1. Networks of Companies from Stock Price Correlations
J. Kertész1,2, L. Kullmann1,
J.-P. Onnela2, A. Chakraborti2, K. Kaski2,
A. Kanto3
1Department of Theoretical Physics
Budapest University of Technology and Economics, Hungary
2Laboratory of Computational Engineering
Helsinki University of Technology, Finland
3Dept of Quantitative Methods in Economics and Management Science
Helsinki School of Economics, Finland

2. Motivation Many groups active: Palermo, Rome, Seoul etc.
Financial market is a self-adaptive complex system; many interacting units, obvious networking.
Networks:
Cooperation Most important and most difficult
Activity, ownership
Similarity
Temporal aspects
Networks generated by time dependencies
Time dependent networks
Revealing NW structure is crucial for understanding
and also for pragmatic reasons (e.g., portfolio opt.)
Many groups active: Palermo, Rome, Seoul etc.

3. Outline Classification by Minimum Spanning Trees (MST) (Mantegna)
Temporal evolution
Relation to portfolio optimization
Correlations vs. noise: Parametric aggregational classification
Temporal correlations: Directed NW of influence

4. Data: price and return
Daily price data for N=477 of NYSE stocks (CRSP of U. of Chicago), such as GE, MOT, and KO
Time span S=5056 trading days: Jan 1980 – Dec 1999
Daily closure price of GE:
PGE(t)
Daily logarithmic price:
lnPGE(t)
Daily logarithmic return:
rGE(t)=lnPGE(t) – lnPGE(t-1)

5. Correlations and distances
For each window a correlation matrix is defined with elements being the equal time correlation coefficients:
where ri ,rj  Rt,  .. denotes time average. Transformation to distance-matrix with elements:
Minimum spanning tree (MST), which is a graph linking N vertices (stocks) with N-1 edges such that the sum of distances is minimum. Efficient algorithms.

6. Central vertex To characterise positions of companies in the tree the
concept of central vertex is introduced:
Reference vertex to measure locations of other vertices, needed to extract further information from asset trees
Central vertex should be a company whose price changes strongly affect the market; three possible criteria:
(1) Vertex degree criterion: vertex with the highest vertex degree, i.e., the number of incident edges; Local.
(2) Weighted vertex degree criterion: vertex with the highest correlation coefficient weighted vertex degree; Local.
(3) Center of mass criterion: vertex vi giving minimum value for mean occupation layer (l(t,vi)); Global.

7. Central vertex: comparison
(1) Vertex degree
criterion (local):
GE: 67.2%
(2) Weighted vertex
degree criterion
(local):
GE: 65.6%
(3) Center of mass
criterion (global):
GE: 52.8%

8. Asset tree and clusters
Business sectors (Forbes)
Yahoo
data

9. Potts superparamagnetic clustering
Kullmann, JK, Mantegna
Antiferromagnetic
bonds

10. Asset tree clustering
Mismatch between tree clusters and business sectors?
Random price fluctuations introduce noise to the system
Business sector definitions vary by institutions (Forbes…)
Historical data should be matched with a contemporary business sector definition
Classifications are ambiguous and less informative for highly diversified companies
MST classification mechanism imposes constraint
Uniformity and strength of correlations vary by business sector (c.f. Energy sector vs. Technology)

11. Mean occupation layer
In order to characterise the spread of vertices on
the asset tree, concept of mean occupation layer
is introduced:
where vc is the central vertex, lev(vi) denotes the
level of vertex vi , such that lev(vc) = 0.
Both static and dynamic central vertex may be
used: exhibit similar behaviour  Robustness

12. Asset tree: topology change
Normal market topology crash topology
Yahoo
data

13. Robustness: single-step survival
Robustness of dynamic asset tree topology measured as
the ratio of surviving connections when moving by one step:
Single-step survival ratio:
T = 4 years, T = 1 month

14. Tree evolution: multi-step survival
Connections survived vs. time
Within the first region decay
is exponential
After this there is cross-over
to power law behaviour:
 (t,k) ~ t--z
Power law
decay: z ≈1.2
Half life vs. window width
T (y) t1/2 (y)
t1/2=0.12T

15. k=0

16. k=1

17. k=2

18. k=4

19. k=6

20. k=12

21. k=24

22. k=36

23. k=48

24. Distribution of vertex degrees
The topological nature of the network is
studied by analysing the distribution of
vertex degrees:
Power law distribution would indicate scale-free topology, a feature unexpected by random network models
Vandewalle et al. find for one year data while we found
Power law fit ambiguous due to limited range of data

25. Distribution of vertex degrees
L: normal
R: crash

26. Portfolio optimisation
In the Markowitz portfolio optimisation theory risks of
financial assets are characterised by standard deviations
of average returns of assets:
The aim is to optimise the asset weights wi so that the
overall portfolio risk is minimized for a given portfolio return
(minimum risk portfolio is uniquely defined)

27. Weighted portfolio layer
How are minimum risk portfolio assets located
on graph?
Weighted portfolio layer is defined
by imposing no short-selling, i.e. wi  0, and it is compared with the mean occupation layer l(t).

28. Portfolio layer No short-selling Short-selling Static c.v. Static c.v.
Dynamic c.v.
Dynamic c.v.
portfolio layer
mean occupation layer

29. Correlations vs. noise
Correlation matrix contains systematics and noise.
MST: Non-parametric, unique classification scheme, but!
Even for uncorrelated random matrix MST would lead to
classification…
Meaningful clustering and robustness already signalize significance.
Different methods to separate noise from information:
Eigenvalue spectra (Boston, Paris)
Independent/principal component analysis (economists)

30. Here: Building up the FCG
Tree condition may ignore important correlations.
(General classification problem)
Visualization through Parametrized Aggregated Classification (PAC): Add links one by one to the graph, according to their rank, started by the strongest and ended with a Fully Connected Graph (FCG). Strongly correlated parts get early interconnected, clustering coefficient becomes high.
Price time series data for a set of 477 companies.
Window width T=1000 business days (4 years),
located at the beginning of the 1980’s
Comparison with random graph (obtained by shuffling the
data)
Ci = # of -s / [k(k-1) / 2] where k is the
degree of node i

31. size=

32. size= 10

33. size= 20

34. size= 30

35. size= 40

36. size= 50

37. size= 60

38. size= 70

39. size= 80

40. size= 90

41. size= 100

42. size= 120

43. size= 140

44. size= 160

45. size= 180

46. size= 200

47. size= 300

48. size= 400

49. size= 500

50. size= 600

51. size= 700

52. size= 800

53. size= 900

54. size=1000

55. Elementary graph concepts
Graph size:
number of edges in the graph (variable)
Graph order:
number of vertices in the graph (constant)
Spanned graph order:
number of vertices in the subgraph spanned by the edges, thus excluding the isolated vertices (variable)
These definitions can be applied also to clusters (two types)
(1) edge cluster
(2) vertex cluster
Edge clusters are more meaningful in the asset graph context

56. Cluster growth
The growth patterns of clusters can be divided into four
topologically different types:
(I) Create a new cluster (two nodes and the incident edge) when neither of the two end nodes are part of an existing edge cluster (spanned cluster order +2, size +1)
(II) Add a node and the incident edge to an already existing edge cluster (spanned cluster order +1, size +1)
(III) Merge two edge clusters by adding an edge between them (combined spanned cluster size +1)
(IV) Add an edge to an already existing edge cluster, thus creating a cycle in it (spanned cluster size +1)

57. Cluster growth
N=477
empirical
random

58. Spanned graph order
N=477
empirical
random

59. Number of vertex clusters
empirical
random

60. Cluster size for edge clusters
N=477
empirical
random

61. Vertex degree distribution
p=0.01
empirical
random

62. Vertex degree distribution
p=0.25
empirical
random

63. Clustering coefficient
empirical
random

64. Mean clustering coefficient
empirical
random

65. NO TIME REVERSAL SYM. ON THE MARKETS
Physics close to equilibrium: Time reversal symmetry (TRS) 
Detailed balance 
Symmetric correlation functions, Fluctuation Dissipation Th. (FDT)
No fundamental principle forcing TRS on the market.
In contrast: The elementary process, a transaction is irreversible:
Though the price is set by equilibrating supply and demand,
both parties (or at least one of them) feel that the
transaction is for their advantage and would not agree
to revert it.
Possibility of
Asymmetry in the cross correlation functions
Differences between the decay of spontaneous fluctuations
and of response to external perturbations

66. Time dependent cross correlations
log return of stock A between t and tt
Correlation fn between returns of company A and B
It depends on t and  Is it symmetric?
Difficulties:
trade not syncronized, frequencies are very different
bad signal/noise ratio
Approptiate averaging

67. Toy model to test the method:
Persistent 1d random walk (increment x  1):
We take two such walks, which are correlated, with increments
x and y
The correlation function can be calculated:
(o=200,  =1000,  =0.99)
We corrupt the data to have similar quality to real ones
Only 1% of the data are kept.

68. The measured correlations on a finite set of data depends
on the averaging procedure (moving average)
The appropriate choice is t min  t  o
DATA set:
Trade And Quote, companies tick by tick
54 days: 195 companies traded more than times
t = 100s but results checked for s.

69. We measure max, C(max), and R = C(max)/noise
Consider Imax I > 100, C(max) > 0.04, and R > 6 as ‘effect’
Results
XON: Exxon
(oil)
ESV: Ensco
(oil wells)
Not all pairs of comp’s show the effect
Peak not only shifted but also asymmetric
Large, frequently traded companies ‘pull’ the smaller ones
Weak effect and short characteristic time (minutes)

70. Many leaders for a follower Many followers for a leader
Directed network of influence
No chains
Many leaders for a follower
Many followers for a leader
Disconnected graph

71. Conclusions Networks constructed from cross correlations of stock
price time series (MST, PAC)
Though Cij noisy, much information content, useful for
portfolio optimization
MST robust, reasonable classification, interesting dyn. at crash-time
Clusters (branches) not equally correlated, PAC reveals differences, separation of noise from info
Asymmetric time dependent cross correlations lead to directed network of influence