Optimizing Social Interaction
A Computational Approach to Support Patient Engagement
Italo Zoppis
1
, Riccardo Dondi
2
, Eugenio Santoro
4
,
Gianluca Castelnuovo
3
, Francesco Sicurello
1
and Giancarlo Mauri
1
1
Department of Computer Science, University of “Milano-Bicocca”, Milano, Italy
2
Department of Letters, Philosophy, Communication, University degli Studi di Bergamo, Bergamo, Italy
3
Department of Psychology, University “Cattolica del Sacro Cuore”, Milano, Italy
4
Laboratory of Medical Informatics, Department of Epidemiology, IRCCS, Mario Negri, Milano, Italy
Keywords:
Social Networks, Optimization, Cohesive Sub-Graphs, Genetic Algorithms.
Abstract:
Social media can directly support disease management by creating online spaces where patients can interact
with clinicians, and share experiences with other patients. Nevertheless, much more work remains to be carried
out for providing and sharing an optimized information content. In this paper we formulate, from a theoretical
perspective, an optimization problem aimed to encourage the creation of a sub-network of patients which,
being recently diagnosed, wish to deepen their knowledge about their pathologies with some other patients,
whose clinical profile turn to be similar, and have already been followed up within specific, even alternative,
care centers. We will focus on the hardness of the proposed problem and provide a Genetic Algorithm (GA-
based) approach to seek faster approximated solutions.
1 INTRODUCTION
The participatory, interactive nature of social media
platforms allows for information to be generated and
shared in a viral fashion, and provide new mecha-
nisms to foster engagement and partnership with users
and patients, to change their behaviors, and to fight
against unhealthy lifestyles.
Due to their possible implications in public health,
a growing number of scientists suggests to incor-
porate social media in health promotion and health
care programs (Burke-Garcia and Scally, 2014). So-
cial media can directly support disease management
by creating online spaces where patients can in-
teract with clinicians, and share experiences with
other patients (Coiera, 2013; Santoro et al., 2015).
For example, cancer patients use Twitter to discuss
treatments and provide psychological support (Tsuya
et al., 2014), and online engagement seems to corre-
late with lower levels of self reported stress and de-
pression (Beaudoin and Tao, 2008).
Similarly, wellness programs frequently incorpo-
rate social media to create a sense of community
(Zoppis et al., 2016), group people around shared
goals, and offer social and emotional support. A trial
reported that adding on line community features to an
Internet-mediated wellness and walking program im-
proves adherence, and did reduce participant attrition
(Richardson et al., 2010).
Nevertheless, much more work remains to be car-
ried out for sharing targeted and optimized informa-
tion content. How can we optimize a procedure which
is able to facilitate the encounter between patients
who want to deepen or share experiences about treat-
ments, care points, and specialists? How to correlate,
for example, similar clinical profiles, while inducing
networks of medical stuff, and treated patients which
offer their availability to share experiences or sugges-
tions? These are exactly the questions we try to an-
swer in this paper.
It is clear that a proper handling of data is funda-
mental in order to convert available social spaces into
useful sub-networks that leads to particular induced
communities. Here, we focus on the problem of cre-
ating a space of individuals and care centers, by con-
sidering the case where recently diagnosed patients
could be interested to meet some other patients (expe-
rience), for sharing information on their own disease
or about the suggested (or available) care center. In
this situation, it would be useful, for example, to en-
courage the diagnosed subjects to socialize, and con-
front with the experience of other patients of similar
clinical profile, who have been already followed up
within the same (or even alternative) proposed care
Zoppis I., Dondi R., Santoro E., Castelnuovo G., Sicurello F. and Mauri G.
Optimizing Social Interaction - A Computational Approach to Support Patient Engagement.
DOI: 10.5220/0006730606510657
In Proceedings of the 11th International Joint Conference on Biomedical Engineering Systems and Technologies (HEALTHINF 2018), pages 651-657
ISBN: 978-989-758-281-3
Copyright
c
2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
Figure 1: An example of a 2-club consisting of 5 vertices,
such that each subgraph of four vertices is not a 2-club. As-
sume that we remove vertex v
5
; then the graph induced by
{v
1
,v
2
,v
3
,v
4
} is not a 2-club (d
G
0
(v
1
,v
4
) = 3).
point.
In the next sections, we first introduce the main
theoretical aspects, focusing on the computational
hardness of the considered problem (Sections 3, 2.1,
and 2.2). Then, we discuss a GA-based approach
to seek faster approximated results to our questions
1
(Section 3). Finally, after reporting the numerical ex-
periments on simulated data, we conclude the paper
(Section 5) by describing future directions of this re-
search.
2 MAIN THEORETICAL
CONCEPTS
Suppose we wish to model the situation where re-
cently diagnosed patients (shortly reported as RD pa-
tients) are fostered to deepen the knowledge of their
diseases with some other patient experience (say, en-
gaged patients or shortly, ED), or they need more in-
formation concerning the suggested (or even alterna-
tive) health-care centers (HC). In this case, RD pa-
tients could benefit from the social interaction with
similarly profiled (ED) patients, which have already
been followed up by specific HC centers. To this aim,
it should be useful for a social platform to encourage,
and optimize, the creation of a sub-network from the
available social data.
From a theoretical perspective, a network is most
commonly modeled using a graph, e.g., (Bollob
´
as,
1998)) which represents relationships between ob-
jects, V (vertices), through a set of edges, E. In
this way, our goal can be formulated by maximiz-
ing, within a defined graph, a cohesive sub-graph (i.e.,
by seeking the largest cohesive sub-graph) with par-
ticular properties. Such structures (i.e., cohesive or
1
See (Mitchell, 1996) for details on genetic algorithms)
dense sub-networks) have been widely applied in sev-
eral contexts. In computational biology, e.g., dense
sub-graphs are sought in protein interaction networks,
as they are considered related to protein complexes
(Bader and Hogue, 2003; Spirin and Mirny, 2003),
and, in gene networks, dense sub-graphs are applied
to detect relevant co-expression clusters (Sharan and
Shamir, 2000).
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 pair-
wise connected by an edge. This definition of dense
sub-graph is often too stringent for particular need.
Indeed, some pair of elements may not be directly
connected in a dense sub-graph, for example due to
missing data that produce a dense sub-graph which
is not, currently, a clique. Alternative definitions of
cohesive subgraphs have been introduced, for exam-
ple by relaxing some constraint of the clique defini-
tion, leading to the concept of relaxed clique (Ko-
musiewicz, 2016).
In this paper, we focus on relaxing the distance
between vertices. In a clique distinct vertices are at
distance of 1, in our case, vertices have to be at dis-
tance 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).
When s = 1, a 1-club is exactly a clique. In Fig. 1 is
represented a 2-club consisting of 5 vertices. 2-clubs
have been extensively applied to social networks anal-
ysis (Mokken, 1979; Alba, 1973; Laan et al., 2016;
Mokken et al., 2016), and biological network anal-
ysis, e.g., protein-protein interaction networks, (Pa-
supuleti, 2008).
2.1 Problem Formulation
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
]
2
. 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,vV
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. We
2
Notice that all the graphs we consider are undirected.
will formulate the problem using 2-clubs whose short-
est path connecting RD patients with specialist cen-
ters/staff (e.g., care center, hospital, or clinical staff)
has to “transit” through, at least one EP patient who
has already been followed up by the considered spe-
cialist “point”.
Formally, for any pair, (d,h), composed by the re-
cently diagnosed patient, d, and, e.g., the health care
center, h, the social platform should suggest for the
patient d, to compare (even to meet) with an identi-
fied (available) patient x’s experience. More specifi-
cally, we are currently seeking (within the input “so-
cial graph”) a 2-Club, G[D X H], where D, X and
H represent the sets of recently diagnosed patients,
experienced patients, and care point centers. respec-
tively. Please note that, when such a structure (i.e.,
a maximum size 2-clubs) exists, within the identi-
fied starting social space, then for any pair of ver-
tices, 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 be also true for any pair, (d, h) where
d D,h H. Indeed, our goal will be to find the
largest-size 2-clubs which has the further property of
providing, for any pair (d,h), a shortest path charac-
terized by the triple of vertices (d,x,h) (D×X ×H).
In this case, the set of edges, modeling the starting so-
cial network, will be defined as follow.
Edges between similar profiled patients.
Edges expressing the fact that an experienced pa-
tient x, has already been properly followed up
from specialists or care centers, h. In this case
the edges in X × H will be constructed by know-
ing both the clinical history of each (experienced)
patient, x, and the clinical staff or hospital h which
has already properly followed up the patients, x.
Edges between two vertices h
1
,h
2
H, for
example because two care centers are similar
(have similar services or are part of the same
institution).
In this situation, the simple path given by the
triple of vertices (d,x,h) (D × X × H) in the
2-club G would suggest for patient d D to contact
the patient, x X , about the health care center (or
specialist staff), h H. For sake of clarity, before
defining computationally the problem, we refer
to any pair of vertices, (d,h) D × H (such that
the minimum path connecting d to h is given by
the vertex sequence (d, x,h), for any x X), as a
”feasible pair”. Considering the above discussion,
we can define the following variant of the 2-clubs
maximization problem
Problem 1. Maximum 2-Club (Max-2-club)
Input: a graph G = (D X H,R F).
Output: a set V
0
D X H such that G[V
0
] is a 2-
club having maximum size, and for each pair of ver-
tices (d,h) D × H in G[V
0
], a minimum path con-
necting d to h is given by the vertex sequence (d,x, h)
for some x X (i.e., (d,h) is feasible).
2.2 Computational Hardness
The complexity of the problem of Maximum s-club
has been extensively studied in literature, and unfortu-
nately it turns to be NP-hard for each s 1 (Bourjolly
et al., 2002); Maximum s-Club is NP-hard even if the
input graph has diameter s + 1, for each s 1 (Bala-
sundaram et al., 2005). The same property holds for
our variant of Maximum 2-club. Indeed, the compu-
tation of a 2-club of maximum size containing a spe-
cific vertex v is also NP-hard. By defining D = {v},
X = N(v) and H the remaining set of vertices, it fol-
lows that the ”feasibility” property holds.
Given an input graph G = (V,E), Maximum s-
club is not approximable within factor |V |
1/2ε
, for
any ε > 0 and s 2 (Asahiro et al., 2010). On the
positive side, polynomial-time approximation algo-
rithms (Asahiro et al., 2010) have been given, with
factor |V |
1/2
for every even s 2, and factor |V |
2/3
for every odd s 3. The parameterized complexity
of Maximum s-Club has also been studied, leading
to fixed-parameter algorithms (Sch
¨
afer et al., 2012;
Komusiewicz and Sorge, 2015; Chang et al., 2013).
Maximum 2-Club has been considered also for spe-
cific graph classes (Hartung et al., 2015; Golovach
et al., 2014).
3 A GENETIC ALGORITHM
The complexity of the problems introduced so far
make optimization potentially impracticable. For
this reason, we designed a Genetic Algorithm
(GA) to seek faster approximation solutions see,
e.g., (Mitchell, 1996) for details.
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, with the property
discussed above) as a binary chromosome c of size
n = |V |, whose ith component is defined as follows:
c[i] = 1, for all v
i
V
0
, else c[i] = 0. During the
offspring generation, chromosomes are interpreted as
hypotheses of feasible solutions, or they can even rep-
resent unfeasible solutions (e.g., s-club with s > 2,
disconnected graphs, or ”unfeasible pairs”, as defined
above), which can evolve into feasible, due to muta-
tion, cross-over, and selection. Moreover, hypotheses
(i.e., chromosomes) are evaluated through the fitness
function defined as follows
f (diam, n,m) =
(
n
2
+ m
2
if 0 diam 2 ;
1
(diam
2
+n
2
+m
2
)
if diam > 2 ,
(1)
where n, m, and diam are, respectively, the number
of vertices of the induced subgraph, G[D X H],
the number of its feasible pairs, (d, h) D × H, and
its diameter. The goal of the fitness function de-
scribed in Eq. 1 is to endorse new populations by
promoting those chromosomes which represent sub-
graphs with high number of vertices, high number of
feasible pairs, and diameter of length at most equal
to 2. Specifically, for any fixed diameter diam 2
the fitness grow proportionally to the number of ver-
tices and feasible pairs, thus promoting dense sub-
graphs. Instead, for any diam exceeding 2, the fit-
ness decreases asymptotically, penalizing in this way,
large subgraphs. Moreover, to allow a proper evolu-
tion (with regard to the Maximum 2-club problem),
we defined the following standard operators.
Mutation. Three type of mutations are considered.
Base Mutation. Similarly to the standard case,
each individual from the current population at
time i is modified with a given probability (see
details in the experimental section). In this
case, mutation flips a bit of the selected chro-
mosome c, in such a way that the correspond-
ing vertex is either removed (i.e., bit flipped to
0) or added (i.e., bit flipped to 1) to the solution
induced by c. Note that, deleting or adding ver-
tices may induce unfeasible sub-graphs, since
the property of being a 2-club is not “heredi-
tary”. On the other hand, such modifications
can introduce the chance to overcoming local
minimum.
Non Standard Mutation 1. In this case mu-
tation has the objective to correct hypotheses
(i.e., chromosomes) consistently and parsimo-
niously. Since any chromosome, by representa-
tion, induce a sub-graph G[V
0
] of G[V ], which
in turn may reflects feasible solutions, such hy-
potheses are verified using the following prin-
ciple. Given a selected chromosome c, a ver-
tex v
0
is (randomly) sampled from the set V
+
=
{v
i
: c[i] = 1} and the minimum length of sim-
ple paths connecting every pair (v
i
,v
0
),v
i
V
+
is checked to be consistent with the chromo-
some representation, i.e., since each chromo-
some “speculates” a feasible 2-club, for such
hypothesis to be true, there must be, at least, a
simple path of size at most equal to 2 connect-
ing any v
i
V
+
with v
0
. If a negative feedback
is observed after this verification, then the sam-
pled vertex v
0
is flipped to 0.
Non Standard Mutation 2. This modification
has the objective to increment (parsimoniously)
the size of a solution. In this case, given a
selected chromosome 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
) is checked to be consistent with the cur-
rent representation of the chromosome c. In
this case, we consider to extend the hypothesis
represented by c, by adding v
0
to V
+
if mini-
mum distances from v
0
to vertices of V
+
are not
larger than 2.
Cross-over. The following operations are pro-
vided.
Standard cross-over. Offspring is generated by
copying and mixing parts of parents’ chromo-
somes.
Logical AND between parents. This operation
has the objective to provide an offspring con-
sistent with the selected parents. For this, pairs
of chromosomes are generated through logical
AND operations between the ascendents.
Logical OR between parents. This operation
has the objective to provide offspring extend-
ing parent hypotheses. Extension is given by
realizing a logical OR operation between two
selected parents.
Elitist selection (or elitism). In order to guarantee
that solution quality does not decrease from one
generation to another (Baluja and Caruana, 1995),
best hypotheses (high fitness values) are allowed
to be part of a new offspring.
4 RESULTS
The genetic algorithm described in Sec. 3 was
coded in R using the Genetic Algorithm pack-
age (Scrucca, 2013) downloadable at https://cran.r-
project.org/web/packages/GA/index.html.
Results are given for synthetic data obtained
by generating Erdos-Renyi (ER) random graphs
ER(n, p) with two free parameters: the number of ver-
tices, n, of the input graph, and the probability, p, to
Table 1: Models (Erdos-Renyi). Input Diameter (InD), Output Diameter (OutD), Input Nodes (InN), Output Nodes (OutN),
Output Feasible Pairs (OutP), Number of Unfeasible Diameters (UnD), Number of Unfeasible Pairs (UnDH), Best Fit (Fit),
Ratio between input and output vertices (Rat.).
Model InD OutD InN OutN OutP UnD UnDH Fit Rat.
ER(45,1/5) 3.2 2 45 13.4 16.6 0 0 466.4 3.36
ER(30,1/5) 4 2 30 10.4 8 0 0 189.6 2.88
ER(15,1/5) 5 2 15 5.2 5 0 0 36.6 2.88
Table 2: CPU time for Models in Tab. 1. CPU User Time (T
1
), CPU System Time (T
2
), CPU Elapsed Time (T
3
) in seconds,
Early stopping with no improvement (Run), Max Number of Generation (Iter).
Model T1 T2 T3 Run Iter
ER(45,1/5) 14658.28 10.132 14696.83 180 700
ER(30,1/5) 9965442 5.4 10003.26 180 700
ER(15,1/5) 5966.29 4.794 5988.46 180 700
Table 3: Models (Erdos-Renyi). In this case results are not averaged.
Model InD OutD InN OutN OutP UnD UnDH Fit Rat.
ER(60,1/5) 3 2 60 15 14 0 0 421 4.00
ER(21,1/10) 8 2 21 8 3 0 0 45 2.63
ER(9,1/10) 4 2 9 6 6 0 0 72 1.50
ER(21,1/2) 3 2 21 14 18 0 0 520 1.50
ER(9,1/2) 9 2 9 6 4 0 0 52 1.50
Table 4: CPU time for Models in Tab. 3
Model T1 T2 T3 Run Iter
ER(60,1/5) 11169.94 8.69 11188.91 180 700
ER(21,1/10) 4971.67 3.91 4980.56 180 700
ER(9,1/10) 4981.36 3.06 5022.58 180 700
ER(21,1/2) 3832.23 3.49 3836.93 180 700
ER(9,1/2) 4707.55 3.1 4713.28 180 700
create edges between two vertices (Bollobas, 2001).
Numerical experiments have the main objective to ob-
tain, as reported above, feasible solutions which have
the further property that, for any pair (d,h) of RD pa-
tient, d, and HC point, h, at least one ED patient is
provided to make his experience available.
In all the experiments we applied a number, n,
of vertices ranging in {9,15, 21,30, 45,60}, while
the probability to create an edge is chosen in
{1/2,1/5,1/10}. Moreover, we randomly labeled
n/3 vertices as ED, n/3 as HC, and finally n/3 as AD
vertices. Tables 1, 2, 3 and 4 report the performances
of the system. First we executed the GAs iteratively
by sampling the corresponding random model (i.e.,
5 observations for each model), and the performance
was averaged on the whole set of experiments (Tab. 1
and 2). The following attributes are reported.
Input and output diameters. Graphs are repre-
sented as discussed in Sec. 3. The best GA so-
lution is re-coded and the resulting diameter is re-
ported (output diameter).
Number of Input and output vertices.
Number of (Final) Feasible Pairs. The resulting
number of feasible pairs, (d,h), in the output (2-
club) graph.
Number of Unfeasible Solutions (Diameters). To-
tal number of graphs (i.e., experiments) whose fi-
nal diameters have dimension greater than 2 (after
running the whole set of experiments).
Number of Unfeasible Solutions (D-H Pairs). To-
tal number of graphs (i.e., experiments) where at
least one pair (d, h) does not provide one ED pa-
tient able to make his experience available.
Fitness value. Fitness as described in Sec. 3.
Ratio between the number of the input graph ver-
tices and the number of vertices of the resulting 2-
club.
CPU User Time, CPU System Time, and CPU
Elapsed Time in seconds.
Early stopping for no improvement. The number
of consecutive generations without improvement
in the best fitness value before the GA is stopped.
Max Number of Generation. The maximum num-
ber of iterations to run before the GA search is
halted.
Another set of experiments was executed without
repetitions by using different values of free parame-
ters for each input graph. Tables 3, 4 reports the ob-
tained performances. The following main considera-
tions emerge from the results.
All models effectively provide feasible 2-clubs
with at least one experienced patient for each pair,
(h,d), considered in the final solution.
The models are able to find combinatorial struc-
tures which actually requires impractical compu-
tational time. This is the case of solutions given
for input graph with high number of vertices (e.g.,
more of 40 vertices). In order to give an idea of
the quality of the returned solutions, we have con-
sidered the ratio between the vertices of the in-
put graphs and of the output graphs. Indeed, due
to the computational hardness of the problem, we
cannot compare the size of the subgraph returned
by the GAs with the size of an optimal solution
for the Maximum 2-club problem. Notice that the
approximation complexity results for Maximum
2-club shows the problem is not approximable
within factor |V |
1/2ε
, for each ε > 0 (Asahiro
et al., 2010), thus the approximability is very hard
to obtain. For this reason, we can say that, solu-
tions which offer a ratio smaller than or close to
two, are effectively compelling and interesting.
5 CONCLUSION
In this paper we focused on the problem of optimiz-
ing the creation of a sub-network of patients aimed to
deepen the knowledge about the available care centers
for their pathologies through the help of other ”expe-
rienced” patients. We considered this problem from a
computational point of view by defining a variant of
the max 2-club problem.
The intrinsic complexity of the introduced formu-
lation requires the use of heuristic algorithms to ob-
tain feasible approximated solutions in a reasonable
time. We showed that the proposed approach (GA-
based) effectively provides empirical approximations
able to find feasible structures (i.e., 2-clubs), which
actually requires impractical computational time. In
fact, while GAs optimization is not new in literature,
a new design of these models is now needed to cope
with the hardness of many computational problems
which actually find new applications in many contexts
(Dondi et al., 2017; Dondi et al., 2016).
From our results, it seems to emerge the possibil-
ity of extending this research using real data sets with
larger instance’s dimension. Moreover, a convergence
analysis, and the use of tuning methods to optimize
some free GAs parameter (e.g., probability values for
choosing the available mutation or cross over opera-
tions, or even the use of alternative parameterized fit-
ness functions) will be one of the future direction for
this research.
Finally, it is important to emphasize that the
framework described in this paper has to be consid-
ered, as discussed in Introduction, a tool to facilitate
and promote the patient engagement. It is not clearly
intended as an instrumentation to constrain the spon-
taneous nature of communication and interaction in a
social network.
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