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Exponential random graph models for multilevel
networks
Peng Wang, Garry Robins, Philippa Pattison, Emmanuel Lazega
To cite this version:
Peng Wang, Garry Robins, Philippa Pattison, Emmanuel Lazega. Exponential random graph models
for multilevel networks. Social Networks, Elsevier, 2013, 35 (1), pp.96 – 115. ffhal-01521673ff
Exponential random graph models for multilevel networks
In: Social Networks
Peng Wanga,∗, Garry Robins a, Philippa Pattison a, Emmanuel Lazega b
a Melbourne School of Psychological Sciences, The University of Melbourne, Australia
b IRISSO-ORIO, University of Paris-Dauphine, France
Abstract
Modern multilevel analysis, whereby outcomes of individuals within groups take into account group membership, has been accompanied by impressive
theoretical development (e.g. Kozlowski and Klein, 2000) and sophisticated methodology (e.g. Snijders and Bosker, 2012). But typically the approach
assumes that links between groups are non-existent, and interdependence among the individuals derives solely from common group membership. It is
not plausible that such groups have no internal structure nor they have no links between each other. Networks provide a more complex representation
of interdependence. Drawing on a small but crucial body of existing work, we present a general formulation of a multilevel network structure. We
extend exponential random graph models (ERGMs) to multilevel networks, and investigate the properties of the proposed models using simulations
which show that even very simple meso effects can create structure at one or both levels. We use an empirical example of a collaboration network about
French cancer research elites and their affiliations (Lazega et al., 2006, 2008) to demon-strate that a full understanding of the network structure requires
the cross-level parameters. We see these as the first steps in a full elaboration for general multilevel network analysis using ERGMs.
1. Introduction
Statistical analysis using multilevel models, or hierarchical linear models (HLMs) (Snijders and Bosker, 2012), is applicable where
the data has a nested hierarchical structure. The units of study are
categorized in two or more levels. Snijders et al. (2012) listed several multilevel examples such as schools and teachers, classes and
pupils, and doctors and patients. The lower levels (i.e. teachers,
pupils and patients) are commonly referred to as the micro- or
individual-levels, and the upper levels (i.e. schools, classes and doctors) are known as macro- or group-levels. The interactions across
levels, known as the meso-level structure, usually describe the
affiliations between the micro- and macro-level units, typically in
a nested structure where the micro-level units are affiliated with
one and only one macro-level unit. The analytical focus is usually on
the outcomes of micro-level units, taking into account this nested
structure and partitioning the variance of the outcome variable
across levels.
To date, multilevel analysis has not been commonly used in
social network analysis, although attention has been drawn to
the theoretical importance of multilevel concepts. In the context of organizational network theory, for example, major reviews
(Kozlowski and Klein, 2000; Borgatti and Foster, 2003; Brass et al.,
2004) have highlighted the relevance of multilevel aspects of organizations. Brass et al. (2004) reviewed organizational network
research at different organizational levels, but the amalgamation
of multilevel and network perspectives has barely begun.
The classic HLM analysis described above assumes that links
between groups at the macro-level are non-existent, and that
interdependence among the individuals derives solely from shared
group membership instead of other forms of endogenous network
processes. Network analysis in general presents a more complex
representation of interdependence. It is not plausible that groupings of individuals have no internal structure beyond the simple
membership structure, nor is it always the case that such groups
have no links among themselves. Yet, it is also not always plausible
that network structure exists at only one level. For instance, organizations are explicitly multilevel in design and operation, yet this
is not taken into account in organizational network analysis.
Network methods focus attention on relationships among individuals. For a one-mode network (nodes of one type – say,
individuals), there is no nesting structure. Network methods may
be used to predict individual outcomes given the network structure: for instance, when individual responses are assumed to relate
to responses by network partners termed as social influence, network diffusion or network contagion (e.g. Friedkin, 2006; Mason
et al., 2007). However, the structure itself may be the research issue.
Network structure may be self-organizing into various endogenous
patterns. Individual qualities may also influence the formation or
dissolution of network ties, known as social selection (e.g. Robins
et al., 2001a). Social influence and selection processes may be
Fig. 1. A multilevel network.
studied simultaneously using the stochastic actor oriented models
(SAOMs) proposed by Snijders et al. (2010) with longitudinal data.
Recent methods to analyze cross-sectional network data include
exponential random graph models (ERGMs) (Snijders et al., 2006;
Robins et al., 2007a,b; Lubbers and Snijders, 2007).
In this paper, we present a general formulation of a multilevel network structure, and extend ERGMs to multilevel networks.
We start with descriptions of the general multilevel data structure in a network context that generalizes all previous data types
hitherto seen in multilevel network studies, and then compare
ERGMs with the classical HLMs. Before introducing the multilevel
ERGMs, we review and investigate some of the current ERGM specifications using simulations. As part of the proposed multilevel
ERGMs, these simulations are crucial for the later model interpretations we give to within-level network structures. We then propose
ERGM specifications for multilevel networks. Through simulation,
we illustrate how simple cross-level associations can create structure at both levels. The proposed models are then applied to an
empirical data set collected among French cancer research elites
and their affiliations (Lazega et al., 2006, 2008). By estimating and
comparing models with and without multilevel effects, we show
the importance of the multilevel effects in both goodness of fit
and model selection. The model interpretations reveal the dependencies among the micro-, macro- and meso-level network, and
provide richer and more detailed descriptions of the empirical data.
The proposed models are implemented in the program MPNet as a
multilevel network extension for PNet (Wang et al., 2006).
2. Multilevel networks
Multilevel network data categorizes nodes into different levels,
and the network ties represent relationships between nodes within
and across different levels. A k-level network has nodes of k different types, and each type represents a different level. A one-mode
network can be defined within each level, and a bipartite (twomode) network can be defined between nodes from two adjacent
levels.
For a two-level network with u nodes at the macro-level, and v
nodes at the micro-level, we label the macro-level network as network A, the micro-level network as network B, and the meso-level
bipartite network as network X. We refer the overall network as
a (u, v) two-level network, labelled as M. This flexible data structure, as shown in Fig. 1, generalizes the multilevel networks in the
literature to date:
1. In a cross-level nested structure, all B nodes have degree one
in X. With both macro- and micro-level networks (A and B)
empty, we have a multilevel modelling structure. With networks
among the at the micro-level B, post hoc multilevel inferences
can bemade after applying ERGMs to each group (Anderson et al.,
1999; Snijders and Baerveldt, 2003; Lubbers, 2003; Lubbers and
Snijders, 2007).
2. A fixed nested structure X, with A and B networks non-empty,
is presented by Lazega et al. (2008). See also Hedström et al.
(2000) and Bellotti (2011). One level is nested in another, but
with networks at one or both levels. We can investigate how
structures in A and B are related.
3. An empty macro-level A network, with non-empty B and X, is the
data structure of Koehly et al. (2003), Torlo et al. (2010), Snijders
(2002, 2009), and Lomi et al. (2011).
4. The most general form presents a range of possibilities depending on whether or not some of the three networks are assumed
exogenous. For instance, with an exogenous meso-level X, we
can ask: given the meso-links, how do micro and macro structures relate? Or when all three networks are endogenous, and we
can examine the interdependencies among the micro-, macroand meso-level networks. The most general form was first discussed by Iacobucci and Wasserman (1990) and Wasserman and
Iacobucci (1991), but not in the context of multiple levels.
Lazega et al. (2008) presented an example with two-level network data collected among French elite cancer researchers and
their affiliated research laboratories. There are individual and collective forms of agency at each level. At the researcher level, the
data captures advice-seeking ties whereas at the laboratory level,
the network represents collaborations among the laboratories, and
the cross-level ties represent researchers’ affiliations with laboratories. Based on various centrality measures, Lazega et al. (2008)
categorized researchers into “little or big fish”, and laboratories into
“small or big pond”, and inferred theory on how network strategies
may affect an individual’s performance; the cross-level structure
is treated as exogenous and understood, whereas the within-level
structure is endogenous and assumed affected by the cross-level
structure. We use the data from Lazega et al. (2008) to demonstrate how the ERGMs proposed in this paper may be applied and
interpreted in an empirical context in the modelling example.
3. Multilevel models
Multilevel modelling using hierachical linear models (HLMs)
(Snijders and Bosker, 2012) has forerunners including “Withinand-between” organizational analysis (Dansereau et al., 1984).
HLMs can be used for nested multilevel data analysis, when there
is an underlying hierarchy that determines the categorization of
the nodes such that nodes from lower level are nested or belong
to nodes in the higher levels. HLMs explain the individual outcome
as a result of explanatory variables at individal level, taking into
account the nested multilevel structure. Variance is apportioned
across both levels because persons are nested within groups. Let j
denote the index for groups and i denote the index for individuals,
the simplest form of HLMs (known as the empty model without any
explanatory variables) can be expressed as
Y
ij = ˇ0 + u0j + eij
where Yij is the dependent variable representing the outcome of
node i from group j; ˇ0 is a general mean; u0j is a random effect at
group level; and eij is a random effect at the individual level. Both u0j
and eij are assumed to be independent of one another and normally
distributed with means 0s and variances 2
u and e2 respectively.
The variance of Yij is therefore decomposed into the variances
at group level and individual level. Taking into account the
explanatory variable (xij) at the individual level and assuming that
the slope associated with the explanatory variable is group dependent, the HLMs have the following general form:
Y
ij = ˇ0 + u0j + ˇ1xij + u1jxij + eij
where ˇ0j = ˇ0 + u0j defines the intercept and ˇ1j = ˇ1 + u1j defines
the slope. Such a model construction assumes both the intercept
and slope are group dependent with group-level residuals u0j and
u1j. One may include several individual level explanatory variables in HLMs; however, direct modelling of dependencies among
micro-level network ties as endogenous effects is not possible. This
approach is not aiming or designed to model network ties directly,
and there are limitations when using HLMs for modelling multilevel networks in general. Firstly, the HLM approach treats the
nested meso-level structure as exogenous and understood, so any
underlying meso-level network processes are not captured. Secondly, the micro- and macro-within level network structure cannot
be captured directly by HLMs, thus the interdependencies between
network ties cannot be tested.
4. Exponential random graph models
Introduced by Frank and Strauss (1986) and Wasserman and
Pattison (1996), exponential random graph models (ERGMs) treat
the network structure as endogenous and a topic of research interest. Treating each network tie as a random variable, ERGMs model
network ties explicitly, and present the overall network structure
as a collective result of various local network processes. The local
network processes are represented by graph configurations such
as edges, stars, and triangles (see, e.g. Frank and Strauss, 1986;
Snijders et al., 2006), where all presented ties in a particular configuration are assumed to be conditionally dependent reflecting
hypotheses that empirical network ties do not form at random, but
that they self organize into various patterns arising from underlying social processes. ERGMs have been applied to different data
structures such as bipartite networks (Skvoretz and Faust, 1999;
Agneessens and Roose, 2008; Wang et al., 2009, 2013) and multiple networks (Pattison and Wasserman, 1999; Robins and Pattison,
2006; Wang, 2013). In the generic cases, where one uni-partite (or
one-mode) network (Y) with n nodes is involved, the ERGMs have
the following form:

Pr(Y = y) =

exp Q Q zQ (y)

1
() where
• y is a network instance.
• Q defines the network configurations which are based on the
dependence assumptions of tie variables. Note that a network
variable Y can be seen as a collection of tie variables (Yij) defined
on each dyad (i, j) of the network. A network configuration of type
Q includes tie variables that are conditionally dependent given
the rest of the network.
• zQ (y) = yYij ∈ Q yij is the network statistic for the corresponding network configuration of type Q.
• Q is the parameter associated with zQ (y).
• () is a normalizing constant defined based on the graph space
of networks of size n and the actual model specification.
Since the normalizing constants of ERGMs become intractable
for networks with even a small number of nodes, maximum likelihood estimates (MLEs) cannot be derived analytically except for
simple Bernoulli models. Snijders (2002) proposed an estimation
algorithm which relies on Markov Chain Monte Carlo (MCMC) simulations of ERGMs. The MCMC simulation also serves as a tool for
model goodness of fit (GOF) test, where simulated graph distributions are compared with the observed networks (Snijders, 2002;
Hunter et al., 2008).
As in Frank and Strauss (1986), the model assumes network
homogeneity such that isomorphic network configurations have
equal parameters, and the corresponding effect is the same across
the network. For networks where such assumptions do not apply, a
homogeneous model may have difficulty in convergence. We may
treat part of such networks as exogenous, or fit conditional models
depending on exogenous covariates.
ERGM specifications for both one-mode and bipartite networks
can be derived from dependence assumptions between network tie
variables. There has been a rich literature on how the network ties
may be conditionally dependent and their inferred ERGM specifications including the dyadic independence assumption (Erdös and
Rényi, 1960; Holland and Leinhardt, 1981) where all the tie variables are considered independent from one another; the Markov
assumption (Frank and Strauss, 1986) where a pair of tie variables are considered conditionally independent unless they have a
node in common; and the social circuit assumptions (Pattison and
Robins, 2002, 2004; Snijders et al., 2006) where tie variables within
a social circuit (four-cycle) are considered conditionally dependent.
Pattison et al. (2009) and Pattison and Snijders (2013) proposed a
hierarchy of dependence assumptions based on a graph theoretical
framework which provides guidance for systematic development
of ERGM specifications. The social circuit specifications proposed
by Snijders et al. (2006) introduced alternating statistics where
geometric weights with parameters () are assigned to the degree
distribution and the two-path (or shared partner) distribution, such
that large changes of graph statistics in simulations are avoided
to alleviate model degeneracy (Handcock, 2003; Rinaldo et al.,
2009). The graph statistics introduced by Snijders et al. (2006)
include alternating-k-stars (AS), alternating-k-triangles (AT) and
alternating-k-two-paths (A2P),1 representing the dispersion of the
degree distribution, the tendency for closure, and the tendency
for sharing multiple partners respectively. For bipartite networks
involving nodes of two distinct sets, Wang et al. (2009) applied
the same geometric weighting technique on the degree distributions with alternating-stars and the two-path distributions with
alternating-4-cycles (ACs) of two different types.
The current available ERGMs may model multilevel networks
by parts. Without considering the cross-level network structure,
ERGMs for one-mode networks can be used to analyze individual
within-level networks one at a time, assuming ties from different
levels are independent from each other. Such analysis are likely
to be inadequate for the obvious reason that the cross level structures are ignored. Explicit multilevel network structure is invoked
through bipartite networks, with nodes of two different types (e.g.
people and groups) and relations (associations) from persons to
groups. For bipartite analysis, ERGMs, SAOMs, and other methods are available (e.g. Latapy et al., 2008; Wang et al., 2009, 2013;
Koskinen and Elding, 2010). In fact, all bipartite networks can be
seen as special cases of two-level networks where within level
ties are absent. This approach treats the cross-level structure as
endogenous but there are no within-level networks. ERGMs for
multivariate network analysis (Pattison and Wasserman, 1999)
provided an approach that models the interdependence of several
networks of different types. However, the different types of ties are
defined on a common type of nodes with no level distinction. The
dependencies between ties of different types from different levels require modelling of the within-level one-mode networks and
1 Also known by Hunter and Handcock (2006) as geometrically weighted degrees,
geometrically weighted edge-wise shared partners, and geometrically weighted
dyadic-wise shared partners.
the cross-level bipartite networks together. We propose ERGMs
for multilevel networks in the simplest form which involves nodes
from two levels. It may be further extended to the general k-level
ERGMs following similar data and model constructions.
5. ERGMs for two-level networks
Based on the description and labels of the two-level network
structure also shown in Fig. 1, ERGMs can then be expressed in the
following form:
Pr(A = a, X = x, B = b) = 1
() exp
Q
{Q zQ (a) + Q zQ (x) + Q zQ (b) + Q zQ (a, x) + Q zQ (b, x) + Q zQ (a, x, b)}
There are several components based on the different networks
involved:
• zQ (a) and zQ (b) are network statistics for the corresponding
within level network configurations. We can apply the current
ERGM specifications for one-mode networks (Snijders et al.,
2006; Robins et al., 2007b, 2009) to look at within level network
structures, given other effects in the model.
• zQ (x) are the network statistics for structural effects within the
bipartite affiliation network. The corresponding ERGM specifications were proposed by Wang et al. (2009, 2013).
• zQ (a, x) and zQ (b, x) are network statistics for configurations
involving ties from one of the unipartite network (A or B) and the
bipartite network (X), representing the interactions between the
two networks. Using zQ (a, x) as an example, it can be expressed
as zQ (a, x) = a,xAij ∈ Q,Xkl ∈ Q aijxkl.
• zQ (a, x, b) = a,x,bAij ∈ Q,Xkl ∈ Q,Buv ∈ Q aijxklbuv are statistics for
configurations involving ties from all three networks, and express
tendencies for structural effects to be associated across both levels simultaneously.
For effects that associate one type of within-level tie (A or B)
with a cross level tie (X), i.e. zQ (a, x) or zQ (b, x), and the cross level
effects zQ (a, x, b), we propose model specifications based on the
various dependence assumptions in the literature (e.g. Erdös and
Rényi, 1960; Frank and Strauss, 1986; Pattison and Robins, 2002,
2004). Although the dependence assumptions applied here are the
same as in ERGMs for one- or two-mode networks, the interpretations are very different as we assume dependencies between
tie-variables of different nature or types. We use the terminology “node of type A” or “A-node” to refer to the nodes that can
be involved in A-ties, and similarly for “node of type B”. For simplicity, we focus on the descriptions and possible interpretations of
models for non-directed multilevel networks in the following simulation studies. These interpretations may be equally applied to the
extensions to directed multilevel models. See Figs. 2 and 3 for a list
of proposed model configurations with possible interpretations for
both non-directed and directed networks.
Following the Markov assumption, several parameters representing interactions between one of the within level network and
the bipartite network may be derived including stars and triangles involving different types of ties. Note that because the Markov
assumption implies cliques of sizes up to three (or triangles), statistics involving all three types of tie-variables, for example (Aij, Bkl
and Xik), cannot be defined. Example Markov configurations including the interaction stars (such as Star2AX and Star2BX) and the
interaction triangles (TXAX, TXBX) are depicted in Fig. 2. Based on
the social circuit assumption, interaction alternating-triangles (e.g.
ATXAX) and cross-level four-cycles (C4AXB) may be included in the
model. A further extension in dependence, the three-path assumption, allows interaction three-paths (e.g. L3XAX) and cross-level
three-paths (L3AXB) to be included in the model. To understand
properties of the various proposed model specifications, simulation studies for the proposed configurations were carried out in the
next section, and we illustrate possible interpretations of the proposed effects in a two-level researcher–laboratory network context
similar to the data we used in our empirical example.
6. Multilevel ERGM simulation studies
The simulation studies were performed on (30, 30) two-level
networks with 65 ties in each of the within-level (A, B) and mesolevel (X) networks. For proposed effects only involving interactions
between one of the within-level network and the meso-level network, network B was kept empty for simplicity, as the interaction
effects between A and X are equally applicable to interactions
between B and X. The densities of the networks are fixed, and we
used strongly positive (+2) and negative (–2) parameter values for
each of proposed configurations and left other effects at 0s to test
the non-zero main effects on the network structure. We present
visualizations of sampled networks, and graph statistics such as the
means and standard deviations of the global clustering coefficients
(GCCs), the standard deviations (SDs) and the skewness (SKs) of
the degree distributions in each of the networks A, B and X. The
GCCs for the within-level networks (A, B) are calculated as the ratio
between closed and open triads; whereas the GCC for the mesolevel network X is calculated as the ratio between the numbers of
four-cycles and three-paths (Robins and Alexander, 2004).
Simulated network samples are shown from Figs. 4–8, where
each row has the tested main effect listed in the first column followed by the decomposition of a simulated multilevel network M
in the order of networks (A, X, B) when they apply. For the simulated graph statistics as listed from Tables 1–4, the main effects
Table 1
Simulated graph statistics for models with Star2AX and AAAXS. (Note: SD and SK stand for standard deviation and skewness of degree distributions. GCC stands for global
clustering coefficient.).
Statistics Random Star2AX+ Star2AX– AAAXS+ AAAXS–
A
SD 1.873(0.240) 6.863(0.008) 3.502(0.184) 2.822(0.258) 1.703(0.224)
SK 0.284(0.425) 3.422(0.010) –0.084(0.168) –0.108(0.253) 0.304(0.414)
GCC 0.144(0.035) 0.147(0.000) 0.334(0.043) 0.229(0.044) 0.136(0.036)
X
SD(A) 1.406(0.186) 7.610(0.000) 3.345(0.194) 1.769(0.190) 1.329(0.169)
SK(A) 0.522(0.429) 3.607(0.000) 1.208(0.193) 0.512(0.364) 0.462(0.410)
SD(B) 1.394(0.178) 0.346(0.000) 1.283(0.160) 1.394(0.178) 1.407(0.180)
SK(B) 0.526(0.430) 2.273(0.000) 0.371(0.403) 0.524(0.409) 0.524(0.415)
GCC 0.066(0.031) 0.896(0.000) 0.207(0.037) 0.083(0.032) 0.064(0.030)

Effects	Configuraons
Density	A	B X
Affiliaon based popularity	Star2AX	Star2BX
AXS1A …	AAS1X …	ABS1X …
AXS1B …
AAAXS
ABAXS … …
Affiliaon based closure, or homophily by shared affiliaons.	TXAX TXBX	ATXAX …	ATXBX …
Interacon between meso-level popularity and within level acvity	L3XAX L3XBX
Cross-level interacons	Within-level acvity assortavity by affiliaon L3AXB
Cross-level alignment C4AXB
… …

Fig. 2. Multilevel ERGM configurations for non-directed networks.
used in the simulation are listed in the header, andwecompared the
statistics from the simulated distributions with a random network
distribution (listed under the header “random”) to understand the
impact on the global network structure. For each of the simulations,
we picked every 10,000th graph from a 10,000,000 iteration simulation after a 1,000,000 iteration burn-in from random starting
networks of the same density.
6.1. Interaction stars
The Star2AX and the Star2BX configurations as shown in Fig. 2
represent the dependence between two tie variables of different
Table 3
Simulated graph statistics for models with three-paths (L3XAX). (Note: SD and SK
stand for standard deviation and skewness of degree distributions. GCC stands for
global clustering coefficient.).
Statistics Random L3XAX+ L3XAX–
A
SD 1.873(0.240) 2.554(0.310) 1.897(0.230)
SK 0.284(0.425) 0.144(0.328) 0.289(0.401)
GCC 0.144(0.035) 0.207(0.047) 0.145(0.036)
X
SD(A) 1.406(0.186) 1.240(0.164) 1.423(0.188)
SK(A) 0.522(0.429) 0.450(0.444) 0.546(0.416)
SD(B) 1.394(0.178) 4.196(0.322) 1.248(0.155)
SK(B) 0.526(0.430) 4.313(0.514) 0.369(0.406)
GCC 0.066(0.031) 0.149(0.071) 0.069(0.033)
Table 2
Simulated graph statistics for models with interaction triangles. (Note: SD and SK stand for standard deviation and skewness of degree distributions. GCC stands for global
clustering coefficient.).
Statistics Random TXAX+ TXAX– ATXAX+ ATXAX–
A
SD 1.873(0.240) 5.379(0.049) 1.892(0.242) 3.628(0.257) 2.466(0.267)
SK 0.284(0.425) 0.437(0.003) 0.297(0.415) 0.730(0.110) 0.557(0.347)
GCC 0.144(0.035) 0.977(0.014) 0.144(0.034) 0.623(0.052) 0.154(0.036)
X
SD(A) 1.406(0.186) 2.535(0.080) 1.420(0.190) 3.377(0.102) 2.337(0.172)
SK(A) 0.522(0.429) 0.428(0.036) 0.539(0.408) 0.987(0.070) 0.660(0.256)
SD(B) 1.394(0.178) 4.541(0.052) 1.258(0.158) 1.282(0.150) 1.359(0.181)
SK(B) 0.526(0.430) 1.856(0.019) 0.364(0.383) 0.359(0.380) 0.481(0.427)
GCC 0.066(0.031) 0.959(0.019) 0.068(0.033) 0.214(0.032) 0.119(0.032)

Effects	Configuraons
Density	A	B X
Affiliaon based popularity or acvity effects.	Out2StarAX In2StarBX Out2StarBX
In2StarAX
AXS1Ain AXS1Aout AXS1Bin AXS1Bout … … … …
AAinS1X …	AAoutS1X ABinS1X ABoutS1X … … …
Affiliaon based closure, or homophily by a common affiliaon.	TXAX arc	TXAXreciprocity TXBXarc TXBXreciprocity
The tendency for nodes sharing mulple affiliaons to connect to each other, or homophily by mulple common affiliaons.	AT_XAX arc AT_XAX reciprocity AT_XBX arc AT_XBX reciprocity … … … …
Assortavity based on popularity or acvity in affiliaon network, or precondion for cross-level closure.	L3XAX arc L3XAX reciprocity L3XBX arc L3XBX reciprocity

Assortavity based on popularity or acvity in within level networks, or precondion for cross-level closure.	L3AXB_in L3AXB_out L3AXB_path L3BXA_path
Cross-level entrainment and exchange	C4AXB entrainment C4AXB exchange
Strong forms of Cross-level entrainment and exchange by reciprocity	C4AXB reciprocity C4AXB exchange reciprocal B C4AXB exchange reciprocal A

Fig. 3. Multilevel ERGM configurations for directed networks.
Fig. 4. Simulated network sample with Star2AX and AAAXS.
types established by a node in common. Fig. 4(a) illustrates
typical networks from the simulated network distributions where
the Star2AX parameter was set at 2 and –2 respectively, and
network B was kept empty, so as to observe only the effects on
network A. The positive Star2AX parameter is an A-hub creator, as
we can see from the plot of the networks (A, X and M), two A-hubs
were generated both connected to all other A-nodes in network A
and all the B-nodes in the affiliation network X. The model created
an almost frozen graph distribution in both network A and the
affiliation network X indicated by the standard deviations close
to 0 in the various graph statistics shown in Table 1. The standard
deviation (SD) and skewness (SK) of A-node degree distributions
in both networks A and X are much higher than the random graph
distribution; the B-node degree distribution in network X is flat,
as all B-nodes are connected to the three A-hubs, hence having
degrees 2 or 3.
When Star2AX is negative, the model tries to avoid any association between the within- and the meso-level ties, and created two
components in the overall multilevel network. This effect created
lots of isolated A-nodes in both networks A and X, as non-isolated
nodes in network A became isolates in network X and vice versa
to form the separated overall structure. Since more isolated nodes
are involved, the available network ties had to constrain themselves to the connected component, so we have higher SDs and
SKs for A-nodes in both networks A and X, and higher clustering coefficients than expected from random (see Table 1). We
expect similar behaviour for the Star2BX effect, and in general, we
may conclude that the within- and meso-level interaction two-star
Table 4
Simulated graph statistics for models with cross-level interaction effects (L3AXB and C4AXB). (Note: SD and SK stand for standard deviation and skewness of degree
distributions. GCC stands for global clustering coefficient.).
Statistics Random L3AXB+ L3AXB– C4AXB+ C4AXB–
A
SD 1.873(0.240) 5.241(0.005) 2.862(0.264) 3.706(0.195) 2.098(0.252)
SK 0.284(0.425) 0.497(0.004) –0.217(0.243) 0.727(0.102) 0.369(0.426)
GCC 0.144(0.035) 0.857(0.003) 0.234(0.045) 0.643(0.025) 0.156(0.037)
B
SD 1.867(0.234) 6.014(0.027) 2.854(0.272) 4.238(0.159) 2.097(0.256)
SK 0.288(0.422) 2.674(0.050) –0.212(0.251) 0.648(0.055) 0.367(0.408)
GCC 0.145(0.035) 0.221(0.007) 0.234(0.046) 0.771(0.027) 0.156(0.036)
X
SD(A) 1.406(0.186) 2.649(0.004) 2.267(0.341) 3.331(0.008) 1.615(0.216)
SK(A) 0.522(0.429) 0.610(0.007) 1.486(0.407) 0.954(0.016) 0.696(0.408)
SD(B) 1.394(0.178) 4.470(0.006) 2.252(0.345) 3.089(0.009) 1.615(0.217)
SK(B) 0.526(0.430) 1.797(0.003) 1.494(0.384) 0.789(0.019) 0.703(0.416)
GCC 0.066(0.031) 0.867(0.002) 0.133(0.036) 0.650(0.006) 0.082(0.032)
Fig. 5. Simulated network samples for models with interaction triangles.
parameters represent popularity effects in both within level and
affiliation network. In a researcher and laboratory network context,
where the B-level network contains advice seeking ties among the
researchers, the A-level network is defined based on the collaboration ties among the laboratories, and the researcher–laboratory
affiliation network X, a positive Star2BX parameter indicates that
researchers popular in the advice seeking network are also popular
in the affiliation network (i.e. have many affiliations).
To alleviate the issues of model degeneracy or the frozen
graph distributions, we can apply the same geometric weighting
techniques on either or both the affiliation star parameters and the
within-level star parameters to form higher-order configurations,
such as alternating-X-stars with one A-tie attached (AXS1A), or
alternating-A-alternating-X-stars (AAAXS). Fig. 2 illustrates some
possible alternating-star effects. The alternating-star effects generally attenuate the overwhelmingly frozen structures created by
the two-star effects seen in the previous simulations, but provide
similar interpretations. We followed the same simulation strategy
with the same parameter values (+2 and –2) for models with
AAAXS. As can been seen from Table 1, when AAAXS is positive, the
graph distribution has higher SD for the A-node degree distribution
than a simple random graph, but much smaller than for the previous model with positive Star2AX. The SD(A) in network X, and SKs
for A-nodes for both networks A and X are not very different from
Fig. 6. Simulated network samples for models with three-paths (L3XAX).
Fig. 7. Simulated network samples for models with three-paths (L3AXB).
Fig. 8. Simulated network samples for models with cross-level four-cycles (C4AXB).
random. From the visualization in Fig. 4(b), we do not see obvious
hubs in the network A but more nodes with degree greater than 2
or 3 compared with positive Star2AX model. There are also more
isolates in network A, as nodes are not forced to be part of hubs anymore; and four out of the six isolates are also isolates in the overall
network M, i.e. the unpopular nodes in within level network are
also unpopular in the affiliation network. When AAAXS is negative,
we actually see marginally smaller A-node SDs in both networks A
and X, and smaller SK in network A than random networks representing more flat A-node degree distributions in both networks A
and X, suggesting a tendency against centralization.
6.2. Interaction triangles
The interaction triangle effects (TXAX, TXBX, ATXAX and ATXBX)
have two connected nodes within the same level, sharing one or
more nodes from the other level through affiliations. They can
be seen as affiliation based closures, or within level homophily
effects based on common affiliations (i.e. the homophily can be
construed as arising from a shared affiliation). We conducted simulations using positive and negative values (+2 and –2) for TXAX
and ATXAX similarly as before, and the results are shown in Fig. 5
and Table 2. When TXAX is positive, it created clique-like structures
in the within-, meso- and overall two-level networks.2 In turn, we
2 Note that for bipartite networks, sub-graphs with two nodes sharing one or
more nodes of the other type are all considered as cliques.
see greater than random SD and SK in network A, and greater a
SD(A) in network X, as there are centralization effects on network
ties among nodes involved in the clique; and the global clustering
coefficients for both networks A and X are also much greater than
expected from random networks. These clique-like formations are
to be expected from positive Markov triangle parameters, given
that similar effects are evident in unipartite models. When TXAX
is negative, the clique-like structures disappear, and most of the
previously isolated nodes merge into the big components of the
networks. Since the negative TXAX is trying to separate bipartite
two-paths from the within level A ties, or in other words, the connected A-nodes tend not to share common affiliations, we can see
long circular-like structures in network X. Compared with random
networks, there is no obvious differences on the degree distribution
and clustering measures we obtained.
The ATXAX attenuates the TXAX effect in a similar fashion as
in the star configurations. When ATXAX is positive, we still have
clique-like core structures in both network A and X. However,
instead of being isolated, the rest of the A-nodes are connected to
the core in network A; and only the core-members in network A
are sharing multiple B-nodes in the bipartite network X and the
overall multilevel network M. The GCCs and SDs for A-node degree
distributions in both networks A and X are higher than random
networks but smaller than the case when TXAX is set at the same
positive value. Comparing the negative effects of TXAX and ATXAX,
the attenuating ATXAX had higher values in most of the statistics in
Table 2, but generally lower compared with the model with positive
ATXAX. In the cases of researcher–laboratory networks for instance,
a positive TXBX or ATXAX effect may indicate that researchers from
the same laboratories tend to seek advice from each other.
6.3. Interaction three-paths
Defined by two X-ties and one within level tie (either A or B), we
label the three-path statistics involving the interactions between
the within- and meso-level networks as L3XAX and L3XBX. They
can be seen as the interaction between meso-level popularity and
within-level activity, or meso-level degree assortativity through
within level ties. From the simulation results as shown in Fig. 6
and Table 3 using the same settings as previously, we can see that
the positive L3XAX parameter creates as many XAX three-paths
as possible such that non-isolated A-nodes in network A are all
involved in the bipartite network X, as an A-tie is an essential part
of the L3XAX. Compared with the random GCCs for both networks
A and X, the networks with positive L3XAX have higher GCCs as
shown in Table 3. Because all bipartite ties are associated with
non-isolated A-nodes in network A, they create more TXAXs, hence
more L3XAXs as the TXAX configuration is also part of L3XAX. However, the effect is not as strong as a positive TXAX effect where the
GCCs are much higher, as shown in Table 2. When L3XBX is negative, there are very long paths in the bipartite network to make
bipartite clustering less likely; and the clustering in network A is
very similar to random networks. Together, they create fewer XAX
three-paths.
In the researcher–laboratory network context, a positive L3XAX
parameter may suggest that researchers (B) popular in the affiliation network tend to be associated with collaborating laboratories
(A); or there is a degree assortativity effect such that labs
tend to collaborate with other labs having a similar number of
researchers.
6.4. Cross-level three-path
The cross-level three-path L3AXB configuration involves ties
from all three (A, B and X) networks, and is very different from
L3XAX or L3XBX; it represents a tendency for nodes that are
popular within levels to be affiliated through the meso-level network X, and can be seen as degree assortativity effect between
networks A and B through meso-level network X. We conducted
simulations for the L3AXB effect using the same positive and negative parameter values. Since L3AXB involve tie variables from all
three networks (A, X and B), network B is now set to have 65 ties
with fixed density; other simulation settings are set the same as
before. The simulated sample plots and graph statistics are presented in Fig. 7 and Table 4. When L3AXB is positive, hubs emerge
from one of the within level networks (in our case, network A);
and ties from the other network (B) form a clique-like strongly
connected component. All nodes that are part of the connected
component in network (A) are connected to the hubs in network (B)
through the bipartite network; and unsurprisingly, all of the SDs,
SKs and GCCs are higher than random networks. Note that starting the simulation from random networks of the same densities,
the structures for networks A and B may swap depending on the
network in which the hubs emerge first, and the other network
in turn forms the clique-like structure. When L3AXB is negative,
to form as few as possible AXB three-paths, the parameter avoids
any affiliations between nodes from the connected components
of the within-level networks. As a result, the isolated nodes in
within level networks become the popular nodes in network X.
We can see from the visualization of the overall multilevel network M that the two within-level networks are loosely connected
by few bipartite ties, and the isolates are working as agents or brokers between the two components. The statistics listed in Table 4
indicate that the global structures of the within-level networks are
very similar.
In a researcher–laboratory network context, a positive L3AXB
parameter suggests high degree researchers are affiliated with high
degree labs.
For directed within level networks, we can define four types
of L3AXB as L3AXB-in, L3AXB-out, L3AXB-path and L3BXA-path
as shown in Fig. 3. The L3AXB-in/-out, where the within level
incoming or outgoing ties represent within level popularity or
expansiveness, may be interpreted similarly to the non-directed
L3AXB as effects indicating the tendency for popular or expansive nodes from the two levels to form affiliations. L3AXB-path
and L3BXA-path however, represent the interaction between popularity of nodes at one level and expansiveness of nodes at the
other level through their affiliations. In the researcher–laboratory
network context, a positive L3AXB-in effect may indicate that the
popular laboratories (A) in the collaboration network tend to have
resourceful researchers (B) from whom lots of other researchers
seek advice; whereas a positive L3AXB-path effect may indicate
that popular laboratories tend to have researchers seeking more
advice from others.
6.5. Cross-level four-cycle
The cross-level four-cycle C4AXB involves ties from all three
networks, representing a cross-level “mirroring” or alignment
effect such that members of connected groups are themselves
connected (Lazega et al., 2010, 2011). Through simulations using
similar settings as before, we can see from Fig. 8 that when C4AXB
is positive, it generated clique-like cores in both networks A and B,
and the members of the cores are affiliated with each other. Statistics from Table 4 suggest there is centralization on ties from both of
the within-level networks with greater than random SDs and SKs;
with the clustering of high-degree nodes reflected by the high GCCs
in all networks A, B and X. The negative C4AXB effect however, is
trying to misalign A– and B-ties through affiliation, and can be generally understood as a decentralization effect, with fewer isolated
nodes especially in the affiliation network X. In terms of degree distributions and global clustering, the SDs, SKs and GCCs for all three
networks are not very different from random.
In the researcher–laboratory network context, a positive C4AXB
may suggest researchers from collaborating laboratories tend to
seek advice from each other.
For directed networks, depending on the directions of the
within-level ties and whether reciprocal ties are involved, C4AXB
may have several forms as shown in Fig. 3.
6.6. Models for caveman graphs
As an illustrative example to show how the proposed multilevel model parameters may interact with each other, we compare
ERGMs with and without the cross level interaction effects to simulate caveman graphs, where small clique like components are
sparsely connected to form a large component. Caveman graphs
were introduced in a discussion of the small world phenomenon
by Watts (1999). In this case, we continue to use the multilevel terminology, although now the ‘levels’ may simply be construed as
two different types of nodes as Wasserman and Iacobucci (1991)
originally described.
To date, there is not a typical set of one-mode ERGM parameters
that could reproduce caveman graphs.3 However, it is not difficult
to reproduce caveman like graphs using two-level ERGMs where
3 Robins et al. (2005) presented some simulation studies involving frozen graphs
of different forms representing small and other worlds.
Fig. 9. ERGM configurations following Snijders et al. (2006).
within level networks form the clique like structures or “caves”, and
the cross level bipartite network can be used to connect the caves.
Simulations were conducted on (20, 20) two-level networks with
fixed within level densities of 0.15, and a fixed bipartite density
of 0.06. For within level networks (A and B), we use the onemode ERGM specification following Snijders et al. (2006), including
alternating-stars (AS), alternating-triangles (AT) and alternatingtwo-paths (A2P) as shown in Fig. 9.
To have the highly clustered but separated components or
“caves” in the within level networks, we use the following set of
parameters for both within level networks A and B where:
AS = –1, AT = 2 and A2P = –1. The positive alternating-triangles
ensure high clustering; the negative alternating-stars decentralize
the degrees, i.e. discourage formation of hubs; and the negative
alternating-two-paths break down the network into separate
components. The bipartite network (X) is left at random, and all
between or cross-level interaction effects are set at 0s. We use this
model as a reference where no between-level interaction effects
are imposed. Fig. 10(a) and (b) show a typical set of within level
networks simulated from this model. Fig. 10(c) is the simulated
random bipartite network (X) that connects the caves from both
networks A and B to form the caveman like two-level graph in
Fig. 10(d). We can observe some random overlapping between
the caves where connected dyads from caves of different types
are forming cross-level four-cycles (C4AXBs). Depending on the
density of the bipartite networks, the random overlapping may
have some of the caves tightly interlocked.
Fig. 10. A caveman graph without cross-level effects.
Fig. 11. A caveman graph with cross-level effects.
For the interaction model, we use the same set of within level
parameter values to create the “caves”. The within bipartite network (X) effects were left at 0s, but the following interaction effects
were imposed where: AAAXS = 2, ABAXS = –2 and C4AXB = –2. A
simulated graph sample is presented in Fig. 11. These parameters
in effect imply that popular nodes in the A network will be popular
with B nodes as well (i.e. centrality in the A network is associated
with centrality in the X network); but that popular B nodes are not
popular with A nodes; and that cross-level 4 cycles are less likely
to be present.
The opposite signs of the interaction stars (AAAXS and ABAXS)
affect the popularities of nodes as well as the sizes of the cliquelike components of different types. To generate more AAAXS but
fewer ABAXS, the bipartite ties are more centralized on nodes of
type A. Comparing Fig. 11(a) to Fig. 10(a), we have fewer but large
clique like components, as well as more isolates, indicating stronger
degree centralization in network A. Compare Fig. 11(c) to Fig. 10(c),
there are more isolates of type A than B, hence the average degree
of A nodes are higher in the bipartite network. As nodes from
larger cliques have higher average degrees than smaller cliques,
and the bipartite ties are centralized on A nodes that are not isolates
(as shown in Fig. 11(d)), the interaction effects effectively created
more AAAXS and less ABAXS than the previous model. The negative
C4AXB limits the alignment or entrainment of ties from different
cliques such that only a single member within a clique may connect
tomultiplemembers of a clique from the other level. In other words,
we have brokerage effects between different types of ‘caves’ where
one member of one cave tends to be the gatekeeper in regra to the
other cave. The combined effect of the three interaction parameters created caveman graphs where smaller caves from one level are
centralized on larger caves of the other level as shown in Fig. 11(d).
Such networks are not uncommon, e.g. in organizational networks,
suppose that larger cliques represent a small number of central
management divisions, and smaller cliques represent other business divisions. This model represent networks where within each
division (large or small) nodes are highly connected, but between
divisions, there is only one or two nodes worked as brokers or
representatives liaising with other divisions.
As we can see from these simulation studies, even very simple
interaction effects can create structure at one or both levels. Our
example in the next section demonstrated the importance of these
parameters in capturing the dependencies between the macro-,
micro- and meso-level networks.
Fig. 12. The multilevel network among French cancer research elites (Lazega et al., 2008).
7. Models for French cancer researchers and their
institutions
7.1. The network data
The network data was collected from French cancer researcher
elites4 and their institutions in 1999. Several studies have been
carried out based on this set of data (Lazega et al., 2004, 2006,
2008, 2010, 2011). The data set consists of 97 researchers and 82
laboratories. The corresponding two-level network consists of a
directed collaboration network among the laboratories (A) as nominated by the laboratory directors; a directed advice network (B)
where researchers nominated from whom they seek advice; and
the bipartite network (X) representing the affiliation between the
researchers and laboratories. Fig. 12 displays the multilevel network as three separate networks.
This data set has several features. First of all, the affiliation
network is based on public information about the researchers’
affiliations, each individual researcher was only affiliated with one
laboratory, and most laboratories only had one researcher who
was a member of these elites. It is hard to infer more interesting
bipartite structures apart from a list of isolated dyads and stars
of sizes two or three, so we treated the bipartite network as
exogenous in our models. This implies that the bipartite network
is fixed throughout the model estimations and goodness of fit
tests. Secondly, there are two laboratories with much higher
4 The selection criterion for inclusion in this population was having published at
least eight papers per yearfor three consecutive years – 1996–1998 – in international
cancer research journals listed in the CANCERLIT database (nowadays included in
the PUBMED database).
out-degrees (29 and 36) compared with the rest of the laboratories
(with a maximum out-degree of 13). Both of these laboratories
were providers of experimental equipment and materials for other
laboratories. Given their special position, we treated ties associated
with these two laboratories as exogenous in our models. Note that
the isolated nodes in the laboratory or researcher networks are
not isolates in the affiliation network.
7.2. Modelling results and interpretations
We firstly present models for within level networks, and
then compare them with a model for the overall multilevel
network in both parameter estimates and model goodness of
fits. In other words, the within-level model simply models the
A and B (researcher and laboratory) network structures under
the assumption that the bipartite affiliations are irrelevant,
whereas the interaction models takes the bipartite structure into
account. Such comparisons can reveal the importance of the
cross level network processes that shape the structure of both
the individual within level networks and the overall multilevel
network.
Both of the within level networks in our data are directed. Fig. 13
shows some of the directed ERGM configurations involved in the
within level models. The alternating-in-stars (AinS) represent the
in-degree centralization or popularity spread; the alternatingout-stars (AoutS) represent out-degree centralization or activity
spread (Robins et al., 2009). Combining the in and out degree
alternating statistics, we can also derive a statistic known as the
alternating-in-alternating-out-star (AinAoutS) representing the
correlation between in and out degrees. While this is seldom used
in unipartite networks (sometimes a simple two-path parameter
is used to control for this correlation), it will be important in
Fig. 13. Directed ERGM configurations used in the modelling example.
the modelling examples. Robins et al. (2009) applied the same
dependence assumption and techniques using geometric weightings and alternating signs as in Snijders et al. (2006) for directed
networks, and introduced alternating-k-triangles/-k-two-paths of
different forms. In our models, we include the AT-T configuration
representing the tendency for multiple transitive paths to form
closure; AT-C for cyclic path closure, which may be interpreted as
a form of generalized exchange; and A2P-U for shared activity.
All models presented here are successfully converged, so convergence statistics are not presented. For model goodness of fit
(GOF) test, simulated graph distributions using the converged
model were compared with the observed statistics, and t-ratios are
used as heuristic GOF test statistics (Snijders, 2002; Hunter et al.,
2008). We tested each within-level models against 54 within-level
graph statistics, and for the overall multilevel goodness of fit, 131
graph statistics were tested. We only present the statistics that
are not well fitted, defined as t-ratios greater than 2.0 in absolute
values.
7.2.1. Model for laboratory collaboration network only
Despite treating the two high out-degree equipment/material
providers as exogenous, the in- and out-degree distributions of the
laboratory remain skewed as shown in Fig. 14, and there are still
a small group of laboratories with high in-/out-degrees compared
with the rest of the nodes, making the degree distributions long
tailed, or even multimodal. Despite the robustness of the social
circuit model specifications (Snijders et al., 2006), the long tailed
degree distributions remain a challenge for ERGMs in model goodness of fit. The model estimates for the laboratory network shown
Table 5
Parameter estimates for laboratory collaboration network only.
Effects Estimates Standard errors
Arc –3.811 0.530*
Reciprocity 1.784 0.345*
Two-path –0.100 0.025*
Isolates 2.051 0.834*
AinS(4.00) 0.702 0.255*
AoutS(4.00) 0.315 0.082*
AinS(2.00) –1.019 0.582
AT-T(2.00) 0.493 0.133*
* indicates significant effect where the estimated effect exceeds twice the
standard error in absolute value.
in Table 5 fitted all 54 graph statistics adequately, including the
standard deviations and skewness of the in-/out-degree distributions. To provide a good fit to the long-tailed in-degree distribution,
we needed two alternating-in-star (AinS) parameters with different
-values (4.0 and 2.0). (Appendix presents some simulation studies
on how combining different star or AS parameters with different
-values within the same model may help capturing the longtailed degree distribution.) Even though the AinS with  = 2.0 is not
Laboratory in-degree distribuon
Standard deviaon = 2.58
Skewness = 1.12
Laboratory out-degree distribuon
(without the two high degree nodes with out-degrees 29 and 36)
Standard deviaon = 4.96
Skewness = 4.20
In/Out degree correlaon = 0.24
Mean degree = 3.34
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 5 10 15
0 1 2 3 4 5 6 7 8 9 10 11
0 5 10 15
Fig. 14. In- and out-degree distributions of the laboratory collaboration network.
Researcher in-degree distribuon
Standard deviaon = 3.76
Skewness = 0.67
Researcher out-degree distribuon.
Standard deviaon = 5.23
Skewness = 1.40
In/Out degree correlaon = 0.50
Mean degree = 5.62
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0 2 4 6 8 10 12 14
0 2 4 6 8 10 12 14 16 18 20 22 24 26
0 2 4 6 8 10 12 14
Fig. 15. In- and out-degree distributions of the researcher advice network.
significant, a GOF comparison between the presented model and a
model without AinS(2.0) showed that by including the AinS(2.0) in
the model, the t-ratio for the skewness of the in-degree distribution
reduced from 2.74 to 1.26. For many practical modelling applications, seeking to obtain such excellent fit may not be necessary, so
these models may seem overly complex. However, we seek good
fit here on many indices in order to illustrate the relevance of the
full multilevel structure.
From the parameter estimates, we can see that the collaborations between laboratories tended to be reciprocated. The positive
and significant alternating in-/out-stars with  = 4.0, indicate both
the in- and out-degree degree distributions are more dispersed
than expected from random networks. In other words, both the
popularity and expansiveness of the laboratory collaborations are
centralized. The positive AT-T suggests the laboratory collaboration tended to be clustered. Combining the clustering effect and
the negative two-path suggests a path-shortening effect whereby
the collaborations are more direct, rather than going through multiple paths. Given the other effects, there are more isolated nodes
than expected (although those isolates are not isolated nodes in the
affiliation network).
7.2.2. Model for the researcher advice network only
The advice network contains advice seeking ties among 97
researchers. Both the in- and out-degree distributions are skewed,
especially for the out-degree distribution with long tails with two
expansive researchers nominating more than 20 ties, as shown in
Fig. 15. To provide adequate fit to the degree distributions along
with the other statistics, we needed several star parameters as
shown in Table 6. Some of the star parameters are not significant, but they are important in fitting the degree distributions and
Table 6
Parameter estimates for researcher advice network only.
Effects Estimates Standard errors
Arc –3.213 1.024*
Reciprocity 3.534 0.213*
2-Out-star 0.358 0.146*
3-Out-star –0.018 0.009
Two-path –0.135 0.010*
AinS(4.00) 0.596 0.159*
AoutS(4.00) –0.722 0.599
AinS(2.00) –1.164 0.450*
AoutS(2.00) 0.384 0.787
AinAoutS(2.00) –0.233 0.369
AT-T(2.00) 0.932 0.067*
* indicates significant effect where the estimated effect exceeds twice the
standard error in absolute value.
the clustering coefficient as well as the correlation between inand out-degrees. The same model specification without two- and
three-out stars underestimated the skewness of the out-degree distribution (t-ratio = 2.12), and failed to capture the global clustering
coefficient (t-ratio = 3.48). The same model specification without
AinAoutS underestimated the in- and out-degree correlation (tratio = –4.66).
Besides the various star parameters, the interpretation of this
model is similar to the laboratory collaboration network. The
strongly positive reciprocity effect indicates advice seeking tended
to be reciprocated in this network. Through the positive AT-T, we
see researchers seek advice in clusters; combined with a negative
two-path parameter, we can infer the path-shortening effect also
existed in the advice network. In other words, network closure is
the predominant effect in this model, suggesting advice seeking
among researchers in a clustered, team-like way.
The interpretations of the various star parameters generally
infer a long-tailed or multi-mode degree distribution, as we have
discussed in the simulation study of AS with different -values.
The positive and significant 2-out-star suggests there are hub-like
advice seekers who ask a lot more advice than others. The positive AinS(4.0) allow some variations for nodes with high in-degrees
but not hubs; and the negative AinS(2.0) suggests there was little
variation among researchers in being popular sources of advice.
7.2.3. Within level model GOF with respect to between-level
statistics
The separate models for laboratories and researcher fitted
within level statistics well with all t-ratios less than 2.0. However,
by testing the models against multilevel interaction statistics, they
fail to capture a list of statistics listed in Table 7.
The cross-level triangulation effects (TXBXarc and TXBXreciprocity) are quite basic: they represent researchers from the same
laboratory seeking advice from each other. Comparing the observed
statistics with the simulated means based on the within level models, the observed advice seeking within laboratories is much higher
than expected from the models that do not explicitly cater for crosslevel effects. The within level models also failed to capture the
Table 7
Poorly fitted statistics for models with within-level effects only.
Configurations Observed Mean Standard
deviations
GOF
(t-ratio)
TXBXarc 30 2.391 1.775 15.55
TXBXreciprocity 14 0.612 0.770 17.38
C4AXBentrainment 202 43.918 14.893 10.62
C4AXBexchange 196 43.532 13.768 11.07
C4AXBexchangeBreciprocity 136 23.700 9.209 12.20
C4AXBexchangeAreciprocity 60 6.844 5.091 10.44
C4AXBreciprocity 11 0.942 0.993 10.13
Table 8
A multilevel ERGM for French cancer research elites.
Effects Estimates Standard errors
Laboratory collaboration network
Arc –3.831 0.556*
Reciprocity 1.679 0.381*
Two-path –0.079 0.029*
Isolates 2.017 0.760*
AinS(4.00) 0.640 0.268*
AoutS(4.00) 0.320 0.086*
AinS(2.00) –0.889 0.614
AT-T(2.00) 0.446 0.127*
Researcher advice network
Arc –4.084 0.118*
Reciprocity 3.313 0.212*
AT-T(2.00) 1.085 0.072*
AT-C(2.00) –0.384 0.068*
A2P-U(2.00) –0.071 0.020*
Laboratory collaboration and affiliation
AXS1Ain(2.00) 0.240 0.131
AXS1Aout(2.00) –0.324 0.129*
Researcher advice and affiliation
TXBXarc 1.958 0.275*
Cross level interactions
L3AXBin –0.006 0.018
L3AXBout –0.012 0.008
L3BXApath –0.003 0.010
L3AXBpath –0.051 0.015*
C4AXBentrainment 0.634 0.104*
C4AXBexchange 0.639 0.109*
C4AXBexchange reciprocal A –0.293 0.065*
C4AXBexchange reciprocal B –0.295 0.136*
* indicates significant effect where the estimated effect exceeds twice the
standard error in absolute value.
cross level exchange and various forms of cross level entrainment
effects. These effects represent tendencies for researchers from collaborating laboratories to ask advice from each other. Evidently,
these tendencies are much stronger than expected from the within
level models. These poorly fitting statistics indicate that the cross
level interaction parameters are required to provide good fit to the
network data.
7.2.4. Model for the overall multilevel network
The model for the overall multilevel network is presented in
Table 8. It has several sections including effects for both within level
networks, interaction effects between the advice network and the
affiliation network, interaction effects between the collaboration
network and the affiliation network, and the cross level interactions
involving all three networks. In this case, all 131 GOF statistics were
less than 2.0 in absolute values, indicating a good fit to the observed
two-level network on a very wide range of graph statistics.
We interpret this model by sections while keeping in mind that
all parameters in the model are dependent on each other, hence
the interpretation of one particular section is always conditional
on the rest of the model. We start with the sections involving cross
level network effects and then move on to the interpretations of
the within level effects. Comparing the changes in the within level
effects with the models without cross level effects, we discuss the
impact and importance of the cross level effects.
The Interaction effects between the laboratory collaborations
and the affiliation network include alternating laboratory affiliation
stars with an incoming collaboration tie (AXS1Ain) and an outgoing collaboration tie (AXS1Aout). Note that the AXS1Ain parameter
is not significant (defined here as an absolute value greater than
twice the size of the estimated standard error) but close to significance. Its removal from the model leads to poor fit on the in-degree
distribution of the within level laboratory collaboration network.
Having the alternating affiliation star as part of the statistics explicitly models the laboratory with more than one researcher, which
are among the popular laboratories or the “big ponds” in the network as described in Lazega et al. (2008). This positive estimate
suggests that bigger laboratories tended to receive more nominations as collaborators from other labs. However, in contrast, they
tended to nominate fewer collaborating labs, as indicated by the
negative and significant AXS1Aout parameter. This interpretation
makes empirical and theoretical sense, as the “big pond” labs might
be the dominant research centres that had good reputations and
more research resources that other labs were chasing, but they
themselves did not feel the same urgency to collaborate with other
laboratories in the same country.
The only significant interaction effect between the advice network and the affiliation network in the model is a triangle:
researchers affiliated with the same laboratories seek advice from
each other (TXBXarc). Not surprisingly, such effect is positive and
strongly significant. From the model goodness of fit test, this effect,
together with other parameters, provide adequate fit to the reciprocated version of TXBXarc, or the TXBXreciprocity which were not
captured by the model without cross level effects.
The cross level interaction parameters involving all three
networks include the in-/out-degree assortativity effects represented by various forms of three-paths, and the cross level
entrainment and exchange effects represented by different fourcycles. Only one of the three-path effects is significant, but
removing the non-significant three-paths made the model convergence difficult. The negative and significant L3AXBpath parameter
estimate suggests there was an anti degree assortativity effect
between the laboratories’ in-degree and researchers’ out-degree. In
other words, the popularity of labs in the collaboration network and
expansiveness of researcher advice seeking are compensating each
other rather than forming a joint core-periphery structure. The negative L3AXBpath does not mean such a configuration does not exist
in the network. By combining the negative L3AXBpath effect with
the positive and significant closure parameter C4AXBexchange, we
see that L3AXBpath is more likely to exist within closure, which
means that researchers from labs who receive more collaboration
tend to seek advice from researchers from collaborating laboratories but not from other researchers.
The different types of cross level four-cycle effects present
an interesting interpretation of the multilevel network structure. (Note that another possible cross level four-cycle parameter
(C4AXBreciprocity) was not included in the final model, as it was
not significant, and the current model fits the statistic well.) The
cross level entrainment and exchange four-cycle effects are both
positive, indicating a general co-occurrence of advice at one level
and collaboration at another, such that researchers within collaborating laboratories were more likely to seek advice from each other.
This is consistent with the ‘fusional’ strategies described in Lazega
et al. (2008) where researchers only work within the boundaries
defined by their affiliated laboratories and the laboratories’ collaborators; or a tendency for researchers not to seek advice from
researchers affiliated with non-collaborating laboratories.
However, such co-occurrence was less likely to happen when
reciprocation occurs in either the collaboration network or the
advice network, as indicated by the negative and significant
(C4AXBexchange reciprocal A) and (C4AXBexchange reciprocal B)
parameters. Note that the reciprocity effects for both the laboratory and researcher within networks are strong and positive. But
these reciprocations are less likely to occur when cross-level closure is involved. As the reciprocal ties can be considered as strong
ties, we suggest a possible interpretation for these negative effects.
It is possible that a strong form of collaboration among laboratories may result in shared knowledge and resources generally, so the
researchers no longer have to seek advice from one another in those
laboratories. On the other hand, reciprocal advice seeking among
researchers may be more likely to occur across non-collaborating
laboratories, perhaps as they may have distinct but complementary
knowledge or skills, such that advice in rather different areas may
provide mutual benefits (Lazega et al., 2011). This is also consistent with the argument that the strong forms of fusional strategies
are not ideal for the researchers’ performance (Lazega et al.,
2008).
The within level effects for the laboratory collaboration network
are largely consistent with the within level only model, as we have
the same parameter specifications and compatible scales on the
parameter values. Therefore, we may use the same interpretations
as previously.
For the within level researcher advice network, however, the
multilevel model greatly simplified the previous specifications. No
star parameters were required to capture the researcher degreedistributions. We may now interpret the apparently multi-modal
or long tailed degree-distributions as largely an artefact of the cross
level interactions. The interpretations of the positive and significant
AT-T confirm that advice seeking in this system tended to happen
in clusters; together with the negative and significant generalized
exchange effect (AT-C), we see the advice network as locally hierarchical. The negative alternating-two-path (A2P-U) representing
shared advice seeking may also help shape the out-degree distribution as it indicates that active advice seekers tended not to seek
advice from common others.
In summary, the multilevel ERGM reveals some interesting
structural features of the researcher multilevel network, and intuitive interpretations consistent with previous work are available.
Collaboration does take place in a clustered manner for both
researchers and laboratories; advice seeking is stronger within laboratories; and collaborating laboratories tend to have affiliated
researchers seeking advice from one another; however, reciprocated or strong collaboration did not encourage advice exchange
between researchers. More importantly, we see that some complicated features of the within-level network structure are explained
solely by the cross-level interactions: in this case the degree distribution of researcher advice-seeking.
8. Conclusion
In this paper, we proposed a generalized data structure which
captures both within-level and meso-level networks. The data
structure is flexible, and accommodates most of the multilevel network studies to date. We proposed a new multilevel ERGM to model
this structure.
Based on the hierarchy of tie dependence assumptions as summarized by Pattison et al. (2009) and Wang et al. (2013), we
have formulated the general form of ERGMs for two-level network
data, and proposed model specifications for both non-directed and
directed networks. They are designed to reveal the interdependencies among the micro-, macro- and meso-level networks. Our
simulation studies demonstrated the properties of each of the proposed configurations, and showed that even simple cross-level
effects can create highly structured within-level networks. When
the cross-level effects are not included, the models are no different from fitting three independent ERGMs, two for within-level
one-mode networks, and one for the bipartite ties. Our empirical example shows that these cross-level effects not only provide
better fits to the data, but also simplified otherwise complicated
within-level models. By comparing the models with and without
the interaction effects, our example showed that an apparently
multi-modal long-tailed degree distribution can be seen as an artefact of the dependence among micro-, macro- and meso-level ties.
Depending on the context and the research question, the definition of the levels in a multilevel network can be determined by
either the different nature of the nodes, individuals and groups, for
example; or the classifications of nodes by their attributes such as
gender, occupation, and education. In the first instance, network
ties of different types may be defined for the nodes of different levels, such as the example we presented in the paper. For the second
case where the levels are defined based on a nodal attribute, but the
same type of relation is defined for nodes within and cross different
categories. This is the type of data originally envisaged by Iacobucci
and Wasserman (1990) and Wasserman and Iacobucci (1991). The
research question is then based on the assumption that the network
effects are heterogeneous among nodes with different attributes.
In fact, any network data involving binary categorization of nodes
can be seen as a “two-level” network. For example, in a friendship
network context, a two-level network may be constructed by the
genders of the nodes assuming that friendship network processes
are different within and between genders. When more than two
categories are defined among a common set of nodes, we can analyze the data as “multilevel” networks assuming network processes
are different within and between categories. The ERGMs proposed
in this paper can be equally applied to both cases.
Multilevel data opens interesting possibilities for sampling. The
cluster sampling method, where one first samples units and then
samples within units, may be relevant to multilevel networks. We
cannot take a simple random sample of nodes in order to estimate a
network model for ties within units, but Handcock and Gile (2010)
and Pattison et al. (2011) have proposed conditional estimation
strategies for snowball sampled network data where ERGM parameters can be obtained based on a snowball sample of the (small or
large) network. More work needs to be done to consider bow best
such approaches may be applied to multilevel networks.
Future elaboration of the proposed multilevel ERGMs may
include social selection models (Robins et al., 2001a) and autologistic actor attribute models (ALAAMs) (Robins et al., 2001b;
Daraganova and Robins, 2013). Social selection models include
actor attributes as covariates, and allow the tests of how nodal
attributes may affect multilevel structure, for example, how individuals’ performance affect their network positions. ALAAMs for
multilevel networks on the other hand assume network structures are exogenous, and test how nodal attributes are affected by
individual’s multilevel network position; they may share similar
configurations or network statistics as in social selection models
but these are treated as exogenous.
Regardless of the forms of the possible models, multilevel
networks have more complicated dependencies among ties from
different levels. These interactionsmake themodels and their interpretations more complex. As there are almost unlimited number
of possible graph configurations, and no formal “step-wise” model
selection strategies, to find the best model for an empirical multilevel network will require theoretical guidance as well as model
fitting experience. We see the proposed models and examples presented in this article as the first steps in a full elaboration of an
ERGM approach to multilevel network analysis.
Acknowledgements
The authors are grateful to Marie-Thérèse Jourda for providing
the network dataset and Tom Snijders for helpful comments.
Appendix.
A.1. Alternating stars and different -values
To fit long-tailed degree distributions well, we may need to
include two or more star parameters with different  values.
We illustrate how different star parameters may be combined
in unipartite models to fit the degree distribution better. The
Fig. 16. Simulation using alternating-stars with different -values.
Fig. 17. Simulation using models with multiple (alternating-)star parameters.
alternating-star (AS) statistic uses geometric weighting parameter
()5 on the graph degree-distribution. For a network with n nodes,
it is defined as
zAS(x, ) =
n–1
k=2
(–1)k zSk(x)
k–2
where zSk(x) is the number of stars of size k, and i+ is the degree of
node i. Different -values may yield different degree distributions.
From experience, the alternating statistics with  = 2.0 (including
AS, AT and A2P, etc.) generally provide robust model convergence.
However, the converged model may still have difficulty in capturing the actual degree- or two-path-distribution for the cases where
the distributions are heavy tailed. Hunter and Handcock (2006) proposed methods for estimating  as a ratio parameter, although they
are computationally intensive in practice. Robins and Lusher (2013)
conducted some simulation studies to demonstrate how different
values of lambda may result in different triangle and two-path
distributions in the cases of alternating triangles and alternating
two-paths. Robins and Lusher (2013) also demonstrated how we
may improve model GOF by including both the alternating star (AS)
and the two-star parameters in the model. In our modelling examples presented in this paper, obtaining excellent fit to the degree
distributions required the two-star parameter as well as two or
more AS parameters with different -values.
To illustrate model interpretation and the properties of AS with
different  values, we present simulations on networks with 30
nodes and a fixed density of 0.1. The AS parameter was set constant at 1.0, but the -values were changed from 2 to 16 in factors
of 2s. We collected every 10,000th sample from a simulation of
10,000,000 iterations which gave a distribution of 1000 graphs. For
each different -value, Fig. 16 presents an example graph from
5 The -values for alternating-k-stars described here are equally applicable to
the geometrically weighted degree distribution as described in Hunter and Handcock
(2006) which uses a ratio parameter ˛ = ln ().
the simulated graph distribution along with a box plot of the
degree-distributions of the 1000 graphs. As the -value increased
from 1 to 8, the range of the degree distributions increased from
the highest maximum degree of 11 ( = 1) to the highest maximum
degree of 23 ( = 8). The number of isolated nodes also increased
from around 3 to about 10. The degree distributions look continuous (not multi-modal), and the tails become longer. With  = 16, the
degree distribution is more discontinuous with two modes, where
most graphs had two or three very high-degree nodes, with the rest
of the nodes having degrees below 5 with some variations. When 
equals infinity, which makes the alternating-star statistic equivalent to the two-star statistic, the graph distribution is almost frozen
across the simulation, with two very high degree nodes. Based on
these observations, we may conclude that a positive AS parameter
with a greater  value is likely to create high-degree nodes or hubs
in a network. A greater  may capture long-tailed degree distributions with greater variances. However, when  is too big, the degree
distribution may become multi-modal and frozen.
Model specifications with two or more star parameters with different  values may be able to capture degree distributions that
are long-tailed, continuous (not two- or multi-modal), and having some variations in both very high-degree nodes as well as low
degree nodes. While a positive two-star parameter, or a positive
AS parameter with a larger  value, creates high-degree hubs, negative alternating star parameters with smaller  values captures
nodes with relatively low degrees and prevents the simulation from
becoming frozen. Fig. 17 illustrates the effects of combining two or
more star parameters in a single model. As in previous simulations,
for networks with 30 nodes and a fixed density at 0.1, we fixed a
positive two-star parameter at 0.25 and a negative AS ( = 2) at –1.
The positive two-star generated graphs with a very high-degree
node connected to most of the other nodes in the network, while
the negative AS created some variability in the low degree nodes
(as compared to the frozen graph distribution generated only by
the positive two-star).
To reduce the degree of the hubs, we replace the positive twostar parameter with an AS parameter with  = 4. The simulation
generated graph distributions with hubs having maximum degree
of 14. Compare the degree distribution to the simulation with only
one positive AS parameter with  = 4, as shown in Fig. 16, the negative AS with  = 2 made the majority of nodes having low degrees
and with little variation, while the positive AS with  = 4 made
some high degree nodes possible. The decision about how many AS
parameters with different -values should be included in a model
depends on the observed degree distribution. In some cases (as
demonstrated in our modelling example section), non-significant
AS parameters are important for adequate fit to the data.
References
Agneessens, F., Roose, H., 2008. Local structural patterns and attribute characteristics
in 2-mode networks: p* models to map choices of theatre events. Journal of
Mathematical Sociology 32, 204–237.
Anderson, C.J., Wasserman, S., Crouch, B., 1999. A p* primer: logit models for social
networks. Social Networks 21, 37–66.
Bellotti, E., 2011. Getting funded. Multilevel network of physicists in Italy. Social
Networks, http://dx.doi.org/10.1016/j.socnet.2011.12.002.
Borgatti, S.P., Foster, P.C., 2003. The network paradigm in organizational research: a
review and typology. Journal of Management 29, 991–1013.
Brass, D.J., Galaskiewicz, J., Greve, H.R., Tsai, W., 2004. Taking stock of networks and
organizations: a multilevel perspective. Academy of Management Journal 47,
795–817.
Dansereau, F., Alutto, J., Yammarino, F., Dumas, M., 1984. Theory Testing in Organizational Behavior: The Variant Approach. Prentice-Hall, Englewood Cliffs, NJ.
Daraganova, G., Robins, G., 2013. Autologistic actor attribute models. In: Lusher, D.,
Koskinen, J., Robins, G.L. (Eds.), Exponential Random Graph Models for Social
Networks: Theories, Methods and Applications. Cambridge University Press,
New York, pp. 102–114.
Erdös, P., Rényi, A., 1960. On the evolution of random graphs. Publications of the
Mathematical Institute of the Hungarian Academy of Science 5, 17–61.
Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81 (395), 832–842.
Friedkin, N., 2006. A Structural Theory of Social Influence, vol. 13. Cambridge University Press, New York.
Handcock, M.S., 2003. Assessing Degeneracy in Statistical Models of Social Networks.
Working Paper No. 39. Centre for Statistics and the Social Sciences, University
of Washington.
Handcock, M.S., Gile, K.J., 2010. Modeling social networks from sampled data. The
Annals of Applied Statistics, 5–25.
Hedström, P., Sandell, R., Stern, C., 2000. Mesolevel networks and the diffusion of
social movements: the case of the Swedish Social Democratic Party. American
Journal of Sociology 106, 145–172.
Hunter, D.R., Handcock, M.S., 2006. Inference in curved exponential family models
for networks. Journal of Computational and Graphical Statistics 15 (3), 565–583.
Hunter, D.R., Goodreau, S.M., Handcock, M.S., 2008. Goodness of fit of social network
models. Journal of the American Statistical Association 103, 248–258.
Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions
for directed graphs. Journal of the American Statistical Association 76 (373),
33–50.
Iacobucci, D., Wasserman, S., 1990. Social networks with two sets of actors. Psychometrika, 707–720.
Koehly, L.M., Peterson, S.K., Watts, B.G., Kempf, K.K.G., Vernon, S.W., Gritz, E.R., 2003.
A social network analysis of communication about hereditary nonpolyposis
colorectal cancer genetic testing and family functioning. Cancer Epidemiology
Biomarkers and Prevention 12, 304–313.
Koskinen, J., Edling, C., 2010. Modelling the evolution of a bipartite network – peer referral in interlocking directorates. Social Networks,
http://dx.doi.org/10.1016/j.socnet.2010.03.001.
Kozlowski, S.W.J., Klein, K.J., 2000. A multilevel approach to theory and research
in organizations: contextual, temporal, and emergent processes. In: Klein, K.J.,
Kozlowski, S.W.J. (Eds.), Multilevel Theory, Research, and Methods in Organizations: Foundations, Extensions, and New Directions. Josse-bass, San Francisco,
pp. 3–90.
Latapy, M., Magnien, C., Vecchio, N.D., 2008. Basic notions for the analysis of large
two-mode networks. Social Networks 30, 31–48.
Lazega, E., Jourda, M.-Th., Lazega, D., Mounier, L., 2011. Network lift from dual alters:
extended opportunity structures from a multilevel and structural perspective.
In: Sunbelt XXXI, The Annual Meeting of the International Network for Social
Network Analysis, Tampa, Florida, February 12, 2011.
Lazega, E., Jourda, M.-Th., Lazega, D., Mounier, L., 2010. Network parachutes from
tetradic substructures: “organizational expansion” and the measurement of the
contribution of laboratories to their scientists’ relational capital and performance, Cahiers de l’ORIO, N◦ 1. Working paper. University of Paris-Dauphine.
Lazega, E., Jourda, M.-Th., Mounier, L., Stofer, R., 2008. Catching up with big fish in the
big pond? Multilevel network analysis through linked design. Social Networks
30, 157–176.
Lazega, E., Mounier, L., Jourda, M.-Th., Stofer, R., 2006. Organizational vs. personal
social capital in scientists’ performance: a multilevel network study of elite
French cancer researchers (1996–1998). Scientometrics 67 (1), 27–44.
Lazega, E., Mounier, L., Stofer, R., Tripier, A., 2004. Discipline scientifique et discipline sociale: Réseaux de conseil, apprentissage collectif et innovation dans la
recherche franc¸ aise sur le cancer (1997–1999). Recherches Sociologiques 35,
3–27.
Lomi, A., 2011. Social settings and social networks: models and ideas. Presentation at
Applications of Social Network Analysis (ASNA), Zurich, Switzerland, September,
2011.
Lubbers, M.J., 2003. Group composition and network structure in school
classes: a multilevel application of the p* model. Social Networks 25 (4),
309–332.
Lubbers, M.J., Snijders, T.A.B., 2007. A comparison of various approaches to the exponential random graph model: a reanalysis of 102 student networks in school
classes. Social Networks 29 (4), 489–507.
Mason, W., Conrey, F., Smith, E., 2007. Situating social influence processes: dynamic,
multidirectional flows of influence within social networks. Personality and
Social Psychology Review 11 (3), 279–300.
Pattison, P.E., Robins, G.L., 2002. Neighborhood-based models for social networks.
Social Methodology 32, 301L 337.
Pattison, P.E., Robins, G.L., 2004. Building models for social space: neighbourhoodbased models for social networks and affiliation structures? Mathematics and
Social Sciences 42 (168), 11–29.
Pattison, P.E., Robins, G.L., Snijders, T.A.B., 2009. A Hierarchy of Dependence Assumptions for Exponential Random Graph Models for Social Networks. Unpublished
Manuscript. Melbourne School of Psychological Sciences, The University of Melbourne, Australia.
Pattison, P.E., Robins, G.L., Snijders, T.A.B., Wang, P., 2011. Conditional Estimation of
Exponential Random Graph Models from Snowball Sampling Designs. Unpublished Manuscript. Melbourne School of Psychological Sciences, The University
of Melbourne, Australia.
Pattison, P.E., Snijders, T.A.B., 2013. Modeling social networks: next steps. In: Lusher,
D., Koskinen, J., Robins, G.L. (Eds.), Exponential Random Graph Models for Social
Networks: Theories, Methods and Applications. Cambridge University Press,
New York, pp. 287–301.
Pattison, P., Wasserman, S., 1999. Logit models and logistic regressions for social
networks. II. Multivariate relationships. British Journal of Mathematical and
Statistical Psychology 52, 169–193.
Rinaldo, A., Fienberg, S.E., Zhou, Y., 2009. On the geometry of discrete exponential random families with application to exponential random graph models.
Electronic Journal of Statistics 3, 446–484.
Robins, G., Pattison, P., Kalish, Y., Lusher, D., 2007a. An introduction to exponential
random graph (p*) models for social networks. Social Networks 29 (2), 173–191.
Robins, G.L., Snijders, T.A.B., Wang, P., Handcock, M., Pattison, P., 2007b. Recent
developments in exponential random graph (p*) models for social networks.
Social Networks 29, 192–215.
Robins, G., Lusher, D., 2013. Illustrations: simulation, estimation and goodness of
fit. In: Lusher, D., Koskinen, J., Robins, G.L. (Eds.), Exponential Random Graph
Models for Social Networks: Theories, Methods and Applications. Cambridge
University Press, New York, pp. 167–185.
Robins, G., Pattison, P., 2006. Multiple Networks in Organizations. Technical Report.
Melbourne School of Psychological Sciences, The University of Melbourne,
Australia.
Robins, G., Pattison, P., Woolcock, J., 2005. Small and other worlds: global network structures from local processes. American Journal of Sociology 110 (4),
894–936.
Robins, G.L., Pattison, P.E., Wang, P., 2009. Closure, Connectivity and Degrees: New
Specifications for Exponential Random Graph (p*) Models for Directed Social
Networks, 31, pp. 105–117.
Robins, G.L., Alexander, M., 2004. Small worlds among interlocking directors: network structure and distance in bipartite graphs. Journal of Computational and
Mathematical Organization Theory 10, 69–94, Invited paper.
Robins, G.L., Elliott, P., Pattison, P., 2001a. Network models for social selection processes. Social Networks 23, 1–30.
Robins, G.L., Pattison, P., Elliott, P., 2001b. Network models for social influence processes. Psychometrika 66, 161–190.
Skvoretz, J., Faust, K., 1999. Logit models for affiliation networks. In: Sobel,
M., Becker, M. (Eds.), Sociological Methodology. Blackwell, New York,
pp. 253–280.
Snijders, T.A., 2002. Markov chain Monte Carlo estimation of exponential random
graph models. Journal of Social Structure 3, 2.
Snijders, T., Baerveldt, C., 2003. A multilevel network study of the effects of delinquent behavior on friendship evolution? Journal of Mathematical Sociology 27
(2–3), 123–151.
Snijders, T.A., 2009. Analyzing the joint dynamics of several networks. In: Sunbelt
XXIX International Social Network Conference, USA, March.
Snijders, T., Van de Bunt, G., Steglich, C., 2010. Introduction to stochastic actor-based
models for network dynamics? Social Networks 32 (1), 44–60.
Snijders., T.A., Bosker, R., 2012. Multilevel Analysis: An Introduction to Basic and
Advanced Multilevel Modelling, 2nd ed. SAGE Publications, London.
Snijders, T.A., Lomi, A., Torlo, V., 2012. Multiplex dynamics of one-mode and twomode networks, with an application to friendship and labor market preference.
Social Networks, http://dx.doi.org/10.1016/j.socnet.2012.05.005.
Snijders, T.A., Pattison, P.E., Robins, G.L., Handcock, M., 2006. New specifications for
exponential random graph models. Sociological Methodology 36, 99L 153.
Torlo, V., Lomi, A., Snijders, T., 2010. The co-evolution of social networks and students’ affiliation preferences. In: Sunbelt XXX International Social Networks
Conference, Italy, July 2010.
Wang, P., 2013. Exponential random graph model extensions: models for multiple
networks and bipartite networks. In: Lusher, D., Koskinen, J., Robins, G.L. (Eds.),
Exponential Random Graph Models for Social Networks: Theories, Methods and
Applications. Cambridge University Press, New York, pp. 115–129.
Wang, P., Pattison, P.E., Robins, G.L., 2013. Exponential random graph model specifications for bipartite networks – a dependence hierarchy. Social Networks,
http://dx.doi.org/10.1016/j.socnet.2011.12.004.
Wang, P., Robins, G.L., Pattison, P.E., 2006. PNet: a program for the simulation and
estimation of exponential random graph models. Melbourne School of Psychological Sciences, The University of Melbourne, Australia.
Wang, P., Sharpe, K., Robins, G.L., Pattison, P.E., 2009. Exponential random graph (p*)
models for affiliation networks. Social Networks 31, 12–25.
Wasserman, S., Pattison, P.E., 1996. Logit models and logistic regression for social
networks. I. An introduction to Markov graphs and p*. Psychometrika 6 (3),
401–425.
Wasserman, S., Iacobucci, D., 1991. Statistical modelling of one-mode and two-mode
networks: simultaneous analysis of graphs and bipartite graphs. British Journal
of Mathematical and Statistical Psychology, 13–43.
Watts, D.J., 1999. Small Worlds: The Dynamics of Networks Between Order and
Randomness. Princeton University Press, Princeton, NJ.

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