Global and local dynamics Global and local dynamics in correlated systemsin correlated systems

T. Di Matteo, T. Aste, F. PozziT. Di Matteo, T. Aste, F. PozziDepartment of Applied MathematicsDepartment of Applied Mathematics

tiziana.dimatteo@anu.edu.autiziana.dimatteo@anu.edu.au

M. Tumminello and R. N. MantegnaM. Tumminello and R. N. Mantegna

Giulia RotundoGiulia Rotundo

Characterization and Visualization Characterization and Visualization of financial markets by means ofof financial markets by means of

Hyperbolic networks Hyperbolic networks

New correlation filtering procedureNew correlation filtering procedurePlanar Maximally Filtered Graph (PMFG)Planar Maximally Filtered Graph (PMFG)

An application to interest rates, 100 stocks An application to interest rates, 100 stocks

of US equity market, 300 stocks NYSEof US equity market, 300 stocks NYSE

Topological properties : degree, betweenness, Topological properties : degree, betweenness, average length of shortest paths at different time average length of shortest paths at different time

horizons (returns) horizons (returns)

Dynamical filtered graphs at different time windows Dynamical filtered graphs at different time windows

Brief overview Brief overview

AAustralianustralian R Researchesearch C Councilouncil Project: Project: “The architecture of networks: “The architecture of networks:

Characterization and Visualization of Characterization and Visualization of complex systems as fluctuating complex systems as fluctuating

networks”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.

gSn

g g (n 3)(n 4)

12

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.

The embedding of Kn is possible on an orientable

surface Sg of genus

2D hyperbolic 2D hyperbolic surfacesurface

•Locally planarLocally planar

•natural hierarchynatural hierarchy

•characterizationcharacterization

•elementary moveselementary moves

WHY NOT?WHY NOT?

WHY SURFACES ?WHY SURFACES ?

any n is a sub-graph ofKn and can be embedded on Sg

Which SURFACES?Which SURFACES?

g = 0 spheresphere0 non-contractible loops1 cut

g = 1torustorus2 non-contractible loops2 cuts

g = 24 non-contractible loops3 cuts

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 isan expansion of K5 or K3,3

WEIGHTSWEIGHTS

The relevance of a link between two node is measured in term of a scalar quantity: the weight or the cost.

Given a weightweight for each of the n(n-1)/2 links in the complete graphcomplete graph,

construct a sub-graphsub-graph of Kn which retains maximal informationmaximal information (minimal

weight) while constraining complexityconstraining complexity.

Construction of graph from the weights:Construction of graph from the weights:

Fix g

If and only if the resulting graph can be embedded on a

surface of genus g

connect two nodes

n unconnected nodes

T. Aste, T. Di Matteo and S. T. Hyde, Complex Networks on Hyperbolic Surfaces, Physica A 346 (2005) 20-26 cond-mat/0408443.

Bottom Up

complete graph Kn

Unfold Sg* into its universal cover H2

Embedding on Sg*

Top Down

Edge pruning H2

Regluing the universal cover on Sg in En

Arbitrary graph on Sg

Glauber dynamics

Local elemetary move

Dynamical

0 500 1000 1500 20002

3

4

5

6

7

8

9

10

Fig.2

1990 - 1996

=3 =15 =30 =48

Inte

rest

rate

s f(t

, ) (

%)

t (days)

0 500 1000 1500 20002

3

4

5

6

7

8

9

10

Fig.1

1990 - 1996

=3 =6 =9 =12 =15 =18 =21 =24 =27 =30 =33 =36 =39 =42 =45 =48

Inte

rest

rate

s f(t

, ) (

%)

t (days)

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

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)

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)

T. Di Matteo, T. Aste, R. N. Mantegna, Physica A 339 (2004) 181

0 200 400 600 800

2

4

6

8

10

12

14

16

18

1982-1997

Inte

rest

rate

s f i (

t) (%

)

t (weeks)

Metric distance )1(2 ,, jiji cd

Correlations

ji

jijiji

ffffc

,1,0, jic

20, jid

Three axioms: 0, jid if and only if i=j

ijji dd ,,

jkkiji ddd ,,,

1)

2)

3)

J. C. Gower, Biometrika 53 (1966) 325-338; R. N. Mantegna, Eur. Phys. J. B (1999) 193-197.

)()()( tfttftf iii

2

1

2

12

))((1 T

Ttii ftf

TT T1 and T2 delimit

the range of t< Δf > is the average over

time of Δfi(t)

Metric graphs

Extending the MSTExtending the MSTHow to construct a graph richer of links but preserving the same hierarchical structure?

R. N. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B (1999) 193-197.

MST retains only (n-1) correlation coefficients from the original n(n-1)/2

MINIMUM SPANNING TREE (MST)Eurodollars 34 US Interest Rates

Graph g=0 embedded on a sphere

Graph g=0 embedded on a sphere

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,j and angles as nearly equal as possible.

Network relaxation procedure)z y (x iiiVertices i,j,k placed at random in Cartesian space

F dz ;F dy ;F dx dz

dE-F ;

dy

dE-F ;

dx

dE-F

)(z)(y)(x :j and i verticesjoining vector theof distance

)2

arccos( :magnitude of

kj,i, vertices threeby the subtended i)on vertex (centered angle thedenotes

length springrest thedenotes d

ly respective edges and angles equalizingfor moduli elastic k and k

)(kE )(kE

EEE

iiiiji ziyixii

zi

yi

x

2j

2j

2j

222

ji,

sb

1ji,

2,slength

2/)1(

1kj,i,

2bangle

lengthangle

iiiijij

ikij

jkikijijk

ijk

n

jiij

nn

ijk

zyx

d

EurodollarsEurodollars

34 US Interest Rates34 US Interest RatesHierarchyHierarchy

jkkiji ddd ,,, 3)jid ,

ˆ}ˆ,ˆmax{ˆ

,,, jkkiji ddd

CLUSTERINGCLUSTERING

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.

Ultra-Metric distance

A Cluster is a set of elements at distances di,j smaller than a given threshold

Disjoined clusters have some elements which are at distances

larger than the threshold.

Three main clusters:1) < 1 year

2) 1-2 years 3) > 2 years

Eurodollar interest rates

1990-1996

1982-1997

Six main clusters and Three isolated

elements

< 1year 1 - 2

years

> 2years

1 month

3 - 6 months(no Tr.)

3 - 6 months

(Tr.)

1 - 3 y.

> 3 years

TBA3-6 m.

FED

CMSLB

T. Di Matteo, T. Aste, S. T. Hyde and S. Ramsden, Interest rates hierarchical structure, Physica A 355 (2005) 21-33.

0 200 400 600 800

2

4

6

8

10

12

14

16

18

1982-1997

Inte

rest

rate

s f i (

t) (%

)

t (weeks)

0 200 400 600 800

2

4

6

8

10

12

14

16

18

1982 - 1997

Inte

rest

rat

es (

%)

t (weeks)

CP3, CP6, FP3, FP6, BA3, BA6, CD3, CD6, ED3M, ED6M

0 200 400 600 800

2

4

6

8

10

12

14

161982 - 1997

Inte

rest

rat

es (

%)

t (weeks)

TC3M, TC6M, TBA3M, TBA6M, TBS3M, TBS6M

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.

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)

Graph richer of links but preserving the MST hierarchical structure

(n-1) 3(n-2)BAC

JPM MER

MOB

XON

CHV ARCA clique of r elements (r-clique) is a complete subgraph that linksall r elements 292 = 3n - 8 97 = n - 3

Such loops and cliques have important and significant relations with the market structure and properties

4-cliques structure31 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

2

,

)(

cliquejij i

ij

s

ciy

cliquejij

iji cs,

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.

300 most capitalized stocks traded at the NYSEJanuary 2001 – December 2003

Return time series sampled at different time horizons:5, 15, 30, 65, 130, 195 and 390 min

1 trading day

Nature and properties of the PMFG associatedto a given financial portfolio as a function of the

time horizon used to record stock return time series

5 min time horizon

Merrill Lynch co inc (MER)

Suntrust banks inc (STI)

PPG industries inc (PPG) Eaton corp (ETN)

Jefferson-Pilot corp (JP)

General Electric (GE)

Wal-Mart stores inc (WMT)

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)

1 day time horizon

Merrill Lynch co inc (MER)

General Electric (GE)

Eaton corp (ETN)

PPG industries inc (PPG)

Suntrust banks inc (STI)

Wal-Mart stores inc (WMT)

Jefferson-Pilot corp (JP)

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)

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.

5 min time horizon

1 day time horizon

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

Degree k(i)number of edges connected to the vertex i

Connection strengthratio between the number of cliques of 3 or 4 elements present among ns stocks belonging to a given set and a normalizing quantity ns – 3 for 4-cliques and 3 ns – 8 for 3-cliques

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.

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.

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.

Degree of GE and PPG evaluated in the PMFG as function of the time horizon

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

PPGhub 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

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

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.

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.

300 most capitalized stocks traded at the NYSEJanuary 2001 – December 2003

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

Booms

Crashes

11/9/2001 19/7/2002 9/10/2002

1 month

2 months3 months

6 months

4 months

12 months

Average distance for 1 month

Complete graph

Planar graph

MST

1 month

2 months3 months

6 months

4 months

12 monthsComplete graph

1 month

2 months

6 months

4 months

12 months

3 months

Planar graph

1 month

2 months 3 months

6 months

4 months

12 months

MST

Persistence of the structure

MSTPlanar

T1 Planar

Characterization and Visualization of Complex systems Characterization and Visualization of Complex systems

by means of Hyperbolic graphs by means of Hyperbolic graphs

A general tool for Information FilteringA general tool for Information Filtering

Measure of complexity looking at the amount of information Measure of complexity looking at the amount of information necessary to describe the systemnecessary to describe the system

Efficient in filtering relevant information about the clustering of the Efficient in filtering relevant information about the clustering of the

system and its hierarchical structuresystem and its hierarchical structure

Generate networks with the same hierarchical structure of the MSTGenerate networks with the same hierarchical structure of the MST

Triangular loops and 4 element cliques have important and Triangular loops and 4 element cliques have important and significant relations with the market structure and propertiessignificant relations with the market structure and properties

The market is progressively structured as a function of the The market is progressively structured as a function of the time horizontime horizon

The market structuring occurs by first connecting stocks The market structuring occurs by first connecting stocks belonging to the same sub-sector and then connecting stocks belonging to the same sub-sector and then connecting stocks

belonging to the same economic sectorbelonging to the same economic sector

Under investigationUnder investigation

Shortest path Shortest path

DegreeDegree

BetweennessBetweenness

Different SectorsDifferent Sectors

Different filtered graphsDifferent filtered graphs

Effect of g on the information filteringEffect of g on the information filtering

Dynamical graphs and elementary movesDynamical graphs and elementary moves