exact solutions for models of evolving networks with addition and deletion of nodes there has been considerable recent interest in the properties of networks such as citation networks and the worldwide web that grow by the addition of vertices and a number of simple solvable models of network growth have been studied in the real world however many networks including the web not only add vertices but also lose them here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest for the case of net growth and so called preferential attachment in which newly appearing vertices attach to previously existing ones in proportion to vertex degree we show that the resulting networks have power law degree distributions but with an exponent that diverges as the growth rate vanishes we conjecture that the low exponent values observed in real world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal were growth to slow in the future for instance in a more mature future version of the web we would expect to see exponents increase potentially without bound

markov chain based numerical method for degree distributions of growing networks in this paper we establish a relation between growing networks and markov chains and propose a computational approach for network degree distributions using the barabasi albert model as an example we first show that the degree evolution of a node in a growing network follows a nonhomogeneous markov chain exploring the special structure of these markov chains we develop an efficient algorithm to compute the degree distribution numerically with a computation complexity of o t sup 2 where t is the number of time steps we use three examples to demonstrate the computation procedure and compare the results with those from existing methods

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

higher order optimality conditions for set valued optimization via studniarski derivatives this paper deals with higher order optimality conditions of a set valued optimiza tion problem sop whose constraint condition is determined by a fixed set by virtue of studniarski derivatives of set valued maps and strict local efficiency higher order necessary and sufficient opti mality conditions for sop are obtained for the sufficient conditions the initial space must be finite dimensional higher order optimality conditions for sop are also investigated under the conditions that the initial and objective spaces are finite dimensional and the multifunction involved is mth order

local preferential attachment model for hierarchical networks abstract in real life networks incomers may only connect to a few others in a local area for their limited information and individuals in a local area are likely to have close relations accordingly we propose a local preferential attachment model here a local area network stands for a node and all its neighbors and the new nodes perform nonlinear preferential attachment k i k i in local areas the stable degree distribution and clustering degree correlations are analytically obtained with the increasing of the clustering coefficient increases while assortativity decreases from positive to negative in addition by adjusting the parameter the model can generate different kinds of degree distribution from exponential to power law the hierarchical organization independent of is the most significant character of this model

scalar multiplier rules in set valued optimization in this work we establish necessary and sufficient conditions for weak minimizers of a constrained set valued optimization problem these conditions are given by means of multiplier rules formulated in terms of contingent epiderivatives of scalar set valued maps we consider that the image spaces are finite dimensional and the set valued maps are convex and stable for these conditions our results provide a scalar version of analogous existing results in the literature under weaker existence conditions moreover we prove that the multiplier rules can be computed in terms of the directional derivative of associated maps of infima

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

design of observers for linear systems with unknown inputs with a straightforward treatment a design of reduced order observers is presented for linear systems with unknown inputs the observers are derived with a physical meaning some explanations are given within the framework of descriptor system observer design principles the conditions for the existence of the observers are presented some illustrative examples are included

a novel approach to the design of unknown input observers a novel state estimator design scheme for linear dynamical systems driven by partially unknown inputs is presented it is assumed that there is no information available about the unknown inputs and thus no prior assumption is made about the nature of these inputs a simple approach for designing a reduced order unknown input observer uio with pole placement capability is proposed by carefully examining the dynamic system involved and simple algebraic manipulations it is possible to rewrite equations eliminating the unknown inputs from part of the system and to put them into a form where it could be partitioned into two interconnected subsystems one of which is directly driven by known inputs only this makes it possible to use a conventional luenberger observer with a slight modification for the purpose of estimating the state of the system as a result it is also possible to state similar necessary and sufficient conditions to those of a conventional observer for the existence of a stable estimator and also arbitrary placement of the eigenvalues of the observer the design and computational complexities involved in designing uios are greatly reduced in the proposed approach

analytical results for stochastically growing networks connection to the zero range process we introduce a stochastic model of growing networks where both the number of new nodes which join the network and the number of connections vary stochastically we provide an exact mapping between this model and the zero range process and calculate analytically the degree distribution for any given evolution rule we argue that this mapping can be used to infer a possible evolution rule for any given network this is being demonstrated for a protein protein interaction network of saccharomyces cerevisiae

a scale free network with limiting on vertices abstract we propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity the random graph process is defined as follows at any time step t i with probability k and independently of other time step each vertex v i i t 1 is inactive which means it cannot be connected by more edges where k is the degree of v i at the time step t ii a new vertex v t is added along with m edges incident with v t at one time and its neighbors are chosen in the manner of preferential attachment we prove that the degree distribution p k of this random graph process satisfies p k c 1 k 3 0 1 0 if is a constant 0 and p k c 2 k 3 if l 0 as l where c 1 c 2 are two positive constants the analytical result is found to be in good agreement with that obtained by numerical simulations furthermore we get the degree distributions in this model with m varying functions by simulation

transportation dynamics on networks of mobile agents most existing works on transportation dynamics focus on networks of a fixed structure but networks whose nodes are mobile have become widespread such as cell phone networks we introduce a model to explore the basic physics of transportation on mobile networks of particular interest is the dependence of the throughput on the speed of agent movement and the communication range our computations reveal a hierarchical dependence for the former while an algebraic power law is found between the throughput and the communication range with the exponent determined by the speed we develop a physical theory based on the fokker planck equation to explain these phenomena our findings provide insights into complex transportation dynamics arising commonly in natural and engineering systems

phase transition in the rich get richer mechanism due to finite size effects the rich get richer mechanism agents increase their wealth randomly at a rate proportional to their holdings is often invoked to explain the pareto power law distribution observed in many physical situations such as the degree distribution of growing scale free nets we use two different analytical approaches as well as numerical simulations to study the case where the number of agents is fixed and finite but large and the rich get richer mechanism is invoked a fraction r of the time the remainder of the time wealth is disbursed by a homogeneous process at short times we recover the pareto law observed for an unbounded number of agents in later times the moving distribution can be scaled to reveal a phase transition with a gaussian asymptotic form for and a pareto like tail on the positive side and a novel stretched exponential decay on the negative side for

effects of accelerating growth on the evolution of weighted complex networks many real systems possess accelerating statistics where the total number of edges grows faster than the network size in this paper we propose a simple weighted network model with accelerating growth we derive analytical expressions for the evolutions and distributions for strength degree and weight which are relevant to accelerating growth we also find that accelerating growth determines the clustering coefficient of the networks interestingly the distributions for strength degree and weight display a transition from scale free to exponential form when the parameter with respect to accelerating growth increases from a small to large value all the theoretical predictions are successfully contrasted with numerical simulations

unveiling community structures in weighted networks random walks on simple graphs in connection with electrical resistor networks lead to the definition of markov chains with transition probability matrix in terms of electrical conductances we extend this definition to an effective transition matrix p ij to account for the probability of going from vertex i to any vertex j of the original connected graph g also we present an algorithm based on the definition of this effective transition matrix among vertices in the network to extract a topological feature related to the manner by which graph g has been organized this topological feature corresponds to the communities in the graph

magnetic models on apollonian networks thermodynamic and magnetic properties of ising models defined on the triangular apollonian network are investigated this and other similar networks are inspired by the problem of covering a euclidian domain with circles of maximal radii maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model different choices for the coupling constants between only nearest neighbors along the lattice are taken into account for both ferromagnetic and antiferromagnetic constants long range magnetic ordering is obtained with exception of a size dependent effective critical behavior of the correlation length no evidence of asymptotic criticality was detected

polynomial growth in branching processes with diverging reproductive number we study the spreading dynamics on graphs with a power law degree distribution pkk with 2 3 as an example of a branching process with a diverging reproductive number we provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern deviating from the standard exponential growth first the population growth is extensive meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit second the temporal evolution is governed by a polynomial growth with a degree determined by the characteristic distance between vertices in the graph these results open a path to further investigation on the dynamics on networks unexpected based on previous mathematical studies we also show that both the characteristic time separating the exponential and polynomial regimes and the polynomial degree depend on the characteristic distance between ver tices more important in the limit of infinite graph sizes the exponential regime is virtually absent indicating that the polynomial regime is a novel and characteristic feature of the spreading dynamics on graphs with degree exponent 2 3 and more generally of branching processes with an unbounded average reproductive number consider a spreading process on a graph with a treelike structure at t 0 a vertex selected at random is infected by a virus which can then propagate to other vertices through the graph edges the causal tree representing the spreading process can be modeled as a branching process each vertex in the causal tree represents an infected vertex in the original graph and each arc in the causal tree represents the generation of a secondary infected vertex from a primary infected vertex the out degree of a vertex in the causal tree gives the number of other vertices it infects i e its reproductive number in turn the length of an arc a b in the causal tree gives the generation time the time elapsed from the infection of the primary case a to the infection of the secondary case b finally the vertex generation coincides with the topological distance from the first infected vertex the root in the original graph we assume that the reproductive numbers are indepen dent random variables with the probability distribution qd k

critical boolean networks with scale free in degree distribution we investigate analytically and numerically the dynamical properties of critical boolean networks with power law in degree distributions and for two choices of update functions when the exponent of the in degree distribution is larger than 3 we obtain results equivalent to those obtained for networks with fixed in degree e g the number of the nonfrozen nodes scales as n 2 3 with the system size n when the exponent of the distribution is between 2 and 3 the number of the nonfrozen nodes increases as n x with x being between 0 and 2 3 and depending on the exponent and on the cutoff of the in degree distribution these and ensuing results explain various findings obtained earlier by computer simulations

simultaneous observation of linear systems a state space interpretation necessary and sufficient conditions for the existence of a simultaneous functional observer for several linear time invariant lti systems are derived using distinguishability concepts that generalize the detectability concept for a single system they generalize also the known results a structure for common functional observers is proposed and for two systems or for several systems without inputs the problem can be constructively solved a relationship between the simultaneous observation problem and the existence of unknown input observers is established

renormalization group for evolving networks we propose a renormalization group treatment of stochastically growing networks as an example we study percolation on growing scale free networks in the framework of a real space renormalization group approach as a result we find that the critical behavior of percolation on the growing networks differs from that in uncorrelated networks

power law accelerating growth complex networks with mixed attachment mechanisms abstract in this paper motivated by the thoughts and methods of the mixture of preferential and uniform attachments we extend the barabasi albert ba model and establish a network model with the power law accelerating growth and the mixture of the two attachment mechanisms in our model the number of edges generated by each newly introduced node is proportional to the power of 0 1 of time t i e m t by virtue of the continuum approach we have deduced the degree distribution p k t of our model with the extended power law form p k t a t k b t when the number of edges k generated by each new node is much greater than the value of b t the degree distribution p k t will converge to the power law form p k t a t k when k is much less than the value of b t the degree distribution p k t will converge to the exponential law form p k t a t b t e k b t by virtue of numerical simulations we also discuss the dependence of the degree distribution p k t on the model s parameters where t is considered as a constant in the simulations finally we investigate the possible application of our model in the spreading and evolution of epidemics in some real world systems

stochastically evolving networks we discuss a class of models for the evolution of networks in which new nodes are recruited into the network at random times and links between existing nodes that are not yet directly connected may also form at random times the class contains both models that produce small world networks and less tightly linked models we produce both trees appropriate in certain biological applications and networks in which closed loops can appear which model communication networks and networks of human sexual interactions one of our models is closely related to random recursive trees and some exact results known in that context can be exploited the other models are more subtle and difficult to analyze our analysis includes a number of exact results for moments correlations and distributions of coordination number and network size we report simulations and also discuss some mean field approximations if the system has evolved for a long time and the state of a random node which thus has a random age is observed power law distributions for properties of the system arise in some of these models

decentralized state estimation in large scale interconnected dynamical systems abstract a simple approach for decentralized state estimation in large scale control systems is proposed given that a large scale dynamical system can be decomposed into a number of interconnected input output decentralized subsystems a method for designing totally decentralized local reduced order estimators is proposed it is shown that totally decentralized state estimation may be possible as long as the number of the subsystem interaction inputs are less than or equal to the locally measured outputs here the interconnection effects among the subsystems are treated as totally unknown inputs to each subsystem the method builds upon a novel approach in designing reduced order unknown input observers uio for dynamical systems and it therefore results in totally decentralized estimators structure the pros and cons of the estimation scheme are pointed out and necessary and sufficient conditions for designing local estimators are stated finally the design approach is verified by a numerical example

strict efficiency in vector optimization the notion of strict minimum of order m for real optimization problems is extended to vector optimization its properties and characterization are studied in the case of finite dimensional spaces multiobjective problems also the notion of super strict efficiency is introduced for multiobjective problems and it is proved that in the scalar case all of them coincide necessary conditions for strict minimality and for super strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set when the objective function is frechet differentiable necessary and sufficient conditions are established for the case m 1 resulting in the situation that the strict efficiency and super strict efficiency notions coincide

sampling properties of random graphs the degree distribution we discuss two sampling schemes for selecting random subnets from a network random sampling and connectivity dependent sampling and investigate how the degree distribution of a node in the network is affected by the two types of sampling here we derive a necessary and sufficient condition that guarantees that the degree distributions of the subnet and the true network belong to the same family of probability distributions for completely random sampling of nodes we find that this condition is satisfied by classical random graphs for the vast majority of networks this condition will however not be met we furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime we conclude by relating the results to real eschericia coli protein interaction network data

exact solution for the time evolution of network rewiring models we consider the rewiring of a bipartite graph using a mixture of random and preferential attachment the full mean field equations for the degree distribution and its generating function are given the exact solution of these equations for all finite parameter values at any time is found in terms of standard functions it is demonstrated that these solutions are an excellent fit to numerical simulations of the model we discuss the relationship between our model and several others in the literature including examples of urn backgammon and balls in boxes models the watts and strogatz rewiring problem and some models of zero range processes our model is also equivalent to those used in various applications including cultural transmission family name and gene frequencies glasses and wealth distributions finally some voter models and an example of a minority game also show features described by our model

aggregation kinetics of popularity abstract the tools of aggregation kinetics are applied to the popularity phenomena of single lane traffic clustering and to the growth of a network that mimics citations of scientific publications in the latter the network is built by introducing papers new nodes one at a time with preferential linking to more popular previously existing nodes from the rate equations the distribution of node degree as well as various global properties can be determined easily a simple extension of the model appears to describe the degree distributions of the world wide web

spectral centrality measures in complex networks complex networks are characterized by heterogeneous distributions of the degree of nodes which produce a large diversification of the roles of the nodes within the network several centrality measures have been introduced to rank nodes based on their topological importance within a graph here we review and compare centrality measures based on spectral properties of graph matrices we shall focus on pagerank pr eigenvector centrality ev and the hub and authority scores of the hits algorithm we derive simple relations between the measures and the in degree of the nodes in some limits we also compare the rankings obtained with different centrality measures

constrained randomization of weighted networks we propose a markov chain method to efficiently generate surrogate networks that are random under the constraint of given vertex strengths with these strength preserving surrogates and with edge weight preserving surrogates we investigate the clustering coefficient and the average shortest path length of functional networks of the human brain as well as of the international trade networks we demonstrate that surrogate networks can provide additional information about network specific characteristics and thus help interpreting empirical weighted networks

spr based markov chain method for degree distributions of evolving networks abstract in this paper we develop a stochastic process rules spr based markov chain method to calculate the degree distributions of evolving networks this new approach overcomes two shortcomings of shi chen and liu s use of the markov chain method shi et al 2005 21 in addition we show how an spr based markov chain method can be effectively used to calculate degree distributions of random birth and death networks which we believe to be novel first spr are introduced to replace traditional evolving rules tr making it possible to compute degree distributions in one sample space then the spr based markov chain method is introduced and tested by using it to calculate two kinds of evolving network finally and most importantly the spr based method is applied to the problem of calculating the degree distributions of random birth and death networks

spanning traceroutes over modular networks and general scaling degree distributions we analyze the class of networks characterized by modular structure where a sequence of l erdos renyi random networks of size nl with random average degrees is joined by links whose structure must remain immaterial we find that traceroutes spanning the entire macronetwork exhibit scaling degree distributions p k approximately k gamma where gamma depends on how the degrees of the joined clusters are distributed we thus suggest that yet another mechanism for the dynamic origin of arbitrary power law degree distributions observed in natural and artificial networks many of which belong to the range 2 or gamma or 3 may be found in random processes on modular networks

anomalous percolation properties of growing networks we describe the anomalous phase transition of the emergence of the giant connected component in scale free networks growing under mechanism of preferential linking we obtain exact results for the size of the giant connected component and the distribution of vertices among connected components we show that all the derivatives of the giant connected component size s over the rate b of the emergence of new edges are zero at the percolation threshold b c and s ensuremath propto mathrm exp ensuremath d ensuremath gamma b ensuremath b c ensuremath 1 2 where the coefficient d is a function of the degree distribution exponent ensuremath gamma in the entire phase without the giant component these networks are in a critical state the probability mathcal p k that a vertex belongs to a connected component of a size k is of a power law form at the phase transition point mathcal p k ensuremath sim 1 k mathrm ln k 2 in the phase with the giant component mathcal p k has an exponential cutoff at k c ensuremath propto 1 s in the simplest particular case we present exact results for growing exponential networks

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

a highly clustered scale free network evolved by random walking abstract in present paper we propose a highly clustered weighted network model that incorporates the addition of a new node with some links new links between existing nodes and the edge s weight dynamical evolution based on weight dependent walks at each time step the analytical approach and numerical simulation show that the system grows into a weighted network with the power law distributions of strength weight and degree the weight dependent walk length l will not influence the strength distribution but the clustering coefficient of the network is sensitive to l particularly the clustering coefficient is especially high and almost independent of the network size when l 2

strict efficiency in set valued optimization in this paper we develop the notion of a phi minimizer or strict efficiency for a set valued map its properties and relations with other similar notions are studied in special circumstances under suitable conditions we prove that a point is a phi minimizer of a vector function if and only if it is a phi minimizer of a certain family of linear scalarizations we also establish a characterization of strict efficiency through a nonlinear scalarization which is a generalization of the gerstewitz function defined on the power set of the image space the final part is focused on minimizers of order one and we provide several necessary or sufficient conditions without convexity assumptions through different kinds of derivatives as contingent radial among others various illustrative examples showing the applicability of our results are also presented

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

markov chains theory for scale free networks abstract this paper proposes a markov chain method to predict the growth dynamics of the individual nodes in scale free networks and uses this to calculate numerically the degree distribution we first find that the degree evolution of a node in the ba model is a nonhomogeneous markov chain an efficient algorithm to calculate the degree distribution is developed by the theory of markov chains the numerical results for the ba model are consistent with those of the analytical approach a directed network with the logarithmic growth is introduced the algorithm is applied to calculate the degree distribution for the model the numerical results show that the system self organizes into a scale free network

clustering coefficients of growing networks abstract in this paper we develop a general analytical method to compute clustering coefficients of growing networks this method can be applied to any network as long as we can construct and solve the dynamic equation for the degree of any node we also verify the accuracy of the method through simulation

disturbance decoupled observers for systems with unknown inputs this note deals with the design of reduced order disturbance decoupled scalar functional observers for linear systems with unknown inputs based on a parametric approach existence conditions are derived and a design procedure for finding reduced order scalar functional observers is given the derived existence conditions are relaxed and the procedure can find first order disturbance decoupled scalar functional observers for some cases where the number of unknown inputs is more than the number of outputs also the observer matching condition which is the necessary requirement for the design of state observers for linear systems with unknown inputs is not required numerical examples are given to illustrate the attractiveness of the proposed design method

higher order mond weir duality for set valued optimization in this paper we introduce a higher order mond weir dual for a set valued optimization problem by virtue of higher order contingent derivatives and discuss their weak duality strong duality and converse duality properties

directed networks different link formation mechanisms causing degree distribution distinction abstract within undirected networks scientists have shown much interest in presenting power law features for instance barabasi and albert 1999 claimed that a common property of many large networks is that vertex connectivity follows scale free power law distribution and in another study barabasi et al 2002 showed power law evolution in the social network of scientific collaboration at the same time jiang et al 2011 discussed deviation from power law distribution others indicated that size effect bagrow et al 2008 information filtering mechanism mossa et al 2002 and birth and death process shi et al 2005 could account for this deviation within directed networks many authors have considered that outlinks follow a similar mechanism of creation as inlinks faloutsos et al 1999 krapivsky et al 2001 tanimoto 2009 with link creation rate being the linear function of node degree resulting in a power law shape for both indegree and outdegree distribution some other authors have made an assumption that directed networks such as scientific collaboration or citation behave as undirected resulting in a power law degree distribution accordingly barabasi et al 2002 at the same time we claim 1 outlinks feature different degree distributions than inlinks where different link formation mechanisms cause the distribution distinctions 2 in outdegree distribution distinction holds for different levels of system decomposition therefore this distribution distinction is a property of directed networks first we emphasize in outlink formation mechanisms as causal factors for distinction between indegree and outdegree distributions where this distinction has already been noticed in barker et al 2010 and baxter et al 2006 within a sample network of oss projects as well as java software corpus as a network second we analyze whether this distribution distinction holds for different levels of system decomposition open source software oss project project dependency within a cluster package package dependency within a project and class class dependency within a package we conclude that indegree and outdegree dependencies do not lead to similar type of degree distributions implying that indegree dependencies follow overall power law distribution or power law with flat top or exponential cut off in some cases while outdegree dependencies do not follow heavy tailed distribution

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

properties of asymmetrical evolving networks abstract introducing utility to describe the attractions of nodes in this work we propose and study a simple asymmetrical evolving network model considering both preferential attachment and random controlled by probability p that is the utility increment u i u j when connecting node i to node j the simulation results show that the model can reproduce power law distributions of utility p u u a a 2 1 p which can be obtained using mean field approximation furthermore the model exhibits exponential networks with respect to small values of p and power law scale free networks with respect to big values of p which is in good agreement with theoretical analysis in other words the model represents a transition between exponential and power law scaling to better understand the degree correlations of our model the clustering coefficient c and degree assortativity r depending on probability p are also discussed

evolving mortal networks we discuss a class of models for the evolution of tree networks in which new nodes are recruited into the network at random times and nodes already in the network may die at random times stochastic mechanisms for growth and death of the network that are either sensitive or insensitive to the coordination number or degree of nodes are studied using simulations and mean field approximations critical behavior is observed in the long time coordination number distribution of the system associated exponents are universal in one part of parameter space but depend on the ratio of birth and death parameters elsewhere

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions

growing networks with mixed attachment mechanisms networks grow and evolve when new nodes and links are added in there are two methods to add the links uniform attachment and preferential attachment we take account of the addition of links with mixed attachment between uniform attachment and preferential attachment in proportion by using numerical simulations and analysis based on a continuum theory we obtain that the degree distribution p k has an extended power law form p k k k0 when the number of edges k of a node is much larger than a certain value k0 the degree distribution reduces to the power law form p k k and when k is much smaller than k0 the degree distribution degenerates into the exponential form it has been found that degree distribution possesses this extended power law form for many real networks such as the movie actor network the citation network of scientific papers and diverse protein interaction networks

preferential attachment in growing spatial networks we obtain the degree distribution for a class of growing network models on flat and curved spaces these models evolve by preferential attachment weighted by a function of the distance between nodes the degree distribution of these models is similar to that of the fitness model of bianconi and barab asi with a fitness distribution dependent on the metric and the density of nodes we show that curvature singularities in these spaces can give rise to asymptotic bose einstein condensation but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature we provide numerical results for spaces of constant curvature sphere flat and hyperbolic space and we discuss the conditions for the breakdown of this approach and the critical points of the transition to distance dominated attachment finally we discuss the distribution of link lengths

solutions of the equation av bw vf and their application to eigenstructure assignment in linear systems two new simple complete analytical and restriction free solutions with complete and explicit freedom of the matrix equation av bw vf are proposed here ab is known and is controllable and f is in the jordan form with arbitrary given eigenvalues based on the proposed solutions of this matrix equation a complete parametric approach for eigenstructure assignment in linear systems via state feedback is proposed and two new algorithms are presented the proposed solutions of the matrix equation and the eigenstructure assignment result are generalizations of some previous results and are simpler and more effective

necessary and sufficient conditions for isolated local minima of nonsmooth functions we consider the mathematical programming problem find inf f x mid x in c where f is an arbitrary extended real valued function and c a subset of a finite dimensional space we give necessary and sufficient optimality conditions for this problem generalizing previous results of a auslendex this journal 22 1984 pp 239 254

second look at the spread of epidemics on networks in an important paper newman phys rev e66 016128 2002 claimed that a general network based stochastic susceptible infectious removed sir epidemic model is isomorphic to a bond percolation model where the bonds are the edges of the contact network and the bond occupation probability is equal to the marginal probability of transmission from an infected node to a susceptible neighbor in this paper we show that this isomorphism is incorrect and define a semidirected random network we call the epidemic percolation network that is exactly isomorphic to the sir epidemic model in any finite population in the limit of a large population i the distribution of self limited outbreak sizes is identical to the size distribution of small out components ii the epidemic threshold corresponds to the phase transition where a giant strongly connected component appears iii the probability of a large epidemic is equal to the probability that an initial infection occurs in the giant in component and iv the relative final size of an epidemic is equal to the proportion of the network contained in the giant out component for the sir model considered by newman we show that the epidemic percolation network predicts the same mean outbreak size below the epidemic threshold the same epidemic threshold and the same final size of an epidemic as the bond percolation model however the bond percolation model fails to predict the correct outbreak size distribution and probability of an epidemic when there is a nondegenerate infectious period distribution we confirm our findings by comparing predictions from percolation networks and bond percolation models to the results of simulations in the appendix we show that an isomorphism to an epidemic percolation network can be defined for any time homogeneous stochastic sir model

spectral dimensions of hierarchical scale free networks with weighted shortcuts spectral dimensions have been widely used to understand transport properties on regular and fractal lattices however they have received little attention with regard to complex networks such as scale free and small world networks here we study the spectral dimension and the return to origin probability of random walks on hierarchical scale free networks which can be either fractal or nonfractal depending on the weight of the shortcuts applying the renormalization group rg approach to a gaussian model we obtain the exact spectral dimension while the spectral dimension varies between 1 and 2 for the fractal case it remains at 2 independent of the variation in the network structure for the nonfractal case the crossover behavior between the two cases is studied by carrying out the rg flow analysis the analytical results are confirmed by simulation results and their implications for the architecture of complex systems are discussed

growing random networks with fitness abstract three models of growing random networks with fitness dependent growth rates are analysed using the rate equations for the distribution of their connectivities in the first model a a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness with rate k 1 for 0 we find the connectivity distribution is power law with exponent 2 in the second model b the network is built by connecting nodes to nodes of connectivity k random additive fitness and random multiplicative fitness with rate k 1 this model also has a power law connectivity distribution but with an exponent which depends on the multiplicative fitness at each node in the third model c a directed graph is considered and is built by the addition of nodes and the creation of links a node with fitness i incoming links and j outgoing links gains a new incoming link with rate i 1 and a new outgoing link with rate j 1 the distributions of the number of incoming and outgoing links both scale as power laws with inverse logarithmic corrections

influence of reciprocal edges on degree distribution and degree correlations reciprocal edges represent the lowest order cycle possible to find in directed graphs without self loops representing also a measure of feedback between vertices it is interesting to understand how reciprocal edges influence other properties of complex networks in this paper we focus on the influence of reciprocal edges on vertex degree distribution and degree correlations we show that there is a fundamental difference between properties observed on the static network compared to the properties of networks which are obtained by simple evolution mechanism driven by reciprocity we also present a way to statistically infer the portion of reciprocal edges which can be explained as a consequence of feedback process on the static network in the rest of the paper the influence of reciprocal edges on a model of growing network is also presented it is shown that our model of growing network nicely interpolates between barabasi albert ba model for undirected and the ba model for directed networks