Optimized Social Explanation for Educational Platforms
Italo Zoppis
1
, Riccardo Dondi
2
, Sara Manzoni
1
, Giancarlo Mauri
1
, Luca Marconi
3
and Francesco Epifania
3
1
Department of Computer Science, University of Milano Bicocca, Milano, Italy
2
Department of Letters, Philosophy, Communication, University of Bergamo, Bergamo, Italy
3
Social Things srl, Milano, Italy
Keywords:
Social Networks, Optimized Social Explanation, Communities Identification, Genetic Algorithms, WhoTeach.
Abstract:
Recommender Systems have became extremely appealing for all technology enhanced learning researches
aimed to design, develop and test technical innovations which support and enhance learning and teaching
practices of both individuals and organizations. In this scenario a new emerging paradigm of explainable
Recommander Systems leverages social friend information to provide (social) explanations in order to supply
users with his/her friends’ public interests as explained recommendation.
In this paper we introduce our educational platform called “WhoTeach”, an innovative and original system
to integrate knowledge discovery, social networks analysis, and educational services. In particular, we re-
port here our work in progress for providing “WhoTeach” environment with optimized Social Explainable
Recommandations oriented to design new teachers’ programmes and courses.
1 INTRODUCTION
Recent approaches in explainable Recommander Sys-
tems (RS) leverage social friend information to sup-
ply users with explanations, concerning his/her so-
cial friends’ public interests as recommended expla-
nations. In fact, user-based RS has shown critical as-
pects, mainly due to the lack of important information
(regulated by privacy) and the difficulty of recogniz-
ing the generated opinions (i.e, explanations) as valid
or correct. In this regard, it is generally more accept-
able to inform the users about social friends’ public
interests on the recommended items. As a result, part
of current literature of recommender systems is fo-
cused on generating social explanations with the help
of social information. For example, in (Papadimitriou
et al., 2012) human-style, item-style, feature style and
hybrid-style explanations in social recommender sys-
tems are considered by reporting geo-social expla-
nations that combine geographical with social data.
Sharma and Cosley (Sharma and Cosley, 2013) stud-
ied the effects of social explanations in music recom-
mendation context by providing the target user with
the number of friends that liked the recommended
items. Chaney et al. (Chaney et al., 2015) presented
social Poisson factorization, a Bayesian model that in-
corporates a user’s latent preferences for items with
the latent influences of her friends, which provides a
source of explainable serendipity (i.e., pleasant sur-
prise due to novelty) to users.
Based on a similar relational context, in (Quijano-
Sanchez et al., 2017) recommendations are explained
on similar users that are friends with the target user.
In this case, social explanation is introduced in a sys-
tem as group recommendation, which significantly in-
crease the user intent to follow the recommendations,
the user satisfaction, and the system efficiency to help
users make decisions.
By following these ideas, we focus on the prob-
lem of modeling an optimized space of users (teach-
ers) and items (resources), where social interaction
is promoted, and Explainable Recommander Systems
(ERS) can benefit from social information to supply
recommended explanations. This perspective follows
an interesting research area where graphs are taken as
models for explainable recommendations (He et al.,
2015; Heckel et al., 2017; Wang et al., 2018). In
particular, we consider the case where new teachers,
(Target Teachers, T ), could be interested to meet some
other colleagues (experienced teachers, X) for sharing
information on recommended items (R). In this situa-
tion, the platform could encourage, for example, new
Zoppis, I., Dondi, R., Manzoni, S., Mauri, G., Marconi, L. and Epifania, F.
Optimized Social Explanation for Educational Platforms.
DOI: 10.5220/0007749500850091
In Proceedings of the 11th International Conference on Computer Supported Education (CSEDU 2019), pages 85-91
ISBN: 978-989-758-367-4
Copyright
c
 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
85

teachers to socialize, and confront with the experience
of specific collegues, who have already held similar
(or alternative) programmes and courses. Similarly,
the Recommender System can supply Target Teachers
with clarifications concerning experienced teachers’
public experiences as recommended explanations.
In this paper, we first introduce our educational
platform called “WhoTeach”; an innovative platform
to integrate knowledge discovery, social networks and
educational services. Then, we report our work in
progress to enhance “WhoTeach” with the capability
described above for the design of new teachers’ pro-
gramme and course material. It is clear that, in such
general situation, a proper handling of procedures and
data is fundamental in order to convert available infor-
mation into useful formulation that leads to particular
(induced) communities. Please notice that the intent
here is to develop the fundamentals for the consid-
ered social optimization problem (future implementa-
tion will be discussed in the conclusion section).
From a theoretical perspective our consideration
can be expressed as the problem of finding cohesive
subgraphs (with particular properties), inside a net-
work. Unfortunately, as we report in this paper, the
intrinsic complexity of the considered computational
problem make optimization potentially impracticable.
For this reason, we designed a specific heuristic (i.e.,
Genetic Algorithms, GAs) to seek faster approxima-
tion solutions.
More in details, we introduce the “WhoTeach”
platform in Section 2. In Section 3.1 we consider
the main theoretical aspects. Then, in Section 4, we
discuss the GA-based approach to seek approximated
results for our formulation. Finally, after reporting
numerical experiments on simulated data (Section 5),
we conclude the paper (Section 6) by discussing our
results and describing future directions of this re-
search.
2 WhoTeach
WhoTeach is an innovative and original system, con-
ceived to promote the development of professional or
academic competences of self-employed, managers
or students by aggregating and disseminating knowl-
edge created by experts. Hereinafter the experts will
be called teachers, while learners will be also called
students, regardless their professional or academic
role in the organization in which they belong.
WhoTeach can be described as a Social Intelli-
gent Learning Management System (SILMS). In fact,
WhoTeach is distinguished from many traditional e-
learning platforms thanks to the presence of two main
components:
1. a recommender system, aimed at suggesting both
teachers the right resources, in any format, to cre-
ate their courses and students the right courses
according to their educational background, their
profile and their target;
2. a social network, aimed at allowing experience
and knowledge exchange among peers, learners
and teachers, so as to create communities of prac-
tice and of interest to empower the learning pro-
cess. The platform is designed for demanding
users, who want to teach or to learn in highly-
dynamic disciplinary environments. Specifically
it is the result of the exploitation of the European
project NETT, with the aim of gathering a Social
Network for improving and promoting the diffu-
sion of the entrepreneurship knowledge and mind-
set.
Teaching requires the organization of effective ad hoc
courses, thus teachers need to get adequate didac-
tic resources to avoid frustration and waste of time
and to create high-quality courses and materials: thus
WhoTeach provides them with suggestions to support
and guide them in the organization of their courses. In
order to obtain the creation of an effective and high-
level course, the recommender system helps by re-
ducing the high number of available resources, thus
time and exhausting attempts to search teaching ma-
terial. Moreover, it provides learners with a suggested
path to improve specific skills, according to their pro-
file, personal interests, usage history and evaluation
check-lists.
A distinguishing approach in the SILMS is the use
of learning algorithms to identify relations between
the features of the platform contents that may prove
suitable to the inquirer. Thanks to the massive use of
metadata, the contents may be homogeneously identi-
fied through a vector of parameters (the metadata rep-
resentation). In addition, thanks to the feedback of the
previous users and the expertise of the system admin-
istrators, each composition of vectors in courses may
be associated to a score. This enables a dynamical
decision tree procedure where, depending on the cur-
rent choice of the user, the system proposes branches
of decision trees that may lead to satisfactory com-
pletion of the course, possibly listed in a monotone
ranking.
The learning material is organised in different
knowledge areas and contents are divided in re-
sources, modules and courses. In particular, the
resources can be different in their structure(wiki,
discussion forum, eBook, etc) and format (word,
pdf,etc). Classification through metadata allows reuse
CSEDU 2019 - 11th International Conference on Computer Supported Education
86

of learning materials. Every user can rate materi-
als and courses found in SILMS; the evaluations are
stored in a table ad-hoc (created by SILMS Develop-
ers) and used to implement the recommender system.
Learning and training are social activities, espe-
cially when the learning objects are relatively new,
hence not yet assessed in well established disciplines.
Thus, the second pillar of the platform structure is
a social network where communities of teachers are
fostered around each disciplinary sector. The objec-
tive is to transform a personal learning experience in a
more collaborative and amazing one, obtaining better
results. To this aim, SILMS platform is equipped with
standard social network tools (like blogs, chat, forum,
messaging), plus the following advanced functionali-
ties tailored for NETT project:
• Definition of Community. Around each disci-
plinary sector it is possible to define specific com-
munities of teachers. These communities are
moderated by the master of the knowledge area
associated to each discipline sector. The objec-
tive of these communities is to transform a per-
sonal learning experience in a more collaborative
and amazing one, obtaining better results. More-
over, thematic communities can be freely created
by teachers.
• Sharing of Didactical Materials. Beside the offi-
cial version of didactic materials published within
the SILMS platform, there is the possibility to
share non official material, without waiting ex-
perts’ or masters’ approvals.
• Informal Communication Among Users. While
sharing, teachers should receive private or pub-
lic feedbacks that can help him/her in improv-
ing his/her materials. Moreover, communications
among contributors/experts and experts/ masters
can be conducted through social network facili-
ties.
• Teacher’s Profile. Teachers are called to create
and edit their own profile, where personal expe-
rience or school education can be reported. This
enables masters to promote contributors in experts
relying on competence and credits. It also fos-
ters social activities of users, who can get in touch
with other teachers beyond the SILMS platform
through either internal tools or external tools, e.g.
Skype.
• Followers. Teachers can create a network com-
posed by people with the same interests or expe-
riences. Among them, people can follow a par-
ticular content of a user and consequently receive
updates and news, keeping in touch with teachers
with either the same skills or working, anyway, in
the same field.
Therefore, the idea is that this kind of social network
can give rise to rich, efficient and fruitful communities
of practice rooted on the common goal of favoring
course design activities.
From a technical perspective, the system consists
of a PHP shell piloting and empowering the cus-
tomization of the Moodle platform, as for a Content
Management System and Mahara as for a nested So-
cial Network. Mahara is a fully-featured web appli-
cation to build an electronic portfolio. A user can
create journals, upload files, embed social media re-
sources from the web and collaborate with other users
in groups. What makes Mahara different from other
ePortfolio systems is that the user can control which
items and what information other users see within
their portfolio.
The Moodle system was chosen because of its
high diffusion within the school and due to the pres-
ence of a wide development community. The plat-
form has been then integrated with social network
features coming from Mahara, in order to introduce
meta-services as previously described.
3 SOCIAL OPTIMIZATION
PROBLEMS
Suppose we wish to model the situation where new
users (i.e., target teachers, or T as they will be referred
in this paper) are interested to design new courses
by applying resources and recommended materials al-
ready used by colleagues, who have held similar (or
even alternative) courses.
In this case, T could benefit from the social in-
teraction with their colleagues, to confront with their
past experience, and to deepen knowledge about their
social friends’ public interests on the recommended
items. As reported above, it should be useful for a
social platform to encourage, and optimize the cre-
ation of a sub-network of users (teachers) and items
(resources) from the available data.
3.1 Problems Formulation
From a theoretical point of view, a network is most
commonly modeled using a graph which represents
relationships between objects, V (vertices), through a
set of edges, E. In this way, our goal can be formu-
lated by maximizing, within a defined graph, a cohe-
sive sub-graph (i.e., by seeking the largest cohesive
Optimized Social Explanation for Educational Platforms
87

sub-graph) with particular properties (as we will de-
tail in the following). Finding cohesive subgraphs in-
side a network is a well-known problem that has been
applied in several contexts (Bader and Hogue, 2003;
Spirin and Mirny, 2003; Sharan and Shamir, 2000).
While a classical approach to compute dense sub-
graphs is the identification of cliques (i.e., complete
sub-graphs induced by a set of vertices which are all
pairwise connected by an edge), this definition is of-
ten too stringent for particular applications. This is
the case, when the knowledge on how an individual
(vertex) is embedded in the sub-network (e.g., some
vertices could act as “bridges” between groups, as in
our case) is a critical issue to take into account. There-
fore alternative definitions of cohesive sub-graphs can
be introduced, for example by relaxing some con-
straints, leading to the concept of relaxed clique (Ko-
musiewicz, 2016). Here, we follow this approach by
relaxing the definition of distance between vertices.
In a clique distinct vertices are at distance of 1, in our
case, vertices can be at distance of at most s = 2. A
sub-graph where all the vertices are at distance of at
most 2 is called a 2-club (or, more in general, s-club
for different values of s).
3.2 Main Definitions
Let us consider a graph G = (V,E), and a subset V
0
⊆
V . We denote by G[V
0
] the subgraph of G induced by
V
0
. Formally G[V
0
] = (V
0
,E
0
), where
E
0
= {{u,v} : u,v ∈ V
0
∧ {u,v} ∈ E}.
Given a set V
0
⊆ V , we say that V
0
induces the graph
G[V
0
]
1
. The distance d
G
(u,v) between two vertices
u,v of G, is the length of a shortest path in G which
has u and v as endpoints. The diameter of a graph
G = (V,E) is max
u,v∈V
d
G
(u,v), i.e., the maximum
distance between any two vertices of V . In other
words, a 2-club in a graph G = (V,E) is a sub-graph
G[W], with W ⊆ V , that has diameter of at most 2.
Moreover, given a vertex v ∈ V , we define the set N(v)
as follows:
N(v) = {u : {v,u} ∈ E}
N(v) is called the neighborhood of v.
We formulate the “social computational problem”
described above, using 2-clubs, in such a way that
paths connecting T with items R has to “transit”
through x ∈ X.
In this way, we are currently seeking (within the
input “social graph”) a 2-Club, G[T ∪X ∪R], where T ,
1
Notice that all the graphs we consider are undirected.
X and R represent the sets of new users, experienced
teachers, and resources, respectively.
Please notice that, if such a structure (i.e., a maxi-
mum size 2-clubs) exists, then for any pair of vertices,
it must exist at least one simple path of length 2, i.e.,
a path composed by a triple of vertices. This, in turn,
will also be true for any pair, (t,r) where t ∈ T, r ∈ R.
Indeed, our goal will be to find a largest-size 2-clubs
which has the further property of providing, the maxi-
mum number of pairs (t, r), characterized by the triple
of vertices (t,x,r) ∈ (T × X × R). In this case, the
following set of fundamentals edges are important to
provide a correct optimization procedure.
• Edges between users, E, (i.e.,, between new
teachers T and experienced teachers X), express-
ing e.g., interest to cooperate, similarity etc.
• Edges between users and items, F, expressing that
an experienced teacher x, has already applied a
course resource, r. In this case the edges in X × R
will be constructed by knowing both the educa-
tional history of each (experienced) teacher, x, and
the resource r, which x has already applied for its
courses.
In this situation, the paths given by the triple of ver-
tices (t,x,r) ∈ (T × X × R) would suggest to teacher
t ∈ T to contact colleagues, x ∈ X, about the recom-
mended resource, r ∈ R. For sake of clarity, before
defining computationally the problem, we refer to any
vertex, x ∈ X, for which there exists at least one pair
(t,r) ∈ T × R within its neighborhood N(x) as “fea-
sible vertex”. Similarly, a set of “feasible vertices”
C will be referred as “feasible set”, and a pair (t,r),
for which there exists the feasible vertex x ∈ X , will
be named “‘feasible pair”. Considering the above dis-
cussion, we can define the following variant of the
2-clubs maximization problem.
Problem 1. Input: a graph G = (T ∪ X ∪ R, E ∪F).
Output: a set V
0
⊆ T ∪ X ∪ R such that G[V
0
] is a 2-
club having both maximum size and the largest num-
ber of feasible pairs.
4 A GENETIC ALGORITHM
The complexity of Maximum s-club has been exten-
sively studied in literature, and unfortunately it turns
to be NP-hard for each s ≥ 1 (Bourjolly et al., 2002).
The same property holds for our variant of Maximum
2-club, thus making optimization potentially imprac-
ticable. For this reason, we designed a Genetic Al-
gorithm (GA) to seek faster approximation solutions
see, e.g., (Mitchell, 1996) for details.
CSEDU 2019 - 11th International Conference on Computer Supported Education
88

In particular, given an input graph G = (V,E), the
proposed GA represents a solution (a subset V
0
⊆ V
such that G[V
0
] is a 2-club of G) as a binary chromo-
some c, of size n = |V |, such that for all v
i
∈ V
0
, c[i] is
either 1 or 0. Note that, with a slight abuse of notation,
we will denote by G[c] the subgraph of G induced by
the representation of chromosome c. Similarly, V[c]
and E[c] will denote the set of vertices (V
0
) and edges
of G[c] = G[V
0
].
During the offspring generation, chromosomes are
interpreted as hypotheses of feasible solutions, under-
going to mutation, crossover and selection. Chro-
mosome evaluation (i.e., hypothesis on a potential
2-club) is then provided, as usually, through the fit-
ness function. For space requirements in the follow-
ing sections, only 2 relevant issue of this approach
are detailed, namely fitness and mutation. Currently,
crossover operator does not differ significantly from
the standard definition.
4.1 Fitness Definition
In this paper, fitness is designed to promote adapta-
tion in such a way that new candidate chromosomes,
able to represent graphs with correct diameter value
(i.e., not grater than 2) will “evolve” through specific
mutation and standard crossover. In particular, given
a chromosome c, and an input graph G = (V,E), the
fitness promote such a mechanism using the following
quantities.
1. An estimation of the number of feasible vertices v
2. The number of vertices, n
V
, of the (sub)graph,
G[c], induced by c.
In particular, for any chromosome c, we observed
a sample S of vertex v ∈ V [c] and by considering the
induced sub-graph representation G[c], we evaluated
the following fitness:
f (n
v
;diam) =
(
(n/|S|)n
v
if 0 ≤ diam ≤ 2 ;
1
n
v
if 2 < diam ,
(1)
where n
v
is the cardinality of G[c], i.e., the sub-graph
induced (speculated) by c, and n is the frequency of
“feasible” vertex observed in S, i.e.,
∑
n
k=1
I(x
k
∈ C ),
with:
I(x)
k
=
(
1 if x
k
∈ C ;
0 otherwise.
(2)
In this way, by using the proportion, n/|S|, the fit-
ness weights (when 0 ≤ diam ≤ 2) the number of ver-
tices n
v
in G[c], thus promoting large (sub)graph. On
the other hand, when diam > 2 (i.e., unfeasible solu-
tions), we have fitness values which decrease asymp-
totically for large (sub)graph size, n
v
, thus penalizing
the corresponding chromosome.
4.2 Mutation
To promote adaptation (with regard to the Maximum
2-club problem), we defined 2 types of mutation, re-
spectively applied with equal probability.
• Mutation Operator 1 has the objective to correct
hypotheses (i.e., chromosomes) consistently and
parsimoniously. Since any chromosome, by con-
struction, induce a sub-graph G[V
0
] of G, which
should reflects feasible solutions, such hypotheses
are partially verified using the following princi-
ple. Given a selected chromosome c, a vertex v
0
is
(randomly) sampled from the set V
+
= {v
i
: c[i] =
1} and the minimum length of simple paths con-
necting every pair (v
i
,v
0
),v
i
∈ V
+
/v
0
is checked to
be consistent with the chromosome “hypothesis”,
i.e., since each chromosome “speculates” a feasi-
ble 2-club, for such hypothesis to be true, there
must be, at least, a simple path of size at most
equal to 2 connecting any v
i
∈ V
+
with v
0
. If a
negative feedback is observed after this verifica-
tion, then the sampled vertex v
0
is flipped to 0.
• Mutation Operator 2. This operator has the ob-
jective to (parsimoniously) increment the size of
a solution. In this case, given a selected chro-
mosome c a vertex v
0
is sampled from V
−
= {v
j
:
c[ j] = 0} and the minimum length of simple paths
connecting every pair (v
i
,v
0
) (v
i
∈ V
+
) is checked
to be consistent with the chromosome c’s hypoth-
esis. In this case, we consider to extend the so-
lution represented by c, by adding v
0
to V
+
if the
minimum distances from v
0
to vertices of V
+
are
not larger than 2.
5 NUMERICAL EXPERIMENTS
The genetic algorithm described in Sec. 4 was coded
in R using the “GA” package (Scrucca, 2013). The
main objective was to evaluate the capability of GAs
to obtain correct solutions for Problem 1 in a reason-
able time.
Results are given for synthetic data, by sampling
Erdos-Renyi random graphs ER(n, p) with different
values of number of vertices, n, and edge probability
p = 0.4 (Bollobas, 2001). To provide correctness at
“reasonable” cost (Lewis and Papadimitriou, 1997),
we followed the standard practice of evolutionary al-
gorithms: keep the tractability of the search operators
and the fitness, and promote, at the same time, evalu-
able chromosomes, which in our case, provide feasi-
ble input diameter values. Moreover, a widely applied
principle for termination has been applied: we set up
Optimized Social Explanation for Educational Platforms
89

Table 1: Performances: Models (Erdos-Renyi); Fitness (Fit); Input Diameter (InD); Output Diameter (OutD); Output Nodes
(OutN), Input Feasible vertices (InFeas); In/out Ratio of Feasible Vertices (IORatio); User (T1) and System Time (T2).
Models Fit InD OutD OutN Ratio IFeas IORatio T1 T2
ER(25,0.4) 10.3 3 2 10.3 0.413 8 0.452 55.6 1.11
ER(50,0.4) 28.3 3 2 29.7 0.593 18.7 0.758 289 3.34
ER(100,0.4) 53.7 2.3 2 83.3 0.833 31.3 0.927 645 2.33
ER(150,0.4) 80.4 2 2 145 0.964 47 0.965 792 1.04
ER(300,0.4) 149 2 2 285 0.951 103 0.955 2776 2.98
both the number of re-evaluation of the fitness over
new populations (equivalently, the number of the GA
iterations), and the number of consecutive generations
without improvement of the best fitness value
2
. Note
that to check the robustness of the solutions, each ER
model has been sampled iteratively 3 times. The re-
sults are reported in Tables 1.
The following main considerations emerge from
the results.
• In all models, we obtained correct 2-clubs (Diam-
eter value ≤ 2).
• Clearly, due to the complexity of the problem, we
cannot compare the optimal solution with the ones
given by the GA. Therefore, to give a qualitative
idea of the approximated solutions, we reported
the output over input ratio of feasible vertices.
Evaluable solutions are those for which this ratio
is close to 1. In these cases, the environment pro-
vides at least one feasible vertex as defined above.
• While the size of the inferred communities does
not seem to differ, the (average) number of itera-
tions of the last level site, for the distributed case,
is much lower than the iterations reported by the
standard, centralized evolution. This is also ev-
ident from the decrease in the average execution
time per level, reported by the distributed case.
Since GA uses the same parameters for the stan-
dard and the distributed evolution, this behavior
can surely be traced back to the initial suggestions
that each lower level site receives from the higher
level.
• System time seems to be reasonable (T2 ≤ 4) on
the considered instances.
6 CONCLUSIONS
In this paper we introduced our work in progress to
enhance the WhoTeach platform with optimized ex-
planations concerning social users’ public interests.
2
See, e.g. (Safe et al., 2004), for a critical analysis of
various aspects associated with the specification of termi-
nation conditions in a simple genetic algorithm.
We considered this issue from a computational point
of view by defining a variant of a well known opti-
mization problem. Due to the hardness of the for-
mulated question, we presented some introductory re-
sults on how we apply GAs for obtaining approxi-
mated solutions (i.e., see also (Dondi et al., 2017;
Dondi et al., 2016)). Our interest for future imple-
mentation of this research will be focused on the fol-
lowing items:
• The platform should be able to not only provide
the user (target teacher) with an optimized “list”
of his social (experienced) friends, but even fo-
cusing on the specific (identification) of (target)
user’s requirement (i.e. items). In this way the
systems should provide the (optimized) enumer-
ated list of items in agreement both with the user
requests, and the “expert” teacher experiences.
• In a further step of generalization, the platform
should be able to provide learners and teachers
with optimised suggestions taking into consid-
eration the weighted requirements coming from
different communities or sub-communities: e.g
teachers enrich their courses through information
and insights coming not only from other teachers
but also from the dynamics of the context.
• Another important issue is related to the possibil-
ity of using WhoTeach as an e-recruitment sys-
tem: the platform then should be able to provide
companies with optimised social recommenda-
tions so as to match users’ and companies’ needs.
That would allow learners to adapt their learn-
ing path so as to keep it up-to-date and to gain
all the necessary professional skills and meta-
competencies required by the labour market.
REFERENCES
Bader, G. D. and Hogue, C. W. V. (2003). An automated
method for finding molecular complexes in large pro-
tein interaction networks. BMC Bioinformatics, 4:2.
Bollobas, B. (2001). Random Graphs. Cambridge Univer-
sity Press.
CSEDU 2019 - 11th International Conference on Computer Supported Education
90

Bourjolly, J., Laporte, G., and Pesant, G. (2002). An ex-
act algorithm for the maximum k-club problem in an
undirected graph. European Journal of Operational
Research, 138(1):21–28.
Chaney, A. J., Blei, D. M., and Eliassi-Rad, T. (2015). A
probabilistic model for using social networks in per-
sonalized item recommendation. In Proceedings of
the 9th ACM Conference on Recommender Systems,
pages 43–50. ACM.
Dondi, R., Mauri, G., and Zoppis, I. (2016). Clique editing
to support case versus control discrimination. In Intel-
ligent Decision Technologies, pages 27–36. Springer.
Dondi, R., Mauri, G., and Zoppis, I. (2017). Orthology
correction for gene tree reconstruction: theoretical
and experimental results. Procedia computer science,
108:1115–1124.
He, X., Chen, T., Kan, M.-Y., and Chen, X. (2015). Tri-
rank: Review-aware explainable recommendation by
modeling aspects. In Proceedings of the 24th ACM In-
ternational on Conference on Information and Knowl-
edge Management, pages 1661–1670. ACM.
Heckel, R., Vlachos, M., Parnell, T., and D
¨
unner, C. (2017).
Scalable and interpretable product recommendations
via overlapping co-clustering. In Data Engineering
(ICDE), 2017 IEEE 33rd International Conference
on, pages 1033–1044. IEEE.
Komusiewicz, C. (2016). Multivariate algorithmics for find-
ing cohesive subnetworks. Algorithms, 9(1):21.
Lewis, H. R. and Papadimitriou, C. H. (1997). Elements of
the Theory of Computation. Prentice Hall PTR.
Mitchell, M. (1996). An introduction to genetic algorithms.
Complex adaptive systems. MIT press, Cambridge
(Mass.).
Papadimitriou, A., Symeonidis, P., and Manolopoulos, Y.
(2012). A generalized taxonomy of explanations
styles for traditional and social recommender systems.
Data Mining and Knowledge Discovery, 24(3):555–
583.
Quijano-Sanchez, L., Sauer, C., Recio-Garcia, J. A., and
Diaz-Agudo, B. (2017). Make it personal: a social ex-
planation system applied to group recommendations.
Expert Systems with Applications, 76:36–48.
Safe, M., Carballido, J., Ponzoni, I., and Brignole, N.
(2004). On stopping criteria for genetic algorithms.
Advances in Artificial Intelligence–SBIA 2004, pages
405–413.
Scrucca, L. (2013). GA: A package for genetic algorithms
in R. Journal of Statistical Software, 53(4):1–37.
Sharan, R. and Shamir, R. (2000). Center CLICK: A clus-
tering algorithm with applications to gene expression
analysis. In Proc. of the Eighth Int. Conf. on Intelli-
gent Syst. for Molecular Biol., August 19-23, 2000, La
Jolla / San Diego, CA, USA, pages 307–316.
Sharma, A. and Cosley, D. (2013). Do social explanations
work?: studying and modeling the effects of social ex-
planations in recommender systems. In Proceedings
of the 22nd international conference on World Wide
Web, pages 1133–1144. ACM.
Spirin, V. and Mirny, L. A. (2003). Protein complexes and
functional modules in molecular networks. Proceed-
ings of the National Academy of Sciences, 100:12123–
12–128.
Wang, X., He, X., Feng, F., Nie, L., and Chua, T.-S.
(2018). Tem: Tree-enhanced embedding model for
explainable recommendation. In Proceedings of the
2018 World Wide Web Conference on World Wide
Web, pages 1543–1552. International World Wide
Web Conferences Steering Committee.
Optimized Social Explanation for Educational Platforms
91