Statistical analysis for partially observed multilayered networks

Multilayered networks have been proposed as a joint representation of associations between multiple types of entities or nodes, such as people and organization, where two types of nodes gives rise to three distinct types of ties. The typical roster data collection method may be impractical or infeasible when the node sets are hard to detect or define or because of the cognitive demands on respondents. Multilayered networks allow us to consider a multitude of different sources of data and to sample on different types of nodes and relations. We consider modelling multilayered networks using exponential random graph models and extend a recently developed Bayesian data-augmentation scheme to allow partially missing data. Abstract Le reti multi-strato sono state proposte come una rappresentazione congiunta di associazioni tra diversi tipi di entità o nodi, quando due tipologie diverse di nodi generano tre diverse combinazioni di legame. I metodi tradizionali di raccolta dati non sempre sono utilizzabili, sia perché l’insieme di nodi potrebbe essere difficile da definire, sia in seguito ad eventuali esigenze cognitive degli intervistati.Le reti multi-strato sono state proposte come una rappresentazione congiunta di associazioni tra diversi tipi di entità o nodi, quando due tipologie diverse di nodi generano tre diverse combinazioni di legame. I metodi tradizionali di raccolta dati non sempre sono utilizzabili, sia perché l’insieme di nodi potrebbe essere difficile da definire, sia in seguito ad eventuali esigenze cognitive degli intervistati. Le reti multi-strato offrono un vantaggio in queste situazioni poiché permettono di utilizzare più fonti d’informazione congiuntamente e semplificano il campionamento di diversi nodi e relazioni. Questo lavoro intende modellare le reti multistrato tramite i modelli esponenziali per reti casuali (ERGM) ed estende una reJohan Koskinen Social Statistics Discipline Area, University of Manchester, Manchester M13 9PL e-mail: johan.koskinen@manchester.ac.uk Chiara Broccatelli Sociology, University of Manchester, Manchester M13 9PLe-mail: chiara.broccatelli@postgrad.manchester.ac.uk Peng Wang, Centre for Transformative Innovation, Faculty of Business and Law, Swinburne University of Technology, Australia e-mail: pengwang@swin.edu.au · Garry Robins,Melbourne School of Psychological Sciences, The University of Melbourne, Australia e-mail: garrylr@unimelb.edu.au


Introduction
Kivelä et al (2014) coined the term 'multilayered networks' as a general framework for jointly designating multiple types of network data, such as one-mode, two-mode, and multiplex networks, where researchers had typically dealt with each instance separately.Here we are primarily considering the extension of the exponential random graph (ERGM) family of distributions proposed by Wang et al. (2013) to the subclass of multilayered networks typically referred to as 'multilevel networks' (Lazega et al., 2008) even though the key ideas in dealing with partially observed data generalises to other extensions of ERGM, such as multiplex networks (Pattison and Wasserman, 1999).In situations where you are likely to have imperfect information on network ties, availing yourself of the full set of tools that may be derived from a wider framework for networks may prove beneficial.

Data Structure
We assume two distinct set of nodes: A = {1, . . ., n} and B = {1, . . ., m} where we might observe ties among all combinations of nodes type.A tie thus belong to either of the sets A 2 , A×B, or B 2 .In the sequel we will use AA, AB, and BB as a notational shorthand for these edge-sets, with the corresponding incidence matrices X AA , X AB , and X BB , respectively.The element X E,v of matrix X E is equal to 1 if the edge v ∈ E belongs to the graph and 0 otherwise.The multilevel network may be represented as a one-mode network with a blocked, symmetric adjacency matrix When extending binary one-mode networks to multiple relations (say 'friendship' and 'advice') it is convention to represent this as a collection of graphs or adjacency matrices, one for each relation.For multilevel networks we by definition have different relations for different combinations of node-sets.Let the number of relations be denoted by R E , for E = AA, AB, BB, with incidence matrices being defined as there is a tie on relation r = 0, . . ., R E − 1 for edgeset E = AA, AB, BB.When the number of relations for E = AA, AB, BB differ, we are not able to unambiguously define the multilayered network as a collection of one-mode network with blocked, symmetric adjacency matrices.
For AA, AB, and BB define the binary indicator matrices D AA , D AB , and D BB , each of which having elements D E,v of D E equal to 1 or 0 depending on whether the corresponding tie-variable v is observed or not, respectively.For each E = AA, AB, BB the indicators extend straightforwardly to account for more than one relation.Thus, for example, if X (0) AA represent friendship ties and X AA would indicate what friendship and advice ties were observed and which ones were not observed.
We follow the convention (Little & Rubin, 1987) of partitioning data X into observed X obs = {X v : D v = 1} and unobserved X miss = {X v : D v = 0} data, conditional on an outcome D. For a given D we take (X obs , X miss ) to denote X reconstructed.

Model Formulation
Frank and Strauss (1986) derived ERGMs for one-mode networks from the so called Markov dependence assumption that posited that for any two pairs {i, j} and {k, } of vertices of a graph, the tie-variables X i, j ⊥X k |X −(i, j),(k, ) if {i, j} ∩ {k, } = / 0. They proved that the Markov dependence assumption implied a log-linear model for the collection of tie-variables that has as its sufficient statistics counts of different network 'configurations' (incidentally echoing the conclusions drawn by Moreno and Jennings, 1938).Snijders et al. (2006) elaborated on the Markov model by proposing parameters derived from the so called social circuit dependence assumption.The general form of ERGM is where the normalising constant ψ(θ ) = ∑ Y ∈X exp{q(Y; θ )} and q(X; θ ) is a potential dependent on the structure of the network and a vector θ of statistical parameters.This general form is agnostic to the specific dependencies we may hypothesis for a particular type of network object.For undirected one-mode network, the model of Frank and Strauss (1986) has the potential written as a weighted sum of sufficient graph statistics log q(X; θ ) = where the statistics correspond to two distinct categories of statistics, namely stars and triangles (in the expression X i+ = ∑ j X i j to explicitly allow for different dependencies depending on what edge-sets are considered.For example, z(X AA ) only involve statistics calculated on AA while z(X AA,BB ) involve crossed statistics, calculated for ties in A 2 × B 2 .With multiple relations, statistics can be further partitioned, so that the linear predictors take into account dependencies between different types of ties between different types of nodes.Considering for example the interactions between ties in AA and AB, we have The interpretation is that a tie of type s among pairs in AA may depend on affiliation of nodes in A with nodes in B of type t.Conditional on a realisation X, we assume an observation process where the parameter ζ is distinct (Little & Rubin, 1987) from θ .The observation process may be thought of equivalently as a missing data generating mechanism or a sampling design, such as snowball sampling, for purposes of inference (Handcock and Gile, 2010).If we assume that tie-variables are observed conditionally independently conditional on X, f (•) can be modelled as a regular log-linear model with a standard link function.Given that D has the same range-space X as X, the observation indicators can also be modelled using an ERGM.Inference for an informative, MNAR process will however be contingent on informative priors.

Empirical illustration
We provide a brief empirical case-study using the so-called 'Noordin Top' Terrorist Network (Everton, 2012) as our assumed true network.The node set A consists of n = 79 individuals and B of m = 129 recorded events.The friendship ties reported in Everton serve as the ties in AA and the participation in events and operations listed by Roberts and Everton (2011) are the ties of the affiliation set AB.To construct ties BB among events, we have elaborated on the time-stamped version of Broccatelli, Everett and Koskinen (2016) and coded up the explicitly mentioned connections between different events and operations in the International Crisis Group Report (International Crisis Group, 2006).For the purposes of illustration, the event-byevent network is considered fixed and exogenous.Furthermore, we condition on the overall activity of the network, fixing the number of ties in both AA and BB.Consequently, all analyses have to be interpreted conditionally on the overall number of event participations and total number of friendship ties.The configurations z(•) are illustrated in Figure 1 and are described in more detail in Wang et al. (2014).For the completely observed network, summaries of the posteriors for the corresponding parameters are provided in Table 2. Typical for one-mode network we find strong support for triadic closure (the 95% CI for ATA is (0.341, 1)) but also strong support for people taking part in events that are functionally related to other events that they take part in (the 95% CI for ATA is (0.786, 1.859)).To provide an example of multilevel snowball sampling, we snowball using Operation 3 as our seed (this is the 2004 Australian embassy bombing that took place on 9 September 2004 in Jakarta, Indonesia, killing 9-11 people and injuring more than 150 people).Anyone who participated in this operation is defined as being in wave 1, and anyone who is not in wave 1 but is tied to anyone in wave 1, belongs to wave 2. The result in Table 2 are qualitatively the same as for the model with completely observed data.
To provide a a brief example of a MNAR observation process, for each tievariable (i, j), we define independently Pr(D i j = 1|X, ζ ) = Pr(D i j = 1|h i j (X), ζ ), where h i j (X) = max{d i (X), d j (X)}, where d i (X) is the distance in X between i ∈ A, B and Noordin Top (all ties in BB are assumed fixed and known).We model the probabilities Pr(D i j = 1|h i j (X), ζ ) as in Table 1, with the interpretation that ties that are further from the leader Noordin Top are less visible that ties close to him.The results in Table 1 indicate that effects corresponding to clustering is attenu-ated but degree-related effects are amplified (with the exception of XASA).These changes are a natural consequence of the observation process respecting distance but not necessarily clustering.We have proposed a statistical approach for analysing the structure of multilayered networks that account for imperfections in data.We provide an illustrative example of analysis of a multilevel network for three types of observation processes.While the approach is consistent when the observation process is known, a MNAR process requires making a number of untestable assumptions and is most likely of use merely as a sensitivity analysis.Further work is needed in order to systematically investigate the sensitivity of MNAR to different plausible MNAR mechanisms.
(t) simplifies to a distribution proportional to f (D|X obs , X miss , ζ )π(ζ ).If the distribution f (•) is not fully tractable, draws of ζ cannot be made directly.Assuming that it is straightforward to draw D from f (•), ζ can be updated using steps (a), (b) and (c), with f (•) playing the role of p(•)

Table 1 :
Detection bias in MNAR observation mechanism for Noordin Top

Table 2 :
Posterior summaries for ERGM fitted to Noordin Top