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Applied Network Science

Espín-Noboa et al. Applied Network Science (2017) 2:16

DOI 10.1007/s41109-017-0036-1

RESEARCH Open Access

JANUS: A hypothesis-driven Bayesian

approach for understanding edge formation in

attributed multigraphs

Lisette Espín-Noboa1,2* , Florian Lemmerich1,2, Markus Strohmaier1,2 and Philipp Singer1,2

*Correspondence:

Lisette.Espin@gesis.org

This article extends a previous

workshop publication (Espín-Noboa

et al. 2016). The main novelties in

this manuscript include the

extension to dyad-attributed

networks (such as as multiplex

networks), additional experimental

results, and a comparison of our

approach to alternative methods.

1GESIS - Leibniz Institute for the

Social Sciences, Unter

Sachsenhausen 6-8, 50667 Cologne,

Germany

2University of Koblenz-Landau,

Universitätstraße 1, 56070 Koblenz,

Germany

Abstract

Understanding edge formation represents a key question in network analysis. Various

approaches have been postulated across disciplines ranging from network growth

models to statistical (regression) methods. In this work, we extend this existing arsenal

of methods with JANUS, a hypothesis-driven Bayesian approach that allows to

intuitively compare hypotheses about edge formation in multigraphs. We model the

multiplicity of edges using a simple categorical model and propose to express

hypotheses as priors encoding our belief about parameters. Using Bayesian model

comparison techniques, we compare the relative plausibility of hypotheses which

might be motivated by previous theories about edge formation based on popularity or

similarity. We demonstrate the utility of our approach on synthetic and empirical data.

JANUS is relevant for researchers interested in studying mechanisms explaining edge

formation in networks from both empirical and methodological perspectives.

Keywords: Edge formation, Bayesian inference, Attributed multigraphs, Multiplex,

HypTrails

Introduction

Understanding edge formation in networks is a key interest of our research commu-

nity. For example, social scientists are frequently interested in studying relations between

entities within social networks, e.g., how social friendship ties form between actors and

explain them based on attributes such as a person’s gender, race, political affiliation or

age in the network (Sampson 1968). Similarly, the complex networks community suggests

a set of generative network models aiming at explaining the formation of edges focus-

ing on the two core principles of popularity and similarity (Papadopoulos et al. 2012).

Thus, a series of approaches to study edge formation have emerged including statistical

(regression) tools (Krackhardt 1988; Snijders et al. 1995) and model-based approaches

(Snijders 2011; Papadopoulos et al. 2012; Karrer and Newman 2011) specifically estab-

lished in the physics and complex networks communities. Other disciplines such as the

computer sciences, biomedical sciences or political sciences use these tools to answer

empirical questions; e.g., co-authorship networks (Martin et al. 2013), wireless networks

of biomedical sensors (Schwiebert et al. 2001), or community structures of political blogs

(Adamic and Glance 2005).

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0

International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and

reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the

Creative Commons license, and indicate if changes were made.

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 2 of 20

Problem illustration Consider for example the network depicted in Fig. 1. Here, nodes

represent authors, and (multiple) edges between them refer to co-authored scientific

articles. Node attributes provide additional information on the authors, e.g., their home

country and gender. In this setting, an exemplary research question could be: “Can

co-authorship be better explained by a mechanism that assumes more collaborations

between authors from the same country or by a mechanism that assumes more collabora-

tions between authors with the same gender?”. These and similar questions motivate the

main objective of this work, which is to provide a Bayesian approach for understanding

how edges emerge in networks based on some characteristics of the nodes or dyads.

While several methods for tackling such questions have been proposed, they come

with certain limitations. For example, statistical regression methods based on QAP

(Hubert and Schultz 1976) or mixed-effects models (Shah and Sinha 1989) do not scale

to large-scale data and results are difficult to interpret. For network growth models

(Papadopoulos et al. 2012), it is necessary to find the appropriate model for a given

hypothesis about edge formation and thus, it is often not trivial to intuitively compare

competing hypotheses. Consequently, we want to extend the methodological toolbox for

studying edge formation in networks by proposing a first step towards a hypothesis-

driven generative Bayesian framework.

Approach and methods We focus on understanding edge formation in attributed multi-

graphs. We are interested in modeling and understanding the multiplicity of edges based

on additional network information, i.e., given attributes for the nodes or dyads in the net-

work. Our approach follows a generative storyline. First, we define the model that can

characterize the edge formation at interest. We focus on the simple categorical model,

from which edges are independently drawn from. Motivated by previous work on sequen-

tial data (Singer et al. 2015), the core idea of our approach is to specify generative

hypotheses about how edges emerge in a network. These hypotheses might be motivated

by previous theories such as popularity or similarity (Papadopoulos et al. 2012)—e.g.,

for Fig. 1 we could hypothesize that authors are more likely to collaborate with each

Fig. 1 Example: This example illustrates an unweighted attributed multigraph. aShows a multigraph where

nodes represent academic researchers, and edges scientific articles in which they have collaborated

together. bShows the adjacency matrix of the graph, where every cell represents the total number of edges

between two nodes. cDecodes some attribute values per node. For instance, node D shows information

about an Austrian researcher who started his academic career in 2001. One main objective of JANUS is to

compare the plausibility of mechanisms derived from attributes for explaining the formation of edges in the

graph. For example, here, a hypothesis that researchers have more collaborations if they are from the same

country might be more plausible than one that postulates that the multiplicity of edges can be explained

based on the relative popularity of authors

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 3 of 20

other if they are from the same country. Technically, we elicit these types of hypothe-

ses as beliefs in parameters of the underlying categorical model and encode and integrate

them as priors into the Bayesian framework. Using Bayes factors with marginal likelihood

estimations allows us to compare the relative plausibility of expressed hypotheses as they

are specifically sensitive to the priors. The final output is a ranking of hypotheses based

on their plausibility given the data.

Contributions The main contributions of this work are:

1. We present a first step towards a Bayesian approach for comparing generative

hypotheses about edge formation in networks.

2. We provide simple categorical models based on local and global scenarios allowing

the comparison of hypotheses for multigraphs.

3. We show that JANUS can be easily extended to dyad-attributed multigraphs when

multiplex networks are provided.

4. We demonstrate the applicability and plausibility of JANUS based on experiments

on synthetic and empirical data, as well as by comparing it to the state-of-the-art

QAP.

5. We make an implementation of this approach openly available on the Web

(Espín-Noboa 2016).

Structure This paper is structured as follows: First, we start with an overview of

some existing research on modeling and understanding edge formation in networks in

Section “Related work”. We present some background knowledge required in this work

in Section “Background” to then explain step-by-step JANUS in Section “Approach”.

Next, we show JANUS in action and the interpretation of results, by running four dif-

ferent experiments on synthetic and empirical data in Section “Experiments”. In Section

“Discussion” we suggest a fair comparison of JANUS with the Quadratic Assignment

Procedure (QAP) for testing hypotheses on dyadic data. We also highlight some impor-

tant caveats for further improvements. Finally, we conclude in Section “Conclusions” by

summarizing the contributions of our work.

Related work

We provide a broad overview of research on modeling and understanding edge formation

in networks; i.e., edge formation models and hypothesis testing on networks.

Edge formation models A variety of models explaining underlying mechanisms of

network formation have been proposed. Here, we focus on models explaining linkage

between dyads beyond structure by incorporating node attribute information. Promi-

nently, the stochastic blockmodel (Karrer and Newman 2011) aims at producing and

explaining communities by accounting for node correlation based on attributes. The

attributed graph (Pfeiffer III et al. 2014) models network structure and node attributes by

learning the attribute correlations in the observed network. Furthermore, the multiplica-

tive attributed graph (Kim and Leskovec 2011) takes into account attribute information

from nodes to model network structure. This model defines the probability of an edge as

the product of individual attribute link formation affinities. Exponential random graph

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 4 of 20

models (Robins et al. 2007) (also called the p∗class of models) represent graph dis-

tributions with an exponential linear model that uses feature-structure counts such as

reciprocity, k-stars and k-paths. In this line of research, p1 models (Holland and Leinhardt

1981) consider expansiveness (sender) and popularity (receiver) as fixed effects associ-

ated with unique nodes in the network (Goldenberg et al. 2010) in contrast to the p2

models (Robins et al. 2007) which account for random effects and assume dyadic indepen-

dence conditionally to node-level attributes. While many of these works focus on binary

relationships, (Xiang et al. 2010) proposes an unsupervised model to estimate continuous-

valued relationship strength for links from interaction activity and user similarity in social

networks. Recently, the work in (Kleineberg et al. 2016) has shown that connections in

one layer of a multiplex can be accurately predicted by utilizing the hyperbolic distances

between nodes from another layer in a hidden geometric space.

Hypothesis testing on networks Previous works have implemented different tech-

niques to test hypotheses about network structure. For instance, the work in (Moreno

and Neville 2013) proposes an algorithm to determine whether two observed networks

are significantly different. Another branch of research has specifically focused on dyadic

relationships utilizing regression methods accounting for interdependencies in network

data. Here, we find Multiple Regression Quadratic Assignment Procedure (MRQAP)

(Krackhardt 1988) and its predecessor QAP (Hubert and Schultz 1976) which permute

nodes in such a way that the network structure is kept intact; this allows to test for signif-

icance of effects. Mixed-effects models (Shah and Sinha 1989) add random effects to the

models allowing for variation to mitigate non-independence between responses (edges)

from the same subject (nodes) (Winter 2013). Based on the quasi essential graph the work

in (Nguyen 2012) proposes to compare two graphs (i.e., Bayesian networks) by testing

and comparing multiple hypotheses on their edges. Recently, generalized hypergeometric

ensembles (Casiraghi et al. 2016) have been proposed as a framework for model selec-

tion and statistical hypothesis testing of finite, directed and weighted networks that allow

to encode several topological patterns such as block models where homophily plays an

important role in linkage decision. In contrast to our work, neither of these approaches

is based on Bayesian hypothesis testing, which avoids some fundamental issues of classic

frequentist statistics.

Background

In this paper, we focus on both node-attributed and dyad-attributed multigraphs with

unweighted edges without own identity. That means, each pair of nodes or dyad can be

connected by multiple indistinguishable edges, and there are features for the individual

nodes or dyads available.

Node-attributed multigraphs We formally define this as: Let G=(V,E,F)be an

unweighted attributed multigraph with V=(v1,...,vn)being a list of nodes, E=

{(vi,vj)}∈V×Va multiset of either directed or undirected edges, and a set of fea-

ture vectors F=(f1,...,fn). Each feature vector fi=(fi[ 1] , ..., fi[c])Tmaps a node

vito c(numeric or categorical) attribute values. The graph structure is captured by an

adjacency matrix Mn×n=(mij ),wheremij is the multiplicity of edge (vi,vj)in E(i.e.,

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 5 of 20

number of edges between nodes viand vj). By definition, the total number of multiedges

is l=|E|=ij mij.

Figure 1a shows an example unweighted attributed multigraph: nodes represent

authors, and undirected edges represent co-authorship in scientific articles. The adja-

cency matrix of this graph—counting for multiplicity of edges—is shown in Fig. 1b.

Feature vectors (node attributes) are described in Fig. 1c. Thus, for this particular case,

we account for n=4nodes,l=44 multiedges, and c=6attributes.

Dyad-attributed networks As an alternative to attributed nodes, we also consider

multigraphs, in which each dyad (pair of nodes) is associated with a set of features ˆ

F=

(ˆ

f11,...,ˆ

fnn). Each feature vector ˆ

fij =(ˆ

fij[ 1] , ..., ˆ

fij[c])Tmapsthepairofnode(vi,vj)to

c(numeric or categorical) attribute values. The values of each feature can be represented

in a separate n×nmatrix. As an important special case of dyad-attributed networks, we

study multiplex networks. In these networks, all dyad features are integer-valued. Thus,

each feature can be interpreted as (or can be derived from) a separate multigraph over the

same set of nodes. In our setting, the main idea is then to try and explain the occurrence

of a multiset of edges Ein one multigraph Gwith nodes Vby using other multigraphs ˆ

G

on the same node set.

Bayesian hypothesis testing Our approach compares hypotheses on edge formation

based on techniques from Bayesian hypothesis testing (Kruschke 2014; Singer et al. 2015).

The elementary Bayes’ theorem states for parameters θ,givendataDand a hypothesis H

that:

posterior

P(θ|D,H)=

likelihood

P(D|θ,H)

prior

P(θ|H)

P(D|H)

marginal likelihood

(1)

As observed data D, we use the adjacency matrix M, which encodes edge counts. θ

refers to the model parameters, which in our scenario correspond to the probabilities of

individual edges. Hdenotes a hypothesis under investigation. The likelihood describes,

how likely we observe data Dgiven parameters θand a hypothesis H.Theprior is

the distribution of parameters we believe in before seeing the data; in other words, the

prior encodes our hypothesis H.Theposterior represents an adjusted distribution of

parameters after we observe D. Finally, the marginal likelihood (also called evidence)

represents the probability of the data Dgiven a hypothesis H.

In our approach, we exploit the sensitivity of the marginal likelihood on the prior to

compare and rank different hypotheses: more plausible hypotheses imply higher evidence

for data D. Formally, Bayes Factors can be employed for comparing two hypotheses. These

are computed as the ratio between the respective marginal likelihood scores. The strength

of a Bayes factor can be judged using available interpretation tables (Kass and Raftery

1995). While in many cases determining the marginal likelihood is computationally chal-

lenging and requires approximate solutions, we can rely on exact and fast-to-compute

solutions in the models employed in this paper.

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 6 of 20

Approach

In this section, we describe the main steps towards a hypothesis-driven Bayesian

approach for understanding edge formation in unweighted attributed multigraphs. To

that end, we propose intuitive models for edge formation (Section “Generative edge for

mation models”), a flexible toolbox to formally specify belief in the model parameters

(Section “Constructing belief matrices”), a way of computing proper (Dirichlet) priors

from these beliefs (Section “Eliciting a Dirichlet prior”), computation of the marginal like-

lihood in this scenario (Section “Computation of the marginal likelihood”), and guidelines

on how to interpret the results (Section “Application of the method and interpretation of

results”). We subsequently discuss these issues one-by-one.

Generative edge formation models

We propose two variations of our approach, which employ two different types of

generative edge formation models in multigraphs.

Global model First, we utilize a simple global model, in which a fixed number of graph

edges are randomly and independently drawn from the set of all potential edges in

the graph Gby sampling with replacement. Each edge (vi,vj)is sampled from a cat-

egorical distribution with parameters θij,1 ≤i≤n,1 ≤j≤n,∀ij :ij θij =1:

(vi,vj)∼Categorical(θij ). This means that each edge is associated with one probability

θij of being drawn next. Figure 2a shows the maximum likelihood global model for the

network shown in Fig. 1. Since this is an undirected graph, inverse edges can be ignored

resulting in n(n+1)/2 potential edges/parameters.

Local models As an alternative, we can also focus on a local level. Here, we model to

which other node a specific node vwill connect giventhatanynewedgestartingfrom

vis formed. We implement this by using a set of nseparate models for the outgoing

edges of the ego-networks (i.e., the 1-hop neighborhood) of each of the nnodes. The

ego-network model for node viis built by drawing randomly and independently a num-

ber of nodes vjby sampling with replacement and adding an edge from vito this node.

Each node vjis sampled from a categorical distribution with parameters θij ,1 ≤i≤

n,1 ≤j≤n,∀i:jθij =1: vj∼Categorical(θij ). The parameters θij can be writ-

ten as a matrix; the value in cell (i,j)specifies the probability that a new formed edge

Fig. 2 Multigraph models: This figure shows two ways of modeling the undirected multigraph shown in

Fig. 1. That is, aglobal or graph-based model models the whole graph as a single distribution. bLocal or

neighbour-based model models each node as a separate distribution

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with source node viwill have the destination node vj. Thus, all values within one row

always sum up to one. Local models can be applied for undirected and directed graphs

(cf. also in Section “Discussion”). In the directed case, we model only the outgoing edges

of the ego-network. Figure 2b depicts the maximum likelihood local models for our

introductory example.

Hypothesis elicitation

The main idea of our approach is to encode our beliefs in edge formation as Bayesian

priors over the model parameters. As a common choice, we employ Dirichlet distribu-

tionsastheconjugate priors of the categorical distribution. Thus, we assume that the

model parameters θare drawn from a Dirichlet distribution with hyperparameters α:

θ∼Dir(α). Similar to the model parameters themselves, the Dirichlet prior (or multiple

priors for the local models) can be specified in a matrix. We will choose the parameters

αin such a way that they reflect a specific belief about edge formation. For that pur-

pose, we first specify matrices that formalize these beliefs, then we compute the Dirichlet

parameters αfrom these beliefs.

Constructing belief matrices

We specify hypotheses about edge formation as belief matrices B =bij .Thesearen×n

matrices, in which each cell bij ∈IR represents a belief of having an edge from node vito

node vj. To express a belief that an edge occurs more often (compared to other edges) we

set bij to a higher value.

Node-attributed multigraphs In general, users have a large freedom to generate belief

matrices. However, typical construction principles are to assume that nodes with spe-

cific attributes are more popular and thus edges connecting these attributes receive

higher multiplicity, or to assume that nodes that are similar with respect to one or more

attributes are more likely to form an edge, cf. (Papadopoulos et al. 2012). Ideally, the

elicitation of belief matrices is based on existing theories.

For example, based on the information shown in Fig. 1, one could “believe” that two

authors collaborate more frequently together if: (1) they both are from the same country,

(2) they share the same gender, (3) they have high positions, or (4) they are popular in

terms of number of articles and citations. We capture each of these beliefs in one matrix.

One implementation of the matrices for our example beliefs could be:

•B1(same country): bij :=0.9 if fi[country]=fj[country]and 0.1 otherwise

•B2(same gender): bij :=0.9 if fi[gender]=fj[gender]and 0.1 otherwise

•B3(hierarchy): bij :=fi[position]·fj[position]

•B4(popularity): bij :=fi[articles]+fj[articles]+fi[citations]+fj[citations]

Figure 3a shows the matrix representation of belief B1, and Fig. 3b its respective

row-wise normalization for the local model case. While belief matrices are identically

structured for local and global models, the ratio between parameters in different rows is

crucial for the global model, but irrelevant for local ones.

Dyad-attributed networks For the particular case of Dyad-Attributed networks, beliefs

are described using the underlying mechanisms of secondary multigraphs. For instance, a

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Fig. 3 Prior belief: This figure illustrates the three main phases of prior elicitation. That is, aamatrix

representation of belief B1, where authors are more likely to collaborate with each other if they are from the

same country. bB1normalized row-wise using the local model interpretation. cPrior elicitation for κ=4; i.e.,

αij =bij

Z×κ+1

co-authorship network—where every node represents an author with no additional infor-

mation or attribute—could be explained by a citation network under the hypothesis that

if two authors frequently cite each other, they are more likely to also co-author together.

Thus, the adjacency (feature) matrices (ˆ

F)of secondary multigraphs can be directly used

as belief matrices B=(bij). However, we can express additional beliefs by transforming

the matrices. As an example, we can formalize the belief that the presence of a feature

tends to inhibit the formation of edges in the data by setting bij :=−sigm(fij),wheresigm

is a sigmoid function such as the logistic function.

Eliciting a Dirichlet prior.

In order to obtain the hyperparameters αof a prior Dirichlet distribution, we utilize

the pseudo-count interpretation of the parameters αij of the Dirichlet distribution, i.e.,

a value of αij can be interpreted as αij −1 previous observations of the respective

event for αij ≥1. We distribute pseudo-counts proportionally to a belief matrix. Con-

sequently, the hyperparameters can be expressed as: αij =bij

Z×κ+1, where κis

the concentration parameter of the prior. The normalization constant Zis computed

as the sum of all entries of the belief matrix in the global model, and as the respec-

tive row sum in the local case. We suggest to set κ=n×kfor the local models,

κ=n2×kfor the directed global case, κ=n(n+1)

2×kfor the undirected global case, and

k={0, 1, ..., 10}. A high value of κexpresses a strong belief in the prior parameters. A sim-

ilar alternative method to obtain Dirichlet priors is the trial roulette method (Singer et al.

2015). For the global model variation, all αvalues are parameters for the same Dirichlet

distribution, whereas in the local model variation, each row parametrizes a separate

Dirichlet distribution. Figure 3 (c) shows the prior elicitation of belief B1forkappa =4

using the local model.

Computation of the marginal likelihood

For comparing the relative plausibility of hypotheses, we use the marginal likelihood. This

is the aggregated likelihood over all possible values of the parameters θweighted by the

Dirichlet prior. For our set of local models we can calculate them as:

P(D|H)=

n

i=1

n

j=1αij

n

j=1αij +mij

n

j=1

(αij +mij)

(αij)(2)

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Recall, αij encodes our prior belief connecting nodes viand vjin G,andmij are the

actual edge counts. Since we evaluate only a single model in the global case, the product

over rows iof the adjacency matrix can be removed, and we obtain:

P(D|H)=

n

i=1n

j=1αij

n

i=1n

j=1αij +mij

n

i=1

n

j=1

αij +mij

(αij)(3)

Section “Computation of the marginal likelihood” holds for directed networks. In the

undirected case, indices jgo from ito naccounting for only half of the matrix including

the diagonal to avoid inconsistencies. For a detailed derivation of the marginal likelihood

given a Dirichlet-Categorical model see (Tu 2014; Singer et al. 2014). For both models we

focus on the log-marginal likelihoods in practice to avoid underflows.

Bayes factor Formally, we compare the relative plausibility of hypotheses by using so-

called Bayes factors (Kass and Raftery 1995), which simply are the ratios of the marginal

likelihoods for two hypotheses H1and H2. If it is positive, the first hypothesis is judged as

more plausible. The strength of the Bayes factor can be checked in an interpretation table

provided by Kass and Raftery (1995).

Application of the method and interpretation of results

We now showcase an example application of our approach featuring the network shown

in Fig. 1, and demonstrate how results can be interpreted.

Hypotheses We compare four hypotheses (represented as belief matrices) B1,B2,B3,

and B4elaborated in Section “Hypothesis elicitation”. Additionally, we use the uniform

hypothesis as a baseline. It assumes that all edges are equally likely, i.e., bij =1 for all

i,j. Hypotheses that are not more plausible than the uniform cannot be assumed to cap-

ture relevant underlying mechanisms of edge formation. We also use the data hypothesis

as an upper bound for comparison, which employs the observed adjacency matrix as

belief: bij =mij.

Calculation and visualization For each hypothesis Hand every κ, we can elicit the

Dirichlet priors (cf. Section “Hypothesis elicitation”), determine the aggregated marginal

likelihood (cf. Section “Computation of the marginal likelihood)”, and compare the plausi-

bility of hypotheses compared to the uniform hypothesis at the same κby calculating the

logarithm of the Bayes factor as log(P(D|H))−log(P(D|Huniform)). We suggest two ways of

visualizing the results, i.e., plotting the marginal likelihood values, and showing the Bayes

factors on the y-axis as shown in Fig. 4a and 4b respectively for the local model. In both

cases, the x-axis refers to the concentration parameter κ. While the visualization show-

ing directly the marginal likelihoods carries more information, visualizing Bayes factors

makes it easier to spot smaller differences between the hypotheses.

Interpretation Every line in Fig. 4a to 4d represents a hypothesis using the local (top) and

global models (bottom). In Fig. 4a and 4c, higher evidence values mean higher plausibility.

Similarly, in Fig. 4b and 4d positive Bayes factors mean that for a given κ, the hypothesis is

judged to be more plausible than the uniform baseline hypothesis; here, the relative Bayes

factors also provide a ranking. If evidences or Bayes factors are increasing with κ,wecan

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 10 of 20

Fig. 4 Ranking of hypotheses for the introductory example. a,bRepresent results using the local model and

c,dresults of the global model. Rankings can be visualized using a,cthe marginal likelihood or evidence

(y-axis), or b,dusing Bayes factors (y-axis) by setting the uniform hypothesis as a baseline to compare with;

higher values refer to higher plausibility. The x-axis depicts the concentration parameter κ. For this example,

from an individual perspective (local model) authors from the multigraph shown in Fig. 1 appear to prefer to

collaborate more often with researchers of the same country rather than due to popularity (i.e., number of

articles and citations). In this particular case, the same holds for the global model. Note that all hypotheses

outperform the uniform, meaning that they all are reasonable explanations of edge formation for the given

graph

interpret this as further evidence for the plausibility of expressed hypothesis as this means

that the more we believe in it, the higher the Bayesian approach judges its plausibility. As

a result for our example, we see that the hypothesis believing that two authors are more

likely to collaborate if they are from the same country is the most plausible one (after

the data hypothesis). In this example, all hypotheses appear to be more plausible than

the baseline in both local and global models, but this is not necessarily the case in all

applications.

Experiments

We demonstrate the utility of our approach on both synthetic and empirical networks.

Synthetic node-attributed multigraph

We start with experiments on a synthetic node-attributed multigraph. Here, we control

the underlying mechanisms of how edges in the network emerge and thus, expect these

also to be good hypotheses for our approach.

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Network The network contains 100 nodes where each node is assigned one of two colors

with uniform probability. For each node, we then randomly drew 200 undirected edges

where each edge connects randomly with probability p=0.8 to a different node of the

same color, and with p=0.2 to a node of the opposite color. The adjacency matrix of this

graph is visualized in Fig. 5a.

Hypotheses In addition to the uniform baseline hypothesis, we construct two intu-

itive hypotheses based on the node color that express belief in possible edge formation

mechanics. First, the homophily hypothesis assumes that nodes of the same color are

more likely to have more edges between them. Therefore, we arbitrary set belief values

bij to 80 when nodes viand vjare of the same color, and 20 otherwise. Second, the het-

erophily hypothesis expresses the opposite behavior; i.e., bij =80 if the color of nodes vi

and vjare different, and 20 otherwise. An additional selfloop hypothesis only believes in

self-connections (i.e., diagonal of adjacency matrix).

Results Figure 5b and 5c show the ranking of hypotheses based on their Bayes factors

compared to the uniform hypothesis for the local and global models respectively. Clearly,

Fig. 5 Ranking of hypotheses for synthetic attributed multigraph. In a, we show the adjacency matrix of a

100-node 2-color random multigraph with a node correlation of 80% for nodes of the same color and 20%

otherwise. One can see the presence of homophily based on more connections between nodes of the same

color; the diagonal is zero as there are no self-connections. In b,cwe show the ranking of hypotheses based

on Bayes factors when compared to the uniform hypothesis for the local and global models respectively. As

expected, in general the homophily hypothesis explains the edge formation best (positive Bayes factor and

close to the data curve), while the heterophily and selfloop hypotheses provide no good explanations for

edge formation in both local and global cases—they show negative Bayes factors

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 12 of 20

in both models the homophily hypothesis is judged as the most plausible. This is expected

and corroborates the fact that network connections are biased towards nodes of the same

color. The heterophily and selfloop hypotheses show negative Bayes factors; thus, they are

not good hypotheses about edge formation in this network. Due to the fact that the multi-

graph lacks of selfloops, the selfloop hypothesis decreases very quickly with increasing

strength of belief κ.

Synthetic multiplex network

In this experiment, we control the underlying mechanisms of how edges in a dyad-

attributed multigraph emerge using multiple multigraphs that share the same nodes with

different link structure (i.e., multiplex) and thus, expect these also to be good hypotheses

for JANUS.

Network The network is an undirected configuration model graph (Newman 2003) with

parameters n=100 (i.e., number of nodes) and degree sequence −→

k=kidrawn from

a power law distribution of length nand exponent 2.0, where kiis the degree of node vi.

The adjacency matrix of this graph is visualized in Fig. 6a.

Fig. 6 Ranking of hypotheses for synthetic multiplex network. In awe show the adjacency matrix of a

configuration model graph of 100 nodes and power-law distributed degree sequence. In b,cthe ranking of

hypotheses is shown for the local and global model respectively. As expected, hypotheses are ranked from

small to big values of since small values represent only a few changes in the original adjacency matrix of

the configuration graph. Both models show that when the original graph changes at least 70% of its edges

the new graph cannot be explained better than random (i.e., uniform)

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 13 of 20

Hypotheses Besides the uniform hypothesis, we include ten more hypotheses derived

from the original adjacency matrix of the configuration model graph where only certain

percentage of edges get shuffled. The bigger the the less plausible the hypothesis since

more shuffles can modify drastically the original network.

Results Figure 6b and 6c show the ranking of hypotheses based on their Bayes fac-

tors compared to the uniform hypothesis for the local and global model respectively. In

general, hypotheses are ranked as expected, from small to big values of . For instance,

the epsilon10p hypothesis explains best the configuration model graph—represented in

Fig. 6a—since it only shuffles 10% of all edges (i.e., 10 edges). On the other hand the

epsilon100p hypothesis shows the worst performance (i.e., Bayes factor is negative and far

from the data curve) since it shuffles all edges, therefore it is more likely to be different

than the original network.

Empirical node-attributed multigraph

Here, we focus on a real-world contact network based on wearable sensors.

Network We study a network capturing interactions of 5 households in rural Kenya

between April 24 and May 12, 2012 (Sociopatterns; Kiti et al. 2016). The undirected

unweighted multigraph contains 75 nodes (persons) and 32, 643 multiedges (contacts)

which we aim to explain. For each node, we know information such as gender and age

(encoded into 5 age intervals). Interactions exist within and across households. Figure 7a

shows the adjacency matrix (i.e., number of contacts between two people) of the network.

Household membership of nodes (rows/columns) is shown accordingly.

Hypotheses We investigate edge formation by comparing—next to the uniform base-

line hypothesis—four hypotheses based on node attributes as prior beliefs. (i) The similar

age hypothesis expresses the belief that people of similar age are more likely to inter-

act with each other. Entries bij of the belief matrix Bare set to the inverse age distance

between members: 1

1+abs(fi[age]−fj[age]). (ii) The same household hypothesis believes that

people are more likely to interact with people from the same household. We arbitrarily

set bij to 80 if person viand person vjbelong to the same household, and 20 otherwise.

(iii) With the same gender hypothesis we hypothesize that the number of same-gender

interactions is higher than the different-gender interactions. Therefore, every entry bij of

Bissetto80ifpersonsviand vjare of the same gender, and 20 otherwise. Finally, (iv)

the different gender hypothesis believes that it is more likely to find different-gender than

same-gender interactions; bij issetto80ifpersonvihas the opposite gender of person vj,

and 20 otherwise.

Results Results shown in Fig. 7b and 7c show the ranking of hypotheses based on Bayes

factors using the uniform hypothesis as baseline for the local and global model respec-

tively. The local model Fig. 7b indicates that the same household hypothesis explains the

data the best, since it has been ranked first and it is more plausible than the uniform. The

similar age hypothesis also indicates plausibility due to positive Bayes factors. Both the

same and different gender hypotheses show negative Bayes factors when compared to the

uniform hypothesis suggesting that they are not good explanations of edge formation in

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 14 of 20

Fig. 7 Ranking of hypotheses for Kenya contact network. aShows the adjacency matrix of the network with

node ordering according to household membership. Darker cells indicate more contacts. b,cDisplay the

ranking of hypotheses based on Bayes factors, using the uniform hypothesis as baseline for the local and

global model respectively. Using the local model bthe same household hypothesis ranks highest followed by

the similar age hypothesis which also provides positive Bayes Factors. On the other hand, the same and

different gender hypotheses are less plausible than the baseline (uniform edge formation) in both the local

and global case. In the global case call hypotheses are bad representations of edge formation in the Kenya

contact network. This is due to the fact that interactions are very sparse, even within households. Results are

consistent for all κ

this network. This gives us a better understanding of potential mechanisms producing

underlying edges. People prefer to contact people from the same household and similar

age, but not based on gender preferences. Additional experiments could further refine

these hypotheses (e.g., combining them). In the general case of the global model in Fig. 7c

all hypotheses are bad explanations of the Kenya network. However, the same-household

hypothesis tends to go upfront the uniform for higher values of κ,butstillfarformthe

data curve. This happens due to the fact that the interaction network is very sparse (even

within same households), thus, any hypothesis with a dense belief matrix will likely fall

below or very close to the uniform.

Empirical multiplex network

This empirical dataset consists of four real-world social networks, each of them extracted

from Twitter interactions of a particular set of users.

Network We obtained the Higgs Twitter dataset from SNAP (SNAP Higgs Twitter

datasets). This dataset was built upon the interactions of users regarding the discovery

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 15 of 20

of a new particle with the features of the elusive Higgs boson on the 4th of July

2012 (De Domenico et al. 2013). Specifically, we are interested on characterizing edge for-

mation in the reply network, a directed unweighted multigraph which encodes the replies

that a person visent to a person vjduring the event. This graph contains 38, 918 nodes

and 36, 902 multiedges (if all edges from the same dyad are merged it accounts for 32, 523

weighted edges).

Hypotheses We aim to characterize the reply network by incorporating other

networks—sharing the same nodes but different network structure—as prior beliefs. In

this way we can learn whether the interactions present in the reply network can be better

explained by a retweet or mentioning or following (social) network. The retweet hypoth-

esis expresses our belief that the number of replies is proportional to the number of

retweets. Hence, beliefs bij aresettothenumberoftimesuserviretweeted a post from

user vj. Similar as before, the mention hypothesis states that the number of replies is

proportional to the number of mentions. Therefore, every entry bij is set to the num-

ber of times user vimentioned user vjduring the event. The social hypothesis captures

our belief that users are more likely to reply to their friends (in the Twitter jargon: fol-

lowees or people they follow) than to the rest of users. Thus, we set bij to 1 if user vi

follows user vjand 0 otherwise. Finally, we combine all the above networks to construct

the retweet-mention-social hypothesis which captures all previous hypotheses at once. In

other words, it reflects our belief that users are more likely to reply to their friends and (at

the same time) the number of replies is proportional to the number of retweets and men-

tions. Therefore the adjacency matrix for this hypothesis is simply the sum of the three

networks described above.

Results The results shown in Fig. 8 suggest that the mention hypothesis explains the

reply network very well, since it has been ranked first and it is very close to the data curve,

in both Fig. 8a and 8b for the local and global models, respectively. The retweet-mention-

social hypothesis also indicates plausibility since it outperforms the uniform (i.e., positive

Bayes factors). However, if we look at each hypothesis individually, we can see that the

combined hypothesis is dominated mainly by the mention hypothesis. The social hypoth-

esis is also a good explanation of the number of replies since it outperforms the uniform

hypothesis. Retweets and Self-loops on the other hand show negative Bayes factors, sug-

gesting that they are not good explanations of edge formation in the reply network. Note

that the retweet curve in the local model has a very strong tendency to go below the

uniform for higher numbers of κ. These results suggest us that the number of replies

is proportional to the number of mentions and that usually people prefer to reply other

users within their social network (i.e., followees).

Discussion

Next, we discuss some aspects and open questions related to the proposed approach.

Comparison to existing method While we have already demonstrated the plausibility of

JANUS based on synthetic datasets, we want to discuss how our results compare to exist-

ing state-of-the-art methods. A simple alternative approach to evaluate the plausibility

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 16 of 20

Fig. 8 Ranking of hypotheses for Reply Higgs Network. a,bRanking of hypotheses based on Bayes factors

when compared to the uniform hypothesis using multiplexes for the local and global models respectively. In

both cases, the mention hypothesis explains best the reply network, since it is ranked first and very close to

the data curve. This might be due to the fact that replies inherit a user mention from whom a tweet was

originally posted. We can see that the combined retweet-mention-social hypothesis is the second best

explanation of the reply network. This is mainly due to the mention hypothesis which performs extremely

better than the other two (social and retweet). The social hypothesis can also be considered a good

explanation since it outperforms the uniform. The retweet hypothesis tends to perform worse than the

uniform in both cases for increasing number of κ. Similarly, the selfloop hypothesis drops down below the

uniform since there are only very few selfloops in the reply network data

of beliefs as expressed by the belief matrices is to compute a Pearson correlation coeffi-

cient between the entries in the belief matrix and the respective entries in the adjacency

matrix of the network. To circumvent the difficulties of correlating matrices, they can be

flattened to vectors that are then passed to the correlation calculation. Then, hypotheses

can be ranked according to their resulting correlation against the data. However, by flat-

tening the matrices, we disregard the direct relationship between nodes in the matrix and

introduce inherent dependencies to the individual data points of the vectors used for Pear-

son calculation. To tackle this issue, one can utilize the Quadratic Assignment Procedure

(QAP) as mentioned in Section “Related work”. QAP is a widely used technique for testing

hypotheses on dyadic data (e.g., social networks). It extends the simple Pearson correla-

tion calculation step by a significance test accounting for the underlying link structure in

the given network using shuffling techniques. For a comparison with our approach, we

executed QAP for all datasets and hypotheses presented in Section “Experiments” using

the qaptest function included in the statnet (Handcock et al. 2008; Handcock et al.

2016) package in R(R Core Team 2016).

Overall, we find in all experiments strong similarities between the ranking provided by

the correlation coefficients of QAP and our rankings according to JANUS. Exemplary,

Table 1 shows the correlation coefficients and p-values obtained with QAP for each

hypothesis tested on the synthetic multiplex described in Section “Syntheticmultiplex

network” as well as the ranking of hypotheses obtained from JANUS for the local

and global model (leaving the uniform hypothesis out). However, in other datasets

minor differences in the ordering of the hypotheses could be observed between the two

approaches.

Compared to QAP, JANUS yields several advantages, but also some disadvantages. First,

by utilizing our belief matrix as priors over parameter configurations instead of fixed

parameter configurations themselves, we allow for tolerance in the parameter specifi-

cation. Exploring different values of tolerance expressed by our parameter κallows for

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 17 of 20

Table 1 QAP on synthetic dyad-attributed network (multiplex): List of correlation coefficients for

each hypothesis tested. Last two columns show ranking of hypotheses according to JANUS for the

local and global models. By omitting the uniform hypothesis in JANUS (rank 7) we can see that the

ranking of hypotheses by correlation aligns with the rankings given by JANUS for the multiplex given

in Section “Synthetic multiplex network”

Hypothesis Correlation Coefficient P-Value JANUS Ranking Local JANUS Ranking Global

Epsilon10p 0.939 0.0** 1 1

Epsilon20p 0.863 0.0** 2 2

Epsilon30p 0.787 0.0** 3 3

Epsilon40p 0.704 0.0** 4 4

Epsilon50p 0.636 0.0** 5 5

Epsilon60p 0.461 0.0** 6 6

Epsilon70p 0.352 0.0** 8 8

Epsilon80p 0.242 0.0** 9 9

Epsilon90p 0.142 0.0** 10 10

Epsilon100p 0.010 0.238 11 11

Statistically highly significant p-values (p<0.001) are marked by (**)

more fine-grained and advanced insights into the relative plausibility of hypotheses. Con-

trary, simple correlation takes the hypothesis as it is and calculates a single correlation

coefficient that does not allow for tolerances.

Second, by building upon Bayesian statistics, the significance (or decisiveness) of results

in our approach is determined by Bayes factors, a Bayesian alternative to traditional

p-value testing. Instead of just measuring evidence against one null hypothesis, Bayes

Factors allow to directly gather evidence in favor of a hypothesis compared to another

hypothesis, which is arguably more suitable for ranking.

Third, QAP and MRQAP, and subsequently correlation and regression, are subject to

multiple assumptions which our generative Bayesian approach circumvents. Currently,

we employ QAP with simplistic linear Pearson correlation coefficients. However, one

could argue that count data (multiplicity of edges) warrants advanced generalized linear

models such as Poisson regression or Negative Binomial regression models.

Furthermore, our approach intuitively allows to model not only the overall network, but

also the ego-networks of the individual nodes using the local models presented above.

Finally, correlation coefficients cannot be applied for all hypotheses. Specifically, it is not

possible to compute it for the uniform hypothesis since in this case all values in the flat-

ten vector are identical. However, our method currently does not sufficiently account

for dependencies within the network as it is done by specialized QAP significance tests.

Exploring this issue and extending our Bayesian approach into this direction will be a key

subject of future work.

Runtime performance A typical concern often associated with Bayesian procedures

are the excessive runtime requirements, especially if calculating marginal likelihoods

is necessary. However, the network models employed for this paper allow to calculate

the marginal likelihoods—and consequently also the Bayes factors—efficiently in closed

form. This results in runtimes, which are not only competitive with alternative methods

such as QAP and MRQAP, but could be calculated up to 400 times faster than MRQAP

in our experiments as MRQAP requires many data reshuffles and regression fits. Fur-

thermore, the calculation (of Bayesian evidence) could easily be distributed onto several

computational units, cf. (Becker et al. 2016).

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Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 18 of 20

Local vs global model In this paper, we presented two variations of our approach,

i.e., a local and a global model. Although both model substantially different generation

processes (an entire network vs. a set of ego-networks), our experiments have shown that

hypothesesintheglobalscenarioarerankedmostlythesameastheonesusingthelocal

model. This is also to be expected to some degree since the constructed hypotheses did

not explicitly expressed a belief that outgoing links are more likely for some nodes.

Inconsistency of local model For directed networks, the local ego-network models can

assemble a full graph model by defining a probability distribution of edges for every source

node. For undirected networks, this is not directly possible as e.g., the ego-network model

for vAgenerated an edge from vAto vB, but the ego-network model for node vBdid not

generate any edge to vA. Note that this does not affect our comparison of hypotheses as

we characterize the network.

Single Edges As mentioned in Section “Background”, JANUS focuses on multigraphs,

meaning that edges might appear more than once. This is because we assume that a given

node vi,withsomeprobabilitypij , will be connected multiple times to any other node vj

in the local models. The same applies to the global model where we assume that a given

edge (vi,vj)will appear multiple times within the graph with some probability pij.For

the specific case of single edges (i.e., unweighted graphs), where mij ∈{0, 1}, one might

consider other probabilistic models to represent such graphs.

Sparse data-connections Most real networks exhibit small world properties such as

high clustering coefficient and fat-tailed degree distributions meaning that the adjacency

matrices are sparse. While comparison still relatively judges the plausibility, all hypothe-

ses perform weak compared to the data curve as shown in Fig. 7. As an alternative, one

might want to limit our beliefs to only those edges that exist in the network, i.e., we would

then only build hypotheses on how edge multiplicity varies between edges.

Other limitations and future work The main intent of this work is the introduction of

a hypothesis-driven Bayesian approach for understanding edge formation in networks.

To that end, we showcased this approach on simple categorical models that warrant

extensions, e.g., by incorporating appropriate models for other types of networks such as

weighted or temporal networks. We can further investigate how to build good hypothe-

ses by leveraging all node attributes, and infer subnetworks that fit best each of the given

hypotheses. In the future, we also plan an extensive comparison to other methods such

as mixed-effects models and p∗models. Ultimately, our models also warrant extensions

to adhere to the degree sequence in the network, e.g., in the direction of multivariate

hypergeometric distributions as recently proposed in (Casiraghi et al. 2016).

Conclusions

In this paper, we have presented a Bayesian framework that facilitates the understand-

ing of edge formation in node-attributed and dyad-attributed multigraphs. The main idea

is based on expressing hypotheses as beliefs in parameters (i.e., multiplicity of edges),

incorporate them as priors, and utilize Bayes factors for comparing their plausibility. We

proposed simple local and global Dirichlet-categorical models and showcased their utility

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Espín-Noboa et al. Applied Network Science (2017) 2:16 Page 19 of 20

on synthetic and empirical data. For illustration purposes our examples are based on small

networks. We tested our approach with larger networks obtaining identical results. We

briefly compare JANUS with existing methods and discuss some advantages and disad-

vantages over the state-of-the-art QAP. In future, our concepts can be extended to further

models such as models adhering to fixed degree sequences. We hope that our work

contributes new ideas to the research line of understanding edge formation in complex

networks.

Acknowledgements

This work was partially funded by DFG German Science Fund research projects “KonSKOE” and “PoSTs II”.

Availability of data and materials

The data sets supporting the results of this article are openly available on the Web. The source code and data for

toy-example and synthetic experiments can be found on GitHub: https://github.com/lisette-espin/JANUS. The rest of

data sets can be found in their respective project websites: Kenya contact network in http://www.sociopatterns.org/

datasets/kenyan-households- contact-network/, and the Higgs Twitter dataset in https://snap.stanford.edu/data/higgs-

twitter.html.

Authors’ contributions

LE, PS and FL conceived and designed the experiments. LE, performed the experiments. LE, PS and FL analyzed the data.

LE, PS and FL contributed reagents/materials/analysis tools: LE PS FL. LE, PS, FL and MS wrote the paper. All authors read

and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 18 March 2017 Accepted: 25 May 2017

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