1. DYNAMICS OF COMPLEX SYSTEMS Self-similar phenomena and Networks Guido Caldarelli CNR-INFM Istituto dei Sistemi Complessi Guido.Caldarelli@roma1.infn.it 3/6

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.BIOLOGY STRUCTURE OF THE COURSE

3. 3.1) SOCIAL NETWORKS: E-Mails 3.2) FINANCIAL NETWORKS 3.3) STOCK CORRELATIONS 3.4) STOCK OWNERSHIPS 3.5) BOARD OF DIRECTORS 3.6) BANK LOAN NETWORKS 3.7) WORLD TRADE WEB STRUCTURE OF THE THIRD LECTURE

4. We introduce a model, based on the idea of preferential exchange, whose applicability can in principle be extended to other contexts. The first data sets available are the mail received by ourselves. Data about traffic is very difficult to obtain. 3.1 SOCIAL NETWORKS: E-mails

5. The data sets that we analyze are five e-mail directories coming from our accounts and the accounts of two other colleagues. They contain 5628 e-mails (corresponding to 393 senders) collected over roughly three years, 19219 e- mails (476 senders, ten years), 16102 e-mails (113 senders, three years), 13385 e-mails (516 senders, five years) and 21782 e-mails (207 senders, five years). 3,1 SOCIAL NETWORKS: E-mails

6. Every agent sends an amount M out of mails at any time-step The mail sent to i are proportional to those received It is possible to derive a rate equation for the number of mails k ij 3.1 SOCIAL NETWORKS: E-mails

7. We defined a model that reproduces the final distribution of mails. The principle is based on PREFERENTIAL EXCHANGE. The probability to send an e-mail to John is proportional to mails we received by John. 3.1 SOCIAL NETWORKS: E-mails

8. WHICH NETWORKS ? The cases of study where graph theory has been applied Stock correlations Stock ownerships Trade between countries Board of Directors Inter-bank Market NOT Network in economics (firm networks (Marro), Production networks (Weisbuch) etc. ). WHAT IN NETWORKS ? Degree distribution Community Structure Robustness WHY NETWORKS ? Do we learn something? 3.2 FINANCIAL NETWORKS

9. Probably the most complex system is human behaviour! Even by considering only the trading between individuals, situation seem to be incredibly complicated. Econophysics tries to understand the basic “active ingredients” at the basis of some peculiar behaviours. For example price statistical properties can be described through a simple model of agents trading the same stock. 3.2 FINANCIAL NETWORKS

10. Logarithmic return of stock i Correlation between returns (averaged on trading days) Distance between stocks i, j 3.3 STOCK CORRELATIONS

11. Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-year period 1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system. The correspondence is: red for mining cyan for construction yellow for manufacturing green for transportation,, light blue for public electric,gas and sanitary services administrationmagenta for wholesale trade black for retail trade purple for finance and insurance orange for service industries Spanning Trees, Bonanno et al. Physical Review E 68 046130 (2003). 3.3 STOCK CORRELATIONS

12. Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-year period 1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system. The correspondence is: red for mining cyan for construction yellow for manufacturing green for transportation, communications, light blue for public electric,gas and sanitary services administrationmagenta for wholesale trade black for retail trade purple for finance and insurance orange for service industries “Topology of correlation based” G. Bonanno, G. C., F. Lillo, R. Mantegna.PRE E 68 046130 (2003). Real Data from NYSE 3.3 STOCK CORRELATIONS

13. Correlation based minimal spanning trees of of an artificial market composed by of 1071 stocks according to the one factor model. The node colour is based on Standard Industrial Classification system. The correspondence is: red for mining cyan for construction yellow for manufacturing green for transportation, communications, light blue for public electric,gas and sanitary services administrationmagenta for wholesale trade black for retail trade purple for finance and insurance orange for service industries Data from Capital Asset Pricing Model In the model it is supposed that returns follow r i (t) = return of stock i r M (t) = return of market (Standard & Poor’s)  i  i = real parameters  i, = noise term with 0 mean 3.3 STOCK CORRELATIONS

14. Without going in much detail about degree distribution or clustering of the two graphs We can conclude that: the topology of MST for the real and an artificial market are greatly different. Real market properties are not reproduced by simple random models 3.3 STOCK CORRELATIONS

15. Graph from Threshold, Onnela et al. Eur. Phys. J. B 38, 353–362 (2004). 3.3 STOCK CORRELATIONS

16. Planar Maximally Filtered Graph, Tumminello et al. PNAS 102, 10421 (2005). 3.3 STOCK CORRELATIONS

17. Stock Ownership Stock Ownership Garlaschelli et al. Physica A, 350 491 (2005). 3.4 STOCK OWNERSHIPS

18. The degree distribution is fat-tailed 3.4 STOCK OWNERSHIPS

19. It is not only the topology that matters. In this case as in many other graphs the weight of the link is crucial For every stock i you compute this quantity. The sum runs over the different holders If there is one dominating holder SI tends to one If all the holders have a similar part SI tends to 1/N For every guy j you compute this quantity. The sum at the denominator runs over the different holders of i Then you sum on the different stocks in the portfolio This gives a measure of the number of stocks controlled 3.4 STOCK OWNERSHIPS

21. Bipartite Graph: two different kind of graphs edges between different groups. Projection Link vertices of the same group Consider Weight ! multiple Interlock Weight Stock Ownership Stock Ownership Various, see Ref in Battiston et al. EPJB, 350 491 (2005). 3.5 BOARD OF DIRECTORS

23. Inter-Bank Loans Inter-Bank Loans De Masi et al. In preparation. Banks exchange money overnightly, in order to meet the customer needs of liquidity as well as ECB requirements 3.6 BANK LOAN NETWORKS

24. EUROPEAN CENTRAL BANK provides LIQUIDITY to European Banks, through weekly auctions. EVERY BANK must DEPOSIT to NATIONAL CENTRAL BANK the 2% of all deposits and debts issued in the last two years. This reserves are supposed to help in the case of liquidity shocks 2% value fluctuates in time and it is recomputed every month. Banks sell and buy liquidity to adjust their liquidity needs and at the same time tend to reduce the value of reserve. We report here the analysis on 196 Italian banks (plus 18 banks from abroad who interact with them) who did 85202 transaction in 2000. 3.6 BANK LOAN NETWORKS

25. Vertices are banks and edges a single loan 3.6 BANK LOAN NETWORKS

26. Most of the activity starts before the 24 the day where the 2% is computed. So dynamics and therefore shape is originated by ECB policy 3.6 BANK LOAN NETWORKS

27. Actually the banks form different groups roughly related to their “size” when considering the average volume of money exchanged. 3.6 BANK LOAN NETWORKS

28. The system is composed by 179 countries connected by trade channels exchanged in the year 2000. World Trade Web World Trade Web Boguña et al. PRE, 68 151011 (2003). 3.7 WORLD TRADE WEB

29. Fitness for WTW Fitness for WTW Garlaschelli et al. PRL 93 188701 (2004). The fitness is given by the normalized Gross Domestic Product of the country. 3.7 WORLD TRADE WEB

30. Financial Networks can help 1.In distinguishing behaviour of different markets 2.In visualizing important features as the chain of control 3.In testing the validity of market models They might be an example of scale-free networks even more general than those described by growth and preferential attachment. CONCLUSIONS

32. + = Random Graphs defined on Pareto distributed vertices Naturally form scale-free networks. (Scale-free networks arise also from Gaussian in specific case) CONCLUSIONS