Optimizing Social Interaction

A Computational Approach to Support Patient Engagement

Italo Zoppis

, Riccardo Dondi

, Eugenio Santoro

Gianluca Castelnuovo

, Francesco Sicurello

and Giancarlo Mauri

Department of Computer Science, University of “Milano-Bicocca”, Milano, Italy

Department of Letters, Philosophy, Communication, University degli Studi di Bergamo, Bergamo, Italy

Department of Psychology, University “Cattolica del Sacro Cuore”, Milano, Italy

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

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

; then the graph induced by

} is not a 2-club (d

) = 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

(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

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

⊆ V .

We denote by G[V

] the subgraph of G induced by V

Formally G[V

] = (V

), where

= {{u,v} : u,v ∈ V

∧ {u,v} ∈ E}.

Given a set V

⊆ V , we say that V

induces the graph

G[V

]

. The distance d

(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

(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

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

∈ 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

⊆ D ∪ X ∪ H such that G[V

] is a 2-

club having maximum size, and for each pair of ver-

tices (d,h) ∈ D × H in G[V

], 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 |

1/2−ε

, 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

⊆ V

such that G[V

] 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

∈ V

, 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) =

(

+ m

if 0 ≤ diam ≤ 2 ;

(diam

)

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

] 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

is (randomly) sampled from the set V

: c[i] = 1} and the minimum length of sim-

ple paths connecting every pair (v

),v

∈ 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

∈ V

with v

. If a negative feedback

is observed after this veriﬁcation, then the sam-

pled vertex v

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

is sampled

from V

−

= {v

: c[ j] = 0} and the minimum

length of simple paths connecting every pair

) 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

to V

if mini-

mum distances from v

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

), CPU System Time (T

), CPU Elapsed Time (T

) 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 |

1/2−ε

, 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 GA’s 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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