distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

complete multipartite graphs are determined by their distance spectra abstract in this paper we prove that the complete multipartite graphs are determined by their distance spectra which confirms the conjecture proposed by lin hong wang and shu 2013 7 although it is well known that the complete multipartite graphs cannot be determined by their adjacency spectra

scaling and memory in volatility return intervals in financial markets abstract for both stock and currency markets we study the return intervals between the daily volatilities of the price changes that are above a certain threshold q we find that the distribution function pq scales with the mean return interval as the scaling function f x is similar in form for all seven stocks and for all seven currency databases analyzed and f x is consistent with a power law form f x x with 2 we also quantify how the conditional distribution pq 0 depends on the previous return interval 0 and find that small or large return intervals are more likely to be followed by small or large return intervals this clustering of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long term correlations known to be present in the volatility econophysics fluctuations extreme values long term correlations long term memory

extreme value statistics in records with long term persistence many natural records exhibit long term correlations characterized by a power law decay of the autocorrelation function c s ensuremath sim s ensuremath ensuremath gamma with time lag s and correlation exponent 0l ensuremath gamma l1 we study how the presence of such correlations affects the statistics of the extreme events i e the maximum values of the signal within time segments of the fixed duration r we find numerically that i the integrated distribution function of the maxima converges to a gumbel distribution for large r similar to uncorrelated signals ii the deviations for finite r depend on the initial distribution of the records and on their correlation properties iii the maxima series exhibit long term correlations similar to those of the original data and most notably iv the maxima distribution as well as the mean maxima significantly depend on the history in particular on the previous maximum the last item implies that conditional mean maxima and conditional maxima distributions with the value of the previous maximum as condition should be considered for an improved extreme event prediction we provide indications that this dependence of the mean maxima on the previous maximum occurs also in observational long term correlated records

method for generating long range correlations for large systems we propose a new method to generate a sequence of random numbers with long range power law correlations that overcomes known difficulties associated with large systems the new method presents an improvement on the commonly used methods we apply the algorithm to generate enhanced diffusion isotropic and anisotropic self affine surfaces and isotropic and anisotropic correlated percolation

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

sharp bounds on the distance spectral radius and the distance energy of graphs the d eigenvalues 1 2 p of a graph g are the eigenvalues of its distance matrix d and form the d spectrum of g denoted by specd g the greatest d eigenvalue is called the d spectral radius of g denoted by 1 the d energy ed g of the graph g is the sum of the absolute values of its d eigenvalues in this paper we obtain some lower bounds for 1 and characterize those graphs for which these bounds are best possible we also obtain an upperbound for ed g and determine those maximal d energy graphs

on the distance spectra of threshold graphs abstract a graph is called a threshold graph if it does not contain induced c 4 p 4 or 2 k 2 such graphs have numerous applications in computer science and psychology and they also have nice spectral properties in this paper we consider the distance matrix of a connected threshold graph we show that there are no distance eigenvalues of threshold graphs lying in the interval 2 1 and all the eigenvalues other than 2 or 1 are simple besides we determine all threshold graphs with distinct distance eigenvalues

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

memory in the occurrence of earthquakes we study the statistics of the recurrence times ensuremath tau between earthquakes above a certain magnitude m in six one global and five regional earthquake catalogs we find that the distribution of the recurrence times strongly depends on the previous recurrence time ensuremath tau 0 such that small and large recurrence times tend to cluster in time this dependence on the past is reflected in both the conditional mean recurrence time and the conditional mean residual time until the next earthquake which increase monotonically with ensuremath tau 0 as a consequence the risk of encountering the next event within a certain time span after the last event depends significantly on the past an effect that has to be taken into account in any effective earthquake prognosis

scaling in a network model of a multispecies ecosystem a new model ecosystem of many interacting species is introduced in which the species are connected through a random matrix with a given connectivity the model is studied both analytically and by numerical simulations a probability distribution derived from the model is in good agreement with simulations and field data it is also shown that the connectivity c and the number of species s are linked through the scaling relation s k c 1 e which is observed in real ecosystems our approach suggests a natural link between log normal and power law distributions of species abundances

on the adjacency matrix of a threshold graph abstract a threshold graph on n vertices is coded by a binary string of length n 1 we obtain a formula for the inertia of the adjacency matrix of a threshold graph in terms of the code of the graph it is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank a formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse when it exists of the adjacency matrix of a threshold graph are obtained results for antiregular graphs follow as special cases

on the normalized spectrum of threshold graphs abstract in this article we investigate normalized adjacency eigenvalues simply normalized eigenvalues and normalized adjacency energy of connected threshold graphs a threshold graph can always be represented as a unique binary string certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix finally we characterize threshold graphs with at most five distinct eigenvalues

power spectrum and detrended fluctuation analysis application to daily temperatures the variability measures of fluctuation analysis fa and detrended fluctuation analysis dfa are expressed in terms of the power spectral density and of the autocovariance of a given process the diagnostic potential of these methods is tested on several model power spectral densities in particular we find that both fa and dfa reveal an algebraic singularity of the power spectral density at small frequencies corresponding to an algebraic decay of the autocovariance a scaling behavior of the power spectral density in an intermediate frequency regime is better reflected by dfa than by fa we apply fa and dfa to ambient temperature data from the 20th century with the primary goal to resolve the controversy in literature whether the low frequency behavior of the corresponding power spectral densities are better described by a power law or a stretched exponential as a third possible model we suggest a weibull distribution however it turns out that neither fa nor dfa can reliably distinguish between the proposed models

statistics of return intervals in long term correlated records we consider long term correlated data with several distribution densities gaussian exponential power law and log normal and various correlation exponents ensuremath gamma 0l ensuremath gamma l1 and study the statistics of the return intervals r j between events above some threshold q we show that irrespective of the distribution the return intervals are long term correlated in the same way as the original record but with additional uncorrelated noise due to this noise the correlations are difficult to observe by the detrended fluctuation analysis which exhibits a crossover behavior but show up very clearly in the autocorrelation function the distribution p q r of the return intervals is characterized at large scales by a stretched exponential with exponent ensuremath gamma and at short scales by a power law with exponent ensuremath gamma ensuremath 1 we discuss in detail the occurrence of finite size effects for large threshold values for all considered distributions we show that finite size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10 6 data points finally in order to quantify the clustering of extreme events due to the long term correlations in the return intervals we study the conditional distribution function and the related moments we find that they show pronounced memory effects irrespective of the distribution of the original data

return intervals of rare events in records with long term persistence many natural records exhibit long term correlations characterized by a power law decay ofthe auto correlation function c s s with time lag s and correlation exponent 0 i i 1 we study how the presence ofsuch correlations a3ects the statistics ofthe return intervals rq for events above a certain threshold value q we 4nd that a the mean return interval rq does not depend on b the distribution of rq follows a stretched exponential ln pq r r rq and c the return intervals are also long term correlated with the exponent yielding clustering of both small and large return intervals we provide indications that both the stretched exponential behaviour and the clustering ofrare events can be seen in long temperature records c

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

statistical volatility duration and complexity of financial dynamics on sierpinski gasket lattice percolation abstract a novel agent based financial dynamics is established based on the percolation system on sierpinski gasket lattice to reproduce the statistical characteristics of financial markets sierpinski gasket lattice is a fractal like graph which corresponds to the fractal well known sierpinski gasket modeling a financial dynamics on sierpinski gasket lattice is a new approach in financial micro mechanism construction in an attempt to investigate the statistical properties of returns and prove the feasibility of the proposed model two new nonlinear statistics maximum and average monotonous volatility duration are introduced in this work for the first time which describe the trend of volatility of return time series in financial markets furthermore a new method cmcid is proposed to measure the synchronization and similarity behaviors of return volatility duration in multiscale the results show that the pairs of real data and simulated data have synchronization and the synchronization becomes stronger when the scale increases and the power law distribution is employed to investigate the corresponding statistical properties which shows that the proposed model shares a common power law distribution with regard to monotonous volatility duration the empirical results show that the simulated series from the financial model and the historical data share common statistical properties which indicate that the proposed model is reasonable in terms of volatility behavior

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

the inertia and energy of the distance matrix of a connected graph abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we show that the distance matrix of a clique tree is non singular moreover we also prove that the distance matrix of a clique tree has exactly one positive d eigenvalue in addition we determine the extremal graphs with maximum and minimum distance energy among all clique trees

effect of nonstationarities on detrended fluctuation analysis detrended fluctuation analysis dfa is a scaling analysis method used to quantify long range power law correlations in signals many physical and biological signals are noisy heterogeneous and exhibit different types of nonstationarities which can affect the correlation properties of these signals we systematically study the effects of three types of nonstationarities often encountered in real data specifically we consider nonstationary sequences formed in three ways i stitching together segments of data obtained from discontinuous experimental recordings or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts a cutting procedure commonly used in preparing data prior to signal analysis ii adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes and iii generating a signal comprised of segments with different properties e g different standard deviations or different correlation exponents we compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities we find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on a the fraction and size of the parts cut out from the signal b the concentration of spikes and their amplitudes c the proportion between segments with different standard deviations or different correlations and d the correlation properties of the stationary signal we show how to develop strategies for preprocessing raw data prior to analysis which will minimize the effects of nonstationarities on the scaling properties of the data and how to interpret the results of dfa for complex signals with different local characteristics

eigenvalue location in threshold graphs abstract let g be a threshold graph of order n with adjacency matrix a we present an o n algorithm for constructing a diagonal matrix congruent to b x a x i for any x an application using sylvester s law of inertia can determine how many eigenvalues lie in an interval allowing efficient approximation we prove that eigenvalues of threshold graphs other than 0 or 1 are simple we give the spectrum for threshold graphs g k j represented by 0 k 1 j when n 3 g n n 3 n 3 has the minimum eigenvalue min n among threshold graphs of order n and a formula for min n is given there is one more graph if n 2 mod 3

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

eigenvalues and energy in threshold graphs abstract assuming a uniform random model of selecting creation sequences we show that almost every connected threshold graph has more negative than positive eigenvalues we show that no threshold graphs have eigenvalues in 1 0 sequences of equienergetic graphs are given including the striking result that for all n 3 there exist n 1 threshold graphs of order n 2 pairwise noncospectral each equienergetic to k n 2 for all n 8 we find an n vertex hyperenergetic threshold graph

correlations in economic time series a financial index of the new york stock exchange the sp here f is called the crossover frequency iii we confirm these results using the dfa detrended fluctuation analysis method see fig 2b which allows accurate estimates of exponents independent of local trends