Mathematical Concepts and Representation of Social Networking Health Site

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1 Mathematical Concepts and Representation of Social Networking Health Site Abhishek Burli, Archit, Mandar Chitale and Prof Anupama Phakatkar Abstract A social network is mathematically defined as set of actors and relational ties between them. The concept of social network has profound significance in the fields of behavioural science, psychology, information science and other social studies. A health site helps its members for sharing various health issues. This Paper lists out Mathematical representation of social networking health site. We also discuss friendship suggestion algorithm which is an important concept in social network. In this paper different concepts and mathematical formulae used in social networking are also discussed. The main aim of this paper is to layout a foundation for further research in mathematical representation and analysis of social networking sites. Keywords cardinality, centrality, clustering, health sites, social network. S I. INTRODUCTION OCIAL network is the set of actors {people or entities} that are connected to each other through certain relationship. That relationship can be of any form may be a friendship, association, behavioral interaction, formal relation, biological relationships. We can represent the relationships between the sets of actors through socio metric notations, graph theoretic notations etc. The above types of representations are very useful in understanding the behavior of the Social network system. Section A deals with the mathematical representation of health sites using necessary mathematical notations, set theory and relevant inputs and outputs. Section B deals with the concepts of the representation of relationship ties using matrix notations. In section C we discuss the concept of centrality. Section D deals with concept of clustering and necessary theorems. In section E we discuss friendship detection algorithm. Section A I. We represent social network by the set S = {A C, R L, I O, O P C R } A C represents the set of actors in System S. R L represents the set of relational ties between actors. Abhishek Burli is with Pune Institute of computer Technology, University of pune.( abhi.burli@gmail.com) Archit is with Pune Institute of computer Technology, University of pune.( archit.rai@gmail.com) Mandar Chitale is with Pune Institute of computer Technology, University of pune.( mandar91688@gmail.com) Prof Anupama Phakatkar is with Pune Institute of computer Technology, University of pune. I O represents the set of inputs. O P represents the set of outputs. C R represents the Constraints. For health site set of actors A C is divided into three subsets. A C = {P, D, U A R } P = {P 0, P 1,..., P N } P represents set of patients. D = {D 0, D 1,..., D N } D represents set of doctors. U = {U 0,U 1,..., U N } U represents set of other users. Relational tie can be measured depending upon type of relation. To generalize consider a set M with K elements in it. M = {m 1, m 2,...,m K } Consider a single relation to be measured on set M for example friendship. Now if a tie is present between m 1, m 2 we can say that this pair is an element of set of collection of unordered pairs R L. R L = {R 1, R 2....R N } R 1 = <m 1, m 2 > We can formulate the number of unordered pairs for a given set by the formula. For the above defined set M. N p = K(K-1) N p is total number of unordered pair. The cardinality of set R L is N P. II. ( output sets) I O = { U N, P w, M g, P v } O P ={ P I, P V } U N ->Username P w ->Password Pw R ->Rules of Password P I ->Profile Information P v ->Profile View M g ->Message T X ->Text I m ->Image V D ->Video U L -> Name of web page or site U N = { [a-za-z]* \ [0-9]* } P w = {[a-za-z]*\[0-9]* PW R } Rule Pw R = {5<Len(P W )<12} 108

2 U N & P w -> P I P I = {U N, P w, I m, T X, V D, U L, files} P v = { I m, T X, V D, U L } P I - ->P V M g = {T X, S L } F b = {Tx,} P I Is An Object with Data P v And A Set Of attributes denotes as A: P v = {text,image, video, url } A=(A 1, A 2, A 3,..., A x ) Each Attribute A x has relationship Rel(A X ) and a set of value V(A X ) = { V 1 (A X ), V 2 (A X ),..., V Y (A X )}. V Y (A X ) is also called metadata of P v III. C R (constraints) Following are Constraint of system 1. As the size of the network increases the interaction between its members reduces. 2. Boundary Set for actors is difficult to determine. Hence researchers define actor set boundaries based on relative frequency of interaction 3. Tie strength between two nodes in the network varies inversely to the distance between them [11]. According to power law the probability of decay is given by P(d)~d -x P(d) defines probability of tie strength. d defines distance between the nodes IV. Mapping cardinality Fig 1 Many to Many relationship between doctors and patients A mapping cardinality is a data constraint that specifies how many entities an entity can be related to in a relationship set. When we consider social networking sites actors are the entities Here if we consider all the actors in the network we can identify that many to many relation exists between them. In health sites we have identified as patients & doctors as actors many to many relation between them is shown in fig 1. Fig 2 one to one relationship between doctors and patients As we are dealing with health sites, at times specific doctors are refered to specific patients. In such cases we can say that one to one relation exists between them. SECTION B Let us begin with some simple definitions. A social network also called a graph in the mathematics Literature is made up of points, usually called nodes or vertices, and lines connecting them are called edges. Mathematically, a social network can be represented by a matrix called the adjacency matrix A, which in the simplest case is an n*n symmetric matrix, where n is the number of vertices in the network. The adjacency matrix has elements 1 if there is an edge between i and j, Ai j = 0 otherwise. { The matrix is symmetric since if there is an edge between i and j then clearly there is also an edge between j and i. Thus Ai j = Aji. To represent the relation between the set of patients and doctors, an adjacency matrix can be used to do so, where the set of sending actors is represented as rows and the set of receiving actors is represented in columns. If the value is 1, it indicates that there is a relationship between two actors. TABLE 1 D 0 D 1 D 2 D 3 P P P P The above matrix can also represented using associativity rule: TABLE II P 0 P 1 P 2 P 3 109

3 D D D D But the above adjacency matrix only gives us the existence of a relationship between two set of actors, but it doesn t talk about the strength of the relationship. To analyze the later aspect, we extend the adjacency matrix to represent the values other than 0 and 1 and each value will represent the number of times two actors have communicated with each other. TABLE III P 0 P 1 P 2 P 3 D D D D The associative rule will also hold true. TABLE IV D 0 D 1 D 2 D 3 P P P P The concept of in-degree and out-degree is used to determine the number of times one particular actor communicates with the other actors. Example: TABLE V In-degree Out-degree P P P P The above table highlights the in-degree and out-degree for the set of patients. From here a threshold value can be set to categorize the actors who communicate constantly to the ones who don t. From the above example, if we set the threshold value as 10, then we can say that actor P 1 falls below that specified value. Such actors are known as isolates or lurkers [13]. SECTION C CENTRALITY: One of the advantages of graph theory in social network analysis is to identify the most important actors in a social network. Centrality is the concept which allows us to identify the most important, actors in social network. Apart from social network centrality is also used in other network systems such as biological networks, traffic networks, urban networks [7][8] Many methods have been used in defining centrality. The relevant ones are degree, closeness and betweenness centrality. Degree centrality: The most fundamental measure of centrality is degree centrality. In degree centrality the degree of the actor is most important. Degree centrality of a node is defined as the Tally of number of nodes to which it is directly linked [3]. Consider a network consisting of g nodes the degree centrality of node n i is Cd(n i ) = d(n i ) (1) d(n i )-. is the degree of node ni To compare centrality of nodes across network of different sizes the above formulation is normalized to CD(n i ) = d(n i )/g (2) One disadvantage of degree centrality is that it takes into account only direct links to that node. Closeness centrality: In this view of actor centrality the closeness is measured. The measure focuses on how close the actor is to the other actors in the network. Closeness centrality is given by the sum of geodesic distance between a particular node and the rest. Geodesic distance between a pair of nodes in a graph is the length of the shortest path between the 2 nodes. For a network consisting of g nodes the closeness centrality is defined as [1] C c (ni) = (3) Closeness centrality uses geodesics for measurement. Hence we can infer that centrality is inversely related to distance. As the node grows farther apart in distance its centrality reduces The above formulation can be normalized to[4]. CC(ni)=(g-1)Cc(ni) (4) Betweenness centrality: In a social network for the interaction of a pair of actors they need not be adjacent. They can be two random non adjacent actors. The interactions between such a pair of actors depends on the actors who lie in the path of these two actors. The idea of betweenness centrality depends on the node being central if it lies between many other nodes. Consider a network consisting of g nodes. For measuring betweenness centrality of node ni, sum of the shortest path that go through ni over the sum of shortest path of all pair of nodes is considered. Again lets consider a network of g nodes. Betweenness centrality is measured using the formulation C b (ni) = (5) The above formula can be standardized to match any scale of network by the formula [4] C B =C b / [(g-1)(g-2)/2] (6) SECTION D Clustrability: According to Davis a signed graph is clustrable, or has a 110

4 clustering, if one can partition the nodes of the graph into finite number of subsets such that each positive line joins two nodes in the same subset and each negative line joins two nodes in different subsets. The subsets derived from the clustering are called clusters. Here Davis only considered complete signed graphs but signed graphs are rarely complete. Thus his idea is relaxed to allow some ties between actors within cluster to be absent [1]. Theorem: A signed graph has a clustering if and only if the graph contains no cycle which have exactly one negative line [1]. negative line [1]. When the signed graph is complete, it is possible that we have a unique Clustering. The lack of some lines between nodes in a signed graph makes it more difficult to check whether the graph is clustrable or not, and if such a graph is clusterable, we have no guarantee that the clustering is unique. A complete signed graph which is clusterable has a unique clustering, and this clustering can be verified by looking just at all the triples. SECTION E Friendship Suggestion Algorithm [2]: Central It is the node which is in focus. Relative: It is the node directly connected to Central. Friend suggestion Algorithm-: Fig 3 G=6 nodes Number of lines=8 Number of positive lines=2 Number of negative lines=7 It is not a complete graph. There are 4 cycles of length 3which are {n1,n2,n6,n1} {n2,n3,n6,n2} {n3,n4,n5,n6} {n3,n5,n6,n3} Since two cycles have a negative sign we can say that the graph is not balanced. None of the cycles contain exactly one negative sign so according to the theorem graph is clustrable. There are 4 clusters in the graph :{n4,n5,n6},{n1},{n2},{n3} But above mentioned clusters are not unique since n2 and n3 can be joined to form a single cluster as they are not connected by a negative line. This lack of uniqueness is due to the fact that graph is not complete. If there are more than one pair of actor with negative lines then these two actors form two different clusters. Following three statements are equivalent for any complete signed graph. Graph is clustrable. Graph has unique clustering. Graph has no cycle(of any length) with exactly one Fig 4 a Graphical representation of friend suggestion algorithm Blue Node represents central Red node represents relative Green node are friends of relative Dynamic friend suggestion feature of social network application provides central (user) a list of users whom central may know. As functionality increases more parameters are added. If user's friend has a friend in common there is a high probability that user might befriend with the person as well. When we have to find the friends who are common among central's friends we construct a graph of the central and friends of central's friends. After that we will do BFS.BFS searches the graph outward from central, then it goes to the relative and after that to the friends of relative. Stepwise execution of algorithm: 1. First the central with the entire relative is drawn. 2. In the next step all friends at distance 2 (disregarding relatives to avoid repetition) is added to the sociogram. 3. Then we use this sociogram to find the nodes at depth 2 that have the greatest degree of cardinality. II. CONCLUSION In this paper we have given the mathematical representation of social networking health sites. We have also given measures of concepts like centrality, cluster-ability and friend 111

5 suggestion algorithm which are helpful in social network analysis,. The documentation concepts and representation provide a solid foundation for rigorously analyzing social networks. REFERENCES [1] S.Wasserman and K. Faust. Social Network Analysis: Methods and Applications. Cambridge University Press, [2] Junhua Ding Cruz, I. ChengCheng Li Dept. of Comput. Sci., East Carolina Univ., Greenville, NC, USA A Formal Model for Building a Social Network, Service Operations, Logistics, and Informatics (SOLI), 2011 IEEE International Conference, July 2011 [3] Lianhong Ding Peng Shi Sch. of Inf., Beijing Wuzi Univ., Beijing, China,Social Network Analysis Application in Bulletin Board Systems Intelligence Science and Information Engineering (ISIE), 2011 International Conference,20-21 Aug [4] John P Scott. Social Network Analysis: A Handbook. SAGE Publications, 2nd edition, [5] L. Freeman, Centrality in social networks: conceptual clarification, Social networks, vol. 1, no. 3, pp , [6] P. Crucitti, V. Latora and S. Porta, Centrality measures in urban networks, Physics, 2006, [7] E. Kohle, R. Mohring and M. Skutella, Traffic networks and fowsover time, TU-Berlin Technical Report, Technical Report, 2002, pp [8] M. Girvan and M. E. J. Newman. Community structure in social and biological networks, Proc Natl Acad Sci USA, 2002, 99, pp [9] A. Hanneman and M. Riddle, "Introduction to social network methods," online at hanneman/nettext/, [10] Carlson, N. Facebook Has More Than 600 Million Users,Goldman Tells Clients. Business Insider. Jan. 5,2011. April 8,2011. [11] Jukka-Pekka Onnela1, Samuel Arbesman1.,Marta C. Gonza lez2, Albert-Laszlo Baraba, Nicholas A.Christakis,Geographic Constraints on Social Network Groups [12] Ding, 1., He, X., Formal Specification and Analysis of an Agent-Based Medical Image Processing System, International Journal of Software Engineering and Knowledge Engineering, Vol. 20, No. 3, pp. 1-35,2010 [13] Erlin, Norazah Yusof, Azizah Abdul Rahman, Department of Software Engineering, Department of Information Systems, Universiti Teknologi Malaysia, Analyzing online asynchronous discussion using content and social network analysis 2009 Ninth International Conference on Intelligent Systems Design and Applications. 112