1. DYNAMICS OF COMPLEX SYSTEMS Self-similar phenomena and Networks Guido Caldarelli CNR-INFM Istituto dei Sistemi Complessi Guido.Caldarelli@roma1.infn.it 2/6

2. 1.SELF-SIMILARITY (ORIGIN AND NATURE OF POWER-LAWS) 2.GRAPH THEORY AND DATA 3.SOCIAL AND FINANCIAL NETWORKS 4.MODELS 5.INFORMATION TECHNOLOGY 6.BIOLOGICAL NETWORKS STRUCTURE OF THE COURSE

3. 2.1 FREQUENCY DISTRIBUTIONS 2.2 ASSORTATIVITY 2.3 BETWEENNESS 2.4 COMMUNITIES (GIRVAN AND NEWMAN) 2.5COMMUNITIES (SPECTRAL ANALYSIS) 2.6 STRUCTURE OF THE LECTURE

4. ·1 Degree frequency density P(k) = how many times you find a vertex with degree k P(k) k ·2 Degree Corr. Knn (k) = average degree of a neighbour of a vertex with degree k ·3 Clustering Coefficient (k) = the average value of c for a vertex whose degree is k 2.1 BASIC OF GRAPH THEORY

5. Assortative networksDisassortative networks Real networks always display one of these two tendencies, “similar” networks display “similar” behaviours. Techological, Biological networks Assortativity coefficient > 0 : Assortative = 0 : Non assortative < 0 : Disassortative Social networks 2.2 BASIC OF GRAPH THEORY

6. M.E.J. Newman, Physical Review E, 67 026126, (2003). Consequences of assortativity: - Resistence to attacks - Percolation - Epidemic spreading 2.2 BASIC OF GRAPH THEORY

7. ·4 Centrality betweenness b(k) = The probability that a vertex whose degree is k has betweenness b ·5 TREES ONLY!!! P(A) = Probability Density for subbranches of size A 1 10 1 1 1 1 8 3 352 1 1 1 5 11 22 33 1 1 Size distribution: P(A ) A Allometric relations: A C(A ) betweenness of I is the number of distances between any pair of vertices passing through I 2.3 BASIC OF GRAPH THEORY

8. 2.4 COMMUNITIES: GN Algorithm

10. Network of e-mails in the University of Tarragona (Spain). On the left the total network, different vertices represent persons and the colour of vertices the various Department. On the right the tree of communities. 2.4 COMMUNITIES: GN Algorithm

11. 0110000000000000000 1011000000000000000 1101010000000000000 0110110000000000000 0100011000000000000 0011101100000000000 0000110000000000000 0000010011110000000 0000000101110000000 0000000110101100000 0000000111011000000 0000000110101000000 0000000001110000000 0000000001000010101 0000000000000101110 0000000000000010110 0000000000000111011 0000000000000011101 0000000000000100110 Undirected graphs → a ij = a ji a ii = 0 2,5 COMMUNITIES: Spectral Analysis

12. Spectral analysis is based on the analysis of the following matrices The Adiacency Matrix A The Laplacian Matrix L= A - K The Normal(ized) Matrix N=K -1 A Note that by definition for every node i COMMUNITIES: Spectral Analysis

13. Laplacian Matrix If  ’ = L  The elements of matrix N give the probability with which one field  passes from a vertex i to the neighbours. Normal Matrix 2.5 COMMUNITIES: Spectral Analysis

14. Given this probabilistic explanation for the matrix N We have a series of results, for example One eigenvalue is equal to one and The eigenvector related is constant. Consider the case of disconnected subclusters: The matrix N is made of blocks and a general eigenvector will be given by the space product of blocks eigenvectors (the constant can be different!) 2.5 COMMUNITIES: Spectral Analysis

15. In general, the probability to pass from one vertex to a neighbour depend upon the nature of the edge. In the case of Internet different cables have different capacity, speed or cost. In social networks, the edge has the strength of the connection. Therefore it is customary to generalize this approach to a case where instead of the a ij we have real numbers (weights) w ij. 2.5 COMMUNITIES: Spectral Analysis

16. To formalize the role between eigenvectors and communities we express the eigen-problem as a research of a minimum under constraint where the x i are values assigned to the nodes, with the constraint expressed by Looking for stationary points of z(x) + constraint (A) → Lagrange multiplier (A) This is a three step procedure 1.Define a ficticious quantity x for the sites of the graph 2.Define a suitable function z on these x’s (a “distance”) 3.Define a suitable constraint on these x’s (to avoid having all equal or all 0) For example 2.5 COMMUNITIES: Lagrange Multipliers

18. Lagrange Multiplier = Normal Eigenvalue problem Lagrange Multiplier = Laplacian Eigenvalue problem 2.5 COMMUNITIES: Lagrange Multipliers

19. In order to test if the model works properly we choose a suitable network Where communities can be easily spotted. The data are collected through a psychological experiment: Persons (about 100) are given as a stimulus a single word i.e. “House” They must answer with the first word that comes on their mind i.e.“Family”. Answer are given as new stimula, so that a network of average associations forms. House Dog Car Mortgage Family Job Road 2.6 COMMUNITIES: Application

20. science1literature1piano1 scientific0.994dictionary0.994cello0.993 chemistry0.990editorial0.990fiddle0.992 physics0.988synopsis0.988viola0.990 concentrat e 0.973words0.987banjo0.988 thinking0.973grammar0.986saxophone0.985 test0.973adjective0.983director0.984 lab0.969chapter0.982violin0.983 brain0.965prose0.979clarinet0.983 equation0.963topic0.976oboe0.983 examine0.962English0.975theater0.982 Therefore we expect similar words to be on the same plateau. We can measure the correlation between the values of various vertices averaged over 10 different eigenvectors. 2.6 COMMUNITIES: Application

21. With Graph Theory we can describe and analyse a series of different systems FINANCIAL SYSTEMS Portfolio Board of Directors Price correlations SOCIAL INTERACTIONS Actors, Scientists Sex TECHNOLOGICAL NETWORKS WWW, Internet e-mail LINGUISTIC NETWORKS Syntactic Networks Word Associazioni di parole. BIOLOGICAL NETWORKS Protein and Metabolic networks Food Webs Taxonomies 2.7 DATA

22. Vertices = Companies Edges = ownership 2.7 DATA: PORTFOLIOS

23. Vertices = companies Edges = to be in the same board Vertices = Boards Edges = share a Director 2.7 DATA: BOARD OF DIRECTORS

24. Vertices = Companies Edges = High correlation in stock prices 2.7 DATA: STOCK CORRELATIONS

25. Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999) MOVIES CITATIONS SEXUAL NETWORKS 2.7 DATA: SOCIAL NETWORKS

26. ΜΗΝΙΝ ΑΕΙΔΕ ΘΕΑ ΠΗΛΙΑΔΕΩ ΑΧΙΛΛΗΟΣ, ΟΥΛΟΜΕΝΗΝ Ή ΜΥΡΙΑ ΑΧΑΙΟΙΣ ΑΛΓΕ' ΕΘΗΚΕ, ΠΟΛΛΑΣ Δ' ΙΦΘΙΜΟΥΣ ΨΥΧΑΣ ΑΙΔΙ ΠΡΟΙΑΨΕΝ ΗΡΩΩΝ, ΑΥΤΟΥΣ ΔΕ ΕΛΩΡΙΑ ΤΕΥΧΕ ΚΥΝΕΣΙΝ ΟΙΩΝΟΙΣΙ ΤΕ ΠΑΣΙ. ΔΙΟΣ ΔΕ ΕΤΕΛΕΙΕΤΟ ΒΟΥΛΗ. ΕΞ ΟΥ ΔΗ ΤΑ ΠΡΩΤΑ ΔΙΑΣΤΗΤΗΝ ΕΡΙΣΑΝΤΕ ΑΤΡΕΙΔΗΣ ΤΕ ΑΝΑΞ ΑΝΔΡΩΝ ΚΑΙ ΔΙΟΣ ΑΧΙΛΛΕΥΣ...... 2.7 DATA: LINGUISTIC NETWORKS

27. 2.7 DATA: TECHNOLOGICAL NETWORKS

28. 2.7 DATA: FOOD WEBS

29. phylum subphylum class subclass order family genus species 2.7 DATA: TAXONOMIES

30. 2.7 DATA: PROTEIN INTERACTION NETWORKS

32. 2.7 DATA: NEURAL NETWORKS

33. Example of a network extracted from a correlation matrix of brain activity Above (1/8 of the total. Colour corresponds to degree yellow = 1, green = 2, red = 3, blue = 4, violet = 5). The plot of the degree distribution for three different values of threshold. 2.7 DATA: NEURAL NETWORKS

34. The web of Human sexual contacts [Lilijeros et al., Nature (2001)] EPIDEMICS IS A FUNDAMENTAL APPLICATION 2.8 EPIDEMICS

35. Individuals are present in different stages HEALTY INFECT IMMUNE DEAD.. Again the network is the most immediate representation Every node is an individual Every edge is a connection 2.8 EPIDEMICS

36.  cc Active phase Absorbing phase Infection Dead of virus In any model WE HAVE A CRITICAL THRESHOLD 2.8 EPIDEMICS

37. Actually all the models assume a regular grid IF THE NETWORK IS SELF- SIMILAR There is no activation!!! Mass vaccination is useless WE MUST ACT ON HUBS!!! 2.8 EPIDEMICS