their degree and multidegree distributions are broadly distributed with fitted power-law exponents.Degrees across layers and multidegrees are positively correlated.
- Weighted properties: Average strength scales linearly with degree in CoCo-PRL/PRE and in the collaboration layer of CoCi-PRE, but super-linearly in its citation layer.Reported exponents are β1 = 0.96 ± 0.04 and β2 = 1.01 ± 0.05 for CoCo-PRL/PRE; CoCi-PRE has β1 = 1.03 ± 0.04 and β2 = 1.14 ± 0.03.
a. Statistical analysis of the average multistrengths in the CoCo-PRL/PRE multiplex network
In CoCo-PRL/PRE, multistrength generally scales with multidegree, but multilinks present in both layers carry systematically greater strength than single-layer multilinks. Their weights are also more heterogeneous.
- Average multistrengths: For multilinks (1,1) and (1,0), fitted multistrength exponents in the PRL layer are not significantly different: β(1,1),1 = 1.06 ± 0.09 and β(1,0),1 = 0.97 ± 0.03.The corresponding proportionality constants are also not significantly different.
- Average multistrengths: 2.90 · 10^-16: the paired t-test rejects equal average multistrengths for matched multidegrees in the PRL layer.The comparison is between multilinks (1,1) and (1,0) at the same multidegree.
- Average multistrengths: s(1,1),1(k) ≈ e^0.53s(1,0),1(k): multilinks present in both layers have higher PRL multistrength at fixed multidegree.The corresponding PRE-layer comparison also rejects the null hypothesis with p = 8.98 · 10^-15 and an exponent difference of 0.57.
- Inverse multiparticipation ratio: λ(1,1),1 = 0.74 ± 0.05 versus λ(1,0),1 = 0.84 ± 0.03: weights on both-layer multilinks are more heterogeneous in PRL.The PRE layer shows the same ordering: λ(1,1),2 = 0.73 ± 0.06 versus λ(0,1),2 = 0.84 ± 0.05.
- Measures: The analysis characterizes average multistrength and inverse multiparticipation ratio conditional on multidegree, with scaling exponents β ≥ 1 and λ ≤ 1.The framework distinguishes incoming and outgoing links where applicable.
a. The statistical analysis of the average multistrengths in the CoCi-PRE multiplex network
In CoCi-PRE, collaboration multistrength is approximately linear in multidegree, while citation multistrength is super-linear and differs by multilink type. Both-layer multilinks also have systematically larger collaboration multistrength.
- Average multistrengths: Collaboration-layer multistrength exponents are approximately linear for both incoming and outgoing links across multilink types.Values range from β(1,1),1,in = 1.03 ± 0.04 to β(0,1),1,out = 0.97 ± 0.05.
- Average multistrengths: Both-layer multilinks have multistrengths related to multidegree linearly and to single-layer multilinks by a multiplicative constant in the collaboration layer.This conclusion follows from the fitted parameters and paired t-tests.
- Average multistrengths: Citation-layer multistrength is super-linear, with β(1,1),2,in = 1.30 ± 0.07 and β(0,1),2,in = 1.11 ± 0.01 for incoming citations.Outgoing exponents are β(1,1),2,out = 1.32 ± 0.08 and β(0,1),2,out = 1.10 ± 0.02.
b. The statistical analysis of the multi inverse participation ratio in the CoCi-PRE multiplex network
The paper compares inverse multiparticipation ratios across multilink types and develops weighted multiplex ensembles that preserve either layerwise or cross-layer structural constraints.
- Inverse multiparticipation ratio: In the collaboration layer, inverse multiparticipation ratios for multilinks (1,1) exceed those for multilinks (1,0).This pattern is supported by paired tests for both incoming and outgoing links.
- Inverse multiparticipation ratio: In the citation layer, λ exponents for multilinks (1,1) and (0,1) are not significantly different for incoming or outgoing citations.The fitted values are λ(1,1),2,in = 0.73 ± 0.05 and λ(0,1),2,in = 0.74 ± 0.04.
- Inverse multiparticipation ratio: The paired tests nevertheless show higher inverse multiparticipation ratios for multilinks (1,1) than (0,1) in the citation layer.The incoming comparison has p = 7.60 · 10^-21; the outgoing comparison has p = 1.12 · 10^-15.
- Weighted ensembles: The weighted multiplex formalism assumes integer link weights, treating them as multiples of a minimal weight.This assumption simplifies the ensemble treatment.
- Weighted ensembles: Canonical weighted multiplex ensembles maximize Shannon entropy subject to soft structural constraints and assign probabilities using Lagrange multipliers and a partition function.The framework includes uncorrelated ensembles with additive entropy and correlated ensembles constrained by expected multidegrees and multistrengths.
- Weighted ensembles: Uncorrelated ensembles factorize across layers, whereas correlated ensembles impose expected multidegree and multistrength constraints across multilinks.These alternatives distinguish information captured by independent layer descriptions from information encoded in multiplex correlations.
Appendix C: Background information on Figure 4 of the main text
The appendix defines correlated and uncorrelated weighted multiplex ensembles, then evaluates entropy-based indicators through finite-size analysis and weight randomization.
- Ensemble construction: The analysis compares correlated and uncorrelated weighted multiplex ensembles across network sizes N = 128, 256, . . ., 2048.The indicators Ψ and Ξ are evaluated as functions of system size because entropy calculations have finite numerical limits.
- Ensemble construction: The correlated ensemble uses power-law multidegree distributions with γ(1,m2) = 2.6 and γ(0,1) = 1.9, with a structural cut-off for multidegree (0, 1).Multidegrees are ranked to avoid effects from fluctuations in the multidegree sequence.
- Ensemble construction: The correlated ensemble also imposes multistrength assumptions, including c⃗m,α = 1, β(1,m2),1 = 1, β(1,1),2 = 1.3, and β(0,1),2 = 1.1.The maximal cut-off is K = N for γ⃗m > 2, with another cut-off specified for γ⃗m < 2.
- Ensemble construction: The uncorrelated ensemble matches each node’s expected layer degree and strength to sums of multidegrees and multistrengths from the correlated ensemble.This preserves corresponding layer-level expectations while removing the specified multiplex correlations.
- Entropy indicators: Ψ compares the entropy of a weighted multiplex ensemble with that of an ensemble whose weights are distributed homogeneously, while Ξ measures additional information in the correlated ensemble relative to the uncorrelated one.The finite-size scaling of Ψcorr, Ψcorr, and Ξ is reported in Figure 4 of the manuscript.
- Entropy indicators: Weights are randomized over links while preserving a minimal link weight wij > 1, and averages use 100 weight randomizations.The resulting ensembles retain their structural properties while distributing the remaining total weight randomly.