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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
[email protected]@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