Correlation analysis among social networks measures for directed scale free network

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Directed scale free network is a network which exists in WWW (World Wide Web), phone call, E.Coli. metabolism, etc. There are many network measures which are used to apply on directed scale free network but it is not known how much they are
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  Correlation analysis among social networks measures for directed scale free network Md. Sarwar Kamal, Rakib Hassan Pran ABSTRACT: Directed scale free network is a network which exists in WWW (World Wide Web), phone call, E.Coli. metabolism, etc. There are many network measures which are used to apply on directed scale free network but it is not known how much they are statistical dependent or correlated (statistically similar) with each other. In our research, we selected 12 popular social network measures which were applied on 48,945 directed scale free network graphs (almost all types of directed scale free network) which are generated by “Bela Bollobas, Christian Borgsy, Jennifer Chayesz, Oliver Riordanx’s directed scale free graph” algorithm [1] and found out how much correlated these 12 social network measures with each other by using PEARSON CORRELATION. Then we found out which is better for computation time among correlated measures .We found out 9 measures are highly correlated among 12 measures. Our Core_i7 processor computer needed more than 112 hours ( 4 days 16 hours) to find out highly correlated measures for directed scale free network . INDEX TERMS: Correlation, directed scale free network, clustering, centrality, network measures, similarity INTRODUCTION: In our world, somehow everything is connected which implies that network exists in every place. It is now an essential part in computer science, physics, in case of micro-biology. From medical science to engineering, network science establish its strong hold to direct the way of victory.  In this paper, we are going to research on directed scale free network by applying 12 popular measures and finding out similarity among them. We are going to solve the two question which are “1.   Is there any statistical dependency (correlation) exists among 12 types of social network measures or analysis (SNA) algorithm if we apply these 12 types of algorithm on directed scale free network?” “2.   If correlation exists then, among correlated algorithms, which algorithm will take less computation time?” So, we used “Bela Bollobas, Christian Borgsy, Jennifer Chayesz, Oliver Riordanx’s directed scale free graph”[1] algorithm and applied 12 popular social network measures on them and uesd core i7 processor compuer to find out correlation among those nework measures. Previously we see, Correlated group exists in undirected scale free network [21]. SCALE FREE NETWORK PROPERTIES: Scale free network is a network which degree distribution follows power law. In scale free network, the fraction P  ( k  ) of nodes in the network having k   connections to other nodes goes for large values of k   as  (  )~   ϒ   Or,  lim → (  )   ϒ  =1  Where, ϒ  is a parameter which value is generally in the range 2< ϒ <3. Here, P(k) =                              Directed scale free network is nothing but scale free network with directed edge which in degree distribution and out degree distribution follows power law. SOCIAL NETWORK MEASURES: Social network measures mean ways of Social Network Analysis (SNA).  There are several types of social network measures. Different types of social network measures are used to measure different types of network attributes like as centrality, clustering , closeness , vitality , density ,etc. Each social network measure follows a specified algorithm to find out particular attribute of network.In this research, we used 12 types of SNA algorithms which are given bellow: 1.   Degree centrality[2] 2.   In-degree centrality[2] 3.   Out-degree centrality[2] 4.   Square clustering[3] 5.   Betweenness centrality[4][5][6][7] 6.   Closeness vitality[8] 7.   Closeness centrality[9] 8.   Load centrality[10][11] 9.   Clustering coefficient[12] 10.   Eigenvector centrality[13][14] 11.   Page rank[15][16] 12.   Average neighbor degree[17] DIRECTED SCALE FREE NETWORK GENARATION ALGORITHM MODEL: If we want to find out correlation among 12 SNA algorithms for directed scale free network then it’s not enough for applying these algorithms for few directed scale free networks. Other hand, it’s impossible to apply these 12 SNA algorithms for all directed scale free networks of the world. So, we need a technique to solve this problem.  We need to categorize directed scale free network by its type and generate possible types of directed scale free network. We used a DIRECTED SCALE FREE NETWORK GENERATION ALGORITHM for categorizing types and generating a large set of directed scale free networks. We selected “Bela Bollobas, Christian Borgsy, Jennifer Chayesz, Oliver Riordanx’s directed scale free graph” algorithm to generate a large set of random directed scale free network. There are 5 parameters in this algorithm. These are α , β , ϒ , δ in and δ out. We used these five parameters to categorized type of directed scale free network. MODEL DESCRIPTION: Here, α  , β , ϒ , δ in and δ out are 5 parameters where  +   +   =1……………  (1)  G(n) is a directed random graph with n edges, N(n) nodes. Set of nodes of G(n) is Vn ; so |Vn| = N(n). Set of edges of G(n) is En = {(u,v) ∈   Vn x Vn : (u,v) ∈  En } and In-degree of v is   (v) and out-degree of v is   (v). With probability  , append to G(n-1),a new node v ∉     and create directed edge v → w ∈    with probability   (  )+     − 1+     ( − 1)    With probability  , append to G(n-1),a new node v ∉      and create directed edge w       → v ∉     with probability   (  )+    − 1+     ( − 1)  With probability  , create new directed edge between existing nodes. Here, created edge v ∈     →  w ∈    with probability (    (  )      (  ) )(    (  )      (  ) ) The proportion of nodes with in-degree i will be   =lim →  (  )  (  )  Where, N(n)=number of nodes in Vn and Ni(n)=number of nodes with in-degree i The proportion of nodes with in-degree j will be   =lim →  (  )  (  )  Where, N(n)=number of nodes in Vn and Nj(n)=number of nodes with out-degree j
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