Numerical Formulation and Simulation of Social Networks Using ...

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 11, Number 3 (2015), pp. 1253-1264 © Research India Publications http://www.ripublication.com

Numerical Formulation and Simulation of Social Networks Using Graph Theory on Social Cloud Platform J.VijayaChandra Research Scholar, DepartmentofComputer Science andEngineering, K.L.University,Guntur Dist., A.P., India. E-mail:[email protected] Dr. NarasimhamChalla Professor & HOD, Department of Computer Science and Engineering, S.R College of Engineering, Ananthasagar, Hasanparthy, Warangal, India. E-mail:[email protected] Dr. SaiKiranPasupuleti Professor, DepartmentofComputer Science andEngineering, K.L.University,Guntur Dist., A.P., India. E-mail: [email protected] Dr. K. Thirupathi Rao Professor, Department of Computer Science and Engineering, K.L. University, Guntur Dist., A.P., India. Email: [email protected] Dr. V. Krishna Reddy Professor, Department of Computer Science and Engineering, K.L. University, Guntur Dist., A.P., India. E-mail: [email protected]

Abstract Social Network is a network where a set of individuals, groups or organizations interact and communicate, explore the social relationshipsand diagrammatically represented as a Graph. Social Networking Service is the service where people can share data resources and communicate in bidirectional with web multimedia. In this paper we discussed metrics in social networking such as clustering coefficient, cohesion, reach and rediality.

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Social network graph theory is related to link analysis, structural analysis and centrality measures; we also concentrated on Numerical formulations and simulations using datasets which are used for analysis of social network.Risks, Threats and vulnerabilities on social network areinterrelated to social engineering and part of Advanced Persistent Threat. Security approaches and defense methods are discussed, developed and proved.Mathematical and Statistical theorems are developed with respect to the Algorithmic approach for social networks security in social cloud platform. Social Patterns are the components and pieces of interactivity that are the building blocks of social experiences. We focused on the electronic connections and social tools that are changing the way that we interact with one another, which are designed and simplified to expand online experiences in social networks. We did considerations as practical approach where we applied statistical tools on network data and data analysis is given. Security is the principle concern; the main objective is to protect the social cloud from advanced risks, threats and attacks. Keywords: Advanced Persistent Threat, vulnerabilities, cloud security, Defense in Depth, Security Management, Social engineering, graph theory, Social Networking, Social Patterns.

Introduction Social Networks are represented by using graph theory as a set of points or vertices, joined by in pairs by lines or edges.A graph G is a finite non-empty set of objectscalled vertices together with a set of unordered pairs of distinctvertices of G called edges. The Social Networks is a set of actors mathematically known as agents or nodes or points or vertices. In the Social Network their relationship is denoted as links or edges or ties. The vertex set and edge set of G aredenoted by V (G) and E(G) respectively. The number of verticesin G is called the order of G and the number of edges in G is calledthe size of G. A graph is trivial if its vertex set is a singleton, if two actors are connected with each other for communication and if the order p and size q is called a (p, q) -graph.If e =( u,v) is an edge of G, we say that u and v are adjacent and each is incident with e. If two distinct edges are incident with a common vertex, then they are said to be adjacent edges. The social network is a map of specified ties;A graph is connected if every pair of vertices are joined assuch as friendship between the nodes known as path. Let u and v be vertices of a graph G. A u-v walkof G is a finite, alternating sequence u = u0,e1,u1,e2,….,ep,up= v of vertices and edges in G beginning with vertex u and ending with vertex v such that ei = ui-1,ui, i= 1,2,….,p. The number p is called the length of the walk. The walk is said to be open if u and v are distinct vertices; It is closed otherwise. A walk u0,e1,u1,e2,….,ep,up is determined by the sequenceu0,e1,u1,e2,….,ep,upof its vertices and hence we specify this walk byW:u0,u1,….,up. A walk is which all the vertices are distinct is called a pathThe nodes which an individual is thus connected are the social contacts of that individual

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for any set S of vertices of G is denoted by G[S] or induced subgraph. The network can also be used to measure social capital – A simple graph that contains every possible edge between all the vertices is called a complete graph. A complete graph with n vertices is denoted by knthe value that an individual gets from the social network. These concepts are often displayed in a social network diagram, mathematics plays a significant role in solving problems related to routing and network using graphic structure in order to understand nodes and ties, where nodes are the points and ties are the lines [1]. Related Work Cloud Computing draw a great attention of industry, researchers, academics and business world, as most talented and emerging paradigm which provides services over the internet where there are great challenges emerged in securingthe sensitive confidential user data against intruders. Cloud Computing provides mobility which is developing very fast, it facilitates the employees workfrom where they want, it provides a complete stack which contains applications, data storage and device management. It provides the employeesthe flexibility to work from where they are, that is to be productive outside the office while addressing very approachable, flexible and secured system. Cloud Computing allows more open accessibility and easier and improved data sharing. Data is uploaded into a Cloud and Stored in a datacenter, for access by users from the data center. Security is a major Issue; these are mainly deal with identity and access management, prevention of data loss and malware attack control management. When an organization moving to a cloud, so many issues will be on screen among them are Is my data secure on cloud? And can others access my confidential data? So Security is the key inhibitor to cloud adoption. Cloud Storage enables users to store data remotely and retrieve on requirementof the cloud user, without maintaining any software or hardware. Cloud storage in Cloud environment is different comparing with other architecture where the user’s data is moved to large data centers, which is remotely located, on which user does not have any control, So Monitoring and Management System plays a great role in Cloud Security. Cloud Computing builds a future-proofed cloud, which increases automation,agility and control, which simplifies Information Technology and empower employees which improves productivity and collaboration and mobility. In theCloud, everything is data, everything can be copied or moved, in the past theservers were physical hardware, now the hardware is virtualized and servicesare software instances. All the devices were virtualized such as routers, switches etc., Virtualization is a fundamental technology in cloud computing which transformed the face of the modern data center and referred the abstraction ofcomputer resources such as processor, storage, network, memory, applicationstack and database. Virtualization is a technology that enables multi –tenancycloud that can share resource platform for all tenants. The major task in Social cloudcomputing is securing the virtual server. Data and Information Security in Cloud monitored and maintained by theconcepts of CIA and AAA where CIA Triad is the Security concept in

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cloudcomputing which means Confidentiality, Integrity and Availability where asAAA is Information Security Mechanism used in Cloud Computing whichmeans Authentication, Authorization and Auditing [2]. The significance of security plays a great role in penetration testing. The major areas of penetration testing are Data Storage, Application, Operating Systems and Networks. It should be a continuous effort to secure the web applications from malicious access. Penetration testing involves testing a web application which is running remotely, with the intention of finding possible vulnerabilities. The best approach to do penetration testing is to do a series of methodological and repeatable tests, while working through all of the different vulnerabilities. Vulnerabilities and intrusion detection techniques should be kept in mind and a continuous monitoring system should be used at the time of developing applications or software [3]. Implementation of Graph Theoretic Algorithms using computer language was done from early ages using FORTRAN, later using by ALGOL. Social Network Analysis is the application for Network Theory modeling and analysis of social system the combined tools analyzing social relations and theory for explaining structures which merge from the social interactions. The idea of studying societies through networks is not new one for computation and massive new data sourcing, the social network analysis is a beginning to apply all types of scales of the social systems, now a days social networks are used from local communities to international politics. Traditionally social networks are formed among different types of societies based on different parameters such as individuals that are age, occupation, habits, likes and dislikes. Where social networks bring all the people whose character and behavior are same, as all the birds of same feathers will fly together. Even different organizations are organizing social network groups to communicate and share data among their employees and persons related to the organization [4]. Graph Theory has strong roots in Mathematics and Computational Sciences;it connects with major areas of Mathematics such as such as algebra, geometry, topology, numerical analysis, matrix theory, combinatorics, operational research and so on. It is majorly used in Computational Sciences for calculating shortest path distance and routing procedures in networks. It has major growth and applied in areas of Electrical Engineering and Computer Engineering [5]. The cloud network topology can be represented graph theory and flow of logic can be explained, It is a very large virtual network linked with different components such as routers, cables, bridges, gateways and other network accessories over a long distance. Thus the computing resources will work together via the network to run the subtasks while accessing necessary data from the data resources[6].

Social Networks In social networks the individuals interact and organize, the result can be configuring the relation among themselves. Social network allows social analytics and scientists’ uses formal language and use to go analysis and compared to go to formal mathematical and sophisticated analysis, compare the value of these properties. The compound based analysis is a powerful mathematical analysis which describes the

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social network systems. A cloud based social network infrastructure allows only authorized users to access its resources and communicate with other users [7]. Social Network is designed to support multiple application models and communication protocols, where new nodes can be connected or deleted to the network which takes support of the distributed paradigms such as cluster, grid and cloud computing [8].

Mathematical Analysis for Social Networks The Social Network Analysis within graph theory is represented with various measures of the centrality are degree centrality, betweenness, closeness, and eigenvector centrality. A graph G : (V , E ) with n vertices, the degree centrality CD (v) for vertex v is: CD (v ) 

deg(v) n 1

Calculating the degree of centrality for all nodes V in a graph takes ϴ(V2) in a dense adjacency matrix representation of the graph, and for edges E in a graph takes ϴ(E) in a sparse matrix representation. Let V* be the node with highest degree centrality in G. Let X :=( Y,Z) be the n node connected graph that maximized the following The highest degree centrality in X | y|

H   CD ( y*)  CD ( y j ) j 1

H is maximized when the graph X contains one node that is connected to all other nodes and all other nodes are connected only to this one central node H  (n  1)(1 

1 )  n2 n 1

So the degree of centrality of G reduces to |V |

 [C C D (G ) 

D

(V *)  CD (Vi )]

i 1

n2

The closeness in a network with n nodes is represented mathematically as:

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n 1

Cc (n j ) 

n



d (ni , n j )

k i , j k

The betweenness centrality is a measure of a vertex within a graph,

 st (v) s  v  tV  st The Closeness centrality of a vertex is in a graph, 1 Cc (v)   tV \v dG (v, t ) CB (V ) 



Eigenvector centrality is a measure of the importance of a node in a network. In calculates all the scores to all nodes in the network based on the principle that connections to high-scoring nodes, for example Google’s PageRank is a variant of the Eigenvector centrality measure. Let Aijbe the adjacency matrix, where Aij= 1, if the ithnode is adjacent to the jthnode, otherwise Aij= 0. Let xi denote the score of the ith node.

xi 

1 



jM ( i )

xj 

1 N A x  j 1 i, j j

C Path Centrality of all vertices of the probability that a message originating from one node and traversing to another node,assuming that the message traversals along random simple paths of at most C edges using shortest path distance. In a network betweeness acts as hub and measure every node from the account. Let P(i,j)be the number of shortest paths between nodes i and j, and Let Pk(i,j)be the number of shortest paths between i and jthat includes nodes K Bk 



( i , j )E

Pk (i, j ) / P (i , j ) (n  1)( n  2) / 2

This gives the fraction of shortest paths that is overall possible pairs of nodes to go through node k and to get maximum flow in a flow network. Algorithmic Analysis for Social Networks Input: Given a Social Network G=(V,E) with flow capacity c, a source node s, sink node t, where P is the Path traveled from s to t Output: Compute a flow f from s to t with maximum flow in a flow network and with shortest path distance.

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Start of Algorithm 1. Let P(i,j)be the path between nodes i and j 2. Let Pk(i,j)be the numberof shortest paths between i and jthat includes nodes K Pk (i, j ) / P (i , j ) 3. Calculate Bk   ( i , j )E ( n  1)( n  2) / 2 4. f (i, j )  0edges (i , j )  P where C f (i , j )  0 5. Create the residual network G(x); While there is some directed path from s to t in G(x) do Let P be the path from s to t in G(x)  :  ( P ) Send  units of flow along P; Update the residual network (r’s); End{while loop} 6. End of Algorithm {The flow X is now Maximum with shortest path}

Experimental Analysis for Social Networks Personal Social Network was designed, implemented and developed with platform of Model View Controller (MVC) using Php Data Object (PDO) and combination of web design technologies such as HTML, Php, Javascript, JQuery, XML and CSS. We investigate the present social networks and algorithms used that implicit data generated through monitoring the users interactions with the underlying system to understand hidden users preferences in a better way. Once the user interests are known, the system provides the personalized recommendations that suits to the needs of the user, depending up on their interests [9]. The major features of developed Social Network are friend system, follower system, photo upload system, albums system, posting system, profile for users, Post like system, messaging system, chatting system, editing profiles, and commenting system. In this work we focus on navigation of webpages, user interfaces and network structures of Advanced Private Social Network [10].

Simulation Results of the social network MATLAB is a platform for scientific calculations and high level Programming through an interactive environment that allows for accurate resolution of complex calculation taskswhich provides the users all the possibilities to find the shortest path with maximum flow in flow networkof social networking. On line social networking getting reputation, to examine we took mostfamous type of social networks that isFacebook. The methodology we used to collect the data is by using public web interface provided by the sites, this methodology provides large data sets from multiple sites.We compare the properties, behaviors and structures of social networks, first we studied the link symmetry in social networks, where the fact is that links are directed can be useful to locating the content in information networks. The target of the link may reciprocate by placing a linkpointing

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back at the source. Analysis of the symmetry in social network is shown in below graphs [11].

Figure 1: Graph of Facebook obtained by using fraction of links Vs fraction of users

The basic concept of a social cloud is secured and feasible within the context of a social network and manageable for an average Facebook user. The Social network of users is captured via the existence of performance between users. A Graph in symmetric nature of social links effect’s the social network; symmetry increases the overall connectivity of the network and reduces its diameter. Symmetry can also make the diameter to identify reputable sources of information just by analyzing the network structure, because the reputed user tends to dilute their importance. When pointing back the arbitrary users who link to them [13].

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Figure 2: Graph of Facebook Users Friends Vs Cumulative degree distribution (%)

Facebook is the best example for large scale social networksat the starting Facebook degree of separation between any two Facebook users is smaller than the commonly cited six degrees, and over the past three years as Facebook grownfinally we observed that while the entire world is only few degrees away [12]. Social Cloud platform Facebook enables the sharing of infrastructure resources between friends via digitally encoded social relationships. The cumulative distribution function provides the distribution in two ways that is continuous distribution and in discrete distribution. Cumulative distribution function at continuous distribution where the x

continuous random variable X is given as F ( x)  Pr( X  x) 

 P( X )dx

, to calculate



d F ( x)  P( x) dx Cumulative distribution function at In-discrete distribution where the discrete random variable is Xiis given as f ( x)  Pr ( xi  x )   P ( x ') P(X) we have to differentiate the F(x) so

x ' x

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Figure 3: Graph of Facebook Active Users in millionsVsYears from 2004-2013

Simulation is the manipulation of the model of a system enabling to observe the behavior of the system in a setting similar to real life. Simulators are widely used to test, verify, and analyze the performances of distributedalgorithm and social networks. We have briefly described simulators that are in use for it verifies the algorithms and protocols used in the social network. It also specifies the topological view and controls the network by using the network links for communication. The performance of the distributed social network is calculated based on metrics such as time complexity, bit complexity, space complexity and message complexity [13].

Conclusion and Feature work Network communities are group of vertices similar to each other, community detection is an assignment of vertices to communities.A Non-overlapping community is used in social network that means every vertex belongs to a single group.The social network focuses on security, different secured distributed algorithms which focuses on scenario which the interacting players know the identity of all nodes in the network, and protects the structural information that is edges of the network [14].Future attempts of modeling human behavior in the Web can benefit from the methodological framework along with social networks, Intelligence systems and ontologies presented in this work to thoroughly investigate such behavior. Social network users should has to provide their preferences as well as methods to detect

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them automatically. If one wants to resort to single model selection techniques, we would recommend usingadvanced social networks based on intelligence systems and ontologies that are computationally feasible [15].

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