FROM THE SIX DEGREES OF SEPARATION TO THE WEIGHTED SMALL-WORLD NETWORKS

Transcription

1 FROM THE SIX DEGREES OF SEPARATION TO THE WEIGHTED SMALL-WORLD NETWORKS Mircea Gligor National College Roman Voda Roman

2 The Stanley Milgram s experiment (1967) The letters path: Nebraska-Boston Criteria: name, residence, field of activity S. Milgram, University of Yale

3 The Stanley Milgram s experiment (1967) The letters path: Nebraska-Boston Criteria: name, residence, field of activity S. Milgram, University of Yale

4 The Stanley Milgram s experiment (1967) CONCLUSION A reasonable number of letters did reach their destination The letters covered an average of six steps to get from Nebraska to Boston. Two randomly chosen individuals can be connected by only a short chain of intermediate acquaintances ( small world effect ) S. Milgram, University of Yale Milgram, S. The small world problem, Psychology Today, 2, (1967)

5 The six degrees of separation in games and talk-show industry During an interview for Premiere in jan about the movie Wild River, the actor Kevin Bacon claimed that he knows everybody at Hollywood or somebody who worked with him/her. In april 1994 appeared on a blog a new game: "Kevin Bacon is the Centre of Universe" Kevin Bacon Jaye Davidson was in THE CRYING GAME with Stephen Rea who was in ANGIE with Geena Davis who was in A LEAGUE OF THEIR OWN with Rosie O'Donnell who will be in THE FLINTSTONES with Elizabeth Perkins who was in HE SAID, SHE SAID with Kevin Bacon

6 The six degrees of separation in games and talk-show industry During an interview for Premiere in jan about the movie Wild River, the actor Kevin Bacon claimed that he knows everybody at Hollywood or somebody who worked with him/her. In april 1994 appeared on a blog a new game: "Kevin Bacon is the Centre of Universe" Kevin Bacon Stockard Channing was in GREASE with John Travolta who was in LOOK WHO"S TALKING with Bruce Willis who was in THE PLAYER with Julia Roberts who was in FLATLINERS with Kevin Bacon.

7 The six degrees of separation in games and talk-show industry During an interview for Premiere in jan about the movie Wild River, the actor Kevin Bacon claimed that he knows everybody at Hollywood or somebody who worked with him/her. In april 1994 appeared on a blog a new game: "Kevin Bacon is the Centre of Universe" Kevin Bacon Omar Sharif who was in FUNNY GIRL with Barbra Streisand who was in NUTS with Richard Dreyfuss who was in MOON OVER PARADOR with Sonia Braga who was in THE MILAGRO BEANFIELD WAR with Daniel Stern who was in DINER with Kevin Bacon.

8 The six degrees of separation in games and talk-show industry

9 Passing over the fashionable account of the phenomenon, we point out that the problem is crucially important for communications. Most human interactions take place directly between individuals. The spread of news, rumours and fashions, as well as various diseases (from simple flues to the HIV virus) can spread far faster in a small world network than in a network where the average degree of separation is, say,

10 The Watts-Strogats model Duncan Watts, Univ. Colombia Steven Strogatz, Univ. Cornell

11 The terminology of the complex networks NETWORK = NODES (VERTICES) + EDGES (LINKS) The edges are usually considered of unit length. The vertex DEGREE = the number of the vertex links <k> = the average degree of the network MINIMAL PATH LENGTH = the length of the shortest trajectory joining two vertices. <l> = the average minimal path length of the network CLUSTERING COEFFICIENT C i = number of actual links / number of the first-order neighbors <C> = the average clustering coefficient

12 The Watts-Strogats model OPORTUNITIES: The development of the techniques for numerical simulations ( ) The possibility of testing the results on new networks www. and Internet.

13 The Watts-Strogats model REWIRING LINKS We consider firstly a 1-dimensional network with periodical boundary conditions. One takes all the edges of the lattice in turn and with probability p rewires to randomly chosen vertices. In such a way, a number of far connections appears.

14 1.2 The transition from a regular lattice to a SW network < l >(p)/l(0) C(p)/C(0) lg(p)

15 The spread of disease in regular and small world networks ni/l t

16 The small-world network of pupils in school

17 network N l C C random pupils

18 Simulation of the pupils small-world network by Watts-Strogatz method network N l C p pupils

19 The overlapping index: O ij 1 2( N 1)( N N O i O ij j 1 2) N l 1 l i, j ( w il w jl ) N p 1 p i w ip N q 1 q j w jq i j v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v

20 Phase transitions in small-world networks during their deconstruction

21 The statistical physics of weighted networks For the weighted small-world network we can define: The partition function; The entropy; The free energy; The average energy / link The thermal susceptibility Some of these physical quantities are expected to exhibit critical points during a thinking process of network deconstruction by removing vertices. The changes of the structure may be studied by the phase transition formalism.

22 1.E+00 Small world network deconstruction Removing the weak connected first 1.E-01 Heat Capacity 1.E-02 1.E-03 1.E-01 1 / Beta 1.E+00

23 Small world network deconstruction Removing the weak connected first VL / Beta

24 1.E+00 Small world network deconstruction Removing the strong connected first 1.E-01 Heat Capacity 1.E-02 1.E / Beta

25 Small world network deconstruction Removing the strong connected first VL / Beta

26 Conclusion As long as the most human interactions take place directly between individuals, we may expect that a large number of social networks to display the small-world effect. The pupils network in school may be easily simulated by Watts-Strogatz method of links rewiring One alternative to the Watts-Strogatz method is to explain the small-world effect by existence of few nodes in the network which have unusually high coordination numbers or which are linked to a widely distributed set of neighbours. In the network there are several nodes whose removal leads to qualitative changes in the network structure. On this way we may conclude that, to a large extent, the small-world effect is due to a few people who are particularly well connected.

27 How small is the world! Thank you!