•• AlonAlon AradArad

Hurst Exponent of ComplexHurst Exponent of ComplexNetworksNetworks

•• Introduction Introduction

•• Random Graph Random Graph

•• Types of Network Models Studied Types of Network Models Studied

•• The Hurst Exponent The Hurst Exponent

•• Linear Algebra and the Adjacent Matrix Linear Algebra and the Adjacent Matrix

•• Results and Conclusion Results and Conclusion

IntroductionIntroduction Used Rescaled Range Analysis and the adjacency

matrix to study the spacing between eigenvalues forthree widely used network models

The Random GraphThe Random Graph Initially proposed to model complex networks

Had well defined properties. Properties of Randomprocesses have been widely studied

Model was not small enough

Clustering Co-efficient was not correct.

The Three ModelsThe 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 ModelsThe 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 ModelsThe 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 QuestionThe 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 Rescaled range analysis studies the distribution ofevents by grouping observed data into clusters ofevents by grouping observed data into clusters ofdifferent sizes and studying the scaling behavior of thedifferent sizes and studying the scaling behavior of thestatistical parameters with the cluster sizes.statistical parameters with the cluster sizes.

•• In 1951, Hurst defined a method to study natural In 1951, Hurst defined a method to study naturalphenomena such as the flow of the Nile River. Processphenomena such as the flow of the Nile River. Processwas not random, but patterned. He defined a constant,was not random, but patterned. He defined a constant,K, which measures the bias of the fractional BrownianK, which measures the bias of the fractional Brownianmotion.motion.

•• In 1968 Mandelbrot defined this pattern as fractal. He In 1968 Mandelbrot defined this pattern as fractal. Herenamed the constant K to H in honor of Hurst. Therenamed the constant K to H in honor of Hurst. TheHurst exponent gives a measure of the smoothness of aHurst exponent gives a measure of the smoothness of afractal object where H varies between 0 and 1.fractal object where H varies between 0 and 1.

•• It is useful to distinguish between random and non- It is useful to distinguish between random and non-random data points.random data points.

•• If H equals 0.5, then the data is determined to be If H equals 0.5, then the data is determined to berandom.random.

•• If the H value is less than 0.5, it represents anti- If the H value is less than 0.5, it represents anti-persistence.persistence.

•• If the H value varies between 0.5 and 1, this If the H value varies between 0.5 and 1, thisrepresents persistence. (what we get)represents persistence. (what we get)

•• Start with the whole observed data set that covers a Start with the whole observed data set that covers atotal duration and calculate its mean over the whole oftotal duration and calculate its mean over the whole ofthe available datathe available data

•• Sum the differences from the mean to get the Sum the differences from the mean to get thecumulative total at each increment point, cumulative total at each increment point, V(N,kV(N,k)), from, fromthe beginning of the period up to any point, the result is athe beginning of the period up to any point, the result is aseries which is normalized and has a mean of zeroseries which is normalized and has a mean of zero

•• Calculate the range Calculate the range

•• Calculate the standard deviation Calculate the standard deviation

•• Plot Plot log-loglog-log plot that is fit Linear Regression plot that is fit Linear Regression YY on on XXwhere where Y=log R/SY=log R/S and and X=log nX=log n where the exponent where the exponent HH is isthe the slope slope of the regression line.of the regression line.

The Adjacent MatrixThe 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 areconnected

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 0otherwise.

We have a problemWe 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.

SolutionSolution 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

ResultsResults Performed rescales analysis on the three models and the

results are as follows

0.6k=10, p=.6200WS

0.73k=10, p=.3200WS

0.59E=2000200ER

0.67E=400200ER

0.83E=5,V_i=5500BA

0.85E=5,V_i=5400BA

Hurst exponentParametersVType of Graph

ResultsResults 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 widelyI have performed R/S analysis on three types of widelyused complex models.used complex models.••I have found that they all exhibit persistent type behaviourI have found that they all exhibit persistent type behaviour

•• If I had more time and available data, I would have If I had more time and available data, I would haveperformed R/S on a real network. One such possibility I wasperformed R/S on a real network. One such possibility I wasinvestigating is the connectivity of international airports.investigating is the connectivity of international airports.

•• University of Melbourne Department of MathematicsUniversity of Melbourne Department of Mathematicsand Statistics Notes for 620-222 Linear and Abstractand Statistics Notes for 620-222 Linear and AbstractAlgebra Semester 2 2005.Algebra Semester 2 2005.•• KazumotoKazumoto Iguchi and Hiroaki Yamada, Exactly Iguchi and Hiroaki Yamada, Exactlysolvable scale-free network model, Physical Review Esolvable scale-free network model, Physical Review E71, 036144 (2005)71, 036144 (2005) O. O. ShankerShanker, Hurst Exponent of spectra of Complex, Hurst Exponent of spectra of ComplexNetworks June 4, 2006 PACS number 89.75.-k.Networks June 4, 2006 PACS number 89.75.-k.Fractal Fractal MaketMaket Analysis, Edgar E. Peters,1994 Analysis, Edgar E. Peters,1994Introductory Graph Theory, Gary Introductory Graph Theory, Gary ChartrandChartrand 1977 1977Introductory Graph Theory , Robin J. Wilso1972Introductory Graph Theory , Robin J. Wilso1972