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J Ambient Intell Human Comput (2017) 8:315–328 DOI 10.1007/s12652-017-0447-y
ORIGINAL RESEARCH
The emergence of inclusive and exclusive virtual communities determined by the preferences of their users
Debora Di Caprio1,2 · Francisco J. Santos‑Arteaga3,4 · Madjid Tavana5,6
Received: 10 August 2016 / Accepted: 20 November 2016 / Published online: 15 February 2017 © Springer-Verlag Berlin Heidelberg 2017
which to build inclusive and exclusive social networks determined by the different expectations and preferences of their users. Social networks are generated using a self-organizing map to cluster the decision makers (DMs) by their friendship acceptance behavior. We analyze the effects on the cluster structure of the resulting social network that follow from modifying the distribution of requesters rela-tive to the preferences of the DMs, the disutility derived from accepting the friendship of an unwanted requester, the costs incurred when searching for potential friends to expand the network of connections, and the minimum net-working capacities of the friendship requesters demanded by the DMs.
Keywords Virtual communities · Expected utility · Preference similarity · Self-organizing map · Social networks
1 Motivation and literature review
The emergence of social media has led to a substantial increase in the amount of personal information available about their users (Adamic and Adar 2003), leading other media users and companies to use this information stra-tegically (Stefanone et al. 2015). At the same time, social media research focuses on identifying the main factors determining the structure of already existing networks, while acknowledging the existence of different types of users in terms of their networking capacities and influence on other users (Kaptein et al. 2010; Guo et al. 2015; Klein et al. 2015).
In particular, the decision to adopt a given social net-work site (SNS) is determined by the will of its users to connect to others by creating valuable content that
Abstract Consider the decision faced by the user of a social network site (SNS) regarding whether or not to accept a friendship request from another user. The user making such a decision is constrained by the limited amount of information available about the requester. There-fore, the decision must be based on incomplete information about the main characteristics and preferences describing the requester. We formalize this decision problem by defin-ing the expected utility tradeoffs derived from the request and simulate the resulting acceptance and rejection incen-tives numerically. These incentives provide the basis on
* Francisco J. Santos-Arteaga [email protected]; [email protected]
Debora Di Caprio [email protected]; [email protected]
Madjid Tavana [email protected] http://tavana.us/
1 Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
2 Polo Tecnologico IISS G. Galilei, Via Cadorna 14, 39100 Bolzano, Italy
3 School of Economics and Management, Free University of Bolzano, Piazza Università 1, 39100 Bolzano, Italy
4 Instituto Complutense de Estudios Internacionales, Universidad Complutense de Madrid, Finca Mas Ferré, Edificio A, Campus de Somosaguas, Pozuelo de Alarcón, 28223 Madrid, Spain
5 Business Systems and Analytics Department, Distinguished Chair of Business Systems and Analytics, La Salle University, Philadelphia, PA 19141, USA
6 Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, 33098 Paderborn, Germany
316 D. Di Caprio et al.
1 3
involves other users. Recent models analyzing the inten-tion of decision makers to use a SNS are generally based on the technology acceptance model (TAM). For exam-ple, Qin et al. (2011) studied the acceptance of online social networks by incorporating the effects of social influence, particularly subjective norm and critical mass, into an enhanced version of the TAM. Similarly, Choi and Chung (2013) developed an extended TAM, incorpo-rating subjective norm and perceived social capital, for predicting the acceptance and usage of SNSs.
Błachnio et al. (2013) reviewed the academic litera-ture on the use of Facebook, the most studied SNS. They found that the analysis of personality and individual dif-ferences among users, together with their motivation to become users of a SNS, constituted the main research topics analyzed in the literature. In this regard, Gianna-kos et al. (2013) asked users to generate phrases describ-ing the way they used a SNS such as Facebook and their perception of these uses. These phrases were clustered into four empirically validated factors: social connec-tion, social network surfing, wasting time and using applications.
Note that, beyond the merely recreational incentives defining the latter factors, the creation of a network of friends requires a formal decision structure that accounts for the preferences and expectations driving the behavior of the users. That is, the decision structure determining the acceptance of friendship requesters must reflect the differ-ences in tastes and incentives across users when defining the evolution of the network. For example, Nadkarni and Hofmann (2012) suggested that Facebook users are mainly motivated by the need to belong and the need for self-pres-entation, with demographic and cultural factors determin-ing the former, while narcissism, shyness, and self-esteem contribute to the latter. Similarly, Błachnio et al. (2016) showed that loneliness and self-promotion are positive pre-dictors of Facebook usage with the need for privacy consti-tuting a negative one.
In the current paper, we consider the above character-istics of social media and their users but take a different research route from that of the existing literature. That is, we formalize the acceptance or rejection decision faced by a decision maker (DM) when receiving a friendship request from another SNS user. The DM has to decide whether to accept the friendship request and generate a link expand-ing his network, or reject it and either find a more suita-ble requester or remain with his current set of friends. In this regard, our novel friendship acceptance (or rejection) model relates to the basic postulates of expected-utility-based decision theory (Kahneman and Tversky 2000), where a DM makes a decision considering the highest expected utility attainable at a given point in time (Tavana et al. 2016a, b).
The decision environment determining the behavior of a DM can be summarized as follows.
1. When receiving a friendship request, some basic but important information becomes available to the DM, indicating the main preferences (i.e. likes, pages fol-lowed) of the person requesting his friendship.
2. If the request is accepted, additional secondary infor-mation becomes available, which can be used by the DM to complete his profile of the requester. Conse-quently, we will assume that the initial information provided to the DM is correlated with the secondary one and, therefore, conditions its expected realization.
3. Finally, the capacity of the requester to increase the network of friends of the DM must also be considered. This capacity should be determined by the connections of the requester and their similarity in preferences with the DM. As a result, the initial (observed) and second-ary (expected) characteristics can be used by the DM to determine the expected networking capacity of the requester.
We formalize this decision environment by defining the expected utility tradeoffs derived from a friendship request and simulate the acceptance incentives of the DM numeri-cally. The resulting decision framework constitutes the base on which to build the corresponding network structures, which will be determined by the different expectations and preferences of its users. In particular, social networks will be generated using a self-organizing map (Kohonen 2001) to cluster the DMs by their friendship acceptance behavior, which, at the same time, is determined by:
• the distribution of the characteristics of the requesters relative to the preferences of the DMs,
• the disutility derived from accepting the friendship of an unwanted requester,
• the costs incurred by the DM when searching for poten-tial friends to expand his network of connections.
We illustrate how the differences between the subjective beliefs used by the DM to define his expectations and the distribution of characteristics across requesters condition the formation of clusters in the resulting network. In addi-tion, we analyze the different clustered structures that result from modifications in the disutility and search costs faced by the DMs. Among the main results obtained, we show how increments in search costs induce a more inclusive behavior among DMs, while higher disutility costs lead to more exclusive environments.
As a result, we conclude that virtual communities with a large base of requesters, i.e. lower search costs, will tend to be more exclusive. Moreover, these communities
317The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
will increase the number of rejections if their mem-bers face high disutility costs from accepting unwanted requesters with substantially different preferences. On the other hand, we will illustrate numerically that the inclu-sive effect induced by higher search costs dominates the exclusive one implied by increments in disutility costs within the framework described in the paper.
Finally, given the importance assigned by the empiri-cal literature to the social networking capacity of the users of SNSs, we will analyze the consequences derived from modifying the networking capacities of the requesters demanded by the DMs. We will illustrate that the DMs who do not prioritize networking obtain a higher utility from their social interactions, while those requiring a large amount of potential contacts from the requester derive a lower utility. However, the friend-ship acceptance behavior of both types of DMs is almost identical.
It should be emphasized that our friendship accept-ance model can be modified to account for the acceptance of new technologies, as has been done in Tavana et al. (2016c). This potential extension relates the current deci-sion environment to the TAM models on which the incen-tives of a DM to become a user of a given SNS are based. It also illustrates the main difference between the stand-ard approaches developed in the literature and the current model, where the characteristics and beliefs of the DMs determine the structure of the resulting social network.
The rest of the paper proceeds as follows. Sections 2 describes the variables and assumptions that will be used to build the accept and reject functions. We introduce these functions in Sect. 3 and 4, respectively. Section 5 studies the clusters of DMs defined in terms of their acceptance behavior, while Sect. 6 analyzes the different clustered structures that result from modifying the disutility and search costs faced by the DMs. Section 7 describes the main consequences derived from modifying the require-ments of the DMs on the networking capacity of the requesters. Section 8 presents some concluding remarks and suggests potential extensions.
2 Basic assumptions
In this section, we describe the main variables determining the friendship acceptance behavior of the DMs. The accept and reject functions designed throughout the next two sec-tions define the novel decision theoretical environment on which different social networks and their corresponding clusters will be built.
The choice made by the DM regarding the friendship request depends on the following variables:
• X1 = [xm1, xM
1]:The characteristics/preferences of the
requester directly observable when receiving a friend-ship request. This variable accounts for publicly avail-able information that describes the main basic tastes of (likes displayed by) the requester. The realization observed is related to the remaining information, which is unavailable at the moment of the request together with the list of friends and, therefore, the networking capacity of the requester.
• X2 = [xm2, xM
2]:The characteristics/preferences of the
requester that become observable after accepting the friendship request. This variable allows the DM to obtain additional information regarding both the tastes of the requester as well as his potential networking capacity. Thus, the distribution of this variable is related to and influenced by the realization of X1, while both X1 and X2 affect and determine the potential networking capacity of the requester consistent with the preferences of the DM.
• X3 = [0, 1]:This characteristic reflects the networking capacity of the requester. The shape of its associated probability function is determined by the realizations of both X1 and X2. It should be noted that the friends of a given social media user can be classified in different cat-egories, with access to different levels of information. However, even if not allowed to access the whole net-work, the DM becomes part of the group of friends of the requester. That is, even though the DM may not have the same status as other friends, who may be used to expand his network but are classified in a different cat-egory by the requester, he may still benefit from the fact that those potential network friends can actually observe him.
The acceptance decision of the DM will therefore be determined by two incentive functions defining the expected utility derived from either accepting a given friendship request or rejecting it. If the DM rejects the request, he must consider the probability of improving upon the current request in the future and compute the cor-responding expected utility that would be derived. At the same time, in order for the DM to actually make a decision, both these functions must be determined by the values of all the potential realizations of X1 that may be observed.
3 Accepting the request
As stated in the previous sections, the information avail-able to the DM when deciding whether to accept the friend-ship request or reject it is limited to the initial observation of X1. If the DM decides to accept the friendship request,
318 D. Di Caprio et al.
1 3
his utility function, in expected terms, will be defined as follows
The following definitions and notations are required to interpret the above equation.
• The utility functions considered through the paper are given by u1
(x1)= x1, u2
(x2)= x2 and
u(x1, x2, x3
)=(x1 + x2
)x3. Note that the first two char-
acteristics are additively separable while the third one is used to generate the expected payoff obtained by the DM based on the networking capacities of his new potential friend.
• �i
(xi)=
1
xMi−xm
i
for xi ∈[xmi, xM
i
], i = 1, 2. The complete
uncertainty of the DM regarding the distribution of potential friends within the population is reflected using uniform density functions, which are endowed with the maximum information entropy value.
• Following the standard economic theory of choice under uncertainty, we assume that the DM elicits the i-th certainty equivalent (CE) value induced by �i(xi) and ui(xi) as the reference point against which to com-pare both the observed and potential characteristics of a requester. Given i = 1, 2, the certainty equivalent of �i and ui, denoted by cei, is a characteristic in Xi that the DM is indifferent to accept in place of the expected one to be obtained through �i and ui.That is, for every i = 1, 2, cei = u−1
i(Ei), where Ei denotes the expected
value of ui. The continuity and strict increasingness of ui can be used to guarantee the existence and uniqueness of the i − th CE value, respectively.
• A direct (subjective) correlation will be defined between the initial realization observed and the expected realiza-tion of the second characteristic of the requester. In this regard, the subjective probability defined by the DM on the set of potential realizations of the second character-istic should become an increasing function of the first characteristic observed. Given the uniform probability assumed on the second characteristic space, its condi-tional density will be defined as follows: with � ∈ [0, 1] weighting the strength of the shift in the mass between the lower and the upper interval domains. Note that if
(1)
Accept; =
1
∫0
xM2
∫x∗2
B3
(x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
�2
(x2|x1
)u(x1, x2, x3
)dx2dx3
+
1
∫0
x∗2
∫xm2
[B3
(x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
�2
(x2|x1
){u(x1, x2, x3
)− c
(x1, x2
)}]dx2dx3.
the characteristic observed is equal to the CE value, there is no shift in mass between the intervals of the dis-tribution. The above definition makes extensive use of the symmetry existing in the risk neutral (linear) case between the CE value and the extremes of the inter-val domain on which x1 is defined. In this case, the CE value is located exactly in the middle of the domain. If this were not the case, the density function should be modified accordingly and the resulting shift in mass between both intervals adapted to the location of the CE value.
• Given the initial realization of x1 obtained from the requester, denoted by xo
1, the minimum value of x2
required by the DM to derive an above-CE expected utility from the new friendship is given by x∗
2, with
x∗2= ce1 + ce2 − xo
1. Whenever x2 < x∗
2, the DM suffers
a disutility of c(x1, x2
) from accepting the friendship
of a requester whose tastes and characteristics differ significantly from his own. These disutility costs may include those derived from any undesired communica-tion or further friendship requests from the network of the requester, together with potential negative effects on the friends from the network to which the DM already belongs.
• B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
) is a Beta
density function that will be used to represent the degree of optimism or pessimism of the DM regarding the net-working capacity of the requester. The density function corresponds to that of the standard Beta distribution but its parameters are defined by: Formally, we have the fol-lowing definition of the Beta density for 0 ≤ x3 ≤ 1
where u−12
(E(x2|�2
(x2|x1
)))= u
−12
(∫ xM
2
xm
2
�2(x2|x1)
u2(x2)dx2
) is the expected realization obtained for the
second characteristic of the requester given the one ini-tially observed.
– The value of x1 together with that of the secondary characteristic expected to be observed, which is deter-mined by �2(x2|x1).
(2)
𝜇2(x2|x1) =1
xM2− xm
2
+ 𝜑
(x1 − ce1
xM1− ce1
)1
xM2− xm
2
if x2 <xM2+ xm
2
2
𝜇2(x2|x1) =1
xM2− xm
2
− 𝜑
(x1 − ce1
xM1− ce1
)1
xM2− xm
2
if x2 ≤ xM2− xm
2
2,
(3)B3(x3; x1 + u
−12(E(x2|�2(x2|x1))), ce1 + ce2)
=xx1+u
−12(E(x2|�2(x2|x1)))−1
3(1 − x3)
ce1+ce2−1
∫ 1
0ux1+u
−12(E(x2|�2(x2|x1)))−1(1 − u)ce1+ce2−1du
,
319The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
– The corresponding CEs, used as reference values for the evaluation performed on the information retrieved from the requester.
The potential capacity of the requester to network the DM with other similar users is based on both the ini-tially observed and the secondary expected character-istic. Note that the shape of the functions �2(x2|x1) and B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
) is deter-
mined by the initial realization of x1. In this regard, Fig. 1 illustrates the B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
density functions considered by the DM for different realiza-tions of x1.
In order to simplify the computations and account for the limited capacity of DMs to assimilate and manage infor-mation (Simon 1997; Samiee et al. 2005), we will consider different potential networking intervals defined by the DM in terms of the value of the initial characteristic observed. That is, similarly to the use of membership functions when defining a fuzzy variable over the interval domain of a set of potential alternatives, we have defined a set of Beta functions based on the potential realization intervals delimited within the domain of the first characteristic, i.e. X1 = [5, 10]. These intervals are described in Fig. 1a, while the correspond-ing Beta functions defined for the x1 reference value of each piecewise interval are presented in Fig. 1b.
It should be noted that Fig. 1 has been introduced to high-light the malleability of the Beta distribution and its capac-ity to account for the networking requirements of the DMs in a natural way. In this regard, the expected utility derived from the networking capacity of the requesters can be eas-ily adjusted to the number of potential contacts that the DM requires from a given friendship.
4 Rejecting the request
The rejection payoff is determined by the following expression
• In the current setting, s(x1, x2
) denotes the search
costs from observing the first characteristic of a new requester, which are incurred by the DM after reject-ing the initial request.
• The first term of Eq. (4) represents the expected payoff derived if the new requester provides a higher expected utility than the initial and the CE-based one. The expres-sion is based on the realization of x1 obtained from the previous requester, xo
1, given the subjective distributions
assigned by the DM to the X1 and X2 variables.• The second term of the equation represents the
expected payoff derived if the new requester provides (4)
Reject =
1
∫0
xM
2
∫x∗2
xM
1
∫xo
1
[B3
(x3;x1 + u
−12
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
�2
(x2|x1
)�1
(x1
){u(x1, x2, x3
)− s
(x1, x2
)}]dx1dx2dx3
+
1
∫0
x∗2
∫xm
2
xM
1
∫xo
1
[B3
(x3;x1 + u
−12
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
�2
(x2|x1
)�1
(x1
){u(x1, x2, x3
)− c
(x1, x2
)− s
(x1, x2
)}]dx1dx2dx3
−
1
∫0
xM
2
∫xm
2
xo
1
∫xm
1
B3
(x3;x1 + u
−12
(E(x2|�2
(x2|x1
))), ce1 + ce2
)
�2
(x2|x1
)�1
(x1
)sc(x1
)dx1dx2dx3.
an expected utility higher than the initial requester but lower than the CE-based one. As in the acceptance setting, c
(x1, x2
) denotes the disutility from accepting
the friendship of a requester whose tastes and char-acteristics differ from those of the DM. In this case, this disutility adds to the search costs incurred when observing the characteristics of a new requester.
• The last term accounts for the search costs sc(x1)
incurred when the first characteristic of the new requester is lower than xo
1. The new request is rejected
and the costs incurred by the DM are assumed to account for the new search together with the disutil-ity derived from not increasing his network of con-nections. Throughout the paper, we will assume that sc(x1)= s
(x1, x2
)+ 1. It should be noted that the main
results obtained are not affected by this simplifying assumption.
The accept and reject functions, together with the resulting acceptance and rejection intervals generated for X1 = [5, 10], X2 = [0, 10], c
(x1, x2
)= s
(x1, x2
)= 0 and
sc(x1)= 1, are represented in Fig. 2. The choice of the
domains for the first and second characteristics empha-sizes the relatively higher importance assigned to the first one by the DM, since it leads to a higher expected utility, and the larger potential spread following from the second one. It could also be assumed that both charac-teristics are defined on [0, 10] but DMs do not consider requesters whose first characteristic is located within the [0, 5] interval. The main qualitative results obtained in the current paper do not depend on the domains on which the characteristics are defined, though the dominance of the first characteristic over the second one constitutes an intuitively reasonable assumption.
Note that we have used the x1 reference value of each piecewise interval generated by the set of Beta functions to define a continuous approximation to the accept func-tion. Clearly, different types of approximations could be
320 D. Di Caprio et al.
1 3
used to define this function though, as can be inferred from the figure, their effect on the subsequent choices made by the DM would not be significant. We should also emphasize that we will analyze the effects derived from modifying the different search and disutility costs on the acceptance behavior of the DM. This analysis will allow us to study the different types of network structures, rang-ing from more inclusive to more exclusive ones, arising from costs differential across DMs.
5 Clustering DMs through self‑organizing maps
In order to generate a network structure where DMs are clustered into different groups by a self-organizing map (see Kohonen (2001) and Sulkava et al. (2015) for a detailed description of the main features of this type of neural network), we consider differences in the distribu-tion of friendship requests in terms of the first characteris-tic observed by the DM. That is, differences in preferences are introduced between the requesters and the DM, who assumes a uniform distribution on the set of requesters due to his uncertainty regarding the distribution of potential friends within the population. We do so by defining a Beta (x1; 2, 4
) and a Beta
(x1; 4, 2
) distribution on the reali-
zations of x1 and accounting for the number of friendship acceptances when 25, 50, 75 and 100 randomly generated requests are received by 100 different DMs.
In other words, we generate a total of 100 DMs per Beta distribution and a given set of parameters. Each one of these DMs is defined by four different realizations stating the number of friendship requests accepted out of a total of 25, 50, 75 and 100 randomly generated ones using a given Beta distribution. Then, given the population of 100 DMs generated per each Beta, we implement a self-organizing map to cluster them in terms of their acceptance behavior. As we will illustrate later on, self-organizing maps allow us to create clustered structures combining populations of DMs generated by different Beta functions.
The threshold value x∗1= 6.54 determining the friend-
ship acceptance behavior of a DM when defined over the [5, 10] domain has been transformed into x∗
1= 0.308 when
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10x1
B 3(x
1,E
(x2|
2(x
2|x
1),
ce1+c
e 2)
B3
(6+E(x2
|2
(x2
|6), )
B3
(5+E(x2
|2
(x2
|5), )
B3
(7+E(x2
|2
(x2
|7), )
B3
(8+E(x2
|2
(x2
|8), )
B3
(9+E(x2
|2
(x2
|9), )
B3
(10+E(x2
|2
(x2
|10), )
(a) Interval values of 1x realizations associated with each
3 3 1 2 2 2 1 1 21
2( ; ( | ( | ))),( )B x x E x x x cu e ce−+ µ +
µ
density.
(b) 3 3 1 2 2 2 1 1 21
2( ; ( | ( | ))),( )B x x E x x x cu e ce−+ µ +density functions for 1 5,6,7,8,9,10=x .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Bet
a D
ensi
ty F
unct
ion
x1=5
x1=6
x1=7
x1=8
x1=9
x1=10
Fig. 1 B3
(x3; x1 + u
−12
(E(x2|�2
(x2|x1
))), ce1 + ce2
) defined by
a DM for the set of potential realizations of x1 when X1 = [5, 10] and X2 = [0, 10]. a Interval values of x1 realizations associated with each B3
(x3; x1 + u
−12
(E(x2|�2
(x2|x1
))), ce1 + ce2
) density.
bB3(x3;x1 + u−12(E(x2|�2(x2|x1))), ce1 + ce2) density functions for
x1 = 5, 6, 7, 8, 9, 10
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10x1
0
2
4
6
8
10
Acc
ept a
nd R
ejec
t Fun
ctio
ns
REJECT ACCEPT
Fig. 2 Accept and reject functions with their corresponding intervals defined by x∗
1= 6.54
321The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
considering the [0, 1] domain on which the Beta distribu-tion is defined.
Figure 3 provides intuition on the acceptance behavior of the DMs based on the distribution of requesters (rela-tive to the preferences of the DM) and the location of the threshold value. Consider the c
(x1, x2
)= s
(x1, x2
)= 0
scenario. Clearly, whenever requesters have similar pref-erences to those of the DM, as is the case in the Beta (x1; 4, 2
) setting, most friendship requests will be accepted.
The behavior of DMs in this case will be quite uniform, with outliers accepting a relatively lower or higher num-ber or requests than the others. In the same way, whenever requesters have dissimilar preferences to those of the DM, as is the case in the Beta
(x1; 2, 4
) setting, a less uniform
behavior would be expected across DMs.
The resulting dispersion in the acceptance behavior of DMs can be observed in Fig. 4, where we plot the number of friendship acceptances when 50, 75 and 100 requests are received by the DMs. Consequently, we should expect to observe two differentiated clusters when implementing a self-organizing map algorithm to categorize the DMs in terms of their acceptance behavior:
• a more cohesive one corresponding to the uniform acceptance of requesters with similar preferences to those of the DM;
• a less cohesive one corresponding to the mixed accept-ance of requesters with dissimilar preferences to those of the DM.
Fig. 3 Beta (x1; 2, 4) and Beta (x1; 4, 2) distributions generating the sets of friendship requesters
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X1
0
0.5
1
1.5
2
2.5
Bet
a de
nsity
Beta(2,4)Beta(4,2)
s=c=2 s=c=0 s=0; c=2
Fig. 4 Acceptance dispersion based on the Beta density and disutility costs considered
5045
4035
50 x 1 obs
302025
30
80 70
40
2060
75 x1 obs
50
50 15
60
40
100
x 1 o
bs
30
70
20 10
80
10
90
100
Beta (2, 4) - c
Beta (2, 4)
Beta (4, 2) - c
Beta (4, 2)
322 D. Di Caprio et al.
1 3
The results obtained after applying a self-organizing map algorithm to the number of friendship acceptances when the set of requesters follows a Beta
(x1; 2, 4
) and a
Beta (x1; 4, 2
) distribution are presented in Figs. 5 and 6,
respectively. As expected, given the location of the thresh-old value, the Beta
(x1; 2, 4
) scenario leads to a more dis-
perse set of weights than the Beta (x1; 4, 2
) one. In both
cases, we observe a central set of weights surrounded by several isolated nodes. In the Beta
(x1; 2, 4
) scenario, these
nodes may correspond to those DMs accepting either a larger or a lower number of friendship requests.
In order to gain additional intuition on whether the requesters described within this latter cluster are being gen-erally accepted or rejected, we use both sets of requesters to generate the self-organizing map described in Fig. 7. This figure illustrates both clusters of DMs differenti-ated by their friendship acceptance behavior. Clearly, the more cohesive cluster defined on the upper right corner of the weight plane describes the generalized acceptance of requesters with similar preferences to those of the DMs. The more dispersed cluster on the lower left corner of the
plane illustrates the less cohesive acceptance behavior to which the Beta
(x1;2, 4
) requesters are subject.
Given the acceptance and rejection incentives determin-ing the x∗
1 threshold values and the subsequent behavior of
the DMs, modifications in the disutility and search costs to which DMs are subject will have a direct effect on the type of clustered structures generated. We analyze these poten-tial scenarios in the following section.
6 Inclusive and exclusive virtual worlds
The current section studies the different clustered structures resulting from modifications in the disutility and search costs faced by the DMs. We must emphasize that observing a more cohesive cluster does not necessarily imply a higher friendship acceptance rate among the DMs. For example, given the location of the c
(x1, x2
)= s
(x1, x2
)= 0 threshold
value, rejecting a larger amount of Beta (x1; 2, 4
) request-
ers would actually result in a more cohesive cluster.
-1 0 1 2 3 4 5-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
SOM Neighbor Weight Distances
-1 0 1 2 3 4 5-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Hits
6 1 3 3 4
5 5 5 3 1
5 4 6 8 7
2 7 4 4 5
1 2 2 3 4
5 10 15 20Weight 1
18
20
22
24
26
28
30
32
34
Wei
ght 2
SOM Weight Positions
Fig. 5 Self-organizing map clusters following from a Beta (x1; 2, 4
) distribution of requesters
-1 0 1 2 3 4 5-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
SOM Neighbor Weight Distances
-1 0 1 2 3 4 5-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Hits
3 1 4 2 2
2 3 8 5 6
4 5 7 3 5
6 6 6 6 3
1 6 2 1 3
23 24 25Weight 1
44
45
46
47
48
49
50
Wei
ght 2
SOM Weight Positions
Fig. 6 Self-organizing map clusters following from a Beta (x1; 4, 2
) distribution of requesters
323The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
It should also be highlighted that the empirical literature has studied the effects of disutility costs on the behavior of SNS users. Oldmeadow et al. (2013) found that users with high attachment anxiety needing reassurance used Face-book more frequently and were more concerned about how other users perceived them. As a result, this type of users, who derive a higher disutility from negative comments, become less willing to accept the friendship of unknown requesters.
With these remarks in mind, we start by considering an increase in the disutility costs faced by DMs when accept-ing an unwanted friendship request. That is, assume that the value of c
(x1, x2
) increases from zero to two. The resulting
accept and reject functions (in orange and dashed), together with their acceptance and rejection areas, are presented in Fig. 8. In order to provide additional intuition, this figure also illustrates the accept and reject functions derived from the initial c
(x1, x2
)= s
(x1, x2
)= 0 setting.
Clearly, an increase in the value of c(x1, x2
) leads to an
increase in the value of the x∗1 threshold. As illustrated in
Fig. 3, such an increase should lead to a small increment in the dispersion of the acceptance behavior of DMs within the Beta
(x1; 4, 2
) setting. At the same time, the accept-
ance behavior of DMs should become more cohesive within the Beta
(x1; 2, 4
) environment. Note, however, that
this cohesion would imply a larger number of friendship
-1 0 1 2 3 4 5 6 7-1
0
1
2
3
4
5
6
SOM Neighbor Weight Distances
-1 0 1 2 3 4 5 6 7-1
0
1
2
3
4
5
6
Hits
7 6 2 0 2 6 1
7 10 5 0 5 4 8
10 7 6 0 8 2 7
5 4 0 7 5 2 6
9 6 0 0 8 1 3
3 4 0 4 2 7 3
3 6 0 1 2 2 4
5 10 15 20 25Weight 1
15
20
25
30
35
40
45
50
Wei
ght 2
SOM Weight Positions
Fig. 7 Self-organizing map clusters generated by Beta (x1; 2, 4
) and Beta
(x1; 4, 2
) requesters
Fig. 8 Accept and reject areas as disutility costs increase from c(x1, x2
)= 0 to c
(x1, x2
)= 2
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10x1
-2
0
2
4
6
8
10
Acc
ept a
nd R
ejec
t Fun
ctio
ns
ACCEPT
REJECT
324 D. Di Caprio et al.
1 3
requests being rejected. Figure 4 illustrates the acceptance dispersion arising from both these settings and allows for a direct comparison with the c
(x1, x2
)= 0 case.
As can be observed in Fig. 4, the increased cohe-sion in the acceptance behavior of DMs within the Beta (x1; 2, 4
)− c setting implies that a lower number of
requests are being accepted. Figure 9 illustrates the clus-ters being formed after a self-organizing map algorithm is applied to the number of friendship acceptances within the c(x1, x2
)= 2 setting.
Note that the distance defined between the clusters in the weight positions plane of Fig. 9 is larger than the distance generated in the c
(x1, x2
)= 0 case. In particular, the lower
number of friendship requests accepted by the DMs within the Beta
(x1; 2, 4
)− c cluster increases both its cohesion
and the distance relative to the acceptance behavior of the DMs within the Beta
(x1; 4, 2
)− c cluster. That is, increas-
ing the disutility derived from an unwanted friendship results in stricter acceptance criteria among DMs, leading to more exclusivity in the corresponding virtual community and a larger number of friendship requesters being rejected.
However, the incentives of DMs can be modified so as to encourage them to accept a larger number of requests within the current theoretical framework. Creating a more inclusive virtual community implies decreasing the value of the x∗
1 threshold, which would lead DMs to increase their
acceptance of both types of requesters. A direct mechanism designed to obtain such a result would consist of increasing the value of the search costs faced by the DMs after reject-ing a given request, i.e. s
(x1, x2
). Figure 10 presents the
accept and reject functions generated by the DMs when the value of s
(x1, x2
) is increased from 0 to 2 in order to com-
pensate for the increase in the value of c(x1, x2
) described
above.
The resulting threshold value has also been illustrated in Fig. 3, while Table 1 describes all the threshold values that have been obtained through the different numerical simu-lations by modifying the disutility and search costs faced by the DMs. Note how increments in search costs induce a more inclusive behavior among DMs, while higher disutil-ity costs lead to more exclusive environments.
That is, virtual communities whose members face rela-tively high search costs will be prone to accept a higher number of requesters, while those communities with a large base of requesters will tend to be more exclusive. At the same time, these latter communities will increase the num-ber of rejections if their members face high disutility costs from accepting requesters with substantially different tastes and characteristics. Finally, note that, within the frame-work described in the paper, the inclusive effect induced by higher search costs dominates the exclusive one implied by increments in disutility costs.
7 Modifying the networking capacity of the requesters
Given the importance assigned by the empirical literature to the social networking capacity of the users of SNSs, we analyze the consequences derived from modifying the networking capacities of the requesters demanded by the DMs. We will illustrate that the DMs who do not prioritize networking derive a higher utility from their social interac-tions while those requiring a large amount of potential con-tacts from the requesters obtain a lower utility.
We start by illustrating the effects of removing B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), ce1 + ce2
) from Equa-
tions (1) and (4). In other words, we assume that DMs
-1 0 1 2 3 4 5 6 7-1
0
1
2
3
4
5
6SOM Neighbor Weight Distances
-1 0 1 2 3 4 5 6 7-1
0
1
2
3
4
5
6Hits
6 5 8 5 0 1 4
6 4 2 0 5 3 2
3 11 12 0 5 9 3
9 8 0 7 4 5 2
6 6 0 0 7 4 4
6 0 0 1 2 6 5
3 0 0 3 10 5 3
0 5 10 15 20 25 30Weight 1
10
15
20
25
30
35
40
45
50
Wei
ght 2
SOM Weight Positions
Fig. 9 Self-organizing map clusters generated by Beta (x1; 2, 4) and Beta (x1; 4, 2) requestres: c(x1, x2
)= 2
325The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
do not consider the potential networking capacity of the requesters when deciding whether or not to accept them as friends. The resulting decision environment is
described in Fig. 11, where the higher expected utility setting (blue and continuous) obtained when the network effects are removed from the decision equations is com-pared with the discontinuous one that was introduced in Fig. 2.
As expected, eliminating the networking capacities of the requesters from the decision environment leads to a higher utility across DMs and to a wider friendship acceptance region. More importantly, this tendency is also observable when modifying the certainty equivalent refer-ence values that define the networking capacity demanded from the requesters through the Beta density.
Fig. 10 Accept and reject areas as disutility and search costs increase from 0 to 2
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
x1
-2
0
2
4
6
8
10
12
Acc
ept a
nd R
ejec
t Fun
ctio
ns
ACCEPT
AREAS
REJECT
AREAS
s=2; c=2
s=0; c=2
s=0; c=0
Table 1 Threshold values determined by the disutility and search costs considered
Cost values x∗1 threshold x
∗1 in
[0, 1]c s
0 0 6.54 0.3082 0 6.89 0.3782 2 6.22 0.244
Fig. 11 Accept and reject func-tions absent network effects. The corresponding intervals are defined by x∗
1= 6.2649
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
x1
-2
0
2
4
6
8
10
12
14
16
18
Acc
ept a
nd R
ejec
t Fun
ctio
ns
REJECT
ACCEPT
326 D. Di Caprio et al.
1 3
That is, consider the initial B3
(x3; x1
+u−12
(E(x2|�2
(x2|x1
))), 7.5 + 5
) environment described
in Fig. 2. We define now two types of DMs, those requir-ing lower networking capacities from the request-ers, B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), 6.5 + 3
),
and those demanding a higher number of poten-tial contacts after accepting the request, B3
(x3; x1 + u−1
2
(E(x2|�2
(x2|x1
))), 8.5 + 7
). The accept
and reject functions determining the behavior of the former type are presented in Fig. 12, while those corresponding to the latter type are described in Fig. 13.
These figures illustrate how those DMs demanding lower networking capacities require also a lower initial characteristic from the requester in order to obtain the same expected utility as those DMs demanding a higher number of potential contacts. Interestingly enough, the friendship acceptance behavior of both types of DMs is almost identi-cal. That is, lower networking requirements do not modify the friendship acceptance behavior of the DMs but increase their expected utility if the request is accepted.
An alternative interpretation of this result would identify the networking capacities of requesters with their suitability
Fig. 12 Accept and reject functions with B3
(x3; x1 + u
−12
(E(x2|�2(
x2|x1)))
, 8.5 + 7)
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
x1
-2
0
2
4
6
8
10
Acc
ept a
nd R
ejec
t Fun
ctio
ns
ACCEPTREJECT
Fig. 13 Accept and reject functions with B3
(x3; x1 + u
−12
(E(x2|�2(
x2|x1)))
, 6.5 + 3)
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
x1
-2
0
2
4
6
8
10
12
Acc
ept a
nd R
ejec
t Fun
ctio
ns
REJECT ACCEPT
327The emergence of inclusive and exclusive virtual communities determined by the preferences…
1 3
as friends or potential relationships, increasing the expected utility derived by those DMs who focus on the recreational use of the SNS. For example, a survey conducted by Tosun (2012) identified ‘maintaining long-distance relation-ships’ as the primary motive for Facebook use. Moreover, the author concluded that those users with a tendency to express their true self on the internet tend to use Facebook for establishing new friendships and initiating or terminat-ing romantic relationships. On the other hand, Trepte et al. (2015) illustrated that SNSs are better designed to leverage informational support but are inferior to offline contexts in terms of emotional or instrumental support.
Consider now the strategic consequences that follow from our decision environment. Basic cost-benefit analysis has been applied to study the incentives determining the intention of users to share context information. Krasnova et al. (2012) concluded that trusting beliefs determine self-disclosure from users while uncertainty avoidance defines the impact of privacy concerns. Lee et al. (2013) built on the privacy calculus framework to analyze the expected benefits and risks derived from different privacy control strategies. Jiang et al. (2013) examined the strategies of self-disclosure and misrepresentation applied by users to protect their privacy and highlighted the roles of privacy concerns and social rewards in online social interactions.
We conclude by emphasizing that our model can be adapted to the strategic framework that follows from the incorporation of trust and reputation problems. In this regard, consider the study of Abell and Brewer (2014), who concentrated on Machiavellianism, a personality trait characterized by cynicism, emotional detachment and a willingness to manipulate others. They showed that Machi-avellianistic women and males engaged in more online self-monitoring. Moreover, the former involved themselves in dishonest self-promotion and relational aggression towards close Facebook friends, while the latter focused on self-promoting behavior.
Consequently, the effects of trust and reputation on the beliefs and acceptance behavior of the DMs should also be studied. For example, Buzzanca et al. (2016) analyzed the assignment of trust to network nodes determined by posi-tive and negative experiences as well as the history of past feedbacks and their aging. The current expected utility set-ting can easily accommodate this type of scenario, applying potential strategic extensions to the analysis of, for exam-ple, recommender systems.
8 Concluding remarks
The current paper has introduced a novel decision theo-retical model of friendship acceptance with incomplete information that has been used to build different network
structures determined by the different expectations and preferences of its users.
Among the potential applications of the current model, we should highlight the following ones. Given its capac-ity to deal with ambiguous environments and its applica-bility to market scenarios, it can complement models such as that of Marey et al. (2015). These authors studied argu-mentation-based agent negotiation in environments charac-terized by incomplete information and uncertainty on the side of the DMs and discussed the implementation of their approach to concrete buyer/seller scenarios. Moreover, the capacity of our model to account for several information sources and their assimilation before DMs make a decision can be exploited to study settings such as the one of Atif et al. (2015), who developed a ubiquitous learning model within a pervasive smart campus environment based on new paradigms such as context-awareness and resource virtualization.
We conclude by emphasizing that several variants of our decision model can be developed in order to consider, for example, multiple sequential observations within a finite set of friendship requests that may be expected to be received by the DM. This feature, together with variations in the degree of risk and uncertainty aversion of the DM, modifications in his subjective formation of beliefs or the x∗2 requirements derived from the potential combinations of
characteristics, and differences in the distribution of friend-ship requests can be easily incorporated in the model and the resulting clustered structures derived from the corre-sponding network analyzed.
In particular, immediate extensions of the model should account for different sets of beliefs on the X1 and X2 charac-teristics defining the requesters and the inclusion of signals reflecting the trust or reputation placed on potential friends. Note that these modifications would shift the accept and reject functions and the corresponding threshold values, generating different network structures from those observed in the current paper.
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