a branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

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

spectral analysis for a family of treelike networks abstract for a network knowledge of its laplacian eigenvalues is central to understand its structure and dynamics in this paper we study the laplacian spectra and their applications for a family of treelike networks firstly in order to obtain the laplacian spectra we calculate the constant term and monomial coefficient of characteristic polynomial of the laplacian matrix for a family of treelike networks by using the vieta theorem we then obtain the sum of reciprocals of all nonzero eigenvalues of laplacian matrix finally we determine some interesting quantities that are related to the sum of reciprocals of all nonzero eigenvalues of laplacian matrix such as kirchhoff index global mean first passage time gmfpt

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

scaling of average sending time on weighted koch networks random walks on weighted complex networks especially scale free networks have attracted considerable interest in the past but the efficiency of a hub sending information on scale free small world networks has been addressed less in this paper we study random walks on a class of weighted koch networks with scaling factor 0 r 1 we derive some basic properties for random walks on the weighted koch networks based on which we calculate analytically the average sending time ast defined as the average of mean first passage times mfpts from a hub node to all other nodes excluding the hub itself the obtained result displays that for 0 r 1 in large networks the ast grows as a power law function of the network order with the exponent represented by log43r 1r and for r 1 in large networks the ast grows with network order as n ln n which is larger than the linear scaling of the average receiving time defined as the average of mfpts for random walks to a given hub node averaged over all starti

scaling of average weighted shortest path and average receiving time on weighted hierarchical networks abstract recent work on the networks has focused on the weighted hierarchical networks that are significantly different from the un weighted hierarchical networks in this paper we study a family of weighted hierarchical networks which are recursively defined from an initial uncompleted graph in which the weights of edges have been assigned to different values with certain scale firstly we study analytically the average weighted shortest path awsp on the weighted hierarchical networks using a recursive method we determine explicitly the awsp the obtained rigorous solution shows that the networks grow unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded then depending on a biased random walk we research the mean first passage time mfpt between a hub node and any peripheral node finally we deduce the analytical expression of the average of mfpts for a random walker originating from any node to first visit to a hub node which is named as the average receiving time art the obtained result shows that art is bounded or grows sublinearly with the network order relating to the number of initial nodes and the weighted factor or grows quadratically with the iteration

eigentime identities for random walks on a family of treelike networks and polymer networks abstract in this paper we investigate the eigentime identities quantifying as the sum of reciprocals of all nonzero normalized laplacian eigenvalues on a family of treelike networks and the polymer networks firstly for a family of treelike networks it is shown that all their eigenvalues can be obtained by computing the roots of several small degree polynomials defined recursively we obtain the scalings of the eigentime identity on a family of treelike with network size n n is n n ln n n then for the polymer networks we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities using the relationship between the generation and the next generation of eigenvalues we obtain the scalings of the eigentime identity on the polymer networks with network size n n is n n ln n n by comparing the eigentime identities on these two kinds of networks their scalings with network size n n are all n n ln n n

the modified box dimension and average weighted receiving time of the weighted hierarchical graph abstract in this paper we study the weighted hierarchical graph which is generated from bipartite graph with n n 1 n 2 vertices in which the weights of edges have been assigned to different values with certain scale firstly we introduce the definition of the modified box dimension then for the weighted hierarchical graph we deduce the modified box dimension dim m b g n n n log r n depending on the weighted factor r and the number n of copies secondly we mainly study their two average weighted receiving times awrts t i n and t ii n of the weighted hierarchical graph on random walk we discuss two cases in the case of n 1 n 2 r n 2 n 1 we deduce both awrts grow as a power law function of the network size v g n with the postive exponent represented by log n n n 1 n 2 or log n r 1 dim m b g n n n which means that the bigger the value of the modified box dimension is the slower the process of receiving information is in the case of n 1 n 2 r n 2 n 1 both awrts tend to constant if n n 1 n 2 the awrts grow with increasing order as log n v g n if n n 1 n 2 and both awrts grow as a power law function of the network size v g n with the exponent represented by log n n n 1 n 2 0 if n n 1 n 2

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

on the generalized adjacency laplacian and signless laplacian spectra of the weighted edge corona networks abstract many problems in real world either natural or man made can be usefully represented by graphs or networks along with a complex topological structure the weight is a vital factor in characterizing some properties of real networks in this paper we define a class of the weighted edge corona product networks the generalized adjacency resp laplacian and signless laplacian spectra with two different structures are determined as applications the number of spanning trees and kirchhoff index of the weighted edge corona product networks are computed

a gradient formula for linear chance constraints under gaussian distribution we provide an explicit gradient formula for linear chance constraints under a possibly singular multivariate gaussian distribution this formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints in smaller dimension and with different distribution parameters this is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e g calculating singular gaussian distributions or regular gaussian probabilities of polyhedra can be employed to calculate gradients at the same time moreover the precision of gradients can be controlled by that of function values which is a great advantage over using finite difference approximations finally higher order derivatives are easily derived explicitly the use of the obtained formula is illustrated for an example of a transportation network with stochastic demands

multifractal analysis of a measure of multifractal exact dimension abstract we focus on the multifractal generalization of the centered hausdorff measure and dimension we analyze the correlation among different approaches to the definition of the multifractal exact dimension of locally finite and borel regular measures on the basis of fractal analysis of essential supports of these measures using characteristic multifractal measures we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures

the mixed vertex packing problem we study a generalization of the vertex packing problem having both binary and bounded continuous variables called the mixed vertex packing problem mvpp the well known vertex packing model arises as a subproblem or relaxation of many 0 1 integer problems whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0 1 integer programming we describe strong valid inequalities for the convex hull of solutions to the mvpp and separation algorithms for these inequalities we give a summary of computational results with a branch and cut algorithm for solving the mvpp and using it to solve general mixed integer problems

the laplacian spectrum and average trapping time for weighted dyson hierarchical network abstract in this paper we consider the weighted dyson hierarchical network that is a weighted fully connected network where the pattern of weights is ruled by a weight factor given that the laplacian operator is intrinsically implied in the analysis of dynamic processes occurring on complex networks laplacian spectrum allows addressing analytically a large class of problems exploiting the deterministic recursive structure we are able to derive explicitly all eigenvalues and their corresponding eigenvectors of laplacian matrix further we derive exact solutions of the trapping time tt and average trapping time att on the weighted dyson hierarchical network with weight dependent walk the obtained results show that tt and att are related to the weight factor of the weighted dyson hierarchical network

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

weighted fractal networks abstract in this paper we define a new class of weighted complex networks sharing several properties with fractal sets and whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension general networks with fractal or hierarchical structures can be set in the proposed framework that moreover could be used to provide some answers to the widespread emergence of fractal structures in nature

scaling of average receiving time and average weighted shortest path on weighted koch networks abstract in this paper we present weighted koch networks based on classic koch networks a new method is used to determine the average receiving time art whose key step is to write the sum of mean first passage times mfpts for all nodes to absorption at the trap located at a hub node as a recursive relation we show that the art exhibits a sublinear or linear dependence on network order thus the weighted koch networks are more efficient than classic koch networks in receiving information moreover average weighted shortest path awsp is calculated in the infinite network order limit the awsp depends on the scaling factor the weighted koch network grows unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded

scaling of average weighted shortest path and average receiving time on the weighted cayley networks abstract in this paper we mainly study two important properties of weighted cayley networks depending on a weight factor r the weight factor makes it more difficult to calculate the topological characteristics such as average weighted shortest path awsp and average receiving time art of the networks we first calculate some basic network measurements such as average degree and average node strength then we derive the expression for awsp and show that it stays bounded with network order growing 0 r 1 in the limit of large network order at last we focus on random walks and trapping issue on the networks that is we calculate the art in more detail for 0 r 1 2 the art grows with increasing size n t as ln n t for r 1 2 the art grows with increasing size n t as ln 2 n t for 1 2 r 1 the art grows sublinearly with the network size n t for r 1 the art grows linearly with the network size n t

typical behavior of mixed lq dimensions abstract previously l olsen had studied the typical behavior in the sense of baire s category of the upper and lower l q dimensions of a single measure in this paper we will focus on the typical behavior of the upper and lower mixed l q dimensions of finitely many measures

generalized adjacency and laplacian spectra of the weighted corona graphs abstract recently corona product graphs have attracted a great deal of attention from scientific communities such as coding theory frequency assignment mathematical chemistry meanwhile weight factor gradually became a research hotpoint because of closing to reality hence we introduce the weighted corona graphs in this paper the weighted corona product graphs are the in depth and promotion of the corona product graphs linking the spectra between the initial graph and the product graph then we deduce the complete information of generalized adjacency and laplacian eigenvalues of the weighted product graph with different structures and we generalize the results obtained previously into product corona graph with m 1 iterations

trapping on modular scale free and small world networks with multiple hubs abstract in this paper we introduce a new kind of scale free and small world networks with multiple hubs based on classic networks and present the trapping problem on them we show that in the multiple hub networks the growth of mfpt with the traps located on the hub or peripheral nodes displays quite a difference from the single hub networks our conclusions are directed to possible applications in rumor or epidemic spreading phenomena

a certain double well potential related to su 2 symmetry in the present paper we analyse a class of partially exactly solvable one dimensional double well potentials we show how the underlying su 2 dynamical symmetry makes it possible analytically to find several energy levels

a review of fractality and self similarity in complex networks we review recent findings of self similarity in complex networks using the box covering technique it was shown that many networks present a fractal behavior which is seemingly in contrast to their small world property moreover even non fractal networks have been shown to present a self similar picture under renormalization of the length scale these results have an important effect in our understanding of the evolution and behavior of such systems a large number of network properties can now be described through a set of simple scaling exponents in analogy with traditional fractal theory