TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190) An Official Journal of Turkish Fuzzy Systems Association Vol.5, No.2, pp. 078-099, 2014.
A New Approach to Social Networks Based on Fuzzy Graphs Sovan Samanta* Department of Mathematics, Joykrishnapur High School (H.S.), Tamluk-721649, India E-mail:
[email protected] *corresponding author Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721102, India E-mail:
[email protected]
Abstract Social networks are the areas in which a huge number of people are connected. In this paper, a new social network called fuzzy social network (FSN) has been introduced based on fuzzy graph. For this network, centrality, single and multi-bridges and transfer value of the bridges are newly defined and illustrated by examples. In this network, the strength of relationship can be graded by different values between 0 and 1 , and we have shown that this representation is more realistic. Also, we have introduced a new concept of registration for a new user so that the chance of fake user may be reduced. Keywords: Social networks, centrality, bridges, fuzzy graphs.
1. Introduction In twenty first century, social networks are new platforms for staying in touch with people in everywhere in the world. These are perfect places of exchanging information of various topics and issues. These help in marketing world, connecting public to clients. These are important tools for public awareness by spreading messages rapidly to a wide audience. Indeed, social networks are important for information diffusion, e-commerce and e-business, influential players (scientists, innovators, employees, customers, companies, etc.), effective social and political campaigns, future events, terrorist/criminal network, alumni, etc.
Facebook, Twitter, LinkedIn, Orkut, ResearchGate are some types of social networks. Facebook was founded by Mark Zuckerberg in collaboration with his college roommates and fellow students to connect the whole world. 901 million monthly active users are connected at the end of March 2012 in Facebook. More than 125 billion friends’ connections exist on Facebook at the end of March 2012 . Twitter is an online social networking service and microblogging service that enables its users to send and read text-based posts. 500 million active users as of 2012 , generating over 340 million tweets daily and handling over 1.6 billion search queries per day exist in Twitter. LinkedIn is a professional social networking website founded in December 2002 and launched in May, 2003. It is mainly used for professional networking. Upto February 9,2012 , more than 150 million registered users in more than 200 countries and territories are active in LinkedIn. ReaserchGate provides its members with a number of tools to facilitate global scientific collaboration. Researchers can create professional profiles, discuss their topics with specific question and answer forums, share papers, search for jobs and discover conferences in their field. A recent calculation of members shows that ResearchGate has so far assembled a user-base of over 1.4 million researchers from 192 countries . Facebook, Orkut, Twitter, etc. have many advantages to exchange pictures, videos, comments, etc. The social networks are places to find old/ lost friends and to connect new friends. These are places to grow consciousness among internet users. Nowadays, political leaders, ministers, film actors, businessmen, etc. are spending time in these social networks to reach people, specially young generation. A lot of advantages are available in social networks, even then some drawbacks in the current social networks are present. These are listed below. (1) In all these social networks, all social units (people, organizations, etc.) are assumed with same importance. In reality, it is not the case where every unit has the same facility/importance. (2) In the present social networks, it is assumed that the strength of friendships between every pair of units are same, but in reality it is not true. (3) In almost all social networks, information of one member can not be found by other registered persons if it is not “public". But the other registered persons to the social network may be his/her friend or acquaintance. (4) In almost all social networks, the relations are represented, initially, as friendship (“add as friend"). But, relationship may be romantic, kinship, friendship, colleagues, parent-child, acquaintance, teacher-student, seller-buyer, etc. (5) Fake profile is one of the big problems of social networks. Now, it has become easier to create a fake profile. People often use fake profile to insult, harass someone, involve in unsocial activities, etc.
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These problems motivate us to introduce a new social network model. We make this model using fuzzy graph theory.
1.1 Review of literature After SixDegrees.com experienced in 1997, hundreds of networks have spread online [3]. There are many popular social networks like Facebook, Twitter, Orkut, MySpace, Friendster, Zorpia, etc. Among them Facebook is most popular. These networks had some disadvantages. Scott [32] removed some of these drawbacks on social networks. Zywica and Danowski [34] investigated about the popularity of Facebook. More investigations are made by Caci et al. [9] and Shanyang [33]. People’s behaviours are one of the topics among new researchers. Brass and Halgin [6] discussed the effect on employee’s attitudes, behaviours, and well-being, in Facebook. Most of the social networks were not secured, but several works have been done for security of these networks. Pham et al. [21] have shown that computer science is becoming more interdisciplinary in social networks. In 2006, Alessandro and Ralph [2] discussed about the privacy and awareness of Facebook. Bhattacharya et al. [4] discussed about the similarity on keywords of users in social networks. Now, the social networks are safe in terms of information sharing, collecting, etc. But the number of false profile users is increasing. Now a lot of researchers are working on privacy of social networks. There are several methods to characterize the social networks. Relations in social networks can be drawn by graph. Probability, game theory, weighted graph theory, etc. are some areas in which social networks have been characterized by several authors [1, 10, 19]. In reality, many systems can be represented in terms of fuzzy graphs. To the best of our knowledge no works are available on social networks based on fuzzy graphs. Here, a social network is proposed which is constructed by fuzzy graphs. In this representation, every social actor and their relations are taken as fuzzy sets. Relation between two actors may be more than one simultaneously (for example friend and relative). This concept helps us to define the model as fuzzy multi-graphs. Fake profile users make anti social activities, dirty works, etc. Some people also create false profiles in the name of their familiar persons. These activities are harmful to these familiar persons. Besides, terrorism is a big problem in all over the world now. The terrorists generally use fake profiles. The teenagers enjoy social networks mostly. The unsocial people direct the teenagers towards wrong way by making fake profile and giving wrong, attractive information. So we should stop making fake friends in social networks. But in present social networks, it is hard to identify the fake profile by observation only. Some rules are adapted to protect creation of fake profiles. In the existing social networks, centre persons are valuable. Some rules have been created to measure the centrality of persons [5, 12]. Here we propose a new formula to measure the centrality of a person on the social network. The strength of any networks depends on units’ centrality. If there is any bridge in a network, then it is a weak network as when the bridge is broken, the communication between two actors may be destroyed. Jones and Volpe [17] have given some concepts on social bridges for existing social networks. Transfer value of bridges is discussed here for different types of bridges. 80
Fuzzy graph theory is introduced by Rosenfeld [24] in 1975. After that several works are going on the topic. Samanta and Pal introduced fuzzy planar graphs [30], fuzzy tolerance graphs [25], fuzzy threshold graphs [26], bipolar fuzzy graphs [27,28], fuzzy competition graphs [29]. Readers may look in [22,23] for further study. 2. Preliminaries Every kind of social group can be represented in terms of units, composing this group and relations between these units. This kind of representation of a social structure is called “ Social Network". In a social network, every unit, usually called “social units" like a person, an organization, a community, and so on, is represented as a node. A relation between two social actors is expressed by a link. So every social network can be represented by a graph. A graph is an ordered pair G = (V , E ) , comprising a set V of vertices or nodes together with a set E of edges or lines. A walk [14] of a graph is an alternating sequence of points and lines v0 , x1 , v1 ,, vn1 , xn , vn , beginning and ending with points, in which each line is incident with the two points immediately preceding and following it. It is a trail if all the lines are distinct, and a path if all the points (and thus necessarily all the lines) are distinct. If the walk is closed, then it is a cycle provided its n points are distinct and n 3 . The length [14] of a walk v0 , x1 , v1 ,, vn1 , xn , vn is n , the number of occurrences of lines in it. Centrality is another important part of a social network. Different terms related to centrality are defined here. Degree centrality [12] is the number of ties incident upon a node. Betweenness centrality [13] is a measure of a node’s centrality in a network equal to the number of shortest paths from all vertices to all others that pass through that node. Structural holes [7] are gap in information flows between alters linked to the same ego but not linked to each other. Structural holes indicate that people on either side of the holes have access to different information flows. A bridge is a line whose removal increases the number of components in the network. In sociology, social capital [8, 18] is the expected collective or economic benefits derived from the preferential treatment and cooperation between individuals and groups. Sometimes, it may be observed that the existence of nodes and edges (links) are not certain. For example, if two nodes represent two friends and the edge between them represents the strength of the friendship (very strong, strong, very weak, etc.), then the corresponding graph is not the classical graph. The entire system can be expressed as fuzzy graph which is defined below. A fuzzy set A on a set X is characterized by a mapping m : X [0,1] , which is called the membership function. A fuzzy set is denoted by A = ( X , m) . A fuzzy graph [22] = (V , , ) is a non-empty set V together with a pair of functions 81
: V [0,1] and : V V [0,1] such that for all x, y V , ( x, y ) ( x) ( y ) and is a symmetric fuzzy relation on . Here (x) and ( x, y ) represent the membership value of the vertex x and of the edge ( x, y ) in respectively. Degree [20] of a node v V of a fuzzy graph = (V , , ) is d (v) = (u, v), for all u V . u v
For the fuzzy graph = (V , , ) , an edge ( x, y ), x, y V 1 min{ ( x), ( y )} ( x, y ) and it is called weak otherwise. 2
is called strong [11] if
As there may be multiple edges between nodes, multi-set is important for this context. A (crisp) multiset over a non-empty set V is simply a mapping d : V N where N is the set of natural numbers. Yager [33] first discussed fuzzy multiset, although he used the term of fuzzy bag, an element of nonempty set V may occur more than once with possibly the same or different membership values. A natural generalization of this interpretation of multiset leads to the notion of fuzzy multiset, or fuzzy bag. The membership values of v V are denoted as j (v), j = 1,2,, p where p = max{ j : j (v) 0}. So the fuzzy multiset can be denoted as
M = {(v, j (v)), j = 1,2,, p | v V } . A multigraph is a graph allowing multiple edges and loops. Let V be a non-empty set and : V [0,1] be a mapping. Also let E = {((x, y), j ( x, y)), j = 1,2,, p | ( x, y) V V } be a fuzzy multiset of V V such that j ( x, y) ( x) ( y) for all x, y V
and for all
j = 1,2,, p where p = max{ j | ( x, y) 0} . Then = (V , , E ) is denoted as j
fuzzy
multigraph where (x) and j ( x, y) represent the membership value of the vertex x and the membership value of the edge ( x, y ) in respectively. In this paper, a new social network called fuzzy social network (FSN) is described.
3. Representation of FSN Here we assume all social actors as fuzzy vertices and all linkages as fuzzy edges. Moreover, linkages may be of different kinds. So we represent the social network as fuzzy multigraph (V , , E ) where V = {P1 , P2 ,, P }, is a large positive integer, be a set of social units.
: V [0,1] be a mapping and E = {((Pi , Pj ), r ( Pi , Pj )), r = 1,2,, g | ( Pi , Pj ) V V } be a fuzzy multiset of V V such that r ( Pi , Pj ) ( Pi ) ( Pj ) for all Pi , Pj V and for all r = 1,2, , g where g = max{r | r ( Pi , Pj ) 0} . (Calculations of ( Pi ), r ( Pi , Pj ) are described
in Sections 3.2,3.3 ). Without loss of generality, we assumed that all linkages are of same kind. In this case, unit membership value of a social unit is same as unit membership value of a social unit, represented in 82
fuzzy multi-graph representation but link membership values are little different. Then the FSN can be represented by a fuzzy graph FSN = (V , , ) where : V [0,1] and : V V [0,1] are mappings such that ( Pi , Pj ) ( Pi ) ( Pj ) for all Pi , Pj V . The registration for a new user in FSN is described below. 3.1 Registration The first step to join a person to FSN is registration. During registration a person has to decide the following steps. Here we add one more setting option, namely automatic authentication, to the existing settings of current social networks. 3.1.1
Basic settings
The newly registered persons are to add some basic information like name, age, images, gender, etc. All the registered persons will get the basic information of a newly registered person. So the newly registered person is to decide what information he/she wants to give publicly. 3.1.2
Automatic authentication
One of the objectives of this step is to avoid creation of fake profile. In registration procedure, we introduce a new step called automatic authentication. In this step, the person has to give some information like his/her (i) school/college/universitiy name of his/her educational life, (ii) location of current job or working place/institution (also previous jobs and working places, if any), (iii) frequent visiting institutions, (iv) best friends’ names. The information of a new registered person is available for those registered persons whose information will match partially/fully with the new person. Now, some such registered persons are required to identify the new user as a valid person. If some consents (at least 3 generally) are available, then the new person is considered as full member. This setting is the only way for a new registered person to find new friends initially (for details see Section 3.2). 3.1.3 Private settings In this settings, the information of the newly registered persons is available for the persons who are selected by the newly registered persons. 3.2 Membership value of a social unit Membership value of a social unit will be assigned between 0 and 1 . If a unit has been given membership value 0 , the unit is taken as non-member (probably a fake profile). If a unit has membership value 1 , the unit is a full member of the network. People will be happy to make friends to full members only. The membership value of a social unit is given in the following way.
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(i) When a social unit (a person, an institution, etc.) is registered to the social network, initially the membership value of this unit is set to 0 . (ii) We divide all the people into two groups. The persons who are working in authentic (viz., government, semi-government, government aided, registered private organization, etc.) organizations and official e-profiles (contains basic personal information and contact information along with email addresses) are available in their respective institute’s web-sites, fall into first group, say group- A . These units can be verified by their official e-profiles (viz., email address verification, etc.). This type of units of group- A , are referred as unit by verification. Other persons fall into the second group, say group B . These units can not be verified by email address verification as these persons’ e-profiles (viz., email addresses, etc) are not available in authentic web-sites. The units can be verified by recognition of registered units. The units of group- B are referred as units by recognition. Initially registered social units of the group, units by recognition will not be given the scope to make direct friends like “add as friend". There will be an automatic choice of registered persons’ list by automatic authentication process. The units can send limited messages to only the listed units. After recognition by both ends they can make friendship, kinship, acquaintance, etc. from the list only. The units can not contact other members, if the information, given in “automatic authentication", do not match with each others. The units by verification initially have the membership value 0 . Then a verification code/ activation link will be sent to the contact addresses of the social units. (iii) Units by recognition Before the discussion of membership values of the social units, a co-related term, recognition number, is defined below. In every society, recognition of each person is measured by some members of the society. In FSN, we take the number of those members as recognition number. It is denoted by n . The membership value of a social unit (P) of the group, unit by recognition, which is connected with d units, is given by d
( P) = n , if d = 1,2,, n 1,
if d > n.
Obviously, n is recognition number for FSN. The social network can be planned for units in rural area with small value of n and for urban area with large value of n . Units by verification After verification of the personal contact information from their respective official e-profiles, the units get the full membership value i.e., 1 directly. 84
(iv) Once a unit gets membership value 1 , it can connect directly to the units with membership value 1 anywhere in the network. 3.3 Link membership value Two social units may be connected by any of the social relations friendship, kinship, acquaintance, etc. In real situation two friends may be very close, close, etc. So the relations between two units are the elements of a fuzzy set. These relations are designated by membership values. Sometimes our minor decisions depend on friends’ decisions. Generally, we accept the decision of a close friend. The closeness depends on the strength of friendship. So strength of friendship is very important. So membership values of links between two units are important for a social network. Relations between two persons may be friendship and kinship together. Again “buyer-seller" relation may be termed as acquaintance or even friendship. So two units may have multiple relations. Let V = {P1 , P2 ,, P } be the fuzzy set of registered social units in FSN such that
( Pi ), i = 1,2,, are the membership values of the social units Pi , i = 1,2,, respectively. Links between two social units form a fuzzy multi-set of V V . Before the definition of membership values of the links between two social units, a co-related term, stable interaction number, is defined below. In FSN, stability of relation between two members is measured by the number of interactions between them. This number, for stable relation, is defined as the stable interaction number and denoted by the fixed positive integer l . The link membership value between Pi , Pj V is defined by the mapping : V V [0,1] where m 1 ( ( Pi ) ( Pj )), ( Pi , Pj ) = l ( Pi ) ( Pj ),
if m [0, l 1] if m > l 1.
For FSN, l is stable interaction number. m stands for the number of interactions between two social units Pi and Pj per unit of time. The social relations are also different types such as friendship, kinship, acquaintance, etc. Let there be g types of relations in the social network. The link membership value between Pi , Pj V be denoted by r ( Pi , Pj ) (where r represents the type of relations) and defined by mr 1 ( ( Pi ) ( Pj )), ( Pi , Pj ) = l ( Pi ) ( Pj ), r
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if mr [0, l 1] if mr > l 1.
r = 1,2, , g . For FSN, l is stable interaction number, mr stands for the number of interactions between two social units Pi and Pj per unit of time.
If s ( Pi , Pj ) denotes the link membership value of a relation (say romantic relation) between two persons Pi and Pj , we assume that r ( Pi , Pj ) = 0, r = 1,2,, s 1, s 1,, g in FSN. 3.3.1 Strong and weak links In 1960s, Mark Granovetter interviewed people who had recently changed their jobs to learn how they discovered their new jobs [15]. He had found out that almost all people found new jobs through acquaintances rather close friends. He termed the close friends as strong ties and acquaintances as weak ties. In 1973, he had described the strength of a weak ties [16]. But he had not specified the measure of strength. We call all the links as strong links if link membership values are greater than 0.5 and weak otherwise. All decisions will be affected by strong links. But only major decisions may be affected by weak links. 4. Stopping of fake profiles in FSN Initially, registered people in unit by recognition can not make direct friends to his/her choices. A list of people, possibly familiar to him/her, is prepared with the help of his/her setting, specially in “automatic authentication". If some of the listed persons verify the new registered person, the new member will be a full-member. So creation of a fake profile is not easy here. Government/ government sponsored institutes / registered companies have personal web sites. Those web sites contain the names, contact details, and information of authorities (for example faculty members of educational institutions, etc). If a person wants to register in FSN in group, units by verification, then he/she has to give his/her institute’s name and web address. Then he/she has to provide his/her email address, phone numbers, etc. which are available in the web address. Then a verification code/activation link is sent to contact addresses. If the information is correctly verified, then it is almost sure that the person is not fake. Here authorized people of various types of institutes like educational, industrial, business, sports, law, companies, etc. get a chance to obtain full membership. But this list of those registered persons is limited. Other people obtain full membership by recognition of his/her familiar persons only like school/collge/university friends, colleagues, people near to home (i.e., neighbours), etc. in the network. On the other hand in Facebook, Twitter, etc. people can not identify which profile is the original profile of a celebrity or common people. So it is just a ridiculous situation sometimes because a profile, created by unknown, need not to be verified or recognized in these social networks. 5. An example of FSN For illustration, a social network by 20 people and their inter-relations are shown. We take 7 days as one unit of time, number of recognition friends, n = 3 and number of stable interaction between two social units, l = 2 . 86
5.1 The status of FSN after 7 days Let, after 7 days, the social network has 20 registered people. Let V = {P1 , P2 ,, P20} . Initially, these 20 people have membership values 0 . P15 is assumed as unit by verification and rests are unit by recognition. So after verification of contact details, P15 will get full membership value 1 . Other units, after sending some enquiry messages to others, they can make recognition friends. The number of recognition friends is 3 here. After connecting 3 recognition friends, they will get full membership value 1 . The list of people with connected friends and number of interactions after 7 days are shown in Table 1. 5.2 The status of FSN after 14 days We observe the linkages of same 20 people after 14 days. Here is the statistics of the people. After 14 days, i.e. after two units of time, the number of interactions between two units is shown in Table 2. From Table 2, the edge (link) membership value is calculated as follows. We see that the number of interaction after 14 days between P1 and P2 is 2 . So the number of interactions per unit m 1 4 ( ( P1 ) ( P2 )) = = 0.44 . All the link membership values time, m is 1 . Now, ( P1 , P2 ) = l 9 are shown in Table 3.
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Table 1: List of people and their connected people. The number within parenthesis represents number of interactions after 7 days. Person P1
Connected people with number of interactions P2 (0), P4 (2)
P2 P3
P1 (0) P4 (2) P1 (2), P3 (2), P5 (2), P17 (2)
P4 P5 P6 P7 P8 P9 P10 P11 P12 P13
P4 (2) P11 (3) P9 (1), P10 (0), P20 (1) P8 (1) P8 (0), P11 (1) P7 (3), P10 (1), P13 (0) P11 (0)
P14 P15 P16 P17 P18 P19 P20
P4 (2), P19 (0) P17 (0) P8 (1)
Figure 1: Status of FSN after 7 days. The number in parenthesis of Pi is unit membership value and number adjacent to link represents the link membership value. 88
Table 2: List of people and their connected people. The number within parenthesis represents number of interactions after 14 days. Person P1
Connected people with number of interactions P2 (2), P4 (4)
P2 P3
P1 (2), P4 (4) P4 (4) P1 (4), P2 (4), P3 (4), P5 (4), P17 (3)
P4 P5 P6 P7 P8 P9 P10 P11 P12 P13
P14 P15 P16 P17 P18 P19 P20
P4 (4) P8 (1) P8 (1), P10 (1), P11 (5) P6 (1), P7 (1), P9 (3), P10 (1), P20 (3) P8 (3), P11 (6), P14 (2) P7 (1), P8 (1), P11 (6), P13 (4), P14 (1) P7 (5), P9 (6), P10 (6), P12 (1), P13 (1), P14 (1) P11 (1), P13 (3) P10 (4), P11 (1), P12 (3), P14 (2) P9 (2), P10 (1), P11 (1), P13 (2) P13 (4), P20 (4) P4 (3), P19 (1) P19 (7), P20 (3) P17 (1), P18 (7), P20 (2) P8 (3), P18 (3), P19 (2)
Figure 2: Status of FSN after 14 days. 89
Table 3: List of links and link membership values after 14 days.
Link
Link membership values 0.44
Link
Link membership values
( P9 , P14 )
0.667
0.667
( P10 , P11 )
1
0.667
( P10 , P13 )
1
( P3 , P4 )
0.333
( P10 , P14 )
0.5
( P4 , P5 )
0.333
0.333
( P4 , P17 )
0.556
( P11 , P12 ) ( P11 , P13 )
( P6 , P8 )
0.167
0.5
( P7 , P8 )
0.5
( P7 , P10 )
0.5
( P7 , P11 )
1
( P8 , P9 )
0.833
( P8 , P10 )
0.5
( P8 , P20 )
0.833
( P13 , P15 )
1
( P11 , P14 ) ( P12 , P1 ) ( P13 , P14 ) ( P17 , P19 ) ( P18 , P19 ) ( P18 , P20 ) ( P19 , P20 ) ( P15 , P20 )
( P9 , P11 )
1
( P1 , P2 ) ( P1 , P4 ) ( P2 , P4 )
0.5
0.556 0.667 0.333 0.667
0.556 0.667 1
6. Centrality of a social unit in FSN Centrality is one of the most studied concepts in social network analysis. Numerous measures have been developed, including degree centrality, closeness, betweenness, eigenvector centrality, information centrality, the rush index, etc. Degree centrality measures the direct linkage of a social unit to the others, betweenness centrality measures the number of paths between any pair of social units through the social unit. In all kinds of networks, central persons are more valuable than others. They can send messages to more people, can collect more information. Degree centrality measures the number of direct friends (or linkages). So it does not measure the number of connected people by a path. It may be noted that friend of a friend in Facebook shares information to the person. So betweenness centrality is also useful. But friend of friend does not share so much information like the direct friend. So importance of the linkage will gradually decrease from a person to another person by a connected path. 90
In FSN, if a unit Pf is directly connected with the unit P , then we say that Pf is distance-1 friend of P . The set of all distance-1 friends of P be denoted by d1 ( P) . That is, d1 ( P) = {Pi V : Pi is a distance 1 friend of P}. Similarly, if there is a shortest path (i.e. minimum number of links) between P and Pf containing
k
egdes or links, then
Pf
is a distance- k
friend of
P . That is,
d k ( P) = {Pi V : Pi is a distance k friend of P}. Now, let d k' ( P) = d k ( P) d k' 1 ( P) , where k = 2,3, and d1' ( P) = d1 ( P) (Note that for classical sets A, B , A B = {x A and x B} ). It is natural that the distance-1 friends are more important than distance-2 friends, distance-2 friends are more important than distance-3 friends, and so on. The linguistic term “more important" can be represented by weights. Let wk , 0 wk 1 be the weight which represents the importance between the distance- k friends. The weights gradually decreases if the distance between the friends increases. Thus w1 w2 wk Let u1 (= Pi ), u2 , u3 ,, uk (= Pj ) be the vertices on the path between Pi and Pj . We define fuzzy k 1
distance D f ( Pi , Pj ) between Pi and Pj along this path as D f ( Pi , Pj ) = (ul , ul 1 ). l =1
In a network, it may be observed that there are multiple paths between two vertices. In FSN, we consider those paths of same length whose fuzzy distance D f is maximum. If there are k edges in this path of maximum fuzzy distance, then we denote this distance by D kf i.e, D kf ( Pi , Pj ) represents the fuzzy distance between the vertices Pi and Pj in FSN along a certain path containing exactly k edges. We assumed that a social unit P has at most distance- p friends. Now we define the centrality C (P ) of a social unit P of FSN as follows
C ( P) =
w1 D1f ( P, u1 )
u1d1' ( P )
w2 D 2f ( P, u2 )
u 2 d '2 ( P )
w p D fp ( P, u p ).
u p d 'p ( P )
In this measurement, the importance of close friend is given more than the next to close friend and gradually decreases the furthest friend. The importance are introduced by incorporating the weight wi , for distance- i friend, i = 1,2,3, A FSN of 20 people after 14 days is shown in Figure 2 . The link membership values are shown in Table 3 . In the definition of centrality of a social unit, p can be taken as fixed for a social network. Here we assumed that p = 3 and measure the centrality of social units. Here, we
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take w1 = 1 and wi1 =
1 wi , i = 1,2, . 2 6.1 Centrality of P1
Here d1 ( P1 ) = {P2 , P4 } = d1' ( P1 ) . d 2 ( P1 ) = {P4 , P3 , P17 , P5} . d 2' ( P1 ) = d 2 ( P1 ) d1' ( P1 ) = {P3 , P17 , P5 } .
d3 ( P1 ) = {P17 , P19} . d 3' ( P1 ) = d 3 ( P1 ) d 2' ( P1 ) = {P19 } . Now,
D1f ( P1 , u1 ) = ( P1 , P2 ) ( P1 , P4 ) = 0.44 0.667 = 1.107.
u1d1' ( P1 )
Also,
D3f ( P1 , u3 ) = D3f ( P1 , P19 ) = { ( P1 , P4 ) ( P4 , P17 ) ( P17 , P19 )} = {0.667 0.556 0.333} = 1.556.
' (P ) u3d3 1
The centrality of the unit P1 is given by
C ( P1 ) = u d' ( P )D1f ( P1 , u1 ) 1
1 1
0.5 D 2f ( P1 , u2 )
u2d'2 ( P1 )
0.25 D3f ( P1 , u3 ) = 3.1075.
' (P ) u3d3 1
Similarly, we calculate centralities of other social units. 6.1.1 Important observation In Figure 2, there are three paths between P15 and P9 of length 3 . They are P15 P20 P8 P9 ,
P15 P13 P11 P9 and P15 P13 P14 P9 . The fuzzy distance of first path is 2.66 , the fuzzy distance of second path is 2.5 and that of third path is 2.334 . Thus to calculate the centrality of P15 , we have to choose the path P15 P20 P8 P9 as its fuzzy distance is most. Calculating we get, C ( P15 ) = 10.1795 . Again the centrality of P4 , C ( P4 ) is 3.7785 . But P4 has 5 direct friends (i.e., distance-1 friends) and P15 has 2 direct friends. So centrality in FSN depends on both direct friends and strong distance- p friends, here p > 1 .
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7. Social bridges in FSN Sometimes two friends are connected by a third friend. If the third friend is lost then two friends may be disconnected. Here the third friend is called a broker or structural hole between two friends. Similarly, if two groups of people are connected by a single person, the person is broker or structural hole between two groups. Opinion and behavior are more homogeneous within the group, so people connected across groups are more familiar with alternative ways of thinking and behaving. People who stand near the holes in a social structure are at higher chance of having good ideas. Social bridges are constructed by at least three units, one from each group and other is broker. We denote a bridge of three units Pi , Pj , Pk by B ( Pi , Pj , Pk ) .
7.1 Social bridges of more than three units Sometimes, one person can not establish a connection between two groups of people. But he has a friend who is connected to one person of another group. If these two persons are connected, two groups will be connected and a bridge will be formed by four social units. Also this kind of bridge can be extended to more people. It is illustrated in Figure 3. For example, P1 is a student of a university A . P2 is a friend of P1 . P3 is brother of P2 . Again
P3 has friend P4 who is a student of university B . So the two universities are connected by a bridge of four units.
Figure 3: A social bridge constructed by more than three units.
7.2 Strong and weak bridges If a broker or structural hole is strongly connected to two groups of people, connected to each other, the transfer of information will be very speedy. But, if the connection is weak, information flow will not be fast. So not only broker or bridge is an important unit in a social network, but also its strength. Depending on the strength, a bridge can be classified as strong and weak bridges. Definition 1 If the link membership values of all the links in a bridge are greater than 0.5 , then the bridge is called strong bridge. On the other hand, if link membership value of at least one link is 93
less than or equal to 0.5 , then the bridge is called weak bridge. Let a bridge be constructed by social units P1 , P2 , P3 between two groups of people A1 (connected with P1 ) and A2 (connected with P3 ). Let the link membership value between P2 and P1 be ( P2 , P1 ) and that of between P3 and P2 be ( P3 , P2 ) . The strong and weak bridges are depicted in Figure 4.
Figure 4: Examples of strong and weak bridges of a network.
7.3 Multi-bridges In real field, two different groups of people want to interact with each other. For example, an Indian people’s group want to interact with the culture of an Australian people’s group and vice-versa. If the two groups have a bridge, then the bridge is the only way to get the information from each other. Sometimes, it may be happened that the social units of the bridge do not share the complete information or not have enough faith on the groups. So another bridge will help to transfer the information between groups. In this situation, alternative bridge is required to get the complete and correct information. If two or more disjoint bridges exist between two groups or units, then these bridges are called multi-bridges (see Figure 5). Here each bridge is called a branch of the multi-bridge. We denote the multi-bridge by {B1 , B2 ,, Bk } where each Bi , i = 1,2,, k is a bridge.
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Figure 5: Example of a multi-bridge of a network In modern day, concept of bridge is not so helpful like multi-bridge and strong bridges. Nobody can believe single person or single social unit. Even a bridge is strong, multi-bridge is required to correct and transfer fast information. Also, multi-bridge network is more reliable than the single bridge network. In bigger area of people, social capital of brokers in multi-bridge will not reduce so much unless number of bridges in a multi-bridge are too large. Sometimes, people confuse about which bridge is good in terms of transfer of information. To remove the confusion, we have introduced the transfer value of bridges in FSN. If a bridge is connected to a person with large centrality value of a group, the bridge will transfer large amount of information to another group. Similarly, if a bridge is connected to a person with small centrality value of a group, bridge will transfer little information to another group. 7.4 Transfer value of bridges in FSN Let a bridge B be constructed by social units Pi , i = 1,2,, z between two groups of people A , connected with P1 and B , connected with Pz . Let the link membership value between Pi and Pj be ( Pi , Pj ) . Let minimum of the link membership values of the links in B be m . The transfer value of B be denoted by T (B ) and defined by the value T ( B) = m C ( P1 ) C ( Pz ) where C (P ) denotes the centrality of a person in FSN. For multi-bridge the transfer value is defined by the sum of the transfer values of each bridges.
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Figure 6: Examples of transfer values of bridges in FSN.
Example 1 Let a bridge B1 in FSN constitutes social units P1 , P2 , P3 , P4 where P1 , P4 are social units connected to two different groups of people (Figure 6(a)). Let ( P1 , P2 ) = 0.8, ( P2 , P3 ) = 0.5, ( P3 , P4 ) = 0.4 . Let C ( P1 ) = 4.2, C ( P4 ) = 5.6 . So the transfer value of B1 , T ( B1 ) = 0.4 C ( P1 ) C ( P4 ) = 0.4 4.2 = 1.68 . In Figure 6(b), a multi-bridge {B1 , B2 } is shown with social units P1 , P2 , P3 , P4 , P5 , P6 , P7 and
( P1 , P2 ) = 0.8, ( P2 , P3 ) = 0.5, ( P3 , P4 ) = 0.4, ( P5 , P6 ) = 0.65, ( P6 , P7 ) = 0.9 . C( P1 ) = 4.2, C( P4 ) = 5.6, C ( P5 ) = 6.2, C ( P7 ) = 8.2 . For the multi-bridge, transfer value
T ({B1 , B2 }) is 1.68 4.03 = 5.71 . Using transfer value people can choose the bridges for faster transformation in FSN. 8. Conclusions This study opens a new area of social networks. A social network model FSN has been introduced with the help of fuzzy graph theory. In the proposed FSN, the registered persons will not be able to create fake profiles easily. So the people identify the valid persons with their membership values. If a person in unit by verification category gets full membership value, the person can not be fake generally. But, if a person in unit by recognition category gets full membership value, then there is a chance that the profile is fake. If a group of persons is involved to recognize each other, then fake 96
profiles can be created. But, if one of the units (those are assigned to recognize the new unit) is a valid unit (i.e. unit by verification), then there is little chance to create a fake profile. Centrality of a person in FSN has been defined. Here fuzzy distance of connected friends of different distances are considered for measurement of centrality. Between two person, if there are multiple paths of same distances, the path whose fuzzy distance is most, is considered for calculation. To measure the centrality we considered the fuzzy path whose fuzzy distance is maximum. A different measure of centrality be given by considering the strength of a fuzzy path. Transfer value of bridges is defined here. So which bridge is better for information transfer, can be identified by the formula of transfer value. Centrality of persons and transfer value of bridges will be helpful in marketing world. In future, this model will help to represent telecommunication systems by fuzzy graphs. Also link prediction in social networks will be introduced in future as an extension of this study.
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