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id	Vol-3194/paper46
wikidataid	Q117344942→Q117344942
title	Attributed Stream-Hypernetwork Analysis: a SocioPatterns Case Study
pdfUrl	https://ceur-ws.org/Vol-3194/paper46.pdf
dblpUrl	https://dblp.org/rec/conf/sebd/FaillaCR22
volume	Vol-3194→Vol-3194
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Attributed Stream-Hypernetwork Analysis: a SocioPatterns Case Study

Attributed Stream-Hypernetwork Analysis:
a SocioPatterns Case Study
Andrea Failla1 , Salvatore Citraro1 and Giulio Rossetti2
1
    Computer Science Dept., University of Pisa, IT
2
    KDD Lab, ISTI, National Research Council, IT


                                         Abstract
                                         Network science comes as a solid framework to describe a multitude of human behaviours. Face-to-
                                         face human interactions, for instance, are often represented by dynamic networks involving time-
                                         varying links. Such temporal models are shown to be effective as proxies for real communications
                                         between individuals. However, networks are intrinsically bounded to pairwise/dyadic connections,
                                         whereas complex human dynamics can naturally involve higher-order organization, namely relationships
                                         between groups of entities. In the last few years, hypergraph and simplicial complex models have been
                                         addressed as promising tools to better understand the dynamics of social groups. In the analysis
                                         of face-to-face interactions, the higher-order organization of temporal networks has been addressed
                                         by investigating collections of datasets initially designed for graph-based analysis. Yet even these
                                         higher-order representations continue to ignore the rich attributes or metadata often carried by the
                                         nodes. Such attributes can offer new interesting perspectives about the dynamics of the higher-order
                                         structure emerging from a stream of social interactions. In this work, we aim to address this gap
                                         by introducing attributed stream-hypernetwork models, i.e., higher-order temporal networks with
                                         attributive information on nodes. Considering the Primary and High School temporal networks from the
                                         well-known SocioPatterns project, we infer the higher-order temporal structure of interactions between
                                         children and high school students, and we characterize their non-trivial relationships with respect to
                                         their gender attribute.

                                         Keywords
                                         High-order Networks, Dynamic Networks, Feature-rich Networks, Data Analysis




1. Introduction
The evolving nature of human interactions is often approached through the lens of dynamic
network analysis [1]. Dynamic networks indeed, built on top of graph theory tools, can
represent a wide set of time-evolving human interactions. Dynamic network analysis, a fruitful
subfield of network science, can shed lights on the complex laws governing such time-evolving
connections as well as their duration. However, the intrinsic nature of dynamic network
interactions can not explicitly go beyond dyadic patterns between pairs of nodes. Such network-
based constraints must be taken into account if we aim to investigate human interactions from
the point of view of groups instead of nodes. Hypernetwork science is the new, cutting-edge line

SEBD 2022: The 30th Italian Symposium on Advanced Database Systems, June 19-22, 2022, Tirrenia (PI), Italy
$ a.failla@studenti.unipi.it (A. Failla); salvatore.citraro@phd.unipi.it (S. Citraro); giulio.rossetti@isti.cnr.it
(G. Rossetti)
€ http://giuliorossetti.net (G. Rossetti)
                                       © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
    CEUR
    Workshop
    Proceedings
                  http://ceur-ws.org
                  ISSN 1613-0073
                                       CEUR Workshop Proceedings (CEUR-WS.org)
�of research emerging in network science community that addresses such higher-order structures
for representing complex systems. Thus, dynamic features and higher-order representations
are the two main ingredients for investigating the complex nature of dynamic systems where
time-varying group relations are involved, e.g., face-to-face and physical proximity interactions.
Finally, a last ingredient we may want to consider is the rich set of attributes that nodes can
carry, with the purpose of analyzing how node-attribute information distributes across groups
and the higher-order contexts they define.
In this work we introduce ASH, an Attributed Stream-Hypernetwork model able to represent
higher-order temporal networks with attributive information on nodes. To test the potentiality
of this new model, we infer the attribute-rich higher-order temporal structure of contact
patterns between children and high school students from the well-known SocioPatterns project
collections.
   The rest of the work is organized as follows. Section 2 sums up the principal literature on
the three main complex network contexts surrounding this work, namely dynamic networks,
higher-order structures and attributed networks. Section 3 introduces our Attributed Stream-
Hypernetwork model and discusses the main results obtained from the two SocioPatterns
datasets in terms of topological, node-features and interactions’ dynamics analyses. Finally,
Section 4 concludes the work and discusses promising lines of research left open for the future.


2. Related work
In the following we will provide a brief overview of the main enriched network models and
higher-order representations for complex systems used throughout the work. Dynamic and
node-attributed graphs are the feature-rich [2] representations on which we will focus pri-
marily; then, we will sum up the main emerging contributions in the analysis of higher-order
interactions in complex networks.

Dynamics of networks. Nowadays temporal information is more and more available from
networks. However, choosing a proper temporal representation is not straightforward. Friend-
ships are persistent over time, whereas face-to-face interactions involve a certain duration,
and e-mails, messages or financial transactions are even instantaneous. Different temporal
semantics imposes different modelling strategies [1]. Among the most suitable models encoding
dynamic features in networks, we can identify those focusing on i) stability, ii) duration, and
(iii) immediacy, where:

  i a dynamic network is represented as a sequence of autonomous and independent graphs;
 ii a set of intervals preserve the time-varying dynamics of connections whose a dynamic
    network is composed;
iii a dynamic network comes as a stream of interactions over time – e.g., the stream graph [3]
    is emerging as one of the most fruitful models able to model both instantaneous and lasting
    links, and to capture the presence/absence of nodes as well.

 Among the most interesting and cutting-edge dynamic network mining tasks we recall com-
munity detection [4], link prediction [5], and mixing pattern estimation [6].
�Networks with metadata. Similarly to temporal information, metadata or attributes de-
scribing the properties and the characteristics of nodes are often available from network data.
Such metadata-enriched models can support new mining and analyses about the relationships
between the structural and the attributive information inferred from complex systems. Gener-
ally speaking, works focusing on the relations between attributes and structure aim to study
their correlation or influence, searching for bridges between tabular and networked data [7].
   Node attributes can be fruitfully used for improving the community detection task, where
both tight internal connectivity and label-homogeneity within communities need to be guar-
anteed [8], and for estimating heterogeneous mixing patterns complex networks [9, 10]. The
distribution of metadata surrounding a single node, e.g., the distribution of features within the
node’s ego-network, can be used in the node classification and in the link prediction tasks [11].

Higher-order networks. Although traditional analyses addressed mostly pairwise interactions,
many network dynamics can be better modelled by higher-order system representations involv-
ing complex interactions between groups of nodes. Being an emerging line of research [12, 13],
the expressive power of such models describing higher-order interactions in complex systems
is yet largely unexplored. The interest in the physics of higher-order interactions is growing
[14], and it has been mainly explored in the context of diffusion analysis, e.g., studying social
contagion with simplicial complexes, [15], as well as in time-varying settings [16]. Higher-order
structures varying in time are an important and emerging trend of research [17].
   In most of such analyses, the higher-order organization of static/dynamic networks is ad-
dressed by investing datasets originally designed for graph-based analysis. One of the most
intriguing and fundamental works in the future will have to focus on the inference of statisti-
cally significant higher-order interactions [18]. Moreover, higher-order-based techniques are
emerging to generalize well-known graph-based techniques or to conservatively shift to them,
as in the case of s-line graph analysis for hypergraph models [19].


3. Hypernetwork Science
To study dynamic high-order social interactions, simply borrowing results from the existing
literature is not enough. Hypergraphs and simplicial complexes, to name the nowadays most
used high-order representation frameworks, have both strengths and weaknesses. None of them
has been adequately defined in the presence of evolving topologies. Indeed, their applicability
to online social environments needs to be carefully analyzed to understand if the constraint
they come with aligns with the semantics expressed by social interaction networks. Moreover,
individuals embedded in a social system can often be characterized by multiple features —
profiles that contextualize some of the key properties playing a role for social interactions (e.g.,
nationality, gender, age. . . ). In order to start filling the existing gap in high-order dynamic and
feature-rich modeling of social systems, here we propose the framework of Attributed Stream
Hypergraphs (henceforth, ASH).

Definition 3.1 (Attributed Stream Hypergraph (ASH)). Let 𝒮 = (𝑇, 𝑉, 𝑊, 𝐸, 𝐿) be a stream
hypergraph, where:
�    • 𝑇 = [A, Ω] is the set of discrete time instants, with A and Ω the initial and final instants;
    • 𝑉 is the set of the nodes of the temporally flattened hypergraph;
    • 𝑊 ⊆ 𝑇 × 𝑉 is the set of temporal nodes;
    • 𝐸 ⊆ 𝑇 × 𝑉 𝑛 is the set of temporal hyperedges such that (𝑡, 𝑁 ) ∈ 𝐸 implies that 𝑁 ⊆ 𝑉
      and ∀𝑢𝑖 ∈ 𝑁, (𝑡, 𝑢𝑖 ) ∈ 𝑊 ;
    • 𝐿 is the set of temporal node attributes such that 𝐿(𝑡, 𝑢) with (𝑡, 𝑢) ∈ 𝑊 and 𝑡 ∈ 𝑇 ,
      identifies the set of categorical values associated to 𝑢 at time 𝑡.

  ASHs are a conservative extension of well known modeling frameworks (namely, Hypergraphs
[12] and Stream Graphs [3]). From them it inherits several analytical peculiarities and, from
their union, it is able to provide novel insights that the original models are not able to unveil
independently. Moreover, integrating time evolving node attributes, it allows to study not only
how individuals’ characteristics changes (e.g., opinions, political leaning) but also how such
changes relate/affect the topological structure surrounding them.
  In the following we provide a first example of how ASH can be used to study real world
complex social systems. Our analysis will focus on some aspects of the three dimensions
modeled by ASH: high-order topology, node semantics and time.

3.1. SocioPatterns data
SocioPatterns1 is a project collecting a variety of physical proximity and face-to-face interactions
across several environments, e.g., hospitals, workplaces and schools. These data were classically
collected to study human behaviours in terms of temporal interactions, confirming, for instance,
the existence of few long-lasting contacts and a multitude of brief contacts while analyzing their
duration in spontaneous human interactions [20], or observing that the contacts are shaped by
the organization of the offices in workplaces [21]. The face-to-face data are also extremely useful
to estimate the transmission of infectious diseases in contexts like schools [22] and workplaces
[21]. Recently, such data have been studied by applying an higher-order temporal perspective
[17], or by enhancing the analysis of human behaviours in terms of temporal assortative mixing
estimation [6].
   In this work we will focus on the two collections expressing face-to-face contacts in primary
[23] and high school [24] contexts. We focus on both days of the primary school and all four
days of the high school. Nodes are enriched with attributive information about the gender of
students. For representing the temporal higher-order structure, we leverage a similar method
as that introduced in [17], namely: if at at time 𝑡 there are 𝑛 * (𝑛 + 1)/2 dyads between the
members of a set of 𝑛 nodes such that they are involved in a fully connected clique, such links
are promoted to form a 𝑛-hyperedge.

3.2. Topological analysis
In the case of hypergraphs modeling social interactions, hyperedges represent groups of in-
dividuals co-acting simultaneously. In this respect, we consider individuals connected by an

   1
       http://www.sociopatterns.org/
�                        (a)                                                 (b)
Figure 1: Hyperedge distribution (left) and node degree distribution in log-log scale (right) of the two
hypernetworks


hyperedge if they interacted within a time frame of 20 seconds. Figure 1a compares the hy-
peredge size distribution of the two hypernetworks. Primary and High School show similar
trends, with smaller-sized (unique) interactions being the most frequent. It should be noted that
Primary records almost twice as many hyperedges as High School. As the number of individuals
increases, the amount of hyperedges rapidly decreases, ranging down to extremely rare 5-way
interactions, which occur less than ten times in each network. A possible explanation is that
smaller interactions are generally the easiest to engage into, with larger ones requiring time
and energy of more and more individuals [25]. This can also be confirmed by analyzing the
frequency of interactions, that is how often the nodes of the same hyperedge interact through
time. Both hypernetworks count nearly 70 000 interactions, which is peculiar considering
that students in High School were monitored for twice as much time as the schoolchildren.
Pairwise relations are once again the most frequent in both networks (∼ 90%), possibly due to
the slightness of the time frame used. Still, multiadic ones take up a non-negligible part of the
whole.
A remarkable difference concerns the nodes’ hyperdegrees. We refer to a node’s hyperdegree as
the number of incident hyperedges, that is the number of hyperedges that include that node. As
shown in Figure 1b, both distributions follow a bell-like trend, with most values concentrating
in the vicinity of the average and a few located near the minimum/maximum. However, degrees
in Primary are generally higher, averaging at 125 connections (𝜎 = 55.16); in contrast, High
School holds lower — the average being 54 — but more clustered values (𝜎 = 27.34).

3.3. Node-Features analysis
Of the 242 nodes in Primary, 115 are males, 112 are females, and the remaining part is of
unknown gender (i.e., teachers). High School, instead, counts 176 males, 146 females and 7
teachers over 329 individuals. Note that despite having less nodes, Primary has more than
double the amount of teachers. Indeed, Primary counts a teacher every 16 students, whereas
High School counts one every 47.
�                       (a)                                                (b)
Figure 2: (a) Purity and (b) Entropy trends in the Primary School contact dataset.


   We use 𝐸𝑛𝑡𝑟𝑜𝑝𝑦 to describe the behaviour of nodes w.r.t. the gender attribute, as it quantifies
the degree of disorder of the attribute. Let 𝐴 be the set of nodes’ attributes, the entropy of an
attribute 𝑎 ∈ 𝐴 is computed as follows:
                                                  |𝐴|
                                                  ∑︁
                              𝐸𝑛𝑡𝑟𝑜𝑝𝑦(𝑎) = −            𝑝(𝑖|𝑎)𝑙𝑜𝑔𝑝(𝑖|𝑎)                        (1)
                                                   𝑖

  We highlight low entropy values in both networks, namely 0.54 in Primary and 0.46 in High
School. This suggests that interpersonal relations in both scenarios are mainly uniform w.r.t.
gender, with younger students having slightly less homogeneous contacts. It should be noted
that these values take into account the existence of teachers, whose gender is unknown, thus
increasing entropy; nonetheless, excluding edges that involve teachers lowers both entropy
values by just ∼ 0.01.
  Another relevant metric is 𝑃 𝑢𝑟𝑖𝑡𝑦, i.e., the relative frequency of the most frequent attribute
value within a hyperedge. Formally, the purity of hyperedge 𝑒 is computed as follows:
                                                       ∑︀
                                            max𝑎∈𝐴 ( 𝑣∈𝑒 𝑎(𝑣))
                               𝑃 𝑢𝑟𝑖𝑡𝑦𝑒 =                                                     (2)
                                                       |𝑒|
  Then, 𝑃 𝑢𝑟𝑖𝑡𝑦 is normalized by the number of hyperedges:
                                                1 ∑︁
                                   𝑃 𝑢𝑟𝑖𝑡𝑦 =         𝑃 𝑢𝑟𝑖𝑡𝑦𝑒                                  (3)
                                               |𝐸|
                                                    𝑒∈𝐸
   Purity values confirm the previous observations with Entropy. Purity scores are high in both
schools, namely 0.76 for Primary and 0.79 for High School, meaning that the majority of nodes
in almost every hyperedge share the same gender. By studying mostly-male and mostly-female
hyperedges separately, we also relieved that the former ones are ∼ 3% purer than the latter
ones, in both scenarios.

3.4. Interactions’ Dynamics analysis
Both Figure 2 and Figure 3 focus on the average temporal trends of hyperedges’ purities and en-
tropies along time. While doing so, purity scores are disaggregated by gender labels, with the aim
�                        (a)                                               (b)
Figure 3: (a) Purity and (b) Entropy trends in the High School contact dataset.


to observe whether different patterns emerge from the labels. On average, both purity/entropy
values are high/low, indicating a strong homogeneity by gender in both hypernetworks. Some
little differences seem to emerge between the two datasets, since (i) primary school gender
entropy is higher than the high school entropy, indicating that a more mixed interaction global
pattern occurs in children contacts; (ii) some periodical patterns emerge in children contacts,
whereas no periodicity is observed in the high school dataset; in detail, the entropy score trend
in children contacts decreases at lunchtime [23], indicating how more homogeneous groups
tend to be shaped by an "external" context. To conclude, no significant differences seem to
emerge between male and female students by looking at the purity trends, even if slightly higher
male scores than female ones could be observed sometimes in high school student contacts;
however, the absence of periodical patterns does not allow us to better interpret/discuss such
behaviours.


4. Discussions and Conclusions
In this work we tested the possibilities of ASH, our new model to represent and analyze
complex streams of higher-order interactions enriched with information about node profiles. We
extracted the temporal higher-order structure of two well-known datasets from the SocioPatterns
project, namely Primary and High School face-to-face contacts. We enriched nodes with
attributive information about the students’ gender. Our main focus was on a qualitative study
on the dynamics of group interactions that a graph-based study on the set of pairwise links
could not highlight. We divided the analysis along three perspectives. On the one hand, we
studied the global properties of the higher-order systems, e.g., highlighting how 𝑛-hyperedges
distribute across group dimensions. On the other hand, we highlight the strong homogeneity of
such groups w.r.t. the gender attribute. Finally, along the temporal dimension we discovered
differences between the two schools: children seem to be characterized by periodical patterns,
�e.g., lower entropy values during lunchtime, and their entropy is on average higher than high
school students.
   In future works, we plan to provide quantitative justifications of such preliminary exploratory
observations. We also plan to define domain-specific measures based on a solid characteriza-
tion of node profiles, leveraging them to better analyze the dynamics of higher-order contact
structures. More insights about the dynamics of groups in face-to-face interaction data will be
provided by extending this preliminary case study to the wide set of SocioPatterns datasets, as
well as other contact patterns databases [26], and by testing different aggregation time frames.


Acknowledgments
This work is supported by the European Union – Horizon 2020 Program under the scheme
"INFRAIA-01-2018-2019 – Integrating Activities for Advanced Communities", Grant Agreement
n.871042, "SoBigData++: European Integrated Infrastructure for Social Mining and Big Data
Analytics" (http://www.sobigdata.eu).


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