spectral properties of the laplacian of multiplex networks omez et al phys rev lett 110 028701 2013 some of the authors proposed a framework for the study of diffusion processes in such networks here we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the laplacian of the full multiplex we derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues this spectrum is used to gain insight into physical phenomena on top of multiplex specifically diffusion processes and synchronizability

matrix fraction construction of linear compensators a construction is given using polynomial matrix fractions of the compensator resulting from a combination of a linear feedback control law and estimator design

mode jumping in the von karman equations we study numerical approximations of bifurcating solution curves of the von karman equations with simply supported and clamped boundary conditions respectively of special interest here is the splitting of a double bifurcation point into two simple bifurcation points and the tracing of the associated secondary solution branches which corresponds to the phenomenon of mode jumping in the buckling of a rectangular plate a continuation unsymmetric lanczos algorithm is utilized for curve tracking our numerical results verify the theoretical prediction of schaeffer and golubitsky namely mode jumping occurs when the boundary is partially clamped but not if it is merely simply supported

brief papers new results in state estimation and regulation an alternative technique to design linear state estimators and regulators is presented this technique is based on transfer matrix considerations the optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices this approach provides further insight displays the duality of estimation and control nicely and bridges the state space and frequency domain techniques the resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form

computational framework for coupling compressible and low mach number codes a hybrid multicode computational method is developed that allows combining compressible and low mach number codes into a single computational solver the proposed methodology can be used for integrated simulations of multicomponent engineering problems the unsteady coupling between the two codes is performed via exchanging time dependent state information through the interfaces using overlapping meshes the proposed coupling procedure is validated on laminar steady and unsteady test problems both constant density and variable density regimes of the low mach number code are investigated results of the test cases performed in frameworks of one way coupling and two way coupling are documented several methods of implementation of interface conditions for the compressible code are compared by looking at the levels of dilatation as well as numerical errors the injection method when all the variables are simply transformed from one code to another through an interpolation procedure is chosen for its superior stability accuracy and simplicity of implementation numerical results for the coupled calculations obtained with the injection method show good agreement with both standalone calculations and analytical solutions

polynomial equations for the linear mmse state estimation the linear minimum mean square error mmse state estimation problem is solved via spectral factorization and a pair of bilateral diophantine equations detectability and or stability requirements in the riccati based solution context are expressed in terms of the stability of the greatest common right and left divisors of polynomial matrices

multiple bifurcation in the von karman equations we numerically investigate nontrivial solutions for some thin plate problems we find that at a corank 2 bifurcation point of the von karman equations with simply supported boundary conditions four nontrivial solution branches bifurcate thus the results of holder schaeffer and golubitsky are numerically verified where two symmetry breaking solution branches are obtained we also show that the result of allgower bohmer and mei in second order semilinear elliptic problems can be extended to certain fourth order elastic plate problems the local bifurcation behaviors of the former and the latter are different the practical implementations are given sample numerical results are reported

designing optimal multiplex networks for certain laplacian spectral properties we discuss the design of interlayer edges in a multiplex network under a limited budget with the goal of improving its overall performance we analyze the following three problems separately first we maximize the smallest nonzero eigenvalue also known as the algebraic connectivity second we minimize the largest eigenvalue also known as the spectral radius and finally we minimize the spectral width maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value however for larger budgets the optimal weights are generally nonuniform the dual formulation transforms the problem into a graph realization embedding problem that allows us to give a fuller picture namely before the threshold budget the optimal realization is one dimensional with nodes in the same layer embedded to a single point while beyond the threshold the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity finally for extremely large budgets the embeddings again revert to lower dimensions minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector before a threshold the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector and the optimal largest eigenvalue of the laplacian remains constant for larger budgets the weight distribution tends to be almost uniform in the dual picture the optimal graph embedding is one dimensional and nonhomogeneous at first with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector while the other layer is embedded at the origin beyond this threshold the optimal embedding expands to be multidimensional and for larger values of the budget the two layers fill the embedding space finally we show how these two problems are connected to minimizing the spectral width

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

finite plateau in spectral gap of polychromatic constrained random networks we consider critical behavior in the ensemble of polychromatic erd s renyi networks and regular random graphs where network vertices are painted in different colors the links can be randomly removed and added to the network subject to the condition of the vertex degree conservation in these constrained graphs we run the metropolis procedure which favors the connected unicolor triads of nodes changing the chemical potential of such triads for some wide region of we find the formation of a finite plateau in the number of intercolor links which exactly matches the finite plateau in the network algebraic connectivity the value of the first nonvanishing eigenvalue of the laplacian matrix 2 we claim that at the plateau the spontaneously broken z 2 symmetry is restored by the mechanism of modes collectivization in clusters of different colors the phenomena of a finite plateau formation holds also for polychromatic networks with m 2 colors the behavior of polychromatic networks is analyzed via the spectral properties of their adjacency and laplacian matrices

analysis of preconditioners for domain decomposition domain decomposition is a class of techniques that are designed to solve elliptic problems on irregular domains and on multiprocessor systems typically a domain is decomposed into many smaller regular subdomains and the capacitance system governing the interface unknowns is solved by some version of the preconditioned conjugate gradient method in this paper we show that for a simple model problem poisson s equation on a rectangle decomposed into two smaller rectangles the capacitance system can be inverted exactly by fast fourier transform an exact eigen decomposition of the capacitance matrix also makes it possible to relate and compare the various preconditioners that have been proposed in the literature for example we show that in the limit as the aspect ratio of the two rectangles tend to infinity the preconditioner proposed by golub and mayers becomes exact but the one proposed by dryja does not both preconditioners however are poor when the aspect ratio is small

effect of the interconnected network structure on the epidemic threshold most real world networks are not isolated in order to function fully they are interconnected with other networks and this interconnection influences their dynamic processes for example when the spread of a disease involves two species the dynamics of the spread within each species the contact network differs from that of the spread between the two species the interconnected network we model two generic interconnected networks using two adjacency matrices a and b in which a is a 2n 2n matrix that depicts the connectivity within each of two networks of size n and b a 2n 2n matrix that depicts the interconnections between the two using an n intertwined mean field approximation we determine that a critical susceptible infected susceptible sis epidemic threshold in two interconnected networks is 1 1 a b where the infection rate is within each of the two individual networks and in the interconnected links between the two networks and 1 a b is the largest eigenvalue of the matrix a b in order to determine how the epidemic threshold is dependent upon the structure of interconnected networks we analytically derive 1 a b using a perturbation approximation for small and large the lower and upper bound for any as a function of the adjacency matrix of the two individual networks and the interconnections between the two and their largest eigenvalues and eigenvectors we verify these approximation and boundary values for 1 a b using numerical simulations and determine how component network features affect 1 a b we note that given two isolated networks g1 and g2 with principal eigenvectors x and y respectively 1 a b tends to be higher when nodes i and j with a higher eigenvector component product xiyj are interconnected this finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics

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

domain decomposition algorithms for fourth order nonlinear elliptic eigenvalue problems we study domain decomposition methods for fourth order plate problems the well known von karman equations are used as our model problem by exploiting the symmetry of the domain the solution of the original problem can be obtained by solving those associated reduced problems which are defined on subdomains with appropriate boundary conditions we show how nonoverlapping and overlapping domain decomposition methods can be used to solve the reduced problems for the linearized von karman equation we present preconditioners using both fourier analysis and probing techniques for the interface systems which are similar to those derived by chan et al finally we compare the efficiency of various domain decomposition preconditioners for solving the von karman equations

dimensionality reduction and spectral properties of multilayer networks network representations are useful for describing the structure of a large variety of complex systems although most studies of real world networks suppose that nodes are connected by only a single type of edge most natural and engineered systems include multiple subsystems and layers of connectivity this new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically one way to address this question is to study the spectral properties of such networks here we apply the framework of graph quotients which occurs naturally in this context and the associated eigenvalue interlacing results to the adjacency and laplacian matrices of undirected multilayer networks specifically we describe relationships between the eigenvalue spectra of multilayer networks and their two most natural quotients the network of layers and the aggregate network and show the dynamical implications of working with either of the two simplified representations our work thus contributes in particular to the study of dynamical processes whose critical properties are determined by the spectral properties of the underlying network

a novel domain decomposition framework for the ground state solution of bose einstein condensates abstract in this study we present a new type of domain decomposition algorithm to obtain the ground state solution for bose einstein condensates using our proposed algorithm instead of directly solving the nonlinear eigenvalue problem one only needs to solve a series of linear boundary value problems on a finite element space sequence using the domain decomposition method and subsequently solve a small scale nonlinear eigenvalue problem on an enriched space simultaneously because solving large scale nonlinear eigenvalue problem directly is time intensive our algorithm can obviously improve the efficiency of producing simulation for bose einstein condensates in addition any domain decomposition algorithm for linear boundary value problems can be applied to our algorithm framework which makes the algorithm considerably flexible two numerical experiments are presented in the paper to demonstrate the efficiency and scalability of our proposed algorithm

strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers abstract doubly coprime factorizations dcf play an important role in the design of compensators for lumped linear time invariant systems they are defined as stable proper rational matrices and they not only constitute a starting point for the fractional approach to all stabilizing compensators but they are also closely related to the frequency domain design of optimal state feedback and estimation and of the resulting observer based compensators when defining dcfs related to reduced order observers some rational matrices become improper this can be prevented by the introduction of artificial stable dynamics the identity elements which cancel in the system and in the compensator transfer matrices such pole zero cancellations however are contrary to the usual notion of coprimeness it is shown here that these additional dynamics which have no meaning in an observer based compensator scheme are superfluous and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements for such factorizations the new definition of strictly doubly coprime factorizations sdcf is proposed

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

algebraic connectivity of interdependent networks the algebraic connectivity un 1 i e the second smallest eigenvalue of the laplacian matrix plays a crucial role in dynamic phenomena such as diffusion processes synchronization stability and network robustness in this work we study the algebraic connectivity in the general context of interdependent networks or network of networks non the present work shows both analytically and numerically how the algebraic connectivity of nons experiences a transition the transition is characterized by a saturation of the algebraic connectivity upon the addition of suffcient coupling links between the two individual networks of a non in practical terms this shows that non topologies require only a fraction of coupling links in order to achieve optimal diffusivity furthermore we observe a footprint of the transition on the properties of fiedler s spectral bisection

exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks an interconnected network features a structural transition between two regimes f radicchi and a arenas nat phys 9 717 2013 one where the network components are structurally distinguishable and one where the interconnected network functions as a whole our exact solution for the coupling threshold uncovers network topologies with unexpected behaviors specifically we show conditions that superdiffusion introduced by gomez et al phys rev lett 110 028701 2013 can occur despite the network components functioning distinctly moreover we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them

integrating cross frequency and within band functional networks in resting state meg a multi layer network approach neuronal oscillations exist across a broad frequency spectrum and are thought to provide a mechanism of interaction between spatially separated brain regions since ongoing mental activity necessitates the simultaneous formation of multiple networks it seems likely that the brain employs interactions within multiple frequency bands as well as cross frequency coupling to support such networks here we propose a multi layer network framework that elucidates this pan spectral picture of network interactions our network consists of multiple layers frequency band specific networks that influence each other via inter layer cross frequency coupling applying this model to meg resting state data and using envelope correlations as connectivity metric we demonstrate strong dependency between within layer structure and inter layer coupling indicating that networks obtained in different frequency bands do not act as independent entities more specifically our results suggest that frequency band specific networks are characterised by a common structure seen across all layers superimposed by layer specific connectivity and inter layer coupling is most strongly associated with this common mode finally using a biophysical model we demonstrate that there are two regimes of multi layer network behaviour one in which different layers are independent and a second in which they operate highly dependent results suggest that the healthy human brain operates at the transition point between these regimes allowing for integration and segregation between layers overall our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum