Table 1.

A nonexhaustive selection of multilayer network approaches for studying questions in behavioural ecology

Research aim	Level (I/G/P/E)	Examples of questions	Multilayer approach	Description	Software package	References
Identify important or influential nodes or edges	I/G	(1) How will a group be affected if certain individuals are removed?	Eigenvector versatility	Multilayer extension of eigenvector centrality; in it, an individual’s importance depends on its connections within and across layers and on the connections of its neighbours	MuxViz (R) (De Domenico, Porter et al„ 2015)	De Domenico, Solé-Ribalta et al. (2015)
		(2) Is social influence determined by Interactions in more than one situation?
		(3) Which relationships are most critical for group cohesion (when applying measures to edges)?
		(4) How stable Is an individual’s importance over time?
		(5) Which individuals link the most Individuals in a group within or across social situations and/or over time?	Betweenness versatility	Multilayer extension of geodesic betweenness centrality; it measures how often shortest paths (including both intralayer and interlayer edges) between each pair of nodes traverse a given node	MuxViz	De Domenico, Sold-Ribalca et al, (2015)
		(6) How important is an individual for group cohesion?
		(7) Does the role of an individual in its social group carry over across social situations?	Multidegree	A vector of the intralayer degrees of each individual across all layers	Pymnet (Python) (Kivelä. n.d.)	Menichatti, Remondini, Panzarasa, Mondragón, & Bianconi (2014)
Quantify network properties at different scales	G/P/E	(1) What are the coherent groups in a network of animals?	Multi slice modularity maximization. Multilayer InfoMap	Identifies communities of individuals in which the same individuals In different layers can be assigned to different communities	MuxViz; GenLouvain (https://github.com/GenLouvain/GenLouvain (Jeub, Bazzi, Jutla, & Mucha (n.d.)); in MATLAB, MathWorks, Natick, MA, U.S.A.)	Mucha et al, (2010)
		(2) Which individuals preferentially interact with each other in different or multiple contexts?
		(3) What are the social communities, core-periphery structures, or other large-scale structures in different types of social situations?	Stochastic block models	Statistical models of arbitrary block structures in networks	Graph-tool (Python)	Peixoto (2015)
		(4) Are there consistent, ‘typical’ types of Interaction patterns across social situations?	Motifs	Interaction patterns between multiple Individuals (e.g. node pairs or triples), within and/or across layers, that appear more often than in a specified null model	MuxViz	Battiston, Nicosia, Chavez, & Latora, 2017; Wernicke & Rasche, 2006
		(5) How similar are the interaction patterns in different social situations?	Global overlap	Number of pairs of nodes that are connected by edges in multiple layers	MuxViz; Multinet R package (Magnani & Dubik, 2018)	Bianconi (2013)
		(6) How often do interactions between Individuals co-occur in multiple situations?
Model statistical properties of a network	G/P/E	(1) Are interaction patterns Influenced by group size?	Randomization for multilayer networks	Construction of randomized ensembles of synthetic multilayer networks for comparison	Pymnet	Kivelä et al., 2014, Section 4.3
		(2) Are relationships or interactions in one social situation related to relationships or interactions in a different social situation?	Exponential random graph model (ERGM)	An extension of ERGMs to multilayer networks	MPNET (Java-based) for two-layer multilayer networks	Heaney, 2014; Wang, Robins, Pattison, & Lazega, 2013
		(3) Are relationships at one time point related to those at a different time point?
		(4) How do network relationships in one social situation or at one point In time affect subsequent relationships In other situations or at other times?	Markov models of coevolving multiplex networks	Models of the probability of an edge existing in a layer at one time as a function of an edge existing between the same pair of nodes in any layer In the previous time	MultiplexMarkovChain (https://github.com/vkrmsv/MultiplexMarkovChain; in Python)	Fisher et al., 2017; Vijayaraghavan, NoSi, Maoz, & D’Souza, 2015
			Stochastic actor- oriented models for multiple networks	Statistical models of what influences the creation and termination of edges over time. The version that we consider can model the coevolution of two networks (or two layers) as a result of their Influence on each other	Code (in R) is available at https://www.stats.ox.ac.uk/~snijders/siena/siena_scripts.htm
Modelling disease or information transmission	I/G/P	(1) What are the roles of different types of social interactions or individual in information or disease transmission?	Compartmental models on networks	Classic epidemiological models that assume that individuals exist in one of several states, with probabilistic transitions between states. For example, SIR models have susceptible infected, and recovered (or removed) states; and SI and SIS models have only susceptible and infected states. These models are sometimes amenable to mathematical analysis, but stochastic simulations are often more accessible	EpiModel (R package) (for temporal network) (Jenness, Goodreau, & Morris, 2018)	Kiss et al., 2017; Pastor-Satorras, Castellano, Van Mieghem, & Vespignani, 2015; Porter & Gleeson, 2016
		(2) Do different types of transmission interact with each other?
		(a) Can the spread of information mitigate the spread of a disease?
		(b) Can the spread of one infection enhance or reduce the spread of a second infection?
		(3) What influence disease transmission in multispecies communities?