Alon Arad Alon Arad Hurst Exponent of Complex Networks

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

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 V 1 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. 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 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. 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. 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 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 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) 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 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 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 range

Calculate the standard deviation 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. 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 = [a ij ] in which a ij =1 if v i is connected to v j, 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 GraphVParametersHurst exponent BA400E=5,V_i=50.85 BA500E=5,V_i=50.83 ER200E= ER200E= WS200k=10, p= WS200k=10, p=.60.6

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

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