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Global and local dynamics in correlated systems

Global and local dynamics in correlated systems. T. Di Matteo, T. Aste, F. Pozzi Department of Applied Mathematics tiziana.dimatteo@anu.edu.au. M. Tumminello and R. N. Mantegna. Giulia Rotundo. Brief overview. Characterization and Visualization of financial markets by means of

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Global and local dynamics in correlated systems

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  1. Global and local dynamics in correlated systems T. Di Matteo, T. Aste, F. Pozzi Department of Applied Mathematics tiziana.dimatteo@anu.edu.au M. Tumminello and R. N. Mantegna Giulia Rotundo

  2. Brief overview Characterization and Visualization of financial markets by means of Hyperbolic networks New correlation filtering procedure Planar Maximally Filtered Graph (PMFG) An application to interest rates, 100 stocks of US equity market, 300 stocks NYSE Topological properties : degree, betweenness, average length of shortest paths at different time horizons (returns) Dynamical filtered graphs at different time windows

  3. Australian Research Council Project:“The architecture of networks: Characterization and Visualization of complex systems as fluctuating networks” Characterize the statistical, geometrical and topological properties of complex systems by mapping the structure of their interactions into graphs in multidimensional spaces, both Euclidean and non-Euclidean.

  4. WHY SURFACES ? 2D hyperbolic surface • Locally planar • natural hierarchy • characterization • elementary moves WHY NOT? The embedding ofKn is possible on an orientable surface Sgof genus any n is a sub-graph of Kn and can be embedded on Sg G. Ringel, Map Color Theorem, Springer-Verlag, Berlin, (1974) cap. 4 P. J. Gilbin, Graphs, Surfaces and Homology, Chapman and Hall, 2nd edition (1981) G. Ringel and J. W. T. Youngs, Proc. Nat. Acad. Sci. USA 60 (1968) 438-445.

  5. Which SURFACES? g = 0 sphere 0 non-contractible loops 1 cut g = 1 torus 2 non-contractible loops 2 cuts g = 2 4 non-contractible loops 3 cuts

  6. Planar graph g=0 K5 K3,3 Kuratowski’s theorem A finite graph is planar if and only if it does not contain a subgraph that is an expansion of K5 or K3,3

  7. WEIGHTS The relevance of a link between two node is measured in term of a scalar quantity: the weight or the cost. Construction of graph from the weights: Given a weight for each of the n(n-1)/2 links in the complete graph, construct a sub-graph of Kn which retains maximal information (minimal weight) while constraining complexity.

  8. Bottom Up Top Down Dynamical Arbitrary graph on Sg Local elemetary move Glauber dynamics complete graph Kn Fix g n unconnected nodes Embedding on Sg* Unfold Sg* into its universal cover H2 connect two nodes Edge pruning H2 Regluing the universal cover on Sg in En If and only if the resulting graph can be embedded on a surface of genus g T. Aste, T. Di Matteo and S. T. Hyde, Complex Networks on Hyperbolic Surfaces, Physica A 346 (2005) 20-26 cond-mat/0408443.

  9. Application to interest rates Eurodollar Interest Rates with maturity dates between 3 to 48 months T. Di Matteo, T. Aste, Int. J. of Theor. and Appl. Finance. 5 (2002) 107

  10. Treasury securities at ‘constant maturity’ (TC) Treasury bill rates (TBA) Treasury bill secondary market rates (TBS) Treasury long-term bond yield (TC10P) Eurodollar interbank interest rates (ED) Corporate bonds Moody’s seasoned rates (AAA, BAA) Conventional mortgages rates (CM) Federal funds rate (FED) State & local bonds (SLB) Commercial Paper (CP) Finance Paper placed directly (FP) Bankers acceptances (BA) Rate on certificates of deposit (CD) T. Di Matteo, T. Aste, R. N. Mantegna, Physica A 339 (2004) 181

  11. Metric graphs Metric distance Correlations T1and T2delimit the range of t < Δf > is the average over time of Δfi(t) Three axioms: 1) if and only if i=j 2) 3) J. C. Gower, Biometrika 53 (1966) 325-338; R. N. Mantegna, Eur. Phys. J. B (1999) 193-197.

  12. MINIMUM SPANNING TREE (MST) R. N. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B (1999) 193-197. Eurodollars 34 US Interest Rates MST retains only (n-1) correlation coefficients from the original n(n-1)/2 Extending the MST How to construct a graph richer of links but preserving the same hierarchical structure?

  13. Graph g=0 embedded on a sphere

  14. Graph g=0 embedded on a sphere

  15. Network relaxation procedure Vertices i,j,k placed at random in Cartesian space In practice, the magnitudes of the elastic moduli are tuned to ensure convergence to a final configuration with all edges of length equal to di,jand angles as nearly equal as possible.

  16. Hierarchy 34 US Interest Rates Eurodollars

  17. CLUSTERING A Cluster is a set of elements at distances di,jsmaller than a given threshold Disjoined clusters have some elements which are at distances larger than the threshold. Ultra-Metric distance 3) Ultra-metric distance between two elements i,j belonging to two different clusters is the maximum metric distance between all couples of elements in the two clusters.

  18. Eurodollar interest rates 1990-1996 Six main clusters and Three isolated elements 1982-1997 Three main clusters: 1) < 1 year 2) 1-2 years 3) > 2 years

  19. CLUSTERING FED < 1 year 1 month 1 - 2 years SLB 3 - 6 months (no Tr.) > 3 years CM > 2 years 1 - 3 y. TBA 3-6 m. 3 - 6 months (Tr.) CP3, CP6, FP3, FP6, BA3, BA6, CD3, CD6, ED3M, ED6M TC3M, TC6M, TBA3M, TBA6M, TBS3M, TBS6M T. Di Matteo, T. Aste, S. T. Hyde and S. Ramsden, Interest rates hierarchical structure, Physica A 355 (2005) 21-33.

  20. 100 stocks in the USA equity markets Basic Materials (B) (Pink) Utilities (U) (Yellow) Financial (F) (Cyan) Consumer Non Cyclical (C) (Purple) Consumer Cyclical (CC) (Orange) Capital Goods (CG) (Magenta) Healthcare (H) (Brown) Services (S) (Red) Technology (T) (Green) Conglomerates (CO) (Gray) Energy (E) (Blue) Transportation (TR) (White) M. Tumminello, T. Aste, T. Di Matteo and R. N. Mantegna, A tool for filtering information in complex systems, Proceedings of the National Academy of Sciences of the United States of America Vol. 102, Num. 30 (2005) 10421-10426.

  21. Graph richer of links but preserving the MST hierarchical structure 3(n-2) (n-1) MOB BAC XON JPM MER CHV ARC A clique of r elements (r-clique) is a complete subgraph that links all r elements 292 = 3n - 897 = n - 3 Such loops and cliques have important and significant relations with the market structure and properties

  22. 4-cliques structure 31 cliques are composed by stocks belonging to the same economic sector 22 are composed by 3 stocks belonging to the same sector 37 have 2 stocks from the same sector 7 have stocks all from different sectors

  23. Nature and properties of the PMFG associated to a given financial portfolio as a function of the time horizon used to record stock return time series 300 most capitalized stocks traded at the NYSE January 2001 – December 2003 Return time series sampled at different time horizons: 5, 15, 30, 65, 130, 195 and 390 min 1 trading day M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) 209-217.

  24. 5 min time horizon Wal-Mart stores inc (WMT) Merrill Lynch co inc (MER) Jefferson-Pilot corp (JP) Suntrust banks inc (STI) General Electric (GE) PPG industries inc (PPG) Eaton corp (ETN) Basic Materials (violet, 24 stocks), Consumer Cyclical (tan, 22 stocks), Consumer Non Cyclical (yellow, 25 stocks), Energy (blue, 17 stocks), Services (cyan, 69 stocks), Financial (green, 53 stocks), Healthcare (gray, 19 stocks), Technology (red, 34 stocks), Utilities (magenta, 12 stocks), Transportation (brown, 5 stocks), Conglomerates (orange, 8 stocks) and Capital Goods (light green, 12 stocks)

  25. 1 day time horizon Suntrust banks inc (STI) Wal-Mart stores inc (WMT) General Electric (GE) Merrill Lynch co inc (MER) Jefferson-Pilot corp (JP) Eaton corp (ETN) PPG industries inc (PPG) Basic Materials (violet, 24 stocks), Consumer Cyclical (tan, 22 stocks), Consumer Non Cyclical (yellow, 25 stocks), Energy (blue, 17 stocks), Services (cyan, 69 stocks), Financial (green, 53 stocks), Healthcare (gray, 19 stocks), Technology (red, 34 stocks), Utilities (magenta, 12 stocks), Transportation (brown, 5 stocks), Conglomerates (orange, 8 stocks) and Capital Goods (light green, 12 stocks)

  26. 5 min time horizon 1 day time horizon M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) 209-217.

  27. Topological properties Shortest path s(i,j) minimum number of edges crossed by connecting vertices i and j in the graph Betweenness btw(i) number of shortest paths traversing the vertex i Degreek(i) number of edges connected to the vertex i Connection strength ratio between the number of cliques of 3 or 4 elements present among nsstocks belonging to a given set and a normalizing quantity ns – 3 for 4-cliques and 3 ns – 8 for 3-cliques

  28. Average length of shortest path as function of the sampling time horizon of return 195 min M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) 209-217.

  29. Betweenness of GE and PPG evaluated in the PMFG as function of the time horizon 130-195 M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) 209-217.

  30. Degree of GE and PPG evaluated in the PMFG as function of the time horizon M. Tumminello, T. Di Matteo, T. Aste and R. N. Mantegna, Correlation based networks of equity returns sampled at different time horizons, The European Physical Journal B 55 (2007) 209-217.

  31. The effect of GE at short time horizons strongly intervenes in the connection between different branches (sectors) of the PMFG whereas at longer time horizon connection between sectors are more complex and the central role of GE progressively disappears GE hub for the whole market at short time horizons its relevance decreases according to the structuring of the market into sectors observed at long time horizon PPG hub for its own economic sector (Basic Materials) it is a local hub both at short and long time horizons sector of basic materials is formed already at short time horizons

  32. Connection strength evaluated by the number of intra-sector 3-cliques (C3)

  33. Conglomerates and capital goods Energy, financial and utilities the connection strength is very close to one already at the shortest time horizon. This behavior indicates that the sectors are well defined and driven by the same factors down to a very short time horizon. Consumer cyclical, healthcare and services clearly showing that the market needs a finite time to produce a profile of correlation compatible with the sector classification. Value smaller than 1 at longer time horizons. Basic materials, consumer non cyclical, and technology sectors show an intermediate behavior characterized by a non marked time dependence and moderately low values of the overall connection strength.

  34. Sub-sectors All the considered sub-sectors show a connection strength greater or at most equal to the connection strength of the economic sector they belong to. They are significantly intra-connected before or at most at the same time horizon as the corresponding economic sector.

  35. Nature and properties of the MST and PMFG at different time series windows: 1, 2, 3, 4, 6, 12 months moving through the time series 300 most capitalized stocks traded at the NYSE January 2001 – December 2003 Booms Crashes 11/9/2001 19/7/2002 9/10/2002

  36. 6 months 12 months 4 months 3 months 2 months 1 month

  37. Average distance for 1 month Complete graph Planar graph MST

  38. Complete graph 12 months 6 months 4 months 1 month 2 months 3 months

  39. 12 months 4 months 1 month 3 months 6 months 2 months Planar graph

  40. 12 months 4 months 1 month 6 months MST 2 months 3 months

  41. Persistence of the structure T1 Planar Planar MST

  42. Characterization and Visualization of Complex systems by means of Hyperbolic graphs A general tool for Information Filtering Measure of complexity looking at the amount of information necessary to describe the system Generate networks with the same hierarchical structure of the MST Efficient in filtering relevant information about the clustering of the system and its hierarchical structure Triangular loops and 4 element cliques have important and significant relations with the market structure and properties The market is progressively structured as a function of the time horizon The market structuring occurs by first connecting stocks belonging to the same sub-sector and then connecting stocks belonging to the same economic sector

  43. Dynamical graphs and elementary moves Under investigation Shortest path Degree Betweenness Different Sectors Different filtered graphs Effect of g on the information filtering

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