Download 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