the dynamics of frictional interfaces are crucial to many situations in mechanical engineering  , geosciences  or biology  . today , after decades of studies , the science of contacts under time - invariant loading conditions , _e.g. _ static contacts or steady sliding contacts  , has reached a high level of advancement which , in many instances , enables quantitative reproduction of global  or local  measurements .in contrast , the dynamics of contacts under rapidly evolving loads or during fast unstable motion like stick - slip  is far less understood . in particular , recent experiments on the transition from static to kinetic friction of side - driven poly(methyl methacrylate ) ( pmma ) rough samples forming line contacts with a rough pmma substrate have revealed unexpected features  .the transition occurs through the fast ( comparable to the speed of sound ) propagation of micro - slip fronts through the contact  .it can also be preceded by a series of fronts that arrest before having ruptured the whole contact , thus denoted as precursors  .these results , which may have important implications for _ e.g. _ the study of earthquakes , have triggered an active modelling activity .et al . _ , using a one - dimensional ( 1d ) spring - block model with a complex time - dependent friction law , produced three types of micro - slip front velocities , analogous to that observed in  .et al . _ , using a 1d spring - block model with a simple amontons - coulomb ( ac ) friction law , showed that the length of precursors is modified when the external normal load is made asymmetric .scheibert and dysthe  , using a quasi - static 1d model with ac friction , showed how the increasing tangential load itself induces an increasing pressure asymmetry which influences the precursors series .due to the intrinsic limitations of 1d models , all these studies yielded only a qualitative agreement with experiments .very recently , trmborg _ et al. _ demonstrated , using a 2d spring - block model with ac friction , that quantitative agreement with the kinematics ( _ i.e. _ the properties of the states in which no micro - slip front is propagating ) of the experiments requires an accurate description of the interfacial stresses , and therefore the use of realistic boundary conditions on the sample  .compared to such a 2d model , the strongest advantage of 1d models is that their results are much easier to analyse and understand , so that they provide opportunities for theoretical approaches .they also require simpler implementation and lower computational power . in most situations ,in which very accurate results are not needed and/or a thorough qualitative understanding of the behaviour of the system is desired , 1d models are preferable . from the pioneering work of burridge and knopoff  , .these models have mainly been used to describe the statistical properties of the series of earthquakes at a seismic fault .inertial blocks are connected in series via internal springs that model the crust s elasticity .a homogeneous tectonic loading is modelled by coupling , via loading springs , all individual blocks to the same rigid driving body .such statistical analysis of homogeneously driven systems contrast with the recent 1d studies by braun _ et al . _ and maegawa _ et al . _ , in which the time evolution of side - driven systems is analysed deterministically , in order to produce data comparable to the experimental measurements .the two models by braun _et al . _ and maegawa _ et al . _are actually very different .both consider an array of blocks connected by internal springs , but the model by braun _et al . _ also considers viscous dissipation and a complex time - dependent friction law emerging from the collective behaviour of interfacial springs with random stiffness , breaking threshold and reattachment time .in contrast the model by maegawa _et al . _ , which only considers blocks and springs and the minimalistic ac friction law , is probably the simplest possible model .  on the one hand, we will see that , due to its extreme simplicity , the model by maegawa _et al . _ yields results that are strongly resolution dependent , which prevents robust comparison with experiments . on the other hand , it is difficult to disentangle the respective roles of the many parameters of the model by braun _et al . _the scope of this article is therefore to ( i ) construct , step by step , a minimal 1d side - driven spring - block model , the results of which are essentially resolution independent and ( ii ) qualify the capabilities of this model to reproduce the main qualitative features of recent experimental observations .this letter is organised as follows . in section 2, we describe the model by maegawa _et al . _ and show its limitations . in section 3 ,we improve this model by introducing successively an internal viscosity , an interfacial stiffness and an initial tangential force distribution . in section 4, we describe an analytic prediction for the length of precursors in both maegawa _ et al ._ s and our improved model .in the model developed by maegawa _ et al . _ , the slider is modelled as a chain of blocks connected by springs ( fig .[ fig : maegawamodel ] ) , with material spring constant and block mass , where is the total mass of the slider and is the number of blocks . in experiments , the base ( also called track )is fixed on a very stiff support , and it is therefore modelled as a rigid surface for simplicity .the tangential force is applied at the trailing edge of the system through a loading spring with stiffness .one end of this spring is attached to the trailing edge block ( block 1 ) , while the other end of the spring moves at a constant velocity .the normal force is imposed on each block , satisfying the criterion , where is the total applied normal force .blocks of mass connected in series through springs of stiffness . the trailing edge of the system ( block 1 ) is slowly driven through a loading spring of stiffness .each block is also submitted to a normal force and to a friction force . ]  the equations of motion are given by where is the position of block as a function of time relative to its equilibrium position and denotes the double derivative with respect to . is the driving force ( or tangential force / load ) given by   a local friction law giving the friction forces is imposed between the blocks and the base .ac friction is used with local kinetic and static friction coefficients and , respectively .the resulting global friction coefficients are similarly denoted by and .the friction force on block , , is then given by where , when , equilibrium of block imposes that balances all other forces acting on block .the material spring constant is chosen such that the elastic deformation of the model is similar to that of a linear elastic medium with young s modulus , which yields : where and are the cross - section area and the length of the slider , respectively .  in the experiments by maegawa _et al . _ , an asymmetric normal loading was used , leading to the following linear model for : where  ] and total number of events ( dashed line ) . (b ) number of precursors longer than any previous one ( solid line ) and total number of precursors ( dashed line ) .all curves are stongly increasing functions of , indicating unphysical resolution - dependence of the model s results.,title="fig : " ] , using .( a ) number of global events ( solid line ) for  ] ( solid line ) and total number of events ( dashed line ) .( b ) number of precursors longer than any previous one ( solid line ) and total number of precursors ( dashed line ) .comparison with fig .[ fig : numberofeventsamontonscoulomb ] shows that introduction of an interfacial tangential stiffness suppresses the resolution dependence of the numbers of global events and of precursors longer than any previous one.,title="fig : " ] , when the interfacial stiffness in taken into account . , and  mm .( a ) number of global events for  ] are assumed to have a tangential force equal to the kinetic friction level .blocks in the interval $ ] , however , now needs to be taken into account for two reasons : both the initial tangential force profile and the springs to the track lead to a non - zero tangential force for . the form of this profile in a static situation has been calculated in app .[ sec : scalingofthemodelwiththespringtotrackstaticfrictionlaw ] , and is given in eq ., where now is the initial tangential force profile .however , it has to be modified to take into account that the loaded block is not located at , but at , and that this block does not have to be loaded up to the static friction level , but may take some other value , say , with a coefficient to be defined .our assumed tangential force profile at the arrest of a precursor of length is therefore given by \\ \left(\alpha p(l_p ) - \tau^0 ( x ) \right ) e^{-\frac{x - l_p}{l_0 } } + \tau^0(x ) , & x \in \left [ l_p , l \right ] \end{array } \right .\label{eq : ours : f_t(l_p)}\ ] ] we have considered that all values of the amplitude of the peak at have the same probability to occur between and , so that we have given its average value in the predictions seen in fig .[ fig : initialshearforce]b. figure [ fig : shearforceprofilewithanalytical ] shows the assumed tangential force profile and the actual tangential force profile at the arrest of an event , and the agreement is seen to be satisfactory .th precursor plotted in fig .[ fig : initialshearforce ] ( red dots ) and the predicted profile used in the analytical prediction of precursor lengths ( black drawn line ) using , , ,  mm and in eq . .]  from eqs ., one can calculate in the same way as was done for the model by maegawa __ this calculation is provided in appendix [ sec : ours : precursorprediction ] . as seen in fig .[ fig : initialshearforce]b , the prediction scheme works well .deviations between the actual precursors and the analytical curve have two contributions : incorrectly assumed tangential force profile and inertial effects , where the former gives the largest contribution .again , the global static friction coefficient is seen to be approximately equal to the local kinetic friction coefficient .the prediction curves appear to bend slightly backwards at . in our continuous prediction scheme, this corresponds to micro - slip fronts that are so close to the leading edge that all blocks at have a tangential force above the kinetic friction level , causing to get smaller as the front moves further . in reality, however , no precursor ever arrests in such a state but propagates all the way to the trailing edge .recent experimental results about the transition from static to kinetic friction in line contacts have triggered the study of the deterministic dynamics of 1d spring - block friction models in which driving is applied at one extremity of the chain of blocks . in this letter , we have improved the simplest of such models  in order to solve its intrinsic unphysical resolution - dependence and to ameliorate its qualitative agreement with experimental results on the kinematics of micro - slip fronts along the contact . in particular, the introduction of a tangential stiffness of the interface , by introducing a new length scale in the model , practically suppresses its resolution dependence and allows for reproduction of realistic numbers of precursory micro - slip fronts .the additional introduction of an initial tangential force distribution at the interface significantly improves the agreement with the evolution of the precursor length with the external tangential load obtained in experiments .our improved model is intented to serve as a framework for more complex friction models when robust comparisons with experiments are desired .we focused on 1d models because they are simple enough to enable deep insights into the qualitative effects of the model s parameters .however , it is known that 2d models  are required to provide quantitative agreement with experiments . in this respect, the improvements brought to the 1d model are effective ways to account for intrinsically 2d effects : first , the length scale introduced through the interfacial stiffness enables coupling between remote points along the interface , analogous to the coupling through the slider s bulk ; second , the initial tangential force distribution accounts for the shear stress arising form the differential poisson expansion of two bodies pressed together .  in analogy with 2d results we developed , based on the well defined force distribution left by an arrested precursor , an efficient analytical prediction for the precursors length as a function of the external tangential load applied .we also find that , like in 2d , the macroscopic static friction coefficient of a side - driven contact is approximately equal to its microscopic kinematic friction coefficient .we thank j. l. vinningland for discussions .we acknowledge funding from the european union ( marie curie grant no .pief - ga-2009 - 237089 ) .this article was supported by a center of excellence grant to pgp from the norwegian research council .if friction forces are ignored , the equation of motion for an infinite chain of blocks connected by springs is given by we then assume a solution of the form where and . inserting eq .  into eq .yields the relation which can be simplified to since the complex parameter is then given by the system is critically damped when eq .  only has one solution for , which occurs when the square root is zero : the oscillations that are to be reduced have a wavelength , _ i.e. _ a wave number . inserting this into eq.  leads to which is the value of the damping coefficient for which waves of wavelength are critically damped .since the absolute value of is always smaller than one , choosing will cause all other waves to be under - damped .an analytical expression for the characteristic length can be found . in order to do so ,the following assumptions are made : , and slow loading compared to the internal dynamics of the system , which enables a static analysis .the system is first placed in a static state with an initial shear force profile given by , and then loaded slowly from the left .the equilibrium of all non - edge blocks writes we introduce a new variable defined by where is the initial position of block . inserting eq .  into eq .yields where the two terms and cancel in eq .since the initial state is static , and thus the above equation can be rewritten to where is the lattice spacing . since , the first term in eq .can be replaced by the second spatial derivative , and replacing with yields which has the general solution the shear force is given by by replacing again finite differences with second order derivatives , and the general expression for the shear force profile can be found by using eq . , which yields the system is loaded from the left , and at the beginning of an event the shear force on block is equal to the static friction threshold . provided , the trailing edge will not be affected by the loading .the latter of these two boundary conditions yields i.e. . the first boundary condition yields and the shear force is therefore given by   the characteristic length is given by eq . , andinserting for given by eq .  and yields and hence eq .  for .we start with eq .  and use the assumed shear force profile in eq . , shown in fig .[ fig : shearforceprofilewithanalytical ] .again we go to the limit , resulting in the substitution the tangential force after a precursor of length is then given by \\ & = \frac{n}{l } \left [ \int_0^{l_p } \mu_k p(x ) \ , { \text{d}}x + \int_{l_p}^{l } \left(\alpha p(l_p ) - \tau^0 ( x ) \right ) e^{-\frac{x - l_p}{l_0 } } + \tau^0(x ) \ , { \text{d}}x \right ] .\label{eq : f_t1}\end{aligned}\ ] ] we limit ourselves to predicting the precursors in fig . [fig : initialshearforce]b , _ i.e. _ using a tangential interfacial stiffness and a linear initial shear forces as depicted in fig .[ fig : initialshearforce]a , but with . the normal and initial shear forceare then given by where the parameter determines the slope in the initial shear force profile .inserting eqs .  and  into eq .yields .\end{gathered}\ ] ] the above integrals can be calculated easily , and the result is the tangential load as a function of the precursor length : .\label{eq : f_t(l_p)_springtotrack}\end{gathered}\ ] ] we observe that again .

Abstract: we study the dynamic behaviour of 1d spring - block models of friction when the external loading is applied from a side , and not on all blocks like in the classical burridge - knopoff - like models . such a change in the loading yields specific difficulties , both from numerical and physical viewpoints . to address some of these difficulties and clarify the precise role of a series of model parameters , we start with the minimalistic model by maegawa _ et al . _ ( tribol . lett . * 38 * 313 , 2010 ) which was proposed to reproduce their experiments about precursors to frictional sliding in the stick - slip regime . by successively adding ( i ) an internal viscosity , ( ii ) an interfacial stiffness and ( iii ) an initial tangential force distribution at the interface , we manage to ( i ) avoid the model s unphysical stress fluctuations , ( ii ) avoid its unphysical dependence on the spatial resolution and ( iii ) improve its agreement with the experimental results , respectively . based on the behaviour of this improved 1d model , we develop an analytical prediction for the length of precursors as a function of the applied tangential load . we also discuss the relationship between the microscopic and macroscopic friction coefficients in the model .


a major problem in the context of complex networks of interacting dynamical systems , which has been considered for many years , is to predict the collective dynamics when the network topology is known .however , in many situations , it is often desirable to address the reverse problem of inferring the topology of the network from available data capturing the collective dynamics .this reverse problem is relevant in fields such as biology ( e.g. reconstructing regulatory networks from gene expression data ) , neuroimaging ( e.g. revealing the structural organization of the brain ) , and engineering ( e.g. localizing failures in power grids or computer networks ) , to list a few .network identification problems have received increasing attention over the past years , and the topic is actively growing in nonlinear systems theory .see e.g. the recent survey .many methods have been developed , exploiting techniques from various fields : linearization , velocities estimation , adaptive control , steady - state control , optimization , compressed sensing , stochastic methods , etc .these methods provide the structural ( i.e. exact ) connectivity of the underlying network and exploit to do so the dynamical nature of the individual units , which is often known , at least partially .in contrast , correlation - based methods using statistical measures or information - theoretic measures have also been developed , but they can only infer the effective ( i.e. statistical ) connectivity of the network .network identification methods developed in the framework of dynamical systems theory are usually not well - suited to the analysis of real networks such as biological networks , social networks , etc .most of them are invasive , requiring the modification of the network connectivity or dynamics .in addition , some of them can not be used `` offline '' for data analysis , since they require to interact dynamically with the network .more importantly , all the methods proposed so far for full network reconstruction require measurements at all the nodes of the network .partial measurements have been considered in in the context of linear time - invariant systems for a partial reconstruction of the network between the measured states , and yet the authors showed that the problem can not be solved without additional information on the system .it can actually be shown that measuring all nodes is necessary for a full network reconstruction , and this is usually out of reach in large real networks . indeed, the number of sensors is limited and typically ( much ) smaller that the number of nodes . some nodes of real networks might also not be accessible , or the only available information might be the averaged activity of a group of nodes lying in a given region of the network ( e.g. electrical activity in a region of the brain ) .all these limitations motivate the network identification framework developed in this paper , which overcomes them .  in this work ,we take the view that identifying the exact complete topology of large networks is not only practically impossible , as mentioned above , but also often unnecessary .the presence or absence of an edge between two specific nodes is for instance often only marginally relevant when analyzing the global structure of a large network .for this reason , we focus instead on the identification of the spectral properties of the network , a framework that we call _ spectral network identification_. note that the idea of estimating the spectral properties of networks has been considered in the control theory community ( e.g. ) , but in the case of specific ( linear ) consensus dynamics imposed at each node . on one hand , spectral properties do not reveal the exact full network topology ( indeed , we can not `` hear the shape '' of a drum ) , so that the identification objective has been relaxed . on the other hand , they are a central theme of study in spectral graph theory where they are typically defined through the so - called laplacian matrix associated with the network . they are shown to provide relevant information on the global network structure such as mean , minimum and maximum node degree , and connectivity , and they are reflected on the network dynamics , see e.g. .for instance , the second smallest eigenvalue of the laplacian matrix also called algebraic connectivity is related to the speed of information diffusion in the network ( e.g. opinion propagation , spreading of epidemics ) and plays a key role in studying network synchronization .more generally , the spectral properties of the network provide simple markers capturing the global network structure .these spectral markers can be used to detect a pathology or a fault and to compare different networks .  while classical full topology identification requires measurements at all the nodes of the network ,we show that spectral network identification requires only sparse measurements in the network .this can be roughly explained by the fact that each node of a strongly connected network `` feels '' the influence of all the other nodes . with the method developed in this paper , measurementscan therefore be performed on a very small subset of nodes ( e.g. only one in some cases ) that might not be representative of the whole set of nodes .they can also be defined by a possibly nonlinear function of the states of several nodes , such as the average dynamics of a group of units . moreover ,the proposed method is not invasive and can be used offline .spectral properties of ( nonlinear ) dynamics are well - defined in a framework based on the so - called koopman operator and can be extracted from data through numerical methods such as the dynamic mode decomposition ( dmd ) algorithm . in this context, our main theoretical contribution is to connect these spectral properties of the collective network dynamics , which are measured , to the spectral properties of the network , which are to be inferred .these results are obtained in the case of a diffusive coupling for networks reaching a synchronized equilibrium , where the states of all units converge to the same value , a behavior which can be observed with excitable neurons , cardiac cells , opinion dynamics , and epidemics . for small networks ,exact spectral identification is achieved . for large networks ,a statistical approach is proposed , which focuses on spectral moments and is well - suited to the case of heterogeneous populations . with a few sparse measurements in the network , this framework allows to estimate the average number of connections , detect the addition of a node to the network , and measure whether two units influence each other .the rest of the paper is organized as follows . in section [ sec : problem_stat ] , the problem of spectral network identification is introduced . in section [ sec : exact ] , theoretical results are presented in the case of linear and nonlinear networks . at the end of the section, we also discuss the limitations encountered for the exact spectral identification of non - identical units .section [ sec : stat ] focuses on large networks and provides statistical results related to the spectral moments of the laplacian matrix , which are related to the statistical distribution of the node degrees .numerical aspects of the methods are discussed in section [ sec : numerical ] and illustrated with several applications in section [ sec : examples ] .concluding remarks and perspectives are given in section [ sec : conclu ] .a networked dynamical system consists of a set of interconnected dynamical systems ( or units ) interacting on a ( weighted ) graph .the system is deterministic since no stochastic perturbation is considered .each unit is attached to a node ( or vertex ) and is directly influenced by another unit if is an edge of the graph , i.e. . the strength of the interaction between units is determined by the function which assigns a weight to each edge .the ( weighted ) degree of a vertex is given by where , with a slight abuse of notation , if and if .we assume for all .the graph is represented by its ( weighted ) adjacency matrix defined with the entries alternatively , the graph is also described by the ( weighted ) laplacian matrix with the degree matrix . in the following , we consider graphs that can be weighted and directed , i.e. does not imply , so that and in general .+ the networked dynamical system is completely defined by  1 .  the graph ; 2 .  the local dynamics of the states , , of the units attached to the vertices ; 3 .the type of coupling between pairs of interacting units .here we are interested in the following network identification problem : under the assumption that ( ii ) and ( iii ) are known , infer the graph topology ( i ) from measurements of the state of the units .in particular , classical and spectral network identification problems are defined as follows .  * * classical network identification .* suppose that ( ii ) and ( iii ) are known . from measurements of the states of _ all the units _ of the network ,infer the set of edges and the weight function . * * spectral network identification .* suppose that ( ii ) and ( iii ) are known . from measurements of the states of _ a small subset of units _, estimate the spectrum of the laplacian matrix ( i.e. the laplacian eigenvalues ) or the first spectral moments in the case of large networks ( see section [ sec : spectral_graph_th ] ) .  in this paper , we will develop the framework of spectral network identification .we focus on networked systems that admit a stable equilibrium corresponding to the synchronization of the units .we also make the standing assumption that the units interact through a diffusive coupling .in contrast to classical network identification , spectral network identification only requires sparse measurements in the network .the price to pay is the relaxed objective of getting only the laplacian eigenvalues .although this spectral information does not reveal the complete graph structure , it captures important topological properties of the network .we will not review the vast literature related to spectral graph theory ( we refer the interested reader to ) , but provide here some basic results connecting the laplacian spectrum to the topological properties of the graph .  in the case of a connected graph ,the second smallest eigenvalue algebraic connectivity captures the connectivity of the graph ; see also the cheeger inequality in undirected graphs .the algebraic connectivity is related to the time constant of the dominant dynamics and to the speed of information propagation in the network .it also provides a bound on the diameter of the graph , i.e. the longest path between any pair of vertices .moreover , the algebraic connectivity and the spectral radius ( i.e. the largest eigenvalue ) can be used to derive bounds on the minimal and maximal vertex degrees and . in particular , for an undirected graph , we have   in the case of large graphs , it is convenient to consider the spectral moments which are related to the moments of the degree distribution .the first spectral moment is equal to the mean vertex degree , i.e. and the first and second spectral moments give bounds on the quadratic mean of the degree distribution : the equality is trivial and a short proof of the inequalities is given in appendix [ sec : app_proofs ] . in the case of undirected and unweighted graphs , other relationships can be derived , which link the spectral moments of to local structural features of the network .in this section , we develop the spectral network identification framework in the case of identical units . we first consider linear systems and then extend the results to nonlinear dynamics .the main results of this section provide the exact connection between the spectral properties of the collective dynamics and the spectral properties of the network .   consider a network of identical units that are each described by states evolving according to the linear dynamics with , , and ( which are assumed to be known ) .the interaction between the units is given by the diffusive coupling considering the state vector ^t \in \mathbb{r}^{mn} ] , or equivalently for all .  in the context of spectral network identification, measurements are performed through the linear observation function , where is a sparse matrix and is the number of measurements .( note that the case of a nonlinear observation function will be treated together with the case of nonlinear dynamics in section [ sec : nonlinear ] . )it is clear that all eigenvalues of appear in the expression of the measurement ( provided that for all , i.e. )=nm ] and )=m ] .the jacobian matrix associated with - linearized at is given by with , , and ( denotes the gradient ) .since is similar to , the following result can be obtained directly from proposition [ prop : net_ident ] .[ prop : net_ident2 ] consider the local dynamics and assume that and are controllable and observable pairs , respectively .then the relationship between the spectrum of the laplacian matrix and the spectrum of the jacobian matrix is a bijection ( i.e. holds ) .moreover , the spectrum of is given by .the proof follows from proposition [ prop : net_ident ] , with , , and .proposition [ prop : net_ident2 ] implies that the laplacian eigenvalues of the network can be obtained from the eigenvalues of the jacobian matrix .moreover , the eigenvalues of can be obtained from sparse measurement of the network dynamics . in the case of nonlinear systems , they are related to spectral properties of the dynamics defined in the framework of the so - called koopman operator .[ [ koopman - operator ] ] koopman operator + + + + + + + + + + + + + + + +  let be a trajectory solution of - associated with the initial condition .we suppose that measurements of are obtained through a possibly nonlinear observation function , which depends on a few local states in our case .the koopman operator(s ) are a semi - group acting on the set of such functions .they are defined by for all and .the value is thus the value that will be observed at time via for a trajectory which is at at time .one can verify that the are always linear and can therefore be characterized by spectral properties .provided that is analytic , the spectral decomposition of the operator yields where are the eigenvalues of the koopman operator and are the so - called koopman modes , which depend on  .in addition , it can be shown that the dmd algorithm extracts the spectral properties of the koopman operator from snapshots of .provided that the dominant koopman modes are nonzero ( see also remark [ rem : nonzero_modes ] ) , the algorithm yields the dominant koopman eigenvalues , which are the eigenvalues of the jacobian matrix ; see e.g. . according to proposition [ prop :net_ident2 ] , these eigenvalues can be used to retrieve the laplacian eigenvalues .[ rem : nonzero_modes ] the dmd algorithm can capture dominant eigenvalues only if the associated dominant koopman modes ( with ) are nonzero in .these dominant modes are given by , where are the right eigenvectors of and with the notation ^t ] is the state vector assigned to vertex .we choose the nonlinear observation function  ] ( with ) .we consider that the functions and are identical for all the units .there is almost no loss of generality , since these functions describe the connections between units , and these connections are already heterogeneous in the case of weighted graphs . in addition , this simplification does not significantly affect the theoretical results developed in the remaining of the paper .the jacobian matrix related to the system ( linearized around ) is where is given by and where is such that with . since the dmd algorithm provides the eigenvalues of ,the spectral identification problem is to compute from .equivalently , since the relationship between and has been established in section [ sec : lin_ident ] and [ sec : nonlinear ] , one has to estimate the eigenvalues of from the perturbed eigenvalues of ( note that since ) . in the case of unweighted graphs, it can be shown that if the perturbation is small enough , so that the spectral identification problem can be solved exactly .however this situation is very restrictive . for more general graphs ,perturbation theory for matrix eigenvalues can provide upper bounds on the difference between the eigenvalues of and , but these bounds are too conservative , especially when the network is large .it is also noticeable that the linearized dynamics of identical agents that do not synchronize but converge to different equilibria are in general equivalent to the ( linear ) dynamics of non - identical agents .in such a case , we would thus encounter similar issues when inferring the spectral properties of the network .  in the next section ,we show that a statistical approach can circumvent these limitations in the case of large networks .in the case of large graphs , most individual laplacian eigenvalues have an insignificant influence on the network dynamics , making them very hard to identify precisely .on the other hand , each of them taken individually only captures a very small amount of information about the network structure . therefore , recovering each individual eigenvalue is on the one hand impractical but on the other hand not really necessary . instead , it is much more relevant and convenient to focus on statistical measures of the spectral density of the laplacian matrix . in this section, we show that one can estimate the first spectral moments of the laplacian matrix from sparse measurements in the network .these spectral moments are related to statistical information on the degree distribution of the vertices ( see section [ sec : exact ] ) .this approach is also well - suited to the case of non - identical units and can be used to obtain some information on the unweighted underlying graph .   from a few eigenvalues of obtained with the dmd algorithm, one can compute the spectral moments of ; see section [ sec : approx_spec_mom ] for details on the numerical method .then the spectral moments of can be obtained from the spectral moments of , as shown in the following proposition .[ prop : moments ] suppose that . then the spectral moments of are given by with .we have and the result follows .note that this result holds for both cases of linear and nonlinear units . in the nonlinear case, we have , , and .  we will focus on the first two moments and it follows from proposition [ prop : moments ] that and   [ example3 ] we consider a random erds - rnyi graph with vertices and a probability ( supposedly not known ) for any two vertices to be connected .the weights of the edges are randomly distributed according to a uniform distribution on $ ] .the local dynamics of the units is given by and the observation function is , corresponding to the measurement of one state of one unit .note that the dynamics are linear , a choice made to provide the best illustration of the theoretical results .examples of identification of large networks with nonlinear dynamics are shown in section [ sec : examples ] .we use a heuristic method to estimate the spectral moments of ( figure [ fig : example3 ] ) , approximating the clusters of eigenvalues by the convex hull of values obtained with the dmd algorithm and assuming a uniform distribution of the eigenvalues within each cluster ; see section [ sec : approx_spec_mom ] for more details on the method . using and , we finally obtain the spectral moments and , which are close to the exact values . from and , one obtains good approximations of the mean degree and quadratic mean degree .the results are summarized in table [ tab : example3 ] .we also performed similar simulations for different random networks and computed in an automatic way the spectral moments of the laplacian matrix .table [ tab : stat_example3 ] shows that the mean relative error is of the order of .   in the case of a large network using measurements at one vertex .simulation parameters are given in appendix [ app_examples ] .the convex hull ( in gray ) of the measured eigenvalues provides a good approximation of the spectral moments of .the inset shows the cluster of eigenvalues lying near the origin.,width=302 ]  .identification of the spectral moments and mean vertex degrees of a large graph . [cols="^,^,^",options="header " , ]   adjacency matrix of the network considered in example [ example1 ] and example [ example2 ] :   authors thank s. kolumban for his insightful comments and helpful discussion on the manuscript .

Abstract: we propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number ( possibly one ) of its nodes . in contrast to classical identification of full graph topology , we focus on the identification of the spectral graph - theoretic properties of the network , a framework that we call _ spectral network identification_.  the main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics , which are well - defined in the context of the so - called koopman operator and can be extracted from data through the dynamic mode decomposition algorithm . these results are obtained for networks of diffusively - coupled units that admit a stable equilibrium state . for large networks , a statistical approach is considered , which focuses on spectral moments of the network and is well - suited to the case of heterogeneous populations .   our framework provides efficient numerical methods to infer _ global _ information on the network from sparse _ local _ measurements at a few nodes . numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node , that need not be representative of the other nodes properties .


capacitively coupled plasmas ( ccps ) have a wide range of applications , most of which require a solid control of the ion properties , i.e. the ion flux and the mean ion energy , at the electrode ( target ) surfaces . during the past decades different schemes , which allow a separate control of these quantities , have been proposed .radio frequency discharges driven simultaneously by two significantly different frequencies have approached this issue by the functional separation of the two components of the driving voltage waveform : the high frequency component plays the dominant role for the generation of the plasma and , thereby , sets the ion flux , while the low frequency component is responsible for accelerating the ions towards the electrodes , i.e. it defines the mean energy of the ions at the surfaces . a more recent approach , based on the electrical asymmetry effect ( eae ) makes use of the generation of a controllable dc self - bias that develops even in geometrically symmetric reactors when the discharge is excited by the sum of two ( or more ) consecutive harmonics of a base frequency .the control over the dc self - bias , , and over the mean ion energy by changing the phase angles of the applied harmonics has been confirmed both experimentally and in simulations , and has been proven benefitial in thin film deposition applications . while these basic approaches and their variants allow controlling these two integral characteristics of the ion flux - energy distribution function ( idf ) , a control over the _ shape _ of the idf would be highly desirable , e.g. to drive energy - selective surface processes at a distinct ion energy . for such applications ,customizing the idf to form a narrow peak at controllable energies would be ideal .under collisionless conditions , the idf in a single - frequency ccp typically exhibits a double peak feature at high energies , whereas collisions lead to a more complex spectrum at lower energies .typically , the peaks are formed by 10 % - 20 % of all ions flowing to the electrode .so far , a control over the idf shape has only been realized in high density remote plasma sources with unmatched arbitrary substrate bias at low repetition rates ( 100 khz ) for ion extraction .while this concept is not applicable for large area ccp sources , the several degrees of freedom in the case of eae - type excitation waveforms may provide a possibility to go beyond the control of the flux and mean energy , i.e. towards the optimization of the complete idf in ccps via a specific tuning of the amplitudes and phases of the individual frequency components of the driving voltage .the work presented here proposes such a tailoring of the idf , via utilizing the effect of charge exchange ( cx ) collisions on the idf in ccps driven by customized voltage waveforms .it will be demonstrated that ( i ) a narrow peak in the idf can be obtained under such conditions via adjusting the ion dynamics for our needs , and ( ii ) the position of this peak ( i.e. the energy ) and the corresponding ion flux can be controlled via the driving voltage waveform .it is of key importance that cx collisions `` convert '' fast ions to slow ions , which can accumulate in the sheath regions until a high electric field drives them to the electrodes .we note that at low pressures a narrow ion energy distribution can also be generated in single - frequency ccps if ( nearly ) collisionless conditions for the sheaths can be reached .the energy of the ions in this case is , however , defined by the mean sheath voltage , so it occurs at the high energy end of the idf .these conditions are far from ideal for deposition processes , which require a high voltage ( high ion flux ) and , at the same time , a peak of the idf at low / intermediate energies to enhance particular surface reactions .this can not be realized in classical ccps , but it is facilitated based on the technique proposed here .we investigate the possibility of the idf tailoring via self - consistent kinetic simulations based on the particle - in - cell technique complemented with monte carlo treatment of collision processes ( pic / mcc ) .we use our 1d3v electrostatic , bounded plasma simulation code to study a geometrically symmetric discharge in helium at 6 pa and 350 k. the plane , parallel , and infinite electrodes are separated by a distance of 5.0 cm . at the electrodes , electrons are reflected with a probability of 0.2 and secondary electrons due to ion impact are generated with a probability of .the cross sections for electron - neutral and ion - neutral collision processes are taken from refs . .  based on the ideas outlined above one can define ideal waveforms , for which the effects will be most pronounced . keeping in mind possible experimental realizations , we approximate the ideal waveforms with signals consisting of five consecutive harmonics , that are applied to one of the electrodes via a blocking capacitor , while the other electrode is grounded . here , = 13.56 mhz . is a voltage factor for all harmonics , while and are the amplitudes and phases of the individual frequency components .such a voltage waveform is feasible in applications using an advanced multi - frequency rf supply system with multiple matching branches and electrical filters . due to the asymmetry of the waveform a dc self - bias develops to equalize the electron and ion fluxes at each electrode , as described by the electrical asymmetry effect .the application of a multi - frequency voltage waveform generally enhances the electron heating .this leads to a high total ion flux , which is benefitial for most processing applications . in the following we present results for two scenariosfirst , we investigate the effect of the driving voltage amplitude and , subsequently , we address the effects of modifying the shape of the driving voltage waveform .) , discharge voltage including dc self - bias ( ) , and voltage drop over the powered electrode sheath ( ) .the grey lines indicate the ideal applied voltage waveform , consisting of three plateaus with voltage values , , and .,scaledwidth=75.0% ]   ions at the powered electrode , for an excitation voltage waveform according to figure [ fig:1 ] , scaled with different amplitude factors .,scaledwidth=75.0% ]  figure [ fig:1 ] presents the driving voltage waveform , the voltage drop across the discharge gap , and the sheath voltage at the powered electrode as a function of time within one fundamental rf period .the applied voltage waveform is generated by setting and the individual amplitudes v , v , v , v , v and phases , , , , in equation ( [ appvol ] ) .these values are obtained from a fourier transform of the ideal applied voltage waveform .thus , they have been chosen , because the resulting voltage waveform is minimum for a short time only , then oscillates around an intermediate value and stays close to the maximum value for almost half of the rf period .the idea is that the powered electrode sheath remains collapsed for a long time , so that ions created by cx collisions ( which we call `` cx - ions '' below ) within the sheath region can accumulate in front of the electrode .after that , these cx - ions are accelerated by the sheath electric field and arrive at the electrode within one rf period , .this field is strong around and then gradually decreases , so that cx - ions are initially accelerated and are , subsequently , located close to the electrode , where the intermediate field is sufficiently large to ensure that they reach the electrode within one rf period ..,scaledwidth=75.0% ]  as shown in figure [ fig:2 ] , utilizing this concept results in a sharp peak at a distinct ion energy within the low / medium energy range of the idf . here, we perform a voltage amplitude variation by changing all the harmonic amplitudes by the same factor , , between 0.5 and 1.5 , while leaving the phases unchanged .we observe that ( i ) for all conditions a sharp peak is present in the spectrum , ( ii ) with changing the voltage factor the position of this peak can be controlled , and ( iii ) this dominant peak of the idf broadens with increasing .the analysis of the properties of the dominant peak is presented in figure [ fig:3 ] , which shows the energy that corresponds to the peak and the ion flux ( integral of the peak ) .the position of the peak can be tuned within a 27 ev 130 ev range by varying the excitation voltage amplitude by a factor of 3 .the flux of the ions that belong to this peak changes by a factor of 6 .the relative contribution of the flux of the ions within the peak to the total ion flux also increases , from 15.1 % at .5 to 21.1 % at 1.5 .this is a significant fraction of the total ion flux and comparable or above such fractions achievable by classical techniques in ccps .however , in contrast to classical techniques our method allows controlling the energy of this peak in the intermediate region of the energy spectrum and generating one single dominant peak .furthermore , the dc self - bias increases from 27.5 v at 0.5 to 91.5 v at 1.5 .the relatively high flux observed in the smallest energy bin ( below 1 ev ) is caused by cx ions that are created very close to the electrode . a large fraction of these ions gains only very little energy , as the sheath electric field is small for almost the entire second half of the rf period .( 1.0 ) .the white dashed box highlights the region of cx - ion accumulation leading to the peak in the idf.,scaledwidth=75.0% ]   ( 1.0 ) .the white dashed box highlights the region of cx - ion accumulation leading to the peak in the idf.,scaledwidth=75.0% ]  the understanding of the formation of the main peak in the idf is aided by analyzing the spatio - temporal distribution of the electric field in the sheath region ( see figure [ fig:4 ] ) and invoking a method based on the est ( ensemble - in - space - time ) approach , where the trajectories of individual ions are traced .we seed cx - ions , which are initially at rest , into the sheath electric field , which is predefined as a result of the pic / mcc simulations .we assign the arrival energy of the ions at the electrode ( neglecting collisions ) to each position in space ( between and ) and time within the sheath region , where the ions started .this construction is shown in figure [ fig:5 ] for the case . a large domain in this plot , where the ion energy is nearly the same ( ev , marked by the white rectangle ) , gives rise to the nearly monoenergetic peak in the idf .this domain establishes because the electric field is strong for a short time , after which it oscillates around an intermediate value and is almost zero between  and  .thus , cx - ions can accumulate in this time interval close to the electrode .these ions are suddenly accelerated by the following peak in the sheath electric field and arrive at the electrode after gaining energy between  and  .therefore , all cx - ions created within the window  and  arrive with about the same energy , resulting in the distinct peak observed in the idf .ions , which start from a position further away from the electrode , need more than one rf period to move through the sheath and , therefore , contribute to the idf at higher energies ; they do not cause another distinct peak due to ( i ) the lower electric field at the starting position , leading to a much longer transition time , so ( ii ) the probability of collisions is high , redistributing these ions in the idf .an increase in the voltage factor simply enhances the sheath electric field and , thereby , enhances both the spatial region , in which ions must be initially located in order to contribute to the peak in the idf , and the energy of this peak due to the stronger force exerted on the ions . in this way the shape and the energy of the peak can be controlled .  ) , discharge voltage including dc self - bias ( ) , and voltage drop over the powered electrode sheath ( ) , for voltage plateau values of ( a ) 0 % , ( b ) 50 % , and ( c ) 100 % , i.e. voltage parameters specified in table [ table1 ] .the grey lines indicate the desired applied voltage waveform.,scaledwidth=75.0% ]  *20c pv & & & & & & & & & & & + }$ ] & [ v ] & [ & [ v ] & [ & [ v ] & [ & [ v ] & [ & [ v ] & [ & [ v ] + 0 & 265 & 180 & 310 & 180 & 154 & 180 & 80.9 & 180 & 14.8 & 180 & 414 + 10 & 268 & 170 & 307 & 182 & 154 & 178 & 80.4 & 188 & 15.5 & 180 & 396 + 20 & 276 & 160 & 300 & 184 & 152 & 175 & 80.4 & 196 & 16.1 & 180 & 363 + 30 & 290 & 150 & 289 & 187 & 149 & 172 & 81.2 & 205 & 16.4 & 180 & 317 + 40 & 309 & 141 & 273 & 190 & 144 & 169 & 82.8 & 213 & 16.6 & 180 & 257 + 50 & 331 & 133 & 255 & 193 & 137 & 166 & 85.5 & 222 & 16.6 & 180 & 186 + 60 & 355 & 126 & 234 & 197 & 131 & 162 & 89.0 & 230 & 16.4 & 180 & 115 + 70 & 381 & 121 & 212 & 201 & 124 & 158 & 93.1 & 236 & 16.1 & 180 & 46.0 + 80 & 406 & 116 & 189 & 205 & 117 & 154 & 97.6 & 242 & 15.7 & 180 & -22.8 + 90 & 431 & 112 & 167 & 210 & 110 & 149 & 102 & 248 & 15.3 & 180 & -63.2 + 100 & 454 & 108 & 145 & 216 & 104 & 144 & 107 & 252 & 14.8 & 180 & -82.0 +  next , we demonstrate the effect of modifying the shape of the applied voltage waveform .the panels of figure [ fig:6 ] display three cases .the `` ideal '' driving voltage waveforms , , are shown by the grey lines .the amplitudes and phases of the individual frequency components are obtained by a fourier transform of the desired voltage waveform .the black dotted curves show the approximation of the desired waveforms by 5 fourier components , which is used in the simulations .the amplitudes and phases of these fourier components are provided in table [ table1 ] .the ideal waveform consists of three plateaus with values between  (  ) and  , between  and  , and between  and  ( ) .furthermore , the voltage waveforms have no dc component , i.e. . here, is varied between and , which are kept at a constant difference .the plateau values ( pv ) of 0 % and 100 % correspond to and , respectively .the voltage amplitudes and phases of the individual frequency components of the applied voltage ( equation ( [ appvol ] ) with ) are provided in table [ table1 ] .this variation causes a change not only in the overall shape of the applied voltage waveform , but also in the asymmetry of the discharge , so that the dc self - bias , ( also included in table [ table1 ] ) , changes as a function of the pv , as well .ions at the powered electrode , for different plateau values ( pvs ) of the driving voltage waveform.,scaledwidth=75.0% ]  the resulting idfs for different pv are depicted in figure [ fig:7 ] . for a pv of 0 %, the idf is very narrow , with a maximum ion energy of about 124 ev .increasing the pv leads to an increase of the voltage drop across the powered electrode sheath and , hence , to an increase of the idf width to up to 318 ev .meanwhile , the peak in the idf due to cx - ions , i.e. caused by the physical mechanisms discussed above , is present for all pvs , but it becomes more narrow for larger pv .it is one of multiple peaks at pvs below 30 % , whereas it is the dominant feature for pvs larger than 30 % . above this plateau value , the total flux within the width of the peak stays almost constant ( it varies by % , not shown ) , whereas the energy of the peak can be tuned by almost a factor of two ( between 43 ev and 78 ev ) by ajusting the pv from 30 % to 100 %.  at small pvs ( e.g. 0% ) , the long time interval of a small sheath voltage and , hence , of a small electric field leads to many cx - ions accumulating in the sheath region at the powered electrode .however , only a small fraction of these ions arrive at the electrode after a sufficiently short transit time .thus , the collection of cx - ions within a transit time of one rf period or less is very inefficient . with an intermediate pv ( e.g. 50 % ) ,this collection is improved , but the longer time interval of a significant sheath electric field also means a shorter accumulation phase .a high pv ( e.g. 100 % ) allows for a larger energy gain of cx - ions , so that the respective peak in the idf is shifted towards higher energies .meanwhile , the relative contribution of the flux of the ions within the peak to the total ion flux slightly decreases , from 17.2 % at pv= 50 % to 16.2 % at pv= 100 % .  in both variations shown in figures [ fig:2 ] and [ fig:7 ] ,the main peak is superimposed on a broad spectrum of the idf .a further increase of the relative flux of the dominant peak would be worthwhile and might be achievable by changing more parameters , such as the gas pressure or the fundamental driving frequency .such a wide parameter variation is beyond the scope of this work , though .furthermore , there are further peaks at higher energies , which are caused by cx ions that are created further away from the electrode and have a transit time of multiple rf periods .therefore , the development of multiple peaks is inherently linked to the periodic behavior of the rf sheath .nevertheless , these peaks are much less pronounced , because the spatial regions , in which cx ions may accumulate , become narrow far away from the electrode ( see figure [ fig:5 ] ) and because the probability of collisions , in which the energy of these ions is redistributed , becomes larger .certainly , the application of the approach discussed here is limited to gases with a symmetrical charge exchange between the primary ion species and the feed gas .in conclusion , in this work we presented a novel method to generate a distinct peak at low / intermediate energies in the flux - energy distribution of the ions impinging on the electrode surfaces in capacitive multi - frequency plasmas .this was made possible by driving the plasma with a tailored voltage waveform that consists of five consecutive harmonics .subject to the electrical asymmetry effect , a dc self - bias develops as a function of the phases between the applied harmonics .we have shown that by adjusting the applied voltage waveform to approximate certain idealized waveforms , the overall shape of the idf , as well as distinct features within the distribution function can be controlled .we found that utilizing the effect of charge exchange collisions of the ions in the sheaths , a dominant peak in the idf can be created .this approach works via a detailed knowledge of the ion dynamics within the sheath and is , in particular , based on a control of the dynamics of cx - ions in the spatio - temporal sheath electric field . the general idea of dividing the rf period into intervals of cx - ion accumulation ( small sheath electric field ) and collection ( high and intermediate sheath electric field ) was tested and the resulting peak in the idf was analyzed .it was found that the position ( i.e. the energy ) and the flux of the peak can be varied at the same time by adjusting the amplitude of the applied voltage waveform , and can be controlled almost independently of one another by changing the pv of the intermediate voltage plateau .these advanced control opportunities allow the generation and control of a single peak at low / medium energies within the idf , so that certain energy - selective reaction process at the substrate surface induced by ion bombardment within a distinct energy window can be specifically enhanced , while the risk of inducing other potentially parasitic reactions initiated by ions at different energies is reduced .certainly , this study is a proof of principle and the method needs to be tested , e.g. for other feed gases , in a future study .this work was supported by the hungarian fund for scientific research ( otka ) , via grants k105467 and nn-103150 .

Abstract: we propose a method to generate a single peak at a distinct energy in the ion flux - energy distribution function ( idf ) at the electrode surfaces in capacitively coupled plasmas . the technique is based on the tailoring of the driving voltage waveform , i.e. adjusting the phases and amplitudes of the applied harmonics , to optimize the accumulation of ions created by charge exchange collisions and their subsequent acceleration by the sheath electric field . the position of the peak ( i.e. the ion energy ) and the flux of the ions within the peak of the idf can be controlled in a wide domain by tuning the parameters of the applied rf voltage waveform , allowing optimization of various applications where surface reactions are induced at particular ion energies .