Contents lists available at ScienceDirect Social Networks journal homepage: www.elsevier.com/locate/socnet Analyzing multiplex networks using factorial methods Giuseppe Giordano a , Giancarlo Ragozini b , Maria Prosperina Vitale c, ⁎ a Dept. of Political and Social Studies, University of Salerno, Italy b Dept. of Political Science, University of Naples Federico II, Italy c Dept. of Political and Social Studies, University of Salerno, Via Giovanni Paolo II, n. 132, IT-84084 Fisciano, SA, Italy ARTICLEINFO MSC: 00-01 99-00 Keywords: Multiplex network data DISTATIS Multidimensional scaling Simulated data ABSTRACT Multiplex networks arise when more than one source of relationships exists for a common set of nodes. Many approaches to deal with this kind of complex network data structure are reported in the literature. In this paper, we propose the use of factorial methods to visually explore the complex structure of multiplex networks. Specifcally, the adjacency matrices derived from multiplex networks are analyzed using the DISTATIS tech- nique, an extension of multidimensional scaling to three-way data. This technique allows the representation of the diferent types of relationships in both separate spaces for each layer and a compromise space. The analytical procedure is illustrated using a real world example and simulated data. 1. Introduction Social relationships are the result of several types of interaction among actors (e.g., friendship, neighborship, kinship, membership) and can be described as complex networks in which these connections act together in a tie formation mechanism. Such complexity can be re- presented by a multilayer network, which arises when there is more than one source of connection for a common set of nodes or diferent sets of nodes. In this framework, multiplex networks consisting of a fxed set of nodes interacting through diferent relationships were in- troduced in the late 1970s (Lazega and Pattison, 1999; Pattison and Wasserman, 1999; White et al., 1976). More recently, multiplex net- works have been considered a particular specifcation of the more general class of multilayer networks (Bianconi, 2018; Kivelaet al., 2014). Many issues have been addressed in exploring and analyzing this type of network. These issues include visualization (Erten et al., 2005; Fatemi et al., 2018; Matsuno and Murata, 2018; Xu et al., 2017), the procedures relating to the aggregation and fattening of layers (De Domenicoet al., 2015a; Kanawati, 2015; Kivela et al., 2014), commu- nity detection (Bothorel et al., 2015; De Bacco et al., 2017; Hmimida and Kanawati, 2015; Kuncheva and Montana, 2015; Mucha et al., 2010), blockmodeling (e.g., see Barbillon et al., 2017; Brusco et al., 2013; Doreian et al., 2005), and statistical network models (ERG or p* models, Shafe, 2015, 2016; Snijders et al., 2013). In parallel, a wide range of software tools 1 have been developed. Empirical studies on real multilayer network data have appeared in several felds (De Stefano and Zaccarin, 2013; Heaney, 2014; Rossi and Magnani, 2015; Santana et al., 2017; Simpson, 2015), as have newer and more sophisticated network measures and models (Battiston et al., 2017; Bródka et al., 2018; Halu et al., 2013; Magnani and Wasserman, 2017; Menichetti et al., 2014; Ostoic, 2017; Solá et al., 2013; Solé-Ribalta et al., 2014). In this paper, we use factorial methods to statistically analyze and visually explore multiplex networks by computing a data driven op- timal weighting system for the layer aggregation and to analyze the hidden structure of this kind of network while preserving its inherent complexity. In general, factorial methods have been proposed in the social network analysis (SNA) framework to explore diferent network struc- tures (see, e.g., D’Esposito et al., 2014a; Faust, 2005; Ragozini et al., 2015; Roberts, 2000), including the attributes of nodes and events (Giordano and Vitale, 2007, 2011), or to analyze network derived measures (Liberati and Zappa, 2013).Inthecaseofmultiplexnetworks, canonical correlation analysis has been adopted to identify the di- mensions along which two networks are related to each other (Carroll, 2006), and an analytical procedure was recently introduced for https://doi.org/10.1016/j.socnet.2019.07.005 ⁎ Corresponding author. E-mail address: mvitale@unisa.it (M.P. Vitale). 1 See,forinstance,theMuxVizplatform(DeDomenicoetal.,2015b),aframeworkforthemultilayeranalysisandvisualizationofnetworks;XPNET,anextensionof PNet (Wang et al., 2006) software incorporating the analysis of multivariate networks; and the R packages Multiplex (Ostoic, 2017) and Multinet (Dickison et al., 2016), developed for the treatment of multiplex and multilayer networks. Social Networks 59 (2019) 154–170 0378-8733/ © 2019 Elsevier B.V. All rights reserved. T