Fractal Financial Market Analysis
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Transcript Fractal Financial Market Analysis
Hurst Exponent of Complex
Networks
• Alon Arad
•
Introduction
•
Random Graph
•
Types of Network Models Studied
•
The Hurst Exponent
•
Linear Algebra and the Adjacent Matrix
•
Results and Conclusion
Introduction
Used Rescaled Range Analysis and the adjacency
matrix to study the spacing between eigenvalues for
three widely used network models
The Random Graph
Initially proposed to model complex networks
Had well defined properties. Properties of Random
processes have been widely studied
Model was not small enough
Clustering Co-efficient was not correct.
The Three Models
Poisson random graph of Erdos and Renyi
Ensemble of all graphs having V vertices E edges
Each pair of vertices connected with probability P
Clearly Graph model needed to be improved
The Three Models
Small world model of Watts and Strogatz
Example widely used is the one dimensional example
A ring with V vertices
Each vertex joined to another k lattice spacing away
Vk edges
Now take edge, with probability P, move to another
point in lattice chosen at random
If P=0 we have regular lattice, if P=1 we have previous
model
Small world is somewhere in between
The Three Models
Preferential attachment of Barbasi and Albert
Start with V1 unconnected nodes
Attach nodes one at a time to existing node with
probability P
Probability is biased
It is proportional to number of links existing node
already has
Gives a power law distribution
Scale free – will have same properties no matter how
many nodes
Most resembles a real network system
The Big Question
Real Networks
Fat tails, power law distributed, scale free
Power Law
We all know that the power law is synonymous with fractal
type behavior
Question
The question we are all asking ourselves is, how fractals are
the graph models described
• Rescaled range analysis studies the distribution of
events by grouping observed data into clusters of different
sizes and studying the scaling behavior of the statistical
parameters with the cluster sizes.
• In 1951, Hurst defined a method to study natural
phenomena such as the flow of the Nile River. Process was
not random, but patterned. He defined a constant, K,
which measures the bias of the fractional Brownian
motion.
• In 1968 Mandelbrot defined this pattern as fractal. He
renamed the constant K to H in honor of Hurst. The Hurst
exponent gives a measure of the smoothness of a fractal
object where H varies between 0 and 1.
• It is useful to distinguish between random and non-
random data points.
• If H equals 0.5, then the data is determined to be
random.
• If the H value is less than 0.5, it represents anti-
persistence.
• If the H value varies between 0.5 and 1, this represents
persistence. (what we get)
• Start with the whole observed data set that covers a total
duration and calculate its mean over the whole of the
available data
• Sum the differences from the mean to get the
cumulative total at each increment point, V(N,k), from
the beginning of the period up to any point, the result is a
series which is normalized and has a mean of zero
• Calculate the range
•
Calculate the standard deviation
• Plot log-log plot that is fit Linear Regression Y on X
where Y=log R/S and X=log n where the exponent H is
the slope of the regression line.
The Adjacent Matrix
Adjacent matrix characterizes the topology of the
network in more usable form
A graph is completely determined by its vertex set and
by a knowledge of which pairs of vertices are
connected
Make a graph with m vertices
The adjacent matrix is an m×m matrix defined by A =
[aij] in which aij =1 if vi is connected to vj, and is 0
otherwise.
We have a problem
The matrix of the graph can be contrived in multiple ways
depending on how the vertices are labeled.
We can show that two unequal matrices in fact represent the
same graph.
Solution
R/S applied to study the distribution of spacing
Not of the actual adjacency matrix
But the eigenvalues of adjacency matrix
This process will be independent of labeling
Results
Performed rescales analysis on the three models and the results
are as follows
Type of Graph
V
Parameters
Hurst exponent
BA
400
E=5,V_i=5
0.85
BA
500
E=5,V_i=5
0.83
ER
200
E=400
0.67
ER
200
E=2000
0.59
WS
200
k=10, p=.3
0.73
WS
200
k=10, p=.6
0.6
Results
All models show persistent behavior
Interesting to note that ER model is also persistent
Clearly at the limit (ie very large system) we would get
H=.5 for ER model
•
I have performed R/S analysis on three types of widely
used complex models.
•I have found that they all exhibit persistent type behaviour
• If I had more time and available data, I would have
performed R/S on a real network. One such possibility I was
investigating is the connectivity of international airports.
• University of Melbourne Department of Mathematics
and Statistics Notes for 620-222 Linear and Abstract
Algebra Semester 2 2005.
• Kazumoto Iguchi and Hiroaki Yamada, Exactly solvable
scale-free network model, Physical Review E 71, 036144
(2005)
O. Shanker, Hurst Exponent of spectra of Complex
Networks June 4, 2006 PACS number 89.75.-k.
Fractal Maket Analysis, Edgar E. Peters,1994
Introductory Graph Theory, Gary Chartrand 1977
Introductory Graph Theory , Robin J. Wilso1972