• Alon AradAlon Arad

Hurst Exponent of Complex Hurst Exponent of Complex NetworksNetworks

• IntroductionIntroduction

• Random Graph Random Graph

• Types of Network Models StudiedTypes of Network Models Studied

• The Hurst ExponentThe Hurst Exponent

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

• Results and ConclusionResults and Conclusion

IntroductionIntroductionUsed Rescaled Range Analysis and the

adjacency matrix to study the spacing between eigenvalues for three widely used network models

The Random GraphThe Random GraphInitially 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 ModelsThe Three ModelsPoisson 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 ModelsSmall world model of Watts and Strogatz

Example widely used is the one dimensional example

A ring with V verticesEach vertex joined to another k lattice spacing awayVk edgesNow take edge, with probability P, move to another

point in lattice chosen at randomIf P=0 we have regular lattice, if P=1 we have

previous modelSmall world is somewhere in between

The Three ModelsThe Three ModelsPreferential attachment of Barbasi and Albert

Start with V1 unconnected nodesAttach nodes one at a time to existing node

with probability PProbability is biasedIt is proportional to number of links existing

node already hasGives a power law distributionScale free – will have same properties no

matter how many nodesMost resembles a real network system

The Big QuestionThe Big QuestionReal 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 Rescaled range analysis studies the distribution of events by grouping observed data distribution of events by grouping observed data into clusters of different sizes and studying the into clusters of different sizes and studying the scaling behavior of the statistical parameters scaling behavior of the statistical parameters with the cluster sizes.with the cluster sizes.

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

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

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

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

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

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

• Start with the whole observed data set that Start with the whole observed data set that covers a total duration and calculate its mean covers a total duration and calculate its mean over the whole of the available dataover the whole of the available data

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

• Calculate the range Calculate the range

• Calculate the standard deviationCalculate the standard deviation

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

The Adjacent MatrixThe Adjacent MatrixAdjacent matrix characterizes the topology of

the network in more usable formA graph is completely determined by its

vertex set and by a knowledge of which pairs of vertices are connected

Make a graph with m verticesThe 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 problemWe have a problemThe 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.

SolutionSolutionR/S applied to study the distribution of

spacing Not of the actual adjacency matrixBut the eigenvalues of adjacency matrix This process will be independent of labeling

ResultsResultsPerformed 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

ResultsResultsAll models show persistent behaviorInteresting 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 I have performed R/S analysis on three types of widely used complex models.widely used complex models.•I have found that they all exhibit persistent type I have found that they all exhibit persistent type behaviour behaviour

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

• University of Melbourne Department of University of Melbourne Department of Mathematics and Statistics Notes for 620-222 Mathematics and Statistics Notes for 620-222 Linear and Abstract Algebra Semester 2 2005.Linear and Abstract Algebra Semester 2 2005.• Kazumoto Iguchi and Hiroaki Yamada, Exactly Kazumoto Iguchi and Hiroaki Yamada, Exactly solvable scale-free network model, Physical solvable scale-free network model, Physical Review E 71, 036144 (2005)Review E 71, 036144 (2005) O. Shanker, Hurst Exponent of spectra of O. Shanker, Hurst Exponent of spectra of Complex Networks June 4, 2006 PACS number Complex Networks June 4, 2006 PACS number 89.75.-k.89.75.-k.Fractal Maket Analysis, Edgar E. Peters,1994Fractal Maket Analysis, Edgar E. Peters,1994Introductory Graph Theory, Gary Chartrand 1977Introductory Graph Theory, Gary Chartrand 1977Introductory Graph Theory , Robin J. Wilso1972Introductory Graph Theory , Robin J. Wilso1972