The final ingredient that completes our multiplex network G = (V , E (1) , E (2) , E (3) , Ẽ ) is
the inter-layer edge set Ẽ . It represents interactions of nodes from different layers and in
this work we use diagonal all-to-all coupling, i.e., each node is connected to itself in both
other layers with weight 1.
For the quantitative analysis, we use the linear algebraic representation of the multiplex network G in terms of the supra-adjacency matrix A ∈ RnL×nL defined as

 (1)



A
0
0
0 I I
A = Aintra + ωAinter =  0 A(2) 0  + ω  I 0 I  ,
I I 0
0
0 A(3)

(1)

where ω ∈ R denotes a scalar layer-coupling parameter, 0 ∈ Rn×n the matrix of all zeros,
and I ∈ Rn×n the identity matrix. Throughout our analysis, we choose the layer-coupling
parameter ω = 1. As an illustrative example, Fig. 1 shows the intra-layer sparsity structure of the three layers for the year 2014. The blank lower blocks in each layer indicate
a set of users that receives user interaction without initiating it. The adjacency matrices
for all ten years are publicly available in Matlab and python formats [54].
Matrix function‑based centrality measures

The identification and ranking of influential individuals in networked complex systems is an important area of research in network science. Prominent examples of
well-established centrality measures have been developed over past decades [30–32,

Bergermann and Wolter ﻿Applied Network Science

(2023) 8:74

Page 5 of 21

Fig. 1 Sparsity structure of the intra-layer adjacency matrices of year 2014

34] with PageRank at the heart of Google’s search engine being an example of a centrality measure impacting everyday life [33, 35].

### Language ###
English

### Assessment ###
The user could ask about the coupling parameter value (omega), which is clearly stated as 1 in the text.

### Query ###
What is the typical value of the coupling parameter in multiplex network analysis?
What is the typical value of the coupling parameter in multiplex network analysis?