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

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

exceedance control of the false discovery proportion multiple testing methods to control the false discovery rate the expected proportion of falsely rejected null hypotheses among all rejections have received much attention it can be valuable to control not the mean of this false discovery proportion fdp but rather the probability that the fdp exceeds a specified bound in this article we construct a general class of methods for exceedance control of fdp based on inverting tests of uniformity the method also produces a confidence envelope for the fdp as a function of rejection threshold we discuss how to select a procedure with good power

convex computation of the region of attraction of polynomial control systems we address the long standing problem of computing the region of attraction roa of a target set e g a neighborhood of an equilibrium point of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints we show that the roa can be computed by solving an infinite dimensional convex linear programming lp problem over the space of measures in turn this problem can be solved approximately via a classical converging hierarchy of convex finite dimensional linear matrix inequalities lmis our approach is genuinely primal in the sense that convexity of the problem of computing the roa is an outcome of optimizing directly over system trajectories the dual infinite dimensional lp on nonnegative continuous functions approximated by polynomial sum of squares allows us to generate a hierarchy of semialgebraic outer approximations of the roa at the price of solving a sequence of lmi problems with asymptotically vanishing conservatism this sharply contrasts with the existing literature which follows an exclusively dual lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative lmi conditions the approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description the approach is demonstrated on several numerical examples

nonlinear optimal control via occupation measures and lmi relaxations we consider the class of nonlinear optimal control problems ocps with polynomial data i e the differential equation state and control constraints and cost are all described by polynomials and more generally for ocps with smooth data in addition state constraints as well as state and or action constraints are allowed we provide a simple hierarchy of lmi linear matrix inequality relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value under some convexity assumptions the sequence converges to the optimal value of the ocp preliminary results show that good approximations are obtained with few moments

distance based vertex identification in graphs the outer multiset dimension abstract the characterisation of vertices in a network in relation to other peers has been used as a primitive in many computational procedures such as node localisation and de anonymisation this article focuses on a characterisation type known as the multiset metric representation formally given a graph g and a subset of vertices s w 1 w t v g the multiset representationof a vertex u v g with respect to s is the multiset m u s d g u w 1 d g u w t a subset of vertices s such that m u s m v s u v for every u v v g s is said to be a multiset resolving set and the cardinality of the smallest such set is the outer multiset dimension we study the general behaviour of the outer multiset dimension and determine its exact value for several graph families we also show that computing the outer multiset dimension of arbitrary graphs is np hard and provide methods for efficiently handling particular cases

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

a moment and sum of squares extension of dual dynamic programming with application to nonlinear energy storage problems abstract we present a finite horizon optimization algorithm that extends the established concept of dual dynamic programming ddp in two ways first in contrast to the linear costs dynamics and constraints of standard ddp we consider problems in which all of these can be polynomial functions second we allow the state trajectory to be described by probability distributions rather than point values and return approximate value functions fitted to these the algorithm is in part an adaptation of sum of squares techniques used in the approximate dynamic programming literature it alternates between a forward simulation through the horizon in which the moments of the state distribution are propagated through a succession of single stage problems and a backward recursion in which a new polynomial function is derived for each stage using the moments of the state as fixed data the value function approximation returned for a given stage is the point wise maximum of all polynomials derived for that stage this contrasts with the piecewise affine functions derived in conventional ddp we prove key convergence properties of the new algorithm and validate it in simulation on two case studies related to the optimal operation of energy storage devices with nonlinear characteristics the first is a small borehole storage problem for which multiple value function approximations can be compared the second is a larger problem for which conventional discretized dynamic programming is intractable

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

on the metric dimension of corona product graphs given a set of vertices s v 1 v 2 v k of a connected graph g the metric representation of a vertex v of g with respect to s is the vector r v s d v v 1 d v v 2 d v v k where d v v i i 1 k denotes the distance between v and v i s is a resolving set for g if for every pair of distinct vertices u v of g r u s r v s the metric dimension of g dim g is the minimum cardinality of any resolving set for g let g and h be two graphs of order n 1 and n 2 respectively the corona product g h is defined as the graph obtained from g and h by taking one copy of g and n 1 copies of h and joining by an edge each vertex from the ith copy of h with the ith vertex of g for any integer k 2 we define the graph g kh recursively from g h as g kh g k 1h h we give several results on the metric dimension of g kh for instance we show that given two connected graphs g and h of order n 1 2 and n 2 2 respectively if the diameter of h is at most two then dim g kh n 1 n 2 1 k 1dim h moreover if n 2 7 and the diameter of h is greater than five or h is a cycle graph then dim g kh n 1 n 2 1 k 1dim k 1 h

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

the degree related clustering coefficient and its application to link prediction abstract link prediction plays a significant role in explaining the evolution of networks however it is still a challenging problem that has been addressed only with topological information in recent years based on the belief that network nodes with a great number of common neighbors are more likely to be connected many similarity indices have achieved considerable accuracy and efficiency motivated by the natural assumption that the effect of missing links on the estimation of a node s clustering ability could be related to node degree in this paper we propose a degree related clustering coefficient index to quantify the clustering ability of nodes unlike the classical clustering coefficient our new coefficient is highly robust when the observed bias of links is considered furthermore we propose a degree related clustering ability path dcp index which applies the proposed coefficient to the link prediction problem experiments on 12 real world networks show that our proposed method is highly accurate and robust compared with four common neighbor based similarity indices common neighbors cn adamic adar aa resource allocation ra and preferential attachment pa and the recently introduced clustering ability ca index

convex computation of the maximum controlled invariant set for polynomial control systems we characterize the maximum controlled invariant mci set for discrete as well as continuous time nonlinear dynamical systems as the solution of an infinite dimensional linear programming problem for systems with polynomial dynamics and compact semialgebraic state and control constraints we describe a hierarchy of finite dimensional linear matrix inequality lmi relaxations whose optimal values converge to the volume of the mci set dual to these lmi relaxations are sum of squares sos problems providing a converging sequence of outer approximations to the mci set the approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description a number of numerical examples illustrate the approach

probabilistic simrank computation over uncertain graphs existing studies on graph based analysis focus on standard graphs that are precise and complete however in many prevalent application domains such as social biological and mobile networks the existence of an edge in a graph is best captured by a likelihood measure or a probability this paper investigates the problem of node similarity computation on large uncertain graphs we first extend the well known simrank to define meaningful similarity measure on probabilistic graphs and then propose a probabilistic framework to compute it based on the observation that the probabilistic transition probabilities computed on the whole graph equal to those computed on multiple sub graphs we design a subg algorithm to compute the probabilistic transition matrix furthermore we propose an efficient dynamic programming algorithm to degrade the time complexity from exponential to polynomial we give the corresponding theoretical justification and extend the proposed algorithm to incremental dynamic programming algorithm which further improves the efficiency and can be applied to dynamic graphs extensive experiments on synthetic and real data sets verify that the proposed methods are efficient and effective

approximate volume and integration for basic semialgebraic sets given a basic compact semialgebraic set mathbf k subset mathbb r n we introduce a methodology that generates a sequence converging to the volume of mathbf k this sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on mathbf k can be approximated as closely as desired which permits the approximation of the integral on mathbf k of any given polynomial the extension to integration against some weight functions is also provided finally some numerical issues associated with the algorithms involved are briefly discussed

tests alternative to higher criticism for high dimensional means under sparsity and column wise dependence we consider two alternative tests to the higher criticism test of donoho and jin ann statist 32 2004 962 994 for high dimensional means under the sparsity of the nonzero means for sub gaussian distributed data with unknown column wise dependence the two alternative test statistics are constructed by first thresholding l 1 and l 2 statistics based on the sample means respectively followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity the two alternative tests can attain the same detection boundary of the higher criticism test in ann statist 32 2004 962 994 which was established for uncorrelated gaussian data it is demonstrated that the maximal l 2 thresholding test is at least as powerful as the maximal l 1 thresholding test and both the maximal l 2 and l 1 thresholding tests are at least as powerful as the higher criticism test

k metric antidimension the study and analysis of social graphs impacts on a wide range of applications such as community decision making support and recommender systems with the boom of online social networks such analyses are benefiting from a massive collection and publication of social graphs at large scale unfortunately individuals privacy right might be inadvertently violated when publishing this type of data in this article we introduce k anonymity a novel privacy measure aimed at evaluating the resistance of social graphs to active attacks k anonymity is based on a new problem in graph theory the k metric antidimension defined as follows let g v e be a simple connected graph and s w 1 w t v an ordered subset of vertices the metric representation of a vertex u v with respect to s is the t vector r u s d g u w 1 d g u w t where dg u v represents the length of a shortest u v path in g we call s a k antiresolving set if k is the largest positive integer such that for every vertex v v s there exist other k 1 different vertices v 1 v k 1 v s with r v s r v 1 s r v k 1 s the k metric antidimension of g is the minimum cardinality among all the k antiresolving sets for g we address the k metric antidimension problem by proposing a true biased algorithm with success rate above 80 when considering random graphs of size at most 100 the proposed algorithm is used to determine the privacy guarantees offered by two real life social graphs with respect to k anonymity we also investigate theoretical properties of the k metric antidimension of graphs in particular we focus on paths cycles complete bipartite graphs and trees

modularity from fluctuations in random graphs and complex networks the mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection we show that finding the modularity of a network is analogous to finding the ground state energy of a spin system moreover we demonstrate that due to fluctuations stochastic network models give rise to modular networks specifically we show both numerically and analytically that random graphs and scale free networks have modularity we argue that this fact must be taken into consideration to define statistically significant modularity in complex networks

partition dimension of certain classes of series parallel graphs abstract the partition dimension problem is to decompose the vertex set of a graph into minimum number of vertex disjoint sets such that the ordered distances between every vertex of the graph and the disjoint sets of that decomposition have different values in the representation this problem has emerging applications in areas such as structure activity issues in drug design navigation of robots pattern recognition and image processing in the present study we provide an upper bound for the partition dimension of parallel composition of any graphs and derive an exact result for parallel composition of paths with different lengths on the other side we find the partition dimension of the series composition of cycles and complete graphs with different sizes and an upper bound is derived for series composition of any graphs further this problem is applied to a graph which has been generated by parallel and series composition together on paths

mixed metric dimension of graphs abstract let g v e be a connected graph a vertex w v distinguishes two elements vertices or edges x y e v if dg w x dg w y a set s of vertices in a connected graph g is a mixed metric generator for g if every two distinct elements vertices or edges of g are distinguished by some vertex of s the smallest cardinality of a mixed metric generator for g is called the mixed metric dimension and is denoted by dim m g in this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds we also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph finally we prove that the problem of determining the mixed metric dimension of a graph is np hard in the general case

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

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

convex duality and nonlinear optimal control problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals in this way methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems the result is a necessary and sufficient condition of optimality that generalizes well known sufficient conditions referred to as verification theorems in dynamic programming as a byproduct we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the hamilton jacobi equation it is a striking illustration of the wide range of problems to which convex analysis and in particular convex duality is applicable the approach applied to parametric problems in the calculus of variations was pioneered by l c young lectures on the calculus of variations and optimal control theory w b saunders philadelphia pa 1969 as recent work has shown however it is equally fruitful when applied in optimal control thi

the linear programming approach to approximate dynamic programming the curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large scale stochastic control problems we study an efficient method based on linear programming for approximating solutions to such problems the approach fits a linear combination of pre selected basis functions to the dynamic programming cost to go function we develop error bounds that offer performance guarantees and also guide the selection of both basis functions and state relevance weights that influence quality of the approximation experimental results in the domain of queueing network control provide empirical support for the methodology

convergence rates of moment sum of squares hierarchies for optimal control problems we study the convergence rate of moment sum of squares hierarchies of semidefinite programs for optimal control problems with polynomial data it is known that these hierarchies generate polynomial under approximations to the value function of the optimal control problem and that these under approximations converge in the l1 norm to the value function as their degree d tends to infinity we show that the rate of this convergence is o 1 log log d we treat in detail the continuous time infinite horizon discounted problem and describe in brief how the same rate can be obtained for the finite horizon continuous time problem and for the discrete time counterparts of both problems