fluctuation analysis in complex networks modeled by hidden variable models necessity of a large cutoff in hidden variable models it is becoming more and more clear that complex networks present remarkable large fluctuations these fluctuations may manifest differently according to the given model in this paper we reconsider hidden variable models which turn out to be more analytically treatable and for which we have recently shown clear evidence of non self averaging the density of a motif being subject to possible uncontrollable fluctuations in the infinite size limit here we provide full detailed calculations and we show that large fluctuations are only due to the node hidden variables variability while in ensembles where these are frozen fluctuations are negligible in the thermodynamic limit and equal the fluctuations of classical random graphs a special attention is paid to the choice of the cutoff we show that in hidden variable models only a cutoff growing as n with 1 can reproduce the scaling of a power law degree distribution in turn it is this large cutoff that generates non self averaging

average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

adaptive estimation for a class of nonlinearly parameterized dynamical systems the technical note presents a new technique for the adaptive parameter estimation in nonlinear parameterized dynamical systems the technique proposes an uncertainty set update approach that guarantees forward invariance of the true value of the parameters in addition it is shown that in the presence of sufficiently exciting state trajectories the parameter estimates converge to the true values and the uncertainty set vanishes around the true value of the parameters the set based approach is also shown to identify when the parameter estimates are biased two simulation examples are presented that demonstrate the effectiveness of the technique

time scale separation redesigns for stabilization and performance recovery of uncertain nonlinear systems in this paper we propose two different time scale separation based robust redesign techniques which recover the trajectories of a nominal control design in the presence of uncertain nonlinearities we first consider additive input uncertainties and design a high gain filter to estimate the uncertainty we then employ the fast variables arising from this filter in the feedback control law to cancel the effect of the uncertainties in the plant we next extend this design to systems with uncertain input nonlinearities in which case we design two sets of high gain filters the first to estimate the input uncertainty over a fast time scale and the second to force this estimate to converge to the nominal input on an intermediate time scale using singular perturbation theory we prove that the trajectories of the respective two time scale and three time scale redesigned systems approach those of the nominal system when the filter gains are increased we illustrate the redesigns by applying them to various physically motivated examples

adaptation and parameter estimation in systems with unstable target dynamics and nonlinear parametrization in this paper we propose a solution to the problem of adaptive control and parameter estimation in systems with unstable target dynamics models of uncertainties are allowed to be nonlinearly parameterized and required to be smooth and monotonic functions of linear functionals of the parameters the mere assumption of existence of nonlinear operator gains for the target dynamics is sufficient to guarantee that system solutions are bounded reach a neighborhood of the target set and the mismatches between the modeled uncertainties and their compensator converge to zero with respect to parameter convergence a standard persistent excitation condition suffices to ensure that it is exponential when a weaker nonlinear version of persistent excitation is satisfied asymptotic convergence is guaranteed the spectrum of possible applications ranges from tyre road slip control to asynchronous message transmission in spiking neural oscillators

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

introduction to a newton type method for solving nonlinear equations in this paper we shall introduce a new newton type method namely rational newton for solving a nonlinear equation this new method is shown to converge cubically we shall examine the effectiveness of the rational newton method by comparing the performance with the well established methods namely the classical newton method the halley rational method the kou et al and the weerakoon and fernando method for approximating the root of a given nonlinear equation the approximate solution of the rational newton method is found to be substantially more accurate than the well established methods

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

on adaptive control of nonlinearly parameterized nonlinear systems towards a constructive procedure abstract a new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized nonlinear systems was recently proposed by the authors the key step is the construction of a monotone mapping via a suitable selection of a controller tuning function which has to satisfy some integrability conditions this translates into the need to solve a partial differential equation pde in this paper this result is extended providing some answers to the questions of characterization of monotonizable systems and solvability of the pde first adding to the design a nonlinear dynamic scaling we obviate the need to solve the pde second for the case of factorizable nonlinearities the following results are established i it is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible ii directly verifiable involutivity conditions that ensure the solution of the pde are presented iii an explicit formula for the required tuning function is given provided the regressor matrix satisfies some rank conditions hence adding a dynamic scaling this yields a constructive solution to the problem

weighted distances in scale free configuration models in this paper we study first passage percolation in the configuration model with empirical degree distribution that follows a power law with exponent tau in 2 3 we assign independent and identically distributed i i d weights to the edges of the graph we investigate the weighted distance the length of the shortest weighted path between two uniformly chosen vertices called typical distances when the underlying age dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time called explosive branching process baroni hofstad and the second author showed in baroni et al j appl probab 54 1 146 164 2017 that typical distances converge in distribution to a bounded random variable the order of magnitude of typical distances remained open for the tau in 2 3 case when the underlying branching process is not explosive we close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary not necessary continuous edge weight distributions that produce a non explosive age dependent branching process with infinite mean power law offspring distributions this sequence tends to infinity with the amount of vertices and by choosing an appropriate weight distribution can be tuned to be any growing function that is o log log n where n is the number of vertices in the graph we show that the result remains valid for the the erased configuration model as well where we delete loops and any second and further edges between two vertices

tight fluctuations of weight distances in random graphs with infinite variance degrees we prove results for first passage percolation on the configuration model with degrees having asymptotic finite mean infinite variance and i i d edge weights with strictly positive support of the form y a x where a is a positive constant and the excess edge weight x is a non negative random variable with zero as the infimum of its support we prove that the weight of the optimal path between two uniformly chosen vertices has tight fluctuations around the asymptotic mean of the graph distance if and only if the following condition holds the random variable x is such that the age dependent branching process describing first passage percolation exploration in the same graph with edge weights from distribution x has a positive probability to reach infinitely many individuals in a finite time this shows that almost shortest paths in the graph distance proliferate in the sense that there even exist paths having tight total excess edge weight for appropriate edge weight distributions our proof makes use of a degree dependent percolation model that we believe is interesting in its own right as well as tightness results for distances in scale free configuration models that we prove to hold under rather weak conditions on the degrees

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

diameters in preferential attachment models in this paper we investigate the diameter in preferential attachment pa models thus quantifying the statement that these models are small worlds the models studied here are such that edges are attached to older vertices proportional to the degree plus a constant i e we consider affine pa models there is a substantial amount of literature proving that quite generally pa graphs possess power law degree sequences with a power law exponent 2

parameter estimation and compensation in systems with nonlinearly parameterized perturbations we consider a class of systems influenced by perturbations that are nonlinearly parameterized by unknown constant parameters and develop a method for estimating the unknown parameters the method applies to systems where the states are available for measurement and perturbations with the property that an exponentially stable estimate of the unknown parameters can be obtained if the whole perturbation is known the main contribution is to introduce a conceptually simple modular design that gives freedom to the designer in accomplishing the main task which is to construct an update law to asymptotically invert a nonlinear equation compensation for the perturbations in the system equations is considered for a class of systems with uniformly globally bounded solutions for which the origin is uniformly globally asymptotically stable when no perturbations are present we also consider the case when the parameters can only be estimated when the controlled state is bounded away from the origin and show that we may still be able to achieve convergence of the controlled state we illustrate the method through examples and apply it to the problem of downhole pressure estimation during oil well drilling

enumeration of graphs with a heavy tailed degree sequence abstract in this paper we asymptotically enumerate graphs with a given degree sequence d d 1 d n satisfying restrictions designed to permit heavy tailed sequences in the sparse case i e where the average degree is rather small our general result requires upper bounds on functions of m k i 1 n d i k for a few small integers k 1 note that m 1 is simply the total degree of the graphs as special cases we asymptotically enumerate graphs with i degree sequences satisfying m 2 o m 1 9 8 ii degree sequences following a power law with parameter 5 2 iii power law degree sequences that mimic independent power law degrees with parameter 1 3 2 732 iv degree sequences following a certain long tailed power law v certain bi valued sequences a previous result on sparse graphs by mckay and the second author applies to a wide range of degree sequences but requires o m 1 1 3 where is the maximum degree our new result applies in some cases when is only barely o m 1 3 5 case i above generalises a result of janson which requires m 2 o m 1 and hence m 1 o n and o n 1 2 cases ii and iii provide the first asymptotic enumeration results applicable to degree sequences of real world networks following a power law for which it has been empirically observed that 2 3

universality for first passage percolation on sparse random graphs we consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density assuming that the degree distribution satises a uniform x 2 logx condition we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices as well the number of edges on this path or hopcount the hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous time branching process centered by a multiple of logn where the constant is the inverse of the malthusian rate of growth of the associated branching process the minimal weight converges in distribution the limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices and a gumbel random variable and thus shows a remarkably universal behavior the proofs rely on a rened coupling between the shortest path problems on these graphs and continuous time branching processes and on a poisson point process limit for the potential closing edges of shortest weight paths between the source and destination the results extend to a host of related random graph models ranging from random rregular graphs inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence

adaptive control of continuous time systems with convex concave parametrization this paper deals with adaptive control of a class of nonlinear dynamic systems with nonlinear parametrization in this class the state variables are assumed to be accessible and the nonlinearity in the parameters is assumed to be either convex or concave by introducing a tuning function and an adaptive law based on a min max strategy it is shown that such a class of dynamic systems can be adaptively controlled in a stable manner global boundedness of the overall adaptive system and tracking to within a desired precision are established with the new adaptive controller the proposed controller is applied to a precise positioning problem in the presence of nonlinearly parametrized friction dynamics it is shown that the controller leads to position errors and friction compensation that are orders of magnitude better than those based on estimation of linear parametrizations the fact that the new parameter estimation strategy used is distinct from the traditionally used gradient schemes permits the expansion of the scope of adaptive control which has been restricted hitherto in most cases to systems with linear parametrization

stable adaptive control of a class of first order nonlinearly parameterized plants stable adaptive control algorithms are designed for a class of first order nonlinearly parameterized plants containing nonlinearities similar to those arising in fermentation process models when the plant nonlinearity is conveniently parameterized and when the controller parameters are adjusted adaptively the output error is shown to converge to zero stability of the overall system is proved using the lyapunov function with a suitably chosen cubic term

some variant of newton s method with third order convergence we present a new modification of the newton s method which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given and the best efficiency in term of function evaluations of two of this new methods is provided

brief growth rate conditions for uniform asymptotic stability of cascaded time varying systems in the last decade several results on cascaded autonomous systems have appeared in the literature the sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system in this paper we analyze three complementary classes of nonlinear time varying systems according to these growth rates for each case we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input to state stability and other growth rate conditions previously reported the advantage of this type of analysis is that it is often easier to verify such conditions than to find a lyapunov function with a negative definite derivative

universal scaling of distances in complex networks universal scaling of distances between vertices of erd ifmmode mbox h o else h o fi s r enyi random graphs scale free barab asi albert models science collaboration networks biological networks internet autonomous systems and public transport networks are observed a mean distance between two nodes of degrees k i and k j equals to l ij a ensuremath b phantom rule 0 2em 0ex mathrm log k i k j the scaling is valid over several decades a simple theory for the appearance of this scaling is presented parameters a and b depend on the mean value of a node degree k nn calculated for the nearest neighbors and on network clustering coefficients

stochastic stability of the discrete time extended kalman filter the authors analyze the error behavior for the discrete time extended kalman filter for general nonlinear systems in a stochastic framework in particular it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough this result is verified by numerical simulations for an example system

adaptive control of a class of nonlinearly parameterized plants a stable adaptive control method is developed for a class of first order nonlinearly parameterized plants the method is based on modified adaptive algorithms that yield a stable system in which the output errors are guaranteed to converge to zero the stability analysis is carried out using suitably constructed lyapunov functions

when is a scale free graph ultra small in this paper we study typical distances in the configuration model when the degrees have asymptotically infinite variance we assume that the empirical degree distribution follows a power law with exponent tau in 2 3 up to value n beta n for some beta n gg log n gamma and gamma in 0 1 this assumption is satisfied for power law i i d degrees and also includes truncated power law empirical degree distributions where the possibly exponential truncation happens at n beta n these examples are commonly observed in many real life networks we show that the graph distance between two uniformly chosen vertices centers around 2 log log n beta n log tau 2 1 beta n 3 tau with tight fluctuations thus the graph is an ultrasmall world whenever 1 beta n o log log n we determine the distribution of the fluctuations around this value in particular we prove these form a sequence of tight random variables with distributions that show log log periodicity and as a result it is non converging we describe the topology and number of shortest paths we show that the number of shortest paths is of order n f n beta n where f n in 0 1 is a random variable that oscillates with n we decompose shortest paths into three segments two end segments starting at each of the two uniformly chosen vertices and a middle segment the two end segments of any shortest path have length log log n beta n log tau 2 tight and the total degree is increasing towards the middle of the path on these segments the connecting middle segment has length 1 beta n 3 tau tight and it contains only vertices with degree at least of order n 1 f n beta n thus all the degrees on this segment are comparable to the maximal degree our theorems also apply when instead of truncating the degrees we start with a configuration model and we remove every vertex with degree at least n beta n and the edges attached to these vertices this sheds light on the attack vulnerability of the configuration model with infinite variance degrees

parameter convergence in nonlinearly parameterized systems a large class of problems in parameter estimation concerns nonlinearly parametrized systems over the past few years a stability framework for estimation and control of such systems has been established we address the issue of parameter convergence in such systems in this paper systems with both convex concave and general parameterizations are considered in the former case sufficient conditions are derived under which parameter estimates converge to their true values using a min max algorithm in the latter case to achieve parameter convergence a hierarchical min max algorithm is proposed where the lower level consists of a min max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie using this hierarchical algorithm a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization in both cases the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions a general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

a modification of newton method with third order convergence in this paper we present a new modification of newton method for solving non linear equations analysis of convergence shows that the new method is cubically convergent per iteration the new method requires two evaluations of the function and one evaluation of its first derivative thus the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself its practical utility is demonstrated by numerical examples

a matrix free quasi newton method for solving large scale nonlinear systems one of the widely used methods for solving a nonlinear system of equations is the quasi newton method the basic idea underlining this type of method is to approximate the solution of newton s equation by means of approximating the jacobian matrix via quasi newton update application of quasi newton methods for large scale problems requires in principle vast computational resource to form and store an approximation to the jacobian matrix of the underlying problem hence this paper proposes an approximation for newton step based on the update of approximation requiring a computational effort similar to that of matrix free settings it is made possible by approximating the jacobian into a diagonal matrix using the least change secant updating strategy commonly employed in the development of quasi newton methods under suitable assumptions local convergence of the proposed method is proved for nonsingular systems numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with newton s chord newton s and broyden s methods

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

a new modified secant like method for solving nonlinear equations in this paper we present a new secant like method for solving nonlinear equations analysis of the convergence shows that the asymptotic convergence order of this method is 1 3 some numerical results are given to demonstrate its efficiency