A LINE GRAPH as a model of a social network

Transcription

1 A LINE GRAPH as a model of a social networ Małgorzata Krawczy, Lev Muchni, Anna Mańa-Krasoń, Krzysztof Kułaowsi AGH Kraów Stern School of Business of NY University

2 outline - ideas, definitions, milestones - line graphs - LiveJournal data R. Albert ea, Nature 46, (27 July 2)

3 Floor-level networing is often staged to loo lie friendly socializing. Nevertheless, it has an instrumental goal: useful contacts. Every networer nows this, so the instrumentality is not latent. It is not completely manifest, either. I presume that there is a mutual implicit contract of not breeching the situation by stating the instrumental goal aloud. The getting of contacts is a conscious but tacit function of socializing. Juha Klemelä, Turu University Managing Mixed Emotions in the Layered Ritual Reality of Networing Events, XVII ISA World Congress of Sociology, Gothenburg, -7 July 2

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5 Cambridge UP, 994 5

6 b. 97 B.Sc. in physics Ph.D. in theoretical and applied mechanics b. 959 B. Sc. in mathematics PhD in applied mathematics D J Watts, S H Strogatz, Collective dynamics of 'small-world' networs, Nature 393 (998)

7 C D Connectivity matrix ( sociomatrix ) A B Clustering coefficient C N i N ( ) i 2L i i 7

8 Mean free path ( diameter ) N L( N) shortest _ path( i, N( N ) i, j Small-world effect j) L( N ) ln( N ) te-in.faceboo.com/group.php?v=info&gid= D J Watts, S H Strogatz, Nature 393 (998)

9 Erdös-Rényi networs -Tae N nodes -Connect each pair with probability p Mean degree <> = (N-)p igraph.sourceforge.net/screenshots2.html Degree distribution P( ) e! 9

10 PhD in complex networs 2 b. 967 MSc in physics and engineering PhD in physics actors www power grid R Albert, A-L Barabasi, Emergence of Scaling in Random Networs, Science 286 (999) 59

11 Growing networs construction for M = Nodes to attach a new node are selected with probability: - constant (exponential graphs) - proportional to their degree (scale-free graphs)

12 networs N # d C actors 45x^4 25x^ www Altavista 2x^9 2x^ /2.7 coauthorship - math 25x^4 5x^ coauthorship - phys 5^^4 25x^ Phone calls 47x^6 8x^ Web chains in water Protein interactions Sexual contacts Words in sentences 46x^4 7x^ M E J Newman, SIAM Review 45, 67 (23)

13 Assortativity r j j( e j 2 q q j q ) where q ( ) p e j fraction of lins between nodes j, MEJ Newman, PRE 67 (23)

14 Why not ALL social networs are scale-free Michael Schnegg, IJMPC 7 (26) 67 4

15 Why social networs are different from other types of networs In this paper we have argued that social and non-social networs differ in two important ways. First, they show distinctly different patterns of correlation between the degrees of adjacent vertices, with degrees being positively correlated (assortative mixing) in most social networs and negatively correlated (disassortative mixing) in most nonsocial networs. Second, social networs show high levels of clustering or networ transitivity, whereas clustering in many non-social networs is no higher than one would expect on the basis of pure chance, given the observed degree distribution. We have shown that both of these differences can be explained by the same hypothesis, that social networs are divided into communities, and non-social networs are not. MEJ Newman, J Par, PRE 68 (23)

16 MEJ Newman, J Par, PRE 68 (23)

17 It is used to assume but it could be

18 Role of line graphs here In the model proposed here, a social networ is the line graph of an initial networ of families, communities, interest groups, school classes and small companies. These groups play the role of nodes, and individuals are represented by lins between these nodes. M J Krawczy et al, arxiv:.246

19 line graph the construction a b c d

20 2 2. If i,j are in the same row or column, then the element C(i,j) of the transformed matrix is A C B D Algorithm. assign numbers to the elements of the connectivity matrix above the diagonal. Mae the matrix symmetric

21 2 2, 2) ( 2) )( ( ) ( ) ( ) ( ) ( ) ( n m n n m m n t n P n n np m mp n np m mp n np P Degree distribution of a line graph n m A. Mańa-Krasoń et al, Comp. Phys. Comm. (2) 8 2

22 Degree distribution P() Erdös-Rényi networs P( ) t e 2 n ( n )!( n )! e 2 (2 )! exponential networs 4 ( c) P t( ) ( )( 2)( 3) c 6 22

23 The degree distribution of a line graph on a scale-free networ P t () <>=6,6 23

24 Clustering coefficient in line graphs n m C 2 n np( n) m mp( m) ( n )( n ( n m 2) ( m 2)( n )( m m 3) 2) C Scale-free Exp E-R <> 24

25 Line graphs - assortativity n m r ' ( ) np( n) mp( m) n m r np( n) mp( m) n m r rp( r)( n m 2) rp( r), m r, m 2 r 2 A. Mańa-Krasoń et al, APPB Proc Suppl 3 (2)

26 Assortativity < ()> Erdös-Rényi networs ' ( ) exponential networs 26

27 Assortativity of a line graph on a scale-free networ < ()> 27

28 LiveJournal LiveJournal is a remarably popular platform for personal blog management, populated with over 8 million blogs and over million of communities. LiveJournal was among the first of such platforms available online and it still remains one of the most active and popular. Its users manage personal blogs where they share their daily experiences, political views or discuss news events. Users can also comment on posts of other users. We defined the networ nodes to correspond to personal blogs. Directional lins connecting these nodes represent the record that a particular user (owning one blog) monitors another blog (owned by another user). M J Krawczy et al, arxiv:

29 LiveJournal: data N=8.* 6 ; #=25* 6 simulations: N=9* 3 P() 29

30 LiveJournal: assortativity < ()> simulations 3

31 LiveJournal simulations C() 3

32 conclusions The degree distribution of a line graph is close to the degree distribution of its initial networ. Line graphs are clusterized and assortative. The degree-dependent clustering coefficient C() indicates the presence of cliques. We have shown that LiveJournal, where P() is scale-free, displays qualitatively the same features. Than you 32