algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

on identification of multidimensional fuzzy systems abstract the paper is concerned with identification of multidimensional fuzzy systems the authors introduce the fuzzy implication operator which ensures in systems with normal fuzzy sets as inputs the minimal value of the average relative hamming distance between calculated and actual fuzzy output a corresponding theorem is proved the approach is illustrated by a catalyst properties catalytic activity model

optimal weighting scheme and the role of coupling strength against load failures in degree based weighted interdependent networks abstract the optimal weighting scheme and the role of coupling strength against load failures on symmetrically and asymmetrically coupled interdependent networks were investigated the degree based weighting scheme was extended to interdependent networks with the flow dynamics dominated by global redistribution based on weighted betweenness centrality through contingency analysis of one node removal we demonstrated that there still exists an optimal weighting parameter on interdependent networks but it might shift as compared to the case in isolated networks because of the break of symmetry and it will be easier for the symmetrically and asymmetrically coupled interdependent networks to achieve robustness and better cost configuration against the one node removal induced cascade of load failures when coupling strength was weaker our findings might have great generality for characterizing load failure induced cascading dynamics in real world degree based weighted interdependent networks

robustness of a network formed of spatially embedded networks we present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks we show results for a treelike and a random regular network of networks each with i unconstrained dependency links and ii dependency links restricted to a maximum euclidean length r analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations for the case of two fully interdependent spatially embedded networks it was found li et al phys rev lett 108 228702 2012 that the system undergoes a first order phase transition only for r r c 8 we find here that for treelike networks of networks composed of n networks r c significantly decreases as n increases and rapidly n 11 reaches its limiting value of 1 for cases where the dependencies form loops such as in random regular networks we show analytically and confirm through simulations that there is a certain fraction of dependent nodes q max above which the entire network structure collapses even if a single node is removed the value of q max decreases quickly with m the degree of the random regular network of networks our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse the theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

biased percolation on scale free networks biased degree dependent percolation was recently shown to provide strategies for turning robust networks fragile and vice versa here we present more detailed results for biased edge percolation on scale free net works we assume a network in which the probability for an edge between nodes i and j to be retained is proportional to kikjwith ki and kj the degrees of the nodes we discuss two methods of network recon struction sequential and simultaneous and investigate their properties by analytical and numerical means the system is examined away from the percolation transition where the size of the giant cluster is obtained and close to the transition where nonuniversal critical exponents are extracted using the generating functions method the theory is found to agree quite well with simulations by presenting an extension of the fortuin kasteleyn construction we find that biased percolation is well described by the q 1 limit of the q state potts model with inhomogeneous couplings

asymptotic and exponential stability of nonlinear two dimensional continuous discrete roesser models abstract sufficient conditions guaranteeing lyapunov stability asymptotic stability and exponential stability of nonlinear two dimensional continuous discrete systems are proposed special attention is paid to neutrally stable systems such as some two dimensional system descriptions of vehicle platoons which may be stable or asymptotically stable but never exponentially stable our conditions for lyapunov stability and asymptotic stability only require the corresponding two dimensional lyapunov function to have a negative semidefinite divergence they are thus suitable for the analysis of non exponential versions of 2d stability examples are given to illustrate the results

cascading failures on correlated interdependent networks with dependency groups abstract the topologies of interdependent networks markedly affect the cascading failures and phase transition however the effects of interlayer degree correlations on cascading failures on interdependent networks with dependency groups have not been investigated in this paper we propose a cascading failure model on correlated interdependent networks with dependency groups and use a percolation theory to describe the final state of the dynamics through extensive numerical simulations and theoretical analysis we find that interlayer degree correlations enhance the robustness of the system on networks with homogeneous degree distribution the system always exhibits a discontinuous phase transition however on networks with heterogeneous degree distributions the phase transition is continuous discontinuous for strong weak interlayer degree correlations the theoretical and numerical predictions agree well with each other

calculation and application of a minimum fuzzy relational matrix abstract this paper compares the maximum and minimum inverse formulations for fuzzy identification of a relational matrix given a series of non normal input output data baboshin and naryshkin 1990 showed for the case of normal input data that identification of the relational matrix using an estimate of the minimum inverse resulted in a smaller hamming distance between actual and predicted output values than the mamdani identification method this paper extends their work to include unrestricted or non normal input data and confirms that the estimate of the minimum inverse for this case is again better than mamdani s method it is then proven that the relative rank of identification algorithms for minimization of the hamming distance between the actual and predicted output values is the maximum inverse calculation of sanchez 1976 the minimum inverse formulation initial formulation by sanchez 1977 and extended to a series of input output data by sessa 1984 the estimate of the minimum inverse and then the mamdani method however this ranking applies only in the fuzzy domain and not to the discrete or defuzzified domain relevant to many practical applications

structure of shells in complex networks we define shell l in a network as the set of nodes at distance l with respect to a given node and define rl as the fraction of nodes outside shell l in a transport process information or disease usually diffuses from a random node and reach nodes shell after shell thus understanding the shell structure is crucial for the study of the transport property of networks we study the statistical properties of the shells of a randomly chosen node for a randomly connected network with given degree distribution we derive analytically the degree distribution and average degree of the nodes residing outside shell l as a function of rl further we find that rl follows an iterative functional form rl phi rl 1 where phi is expressed in terms of the generating function of the original degree distribution of the network our results can explain the power law distribution of the number of nodes bl found in shells with l larger than the network diameter d which is the average distance between all pairs of nodes for real world networks the theoretical prediction of rl deviates from the empirical rl we introduce a network correlation function c rl identical with rl phi rl 1 to characterize the correlations in the network where rl is the empirical value and phi rl 1 is the theoretical prediction c rl 1 indicates perfect agreement between empirical results and theory we apply c rl to several model and real world networks we find that the networks fall into two distinct classes i a class of poorly connected networks with c rl 1 where a larger smaller fraction of nodes resides outside inside distance l from a given node than in randomly connected networks with the same degree distributions examples include the watts strogatz model and networks characterizing human collaborations such as citation networks and the actor collaboration network ii a class of well connected networks with c rl 1 examples include the barabasi albert model and the autonomous system internet network

the effect of interdependence on the percolation of interdependent networks abstract two stochastic models are proposed to generate a system composed of two interdependent scale free sf or erd s renyi er networks where interdependent nodes are connected with an exponential or power law relation as well as different dependence strength respectively each subnetwork grows through the addition of new nodes with constant accelerating random attachment in the first model but with preferential attachment in the second model the two subnetworks interact with multi support and undirectional dependence links the effects of dependence relations and strength between subnetworks are analyzed in the percolation behavior of fully interdependent networks against random failure both theoretically and numerically and as a result for both relations interdependent sf networks show a second order percolation phase transition and the increased dependence strength decreases the robustness of the system whereas interdependent er networks show the opposite results in addition the power law relation between networks yields greater robustness than the exponential one at the given dependence strength

stability analysis of 2 d digital filters with saturation arithmetic an lmi approach an improved lmi based criterion for the nonexistence of overflow oscillations in two dimensional 2 d digital filters described by the roesser model employing saturation arithmetic is presented the criterion makes use of the structural properties as prevailing in the system under consideration of the saturation nonlinearities in a greater detail than the usual sector restriction 0 1

on the absence of limit cycles in a class of state space digital filters which contains minimum noise realizations this paper presents a theorem for the absence of limit cycles in a class of state space digital filters such a class contains minimum noise realizations of any order as a special case that is it is proved that the transition matrix a of minimum noise realizations of any order satisfies i a t a geq 0 and thus the realizations are free of limit cycles

a preferential attachment strategy for connectivity link addition strategy in improving the robustness of interdependent networks abstract given the same two networks and only one to one interlinks are allowed apparently interdependent networks coupled by these two networks has the optimal robustness when we connect every pair of the same nodes in these two networks according to the structure of this interdependent network with the optimal robustness we propose a preferential attachment strategy and by applying this preferential attachment strategy to three existing connectivity link addition strategies ra random addition strategy ld low degree addition strategy and lidd low inter degree degree difference addition strategy we find that each improved strategy is obviously better than before in improving the robustness of interdependent networks our findings can provide guidance on connectivity link addition strategy to improve robustness of interdependent networks against cascading failures

stability of linear discrete dynamics employing state saturation arithmetic this note is concerned with the stability of discrete time dynamical systems employing saturation arithmetic in the state space a matrix measure is introduced so that it can administer the proximity evaluation of a matrix to the set of diagonal matrices and the measure is utilized for making an additional condition to the lyapunov stein matrix inequality the solvability of the modified matrix inequality ensures not only the stability but also the absence of overflow oscillation under the state saturation arithmetic and this approach has the advantage of being free from auxiliary parameters as an application the obtained result is applied to the stability analysis of two dimensional dynamics numerical examples are given to illustrate the results

a software routine to solve the generalized inverse problem of fuzzy systems abstract a software routine to produce the solution of the generalized inverse problem of fuzzy relational equations is presented given the fuzzy matrices r and b defined on u v and w v respectively the routine produces all fuzzy matrices a defined on the product space w u which satisfy a r b where denotes maxmin composition

on the fuzziness of solutions of fuzzy relation equations on finite spaces let be the set of solutions of a fuzzy relation equation defined on finite spaces in this paper we use some algorithms already introduced in our foregoing work in order to determine some relations of which have a minimal value of fuzziness measure by taking into account the fuzzy entropy concept we build a new lattice structure and some fuzzy topological results are also presented

new fuzzy learning model with recursive estimation for dynamic systems abstract a systematic design technique for a discrete time self learning fuzzy model based on fuzzy relational equations is presented with some novel features firstly the model utilizes a probabilistic fuzzy relation obtained from referential fuzzy sets on the input and output spaces secondly the conventional input output formulation of the fuzzy relation is extended by introducing internal fuzzy states in order to model the internal dynamics of the given dynamic system thirdly the fuzzy relation is used to estimate the output as a response to a given stimulus by means of the compositional rule of inference extended to include recursive estimation of the internal fuzzy state as a result the new model is shown to be a fuzzy automaton first and second order versions of the fuzzy model are validated by means of a well known set of industrial data

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

robustness of a network formed by n interdependent networks with a one to one correspondence of dependent nodes many real world networks interact with and depend upon other networks we develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks each composed of the same number of nodes n the dependency links connecting nodes from different networks establish a unique one to one correspondence between the nodes of one network and the nodes of the other network we study the dynamics of the cascades of failures in such a network of networks non caused by a random initial attack on one of the networks after which a fraction p of its nodes survives we find for the fully interdependent loopless non that the final state of the non does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the non which generalizes both the giant component of regular percolation of a single network n 1 and the recently studied case of the mutual giant component of two interdependent networks n 2 we also find that the mutual giant component does not depend on the topology of the non and express it in terms of generating functions of the degree distributions of the network our results show that for any n 2 there exists a critical p p c 0 below which the mutual giant component abruptly collapses from a finite nonzero value for p p c to zero for p 2 a rr non is stable for any n with p c 1 this results arises from the critical role played by singly connected nodes which exist in an er non and enhance the cascading failures but do not exist in a rr non

percolation of partially interdependent networks under targeted attack we study a system composed of two partially interdependent networks when nodes in one network fail they cause dependent nodes in the other network to also fail in this paper the percolation of partially interdependent networks under targeted attack is analyzed we apply a general technique that maps a targeted attack problem in interdependent networks to a random attack problem in a transformed pair of interdependent networks we illustrate our analytical solutions for two examples i the probability for each node to fail is proportional to its degree and ii each node has the same probability to fail in the initial time we find the following i for any targeted attack problem for the case of weak coupling the system shows a second order phase transition and for the strong coupling the system shows a first order phase transition ii for any coupling strength when the high degree nodes have higher probability to fail the system becomes more vulnerable iii there exists a critical coupling strength and when the coupling strength is greater than the critical coupling strength the system shows a first order transition otherwise the system shows a second order transition

two dimensional digital filters without overflow oscillations and instability due to finite word length new criteria for the absence of finite word length effects such as overflow oscillations and instability in two dimensional digital filters are presented the criteria are formulated using the state space representations and are based on results concerning the 2d lyapunov equation several examples illustrate the theoretical results

some remarks on the identification problem in fuzzy systems abstract several identification problems in fuzzy systems are considered which are described by means of fuzzy relational equations the determination of a family of fuzzy relations of the system is described in detail

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

comments on stability and absence of overflow oscillations for 2 d discrete time systems in their article xiao and hill see ibid vol 44 p 2108 10 1996 show via an example that the conditions of the theorem 3 in an earlier paper by tzafestas et al 1992 may not guarantee stability of 2 d linear systems the authors incorrectly conclude that the bibo unstable 2 d linear system may have a stable finite wordlength implementation using saturation arithmetic the authors comment that this incorrect conclusion is due to an erroneous result in the earlier paper by tzafestas et al

robust extended kalman filtering linearization errors inherent in the specification of an extended kalman filter ekf can severely degrade its performance this correspondence presents a new approach to the robust design of a discrete time ekf by application of the robust linear design methods based on the h sub spl infin norm minimization criterion the results of simulations are presented to demonstrate an advantage for signal demodulation and nonlinear equalization applications

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

robustness of network of networks under targeted attack the robustness of a network of networks non under random attack has been studied recently gao et al phys rev lett 107 195701 2011 understanding how robust a non is to targeted attacks is a major challenge when designing resilient infrastructures we address here the question how the robustness of a non is affected by targeted attack on high or low degree nodes we introduce a targeted attack probability function that is dependent upon node degree and study the robustness of two types of non under targeted attack i a tree of n fully interdependent erd s renyi or scale free networks and ii a starlike network of n partially interdependent erd s renyi networks for any tree of n fully interdependent erd s renyi networks and scale free networks under targeted attack we find that the network becomes significantly more vulnerable when nodes of higher degree have higher probability to fail when the probability that a node will fail is proportional to its degree for a non composed of erd s renyi networks we find analytical solutions for the mutual giant component p as a function of p where 1 p is the initial fraction of failed nodes in each network we also find analytical solutions for the critical fraction p c which causes the fragmentation of the n interdependent networks and for the minimum average degree k over min below which the non will collapse even if only a single node fails for a starlike non of n partially interdependent erd s renyi networks under targeted attack we find the critical coupling strength q c for different n when q q c the attacked system undergoes an abrupt first order type transition when q q c the system displays a smooth second order percolation transition we also evaluate how the central network becomes more vulnerable as the number of networks with the same coupling strength q increases the limit of q 0 represents no dependency and the results are consistent with the classical percolation theory of a single network under targeted attack

cascade of failures in coupled network systems with multiple support dependence relations we study both analytically and numerically the cascade of failures in two coupled network systems a and b where multiple support dependence relations are randomly built between nodes of networks a and b in our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network we assume that networks a and b have i sizes n a and n b ii degree distributions of connectivity links p a k and p b k iii degree distributions of support links p a k and p b k and iv random attack removes 1 r a n a and 1 r b n b nodes form the networks a and b respectively we find the fractions of nodes a and b which remain functional giant component at the end of the cascade process in networks a and b in terms of the generating functions of the degree distributions of their connectivity and support links in a special case of erd s renyi networks with average degrees a and b in networks a and b respectively and poisson distributions of support links with average degrees a and b in networks a and b respectively a r a 1 exp a b 1 exp a a and b r b 1 exp b a 1 exp b b in the limit of a and b both networks become independent and our model becomes equivalent to a random attack on a single erd s renyi network we also test our theory on two coupled scale free networks and find good agreement with the simulations

tomography of scale free networks and shortest path trees in this paper we model the tomography of scale free networks by studying the structure of layers around an arbitrary network node we find both analytically and empirically that the distance distribution of all nodes from a specific network node consists of two regimes the first is characterized by rapid growth and the second decays exponentially we also show analytically that the nodes degree distribution at each layer exhibits a power law tail with an exponential cutoff we obtain similar empirical results for the layers surrounding the root of shortest path trees cut from such networks as well as the internet

criteria for the absence of limit cycles in two dimensional discrete systems sufficient conditions for the nonexistence of zero input limit cycles in two dimensional systems which contain a finite number of memoryless nonlinearities are presented two different conditions are given the first one is based on the frequency domain representation of the linear part of the discrete system and it is shown to be more general than the one already presented in 1 the second condition is formulated using the state space representation and is based on the properties of quasidominant matrices and on partial results on the 2 d lyapunov equation

stability and absence of overflow oscillations for 2 d discrete time systems it is shown that a 2 d linear time invariant discrete time system could be bounded input bounded output bibo unstable even if the 2 d system which is implemented by finite word length is free of overflow oscillations with zero input according to a previous criterion for the absence of overflow oscillations in 2 d discrete time systems an improved criterion is presented to ensure both stability properties

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

robustness of interdependent and interconnected clustered networks abstract in real world most systems show significant clustering and it is more practical to investigate the behaviors of clustered network previous studies are mostly focused on single clustered network and coupled clustered networks with dependency links here we study two clustered networks coupled with both interdependent and interconnected links by introducing generating function of the joint degree distribution when the networks are fully dependent we obtain the analytical solution of giant component p we show rich phase transition phenomena and analyze their behaviors we find that as dependency coupling strength increases the system changes from second order phase transition through hybrid transition to first order phase transition for weak dependency coupling strength q a corresponding to second order phase transition we find that clustering has almost no effect on the robustness of network but for strong dependency coupling strength q a corresponding to first order transition the more clustered system is more vulnerable at the same time we notice that when the system is more clustered the hybrid order region is almost unchangeable the first order region becomes smaller and the second order region is larger additionally we can see that the bigger the clustering coefficient c is the bigger the second order region becomes for the same c the density of connectivity links between networks is higher the second order region becomes smaller and the density of connectivity links within each network is higher the second order region becomes bigger

stability analysis of 2 d state space digital filters using lyapunov function a caution a previously proposed lyapunov based approach for the stability analysis of two dimensional 2 d state space digital filters is reviewed it is shown that the approach as it is may lead to an erroneous conclusion

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

percolation of a general network of networks percolation theory is an approach to study the vulnerability of a system we develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks netonet typically percolation of a single network shows that the damage in the network due to a failure is a continuous function of the size of the failure i e the fraction of failed nodes in sharp contrast in netonet due to the cascading failures the percolation transition may be discontinuous and even a single node failure may lead to an abrupt collapse of the system we demonstrate our general framework for a netonet composed of n classic erd ifmmode mbox h o else h o fi s r enyi er networks where each network depends on the same number m of other networks i e for a random regular network rr formed of interdependent er networks the dependency between nodes of different networks is taken as one to one correspondence i e a node in one network can depend only on one node in the other network no feedback condition in contrast to a treelike netonet in which the size of the largest connected cluster mutual component depends on n the loops in the rr netonet cause the largest connected cluster to depend only on m and the topology of each network but not on n we also analyzed the extremely vulnerable feedback condition of coupling where the coupling between nodes of different networks is not one to one correspondence in the case of netonet formed of er networks percolation only exhibits two phases a second order phase transition and collapse and no first order percolation transition regime is found in the case of the no feedback condition in the case of netonet composed of rr networks there exists a first order phase transition when the coupling strength q fraction of interdependency links is large and a second order phase transition when q is small our insight on the resilience of coupled networks might help in designing robust interdependent systems

some results in the setting of fuzzy relation equations theory abstract in this paper we point out certain theoretical aspects of fuzzy relation equations connecting some known results between composite and o fuzzy equations several theorems are obtained and some examples are provided too

stability and overflow oscillations in 2 d state space digital filters an important theorem relating to the lyapunov stability of two dimensional discrete systems is proven using this theorem it is shown that for any 2 d digital filter satisfying shanks criterion there exists a realization that cannot support overflow oscillations in the process of proving the theorem some interesting results on the multi dimensional bilinear transformation are developed one of these results yields a simple test that can be used to check the stability of a 2 d discrete transfer function that has been obtained from the bilinear transform of a 2 d continuous transfer function with a 2 d hurwitzian denominator polynomial a technique is given for determining whether a normal realization exists for a given 2 d discrete system also a theorem is presented that allows the determination of the norm of the minimum norm realization of a given transfer function a noniterative technique for obtaining a low norm realization and an iterative technique for obtaining a minimum norm realization are developed

minimum vertex cover problem for coupled interdependent networks with cascading failures abstract this paper defines and analyzes a generalization of the classical minimum vertex cover problem to the case of two layer interdependent networks with cascading node failures that can be caused by two common types of interdependence previous studies on interdependent networks mainly addressed the issues of cascading failures from a numerical simulations perspective whereas this paper proposes an exact optimization based approach for identifying a minimum cardinality set of nodes whose deletion would effectively disable both network layers through cascading failure mechanisms we analyze the computational complexity and linear 0 1 formulations of the defined problems as well as prove an lp approximation ratio result that generalizes the well known 2 approximation for the classical minimum vertex cover problem in addition we introduce the concept of a depth of cascade i e the maximum possible length of a sequence of cascading failures for a given interdependent network and show that for any problem instance this parameter can be explicitly derived via a polynomial time procedure

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

targeted attack on networks coupled by connectivity and dependency links abstract coupled systems used to increase capacity were shown beneficial as long as it does not open pathways to cascades previous studies on the robustness of coupled networks except for interdependent networks are almost the cases of random attack many challenges remain exist in targeted attack problem of coupled networks since nodes within coupled networks show different functions for each network this paper both analytically and numerically analyzed the robustness of coupled networks under three types of targeted attacking strategies including attacking on nodes by considering internal and external degree internal degree only and external degree only for coupled network with both interdependent and interconnected links all degree distributions of intra and inter connectivity links are poissonian we find the system undergoes from second to first order phase transition as coupling strength q increases the fraction of nodes in the giant component p at stable state the critical phase transition threshold p c first order threshold p c i and second order threshold p c i i and the critical point p c q c separating the first and second order phase transitions are analytically obtained for three types of attacking strategies with attacking probability parameter 0 1 we also find the system becomes more vulnerable as the average degree of intra links k or inter links k decreases therefore the minimum average degrees k m i n and k m i n to maintain the system stable are obtained for the case of 0 1 moreover we discussed three typical cases of coupled networks interdependent networks k 0 interacting networks q 0 and bipartite network k 0 q 0 the analytical expressions of p p c and p c q c only for interdependent are given respectively besides we study the equivalence relations between interdependent networks and coupled networks with connectivity and dependency links for the same p c the results imply that we can adjust the parameters q k and k to make the two systems have the same robustness moreover the results imply that one should not only protect nodes with high degree of intra links or inter links but also defend nodes the sum of whose degree of intra links and inter links is big can avoid the system becoming vulnerable

analysis of percolation behaviors of clustered networks with partial support dependence relations we carry out a study of percolation behaviors of clustered networks with partial support dependence relations by adopting two different attacking strategies attacking only one network and both networks which help to further understand real coupled networks for two different attacking strategies we find that the system changes from a second order phase transition to a first order phase transition as coupling strengthq increases we also notice that the first order region becomes smaller and the secondorder region becomes larger as average degree or clustering coefficient increases and as the average supported degree approaches infinity coupled clustered networks become independent and only the second order transition is observed which is similar toq 0 furthermore we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength but for weak coupling strength it has little influence especially for attacking both networks the study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength however for weak coupling strength a more robust network is obtained only by increasing average degree for the same support average degree

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature