building on the classical fields of graph theory , statistical physics , and nonlinear dynamics , as well as leveraging the recent advances in collecting large - scale network data , the field of network dynamics has flourished over the last 15 years .much of the current effort in this area is driven by the premise that understanding the structure , the function , and the relation between the two will help explain the workings of natural systems and facilitate the design of engineered systems with expanded capability , optimized performance , and enhanced robustness  .there have been extensive studies on this structure - dynamics relation  in the context of network synchronization  , reaction and/or diffusion dynamics  , as well as other types of dynamical processes on networks  .many of these studies have led to systematic methods for enhancing the dynamics through network - structural modifications .examples include controlling synchronization patterns of oscillators  or target wave patterns of excitable elements  , suppressing epidemic spreading  , and enhancing the stability of synchronous oscillations  .the latter can be achieved by adjusting coupling strengths , removing nodes , rewiring links , and even removing links and adding nodes , and has been demonstrated in applications , with examples including systems of space - filling bearings  and coupled laser networks  .a fundamental question at the core of the structure - dynamics relation is that of optimization : what network structure optimizes the dynamics of the system for a given function and what are the properties of such a network ?the significance of addressing this question is two fold .first , knowledge on the properties of optimized structures informs system architecture design . in power - grid networks , for example , the structures that maximize the stability of synchronous operation of power generators a required condition for power grids to operate could potentially be used to devise effective strategies for upgrading the system  .second , the identification of the network structures that guarantee the best fitness of natural complex systems can provide insights into the mechanisms underlying their evolution .examples of such systems include neuronal networks , whose synaptic connectivity structure is believed to have been optimized through evolution or learning for categorization tasks  , synchronization efficiency  , dynamical complexity  , information transfer efficiency  , and/or wiring cost  .the question of optimizing the network structure can be conceptualized as the problem of maximizing / minimizing a measure of dynamical stability , robustness , or performance over all possible configurations of links connecting the dynamical units .  herewe demonstrate that optimized dynamics is often highly sensitive to perturbations applied to the structure of the network . for concreteness , we focus on optimizing the stability against small changes in the network s dynamical states , among all networks with a given number of nodes and links .we consider dynamical states in which the ( time - dependent ) states of the individual nodes are identical across the network , such as the states of agreement in consensus dynamics , states of synchronized periodic or chaotic oscillations , and states of equilibrium in diffusion processes .the appearance of sensitivity to structural perturbations generally depends on the class of networks and the nature of perturbations considered ( fig .[ fig : summary ] ) , and here we establish the following two types of sensitivity :  1 .  _ sensitivity of undirected networks to changing the number of nodes and links within the class of networks with optimized structure _ ( lower right red box in fig .[ fig : summary ] ) .this class consists of those networks characterized as having the inimum possible size of the largest omponents in their omplement ( mcc ) among all networks with the same number of nodes and links .we provide a full analytical characterization of the mcc networks of arbitrary finite size and study their behavior as the network size approaches infinity .for many combinations of the number of nodes and links , an mcc network actually satisfies a stronger condition , under which node groups of _ equal _ sizes are fully connected to each other , but have no internal links .such networks are referred to as _ uniform complete multi - partite _ ( ucm ) networks , and here we show that the ucm networks are optimal .sensitivity of optimal directed networks to generic structural perturbations , i.e. , small random changes to the link weights _( upper left red box in fig .[ fig : summary ] ) .we show that the tendency of optimization to induce spectral degeneracy underlies this sensitivity , which we expect to observe in many applications because spectral degeneracy appears to be common in real networks  .we also show that , even when the network has only approximate degeneracy ( and thus approximate optimality ) , it exhibits the same sensitivity , at least approximately .in addition to these two cases of sensitivity , we have results on the absence of sensitivity in the other two cases ( blue boxes in fig . [fig : summary ] ) . for networks under non - generic structural perturbations ( bottom row in fig .[ fig : summary ] ) , our results provide a basis for a comprehensive theory of their spectral sensitivity / non - sensitivity . for networks under generic structural perturbations ( top row in fig . [ fig : summary ] ) , we have a unified theory that determines the scaling of their spectrum with respect to the size of perturbations .we illustrate the implications of our results using a general class of diffusively coupled systems for which we establish that their network spectrum determines the stability and other aspects of the dynamics to be optimized .specific cases we analyze include the rate of diffusion over networks , the critical threshold for turing instability in networks of activator - inhibitor systems , and the synchronization stability in power grids and in networks of chaotic oscillators .we aim to address a wide range of network dynamics in a unified way . for this purposewe consider the dynamics of a network of coupled dynamical units governed by the following general equation with pairwise interactions : for , where is the number of dynamical units ( nodes ) , is the column vector of state variables for the -th unit at time , and denotes the time derivative of .the function is generally nonlinear and describes how the dynamics of unit is influenced by the other units through intermediate variables , where indicates no interaction .this means that the dynamics of an isolated unit is described by .we assume that the dependence of on is the same for all ( or more precisely , that is invariant under any permutation of ) .thus , the topology of the interaction network and the strength of individual pairwise coupling are not encoded in , but rather in the -dependence of the coupling function .this extends the framework introduced in ref . and can describe a wide range of dynamical processes on networks , including consensus protocol  , diffusion over networks , emergence of turing patterns in networked activator - inhibitor systems  , relaxation in certain fluid networks  , and synchronization of power generators  as well as other coupled identical and non - identical oscillators  .  for the class of systems described by eq ., we consider _ network - homogneous states _ given by , where satisfies the equation for an isolated unit , .each of the example systems mentioned above exhibits such a state : uniform agreement in consensus protocols , a synchronous dynamics in oscillator networks , uniform occupancy in network diffusion , uniform concentration in coupled activator - inhibitor systems , and the equilibrium in fluid networks .note that certain non - homogeneous states can also be represented by such a solution by changing the frame of reference .  to facilitate the stability analysis ,we make two general assumptions on the nature of node - to - node interactions when the system is close to the network - homogeneous state : ( a-1 ) the interactions are `` diffusive , '' in the sense that the coupling strength between two nodes , , is to first order proportional to the difference between their states , .in particular we assume that the coupling strength vanishes as the node states become equal .( a-2 ) there is a constant coupling matrix encoding the structure of the network of interactions , in the sense that the proportionality coefficient ( the `` diffusion constant '' ) in assumption ( a-1 ) can be written as , where the scalar represents the strength of coupling from node to node , and the vector - valued function is independent of and . under these assumptions ,we define a stability function for each complex - valued parameter ( see appendix [ sec : stab_func ] ) , which captures the factors determining the stability of the network - homogeneous state but is independent of the network structure .this function , referred to as a master stability function in the literature , was originally derived for a general class of systems that is different from the one we consider here  .the influence of the network structure on the stability is only through the ( possibly complex ) eigenvalues of the laplacian matrix , defined by note that always has a null eigenvalue associated with the eigenvector , which corresponds to the mode of instability that does not affect the condition . the maximum lyapunov exponent measuring the stability of the network - homogeneous state is then given by i.e., it is stable if , and unstable if .in addition , gives the asymptotic rate of exponential convergence or divergence .we first consider the following fundamental question : _ for a given number of nodes representing dynamical units , and a given number of links with identical weights , what is the assignment of links that maximize the rate of convergence to a network - homogeneous state ? _ we may assume to be binary ( or ) without loss of generality , since any link weight can be accounted for by the stability function ( see appendix [ sec : stab_func ] ) .we additionally assume that there is no self - interaction , i.e. , .the binary - weight assumption is used in the following unless explicitly relaxed to extend our results to weighted networks .   for the class of networks with a fixed number of undirected links , we have the additional constraint that the matrix is symmetric .this constraint can arise from the symmetry of the physical processes underlying the interaction represented by a link , such as the diffusion of chemicals through a channel connecting reactor cells in a chemical reaction network . in this case, the optimization problem reads \text{subject to } & a_{ij}\in\{0,1\ } , \,\ , a_{ii } = 0 , \,\, a_{ij } = a_{ji } , \\[2 mm ] & \displaystyle\sum_i\sum_{j > i } a_{ij } = m. \end{array}\ ] ] if the stability function on the real line is strictly decreasing for ( which is satisfied in most cases ) , maximizing the convergence rate to the network - homogenous state for undirected networks is equivalent to maximizing , the smallest eigenvalue excluding the null eigenvalue that exists for any networks . for ,the only network with nodes and links is the complete graph , resulting in the ( maximum ) value . for ( implying that the network is a tree ) , the maximum possible value of is achieved if and only if the network has the star configuration  . for other values of ( assuming to ensure that the network is connected ) , it is challenging even numerically , mainly because each is constrained to be either or , which makes it a difficult non - convex combinatorial optimization .we note that the problem is also equivalent to minimizing a bound on the deviations from a network - homogeneous state in a class of networks of non - identical oscillators  .the maximization of has been a subject of substantial interest in graph theory , with several notable results in the limit , assuming that each node in the network has the same degree and keeping this common degree constant  or assuming a fixed maximum degree  .in contrast to these sparse - network results , below we address the maximization of in a different limit , , keeping the link density  ] ( where denotes the largest integer not exceeding ) . to prove this bound , we make use of the complement transformation , which maps a network with adjacency matrix into the network with the adjacency matrix defined by this transformation maps each nonzero eigenvalue to where we denote the laplacian eigenvalues of the complement as  .the bound in eq .can then be proved by applying proposition 3.9.3 of ref . to the complement of the network , which gives where and denote the maximum and mean degree of the complement , respectively .thus , we have , verifying eq . .  for special combinations of and , namely , and , with arbitrary positive integers and , this boundcan actually be attained by the ucm network . herethe _ ucm network _ is defined ( for the given and ) as the network in which ( i ) the nodes are divided into groups of equal size , ( ii ) all pairs of nodes from different groups are connected , and ( iii ) no pair of nodes within the same group are connected ( see fig .[ fig : ucm]_a _ for an example ) .moreover , we can show that the ucm network is the only one that attains the bound , and thus is optimal , among all networks with the same number of nodes and links .we in fact have a more general result : for each and such that the mean degree of the network is a nonnegative integer , attains the upper bound if and only if the network is a ucm network ( see appendix [ sec : ucm_optimality ] for a proof ) . for the cases and , the condition on and is satisfied because , and our result implies that the ucm network is the only one achieving the upper bound . for other combinations of and for which is a nonnegative integer but a ucm network is not possible , the result implies that no network can achieve the upper bound .  _mcc networks_for values of and that do not allow forming a ucm network , we consider the class of mcc networks , which we now define using a key property that makes the ucm networks optimal . to describe this property, we again make use the complement transformation . in particular , eq .implies that the maximization of is equivalent to the minimization of . in terms of its complement ,a ucm network consists of isolated , fully - connected clusters of equal sizes [ see fig . [fig : ucm]_c _ for an example with and fig .[ fig : construction ] for examples with . fora given , let denote the maximum number of links allowed for any -node network whose connected components have size ( see eq .  in appendix [ si: sec : construction_mcc ] for an explicit formula ) . given ,we define to be the smallest ( necessarily positive ) integer for which ; thus , is the minimum size of the largest connected components of any network with nodes and links .the key property of a ucm network we referred to above can now be expressed in terms of its complement : the size of the ( largest ) components equals the smallest possible value , where is the number of links in its complement . for any network in general ,the smaller the largest component size in the complement , the smaller we expect the eigenvalue to be , since is bounded by the largest component size .however , the smaller the components , the smaller the number of links they can accommodate .we thus define an mcc network as one that achieves the best possible , namely , as a network whose largest component of the complement is as small as possible among all networks with the given and .formally , we call a network with nodes and links an _ mcc network _ if the largest connected component of the complement is of size , where is the number of links in the complement .  for given and , there can be multiple networks satisfying the above property , forming the class of mcc networks ( see fig .[ fig : ucm]_b _ for an example ) . one can explicitly construct an mcc network by forming as many isolated , fully - connected clusters of size as possible in the complement of the network ( see appendix [ si : sec : construction_mcc ] ) .similar strategies that suppress the size of largest connected components , when incorporated into a network growth process , have been observed to cause discontinuous , or continuous but `` explosive '' percolation transitions  . the deterministic growth process defined in ref . is particularly close to the definition of mcc networks in that the process explicitly minimizes the product of the sizes of the components that the new link connects in each step .ucm and mcc networks of size for the maximization of smallest non - zero laplacian eigenvalue . for a given link density , the orange dot indicates for mcc networks .a ucm network , when possible for that value of , is indicated by a green dot . as increases , the eigenvalue experiences discrete jumps , corresponding to sudden changes in the structure of the network .the changes in the link configuration of the network s complement , as well as the associated jumps in the size of their largest clusters , are illustrated in the circles . at values just above and below the jumps ,the complement has links and links , respectively . ]as anticipated above , the class of mcc networks inherits part of the optimality of the ucm networks : the values of for mcc networks are locally optimal in the space of all networks with the same and .indeed , we can show that the formula is valid for any mcc network with link density , and that we have for any network that can be obtained from such an mcc network by rewiring a single link .we can also show that mcc networks are globally optimal in several cases of lowest and highest link densities , as well as for a range of link density around each value corresponding to a ucm network .the optimality of the networks with fully - connected clusters in the complement suggests potential significance of other , more general network motifs  , whose statistics has been studied in the context of network optimization  .we can also show that , whenever an mcc network is globally ( locally ) optimal , it is also globally ( locally ) optimal when the link weights are allowed to be any positive number ( thus relaxing one of the assumptions we made earlier ) .in fact , we now prove a more general result : for any undirected networks with all link weights equal to one , the eigenvalue of the laplacian matrix can not increase as the link weights are decreased .let denote the laplacian matrix of the network with the same set of links but with positive weights .it can be shown that the matrix is the laplacian matrix of another weighted network with positive link weights .for each of the three ( symmetric ) laplacian matrices , , , and , denote by , , and , respectively , the smallest eigenvalue excluding the identically null eigenvalue .courant s theorem implies that , , and , where the minimum is taken over all that are perpendicular to the vector . since , it follows that \\ & \geq \min_{\mathbf{x}}\mathbf{x}^t l_{w } \mathbf{x } + \min_{\mathbf{x}}\mathbf{x}^t l_{w}^c\mathbf{x}\\ & = \lambda_2(l_{w } ) + \lambda_2(l_{w}^c)\\ & \geq \lambda_2(l_{w } ) .\end{split}\ ] ] thus , of the network with decreased link weights can not be larger than of the original network . applying this result to the mcc networks, we see that these networks serve as the optimal networks within the larger class of ] . for both the er random networks ( magenta crosses ) and the networks obtained by sa ( blue circles ) , we observe the scaling with significantly different from . for the er random networks , we averaged over realizations , which are indicated by the gray dots .similar results are obtained for other values of . ]given that is intimately related to the lengths of shortest paths between nodes  ( as they both reflect the connectivity of the network ) , it is natural to study path length statistics ( and other common network statistics ) of ucm and mcc networks . for ucm networks we can explicitly calculate the average length of shortest paths over all pairs of nodes . in the limit , with fixed , the average path length approaches , which decreases linearly from two to one as the link density increases from zero to one .thus , the ucm networks have very small path lengths .the clustering coefficient of any given node , defined as the fraction of pairs of its neighbors that are directly connected , can also be computed for the same limit and equals for ( for any node ) , which spans the entire range of possible values from zero to one as increases from  to one .the singular behavior seen in fig .[ fig : comparison ] on the left of the points implies that , for the mcc networks with fixed number of nodes , decreasing the link density from can dramatically decrease and thus the convergence rate to the network - homogeneous state .in addition to this sensitivity , for the mcc networks has similar sensitivity with respect to another network parameter , the number of nodes .when is fixed , the expression for given in eq ., considered now as a function of , has a square - root dependence on the right of the points /2 ] as with fixed link density .this in particular implies that _ the normalized maximum _re _ has no sensitive dependence on . moreover , this result also applies to two classes of weighted directed networks : ( i ) networks with a fixed number of links , each with ( possibly different ) weight in the interval $ ] , and ( ii ) integer - weighted networks with a fixed sum of link weights .thus , the networks optimizing re in these classes exhibit no sensitivity under structural perturbations in which link density is varied while keeping the network optimized . in sharp contrast to such non - generic perturbations , we see below that random ( and thus generic ) perturbations to the network structure can reveal sensitivity in many of these optimal networks .we now study how the convergence rate behaves when a small generic perturbation is applied to the network structure , particularly when the initial network is optimal or close to being optimal . since the convergence rate is determined by the laplacian eigenvalues through the stability function and eq ., it suffices to analyze how the laplacian eigenvalues respond to generic structural perturbations , which we formulate as perturbations of the adjacency matrix in the form , where is a small parameter and is a fixed matrix with each chosen randomly from a continuous distribution .this type of structural perturbations can represent imperfections in the strengths of couplings in real networks , such as power grids and networks of chemical  , electrochemical  , or optoelectronic  oscillators .here we show that for a given laplacian eigenvalue of a directed network and a generic choice of , the change of the eigenvalue due to the perturbation generally follows a scaling relation , ; we also provide a rigorous bound for the scaling exponent .this scaling exponent determines the nature of the dependence of the perturbed eigenvalue on : if , the dependence is sensitive and characterized by an infinite derivative at , and if , it is non - sensitive and characterized by a finite derivative ._ bound on scaling exponent_we can prove the following general result on matrix perturbations , which provides an informative bound on .suppose is an eigenvalue of an arbitrary matrix with geometric degeneracy  , defined as the largest number of repetitions of associated with the same eigenvector ( i.e. , the size of the largest jordan block associated with ) .for perturbations of the form with an arbitrary matrix , there exists a constant such that the corresponding change in the eigenvalue , as a function of , satisfies applying this result to an eigenvalue of the laplacian matrix , we see that , implying that the set of perturbed eigenvalues that converge to as do so at a rate no slower than .  _sensitivity to random changes in link weights_the bound established above suggests the scaling for each eigenvalue of the laplacian .in fact , our numerics supports a more refined statement for networks under generic structural perturbations : for each eigenvector ( say , the -th one ) associated with , there is a set of perturbed eigenvalues that converge to as and follows the scaling , latexmath:[\[\label{eqn : generic_scaling }  is the number of repetitions of associated with the -th eigenvector ( i.e. , the size of the -th jordan block associated with ) .we verify this individual scaling using random perturbations applied to all off - diagonal entries of ( see fig .[ fig : gen - pert ] for the results and simulation details ) . since larger means more sensitive dependence on in eq ., the sensitivity of the degenerate eigenvalue is determined by the largest of the associated s , which is by definition .thus , we see that _ a laplacian eigenvalue is sensitive to generic perturbations if and only if the associated eigenvector is degenerate _ ,i.e. , the geometric degeneracy .    to provide evidence for the tendency of network optimization to induce geometric degeneracy ,we compute the laplacian eigenvalues and the corresponding symbolically ( and thus exactly ) for all possible directed networks of size . we find that a large fraction of the re-maximizing networks are indeed sensitive due to geometric degeneracy : % , % , and % of them have for , , and , respectively .these fractions are significantly higher than the corresponding fractions among all directed networks ( including non - optimal ones ) : % , % , and % , respectively .since is bounded by the algebraic degeneracy ( multiplicity ) of , an interesting question is to ask how often attains this bound , giving the network the maximum possible level of sensitivity . among those networks that are both optimal and sensitive , % and %achieve the maximal sensitivity for and , respectively .( the fraction is trivially % for . )figure [ fig : generic_perturbation ] illustrates the range of sensitivity possible for the subclass of optimal networks satisfying , for which the maximum geometric degeneracy is .non - sensitive networks in this subclass include simple cases such as the fully - connected networks and directed star networks , but also include other networks with more complicated structure , such as the network in fig . [fig : generic_perturbation ] . among the sensitive networks ,there is a hierarchy of networks having different levels of sensitivity , from ( e.g. , network in fig . [ fig : generic_perturbation ] ) all the way up to the maximum possible value ( e.g. , network in fig . [ fig : generic_perturbation ] ) , including all intermediate cases ( e.g. , network in fig .[ fig : generic_perturbation ] ) .interestingly , the scaling behavior of the laplacian eigenvalues also depends on the exact nature of the perturbation , and in particular the exponent can be different from in eq . if the perturbation is constrained to a subset of the off - diagonal entries of .for example , when perturbing only the existing links of a directed tree ( which is optimal with ) , the exponent is one , and thus the network is not sensitive to this type of perturbations even if the degeneracy ( see fig . [fig : gen - pert]_a _ ) .this follows from the fact that the laplacian matrix of a directed tree is triangular under appropriate indexing of its nodes , which remains true after perturbing the existing links .such non - sensitivity to perturbations of existing links can also be shown for certain other cases , e.g. , when is a triangular matrix , where is the non - singular matrix in the jordan decomposition of and is the perturbation of the laplacian matrix .we note that there are also examples in which perturbing only the existing links leads to the scaling exponent ( see fig .[ fig : gen - pert]_b _ ) .   for an arbitrary undirected network ,for which we have the constraint that the matrix is symmetric , _ all of its laplacian eigenvalues are non - sensitive to any ( generic or non - generic ) perturbation of the form _ , since symmetric matrices are diagonalizable  and thus .this implies that the rate of convergence to network - homogeneous states in any system governed by eq .  with symmetric individual interactionsare non - sensitive to any infinitesimal structural perturbations , independently of the optimality of the network structure .in particular , the convergence rate ( as well as itself ) is non - sensitive for all ucm and mcc networks . note that this is not in contradiction with the sensitivity of the ucm networks we discussed in sec . [sec : sensitivity - undirected ] . indeed , consistent with eq ., we do observe that depends linearly on when interpolating between any ucm network of finite size ( with nodes and links ) and any mcc network having one additional link .however , the derivative of as a function of link density on this interpolation interval is equal to , which diverges as , consistent with the square - root singularity in the large - network limit established in sec . [sec : sensitivity - undirected ] .the lack of sensitivity under infinitesimal perturbations is curious since the algebraic degeneracy of appears to be a key feature that gives these networks advantage over other undirected networks [ except for some cases with and leads to the sensitivity in the large - network limit .here we show that the scaling in eqs .  andis observed even when the eigenvector is only approximately degenerate , which is likely the case for real - world applications where exact degeneracy is practically impossible  .more precisely , we show that , when the matrix is close to one with exact degeneracy , the scaling remains valid over a range of much larger than the distance between the two matrices .suppose that a matrix has an eigenvalue with exact geometric degeneracy .we consider a perturbation of in the form where is a fixed matrix satisfying .thus , the distance between and is , and for small ( and a generic choice of ) the matrix is approximately degenerate .we now apply a perturbation of size to in the form , where is another fixed matrix satisfying . denoting , we can write as a perturbation of rather than , namely , .when taking the limit with fixed , matrices and are both fixed , so we can apply the result in eq . .this leads to for some constants , where and denote eigenvalues of and , respectively , that approach as .this means that for an arbitrary , we can find and ( which can depend on ) such that then , -[\lambda(m_1)-\lambda(m_0)]|}{\delta^{1/\beta}}\\ & \le \frac{|\lambda(m_2)-\lambda(m_0)|}{\delta^{1/\beta}}+\frac{|\lambda(m_1)-\lambda(m_0)|}{(\eta\delta)^{1/\beta}}\cdot\eta^{1/\beta}\\ & < \left(c_2+\frac{\xi}{2}\right)+\left(c_1+\frac{\xi}{2\eta^{1/\beta}}\right)\cdot\eta^{1/\beta}\\ & = c_2+c_1\eta^{1/\beta}+\xi , \end{split}\ ] ] if .since can be made arbitrarily small by making sufficiently small , we have thus , eq .  and the corresponding bound on the scaling exponent , , remain valid for any fixed ( i.e. , with as while holding constant ) . for finite and , this result suggests that we should observe the scaling with when .now consider the stronger scaling property in eq . , which can be formalized for and as replacing eq .  with eq.  and using the resulting lower bounds analogous to those in eq ., we obtain a lower bound analogous to that in eq . .combining this with eq . , we obtain since can be made arbitrarily small by making sufficiently small , we see that .this implies the scaling in eq . , or more precisely , with a prefactor that can vary with butis bounded between as .the ratio of perturbation sizes thus determines the range of variation of this scaling prefactor . in the limit of both and , eq.  implies .therefore , we have the scaling when .  altogether , we have shown that the scaling properties in eqs .  and are observed for the eigenvalues of when the size of the perturbation applied to is much larger compared to the distance between and the exactly degenerate matrix .the scaling bound in eq .is general and applies to both directed and undirected networks , both with weighted and unweighted links .we also expect the scaling in eq .  to generically hold true across these classes of networks . these results are not even limited to laplacian matrices and cover all matrices , including the adjacency matrix and any other matrix that may characterize a particular system .the degree to which the scaling holds is likely related to the normality of the laplacian matrix , which can range from completely normal matrices with orthogonal eigenvectors ( as in undirected networks ) , to highly non - normal matrices with parallel , degenerate eigenvectors ( as in many optimal networks )  . combining all these with the tendency of optimization to cause geometric degeneracy and the results for approximately optimal networks in sec .[ sec : approximate - optimal ] ( as well as the wide range of systems that eq .  can describe ) , we expect to observe sensitivity to generic perturbations in many applications .as summarized in fig . [ fig : summary ] , we have established two cases in which sensitive dependence on network structure arises : undirected networks under non - generic , optimized perturbations ( sec .[ sec : sensitivity - undirected ] ) and directed networks under generic perturbations ( sec . [sec : sensitivity - directed ] ) . belowwe discuss implications of these cases for concrete examples of physical networked systems .  for undirected networks ,the sensitivity of observed for ucm networks is relevant for a wide range of networked systems , since the stability function formalism establishes that in many systems determines the stability properties of relevant network - homogeneous states .typically the asymptotic rate of convergence is a smooth , monotonically increasing function of , and thus the maximized convergence rate exhibits the cusp - like dependence on the number of nodes and links .below we list specific cases in which sensitivity is observed in or a related quantity :  1 .  for networks of phase oscillators , including models of power - grid networks , the convergence rate to a frequency - synchronized , phase - locked state is a function of the laplacian eigenvalue associated with the matrix of effective interactions between generators . while is generally different from , it is strongly correlated with , and hence to .we thus expect to observe sensitive dependence of , which is indeed confirmed in fig .[ fig : example - systems]_a _ for power - grid networks with prescribed network topology and realistic parameters for the generators and other electrical components in the system .2 .  in addition to the asymptotic convergence rate , sensitive dependence can be observed for the convergence rate in the transient dynamics of the network , which depends not only on but on all laplacian eigenvalues .this is illustrated in fig .[ fig : example - systems]_b _ using the example of coupled optoelectronic oscillator networks  .another physical quantity that can exhibit sensitive dependence is the critical coupling threshold for the stability of the network - homogeneous state in systems with a global coupling strength . in such systems , the functions are proportional to . for identical oscillators capable of chaotic synchronization ,the minimum coupling strength for stable synchronization is inversely proportional to . for the activator - inhibitor systems , the parameter is interpreted as the common diffusivity constant associated with the process of diffusion over individual links . as is decreased from a value sufficiently large for the uniform concentration state to be stable , there is a critical diffusivity , , corresponding to the onset of turing instability , and this is inversely proportional to .such a critical threshold thus depends sensitively on the link density of the network ( as illustrated for the activator - inhibitor systems in fig .[ fig : example - systems]_c _ ) as well as on the number of nodes .  for directed networks , the sensitivity of laplacian eigenvalues under generic perturbations is typically inherited by the convergence rate for many systems governed by eq . .in fact , would have the same sensitivity as the laplacian eigenvalue whenever has a smooth ( non - constant ) dependence on near the unperturbed values of .figure [ fig : generic_perturbation_phys ] illustrates the sharp contrast between sensitive and non - sensitive cases using the example of synchronization in networks of chaotic optoelectronic oscillators  .the sensitive dependence of collective dynamics on the network structure , characterized here by a derivative that diverges at the optimal point , has several implications . on the one hand, it implies that the dynamics can be manipulated substantially by small structural adjustments , which we suggest has the potential to lead to new control approaches based on modifying the effective structure of the network in real time ; indeed , the closer the system is to being optimal , the larger the range of manipulation possible with the same amount of structural adjustment . on the other hand , the observed cusp - like behavior imposes constraints on how close one can get to the ultimate optimum in practice ;given an optimal network configuration at the tip of a cusp , a small parameter mismatch or deactivation / activation of nodes or links can have a significant impact on the optimality .it is insightful to interpret our results in the context of living systems .the apparent conundrum that follows from this study is that biological networks ( such as genetic , neuronal , and ecological ones ) are believed to have evolved under the pressure to both optimize fitness and be robust to structural perturbations  .the latter means that the networks would not undergo significant loss of function ( hence of optimality ) when perturbed .for example , a mutation in a bacterium ( i.e. , a structural change to a genetic network ) causes the resulting strain to be nonviable in only a minority of cases  .a plausible explanation is that much of the robustness of living systems comes from the plasticity they acquire from optimizing their fitness under varying conditions . in the case of bacterial organisms , for example, it is believed that the reason most of their genes are not essential for a given environmental condition is because they are required under different conditions .bacteria kept under stable conditions , such as intracellular ones , have evolved to virtually have only those genes essential under that condition  ; they are a close analog of optimization of a fixed objective function considered here .while there is therefore no conflict between our results and the optimization - robustness trade - off expected for biological networks , investigating the equivalent of the sensitive dependence on network structure in the case of varying conditions or varying objective function would likely provide further insights .a complementary example of interplay between optimization and robustness is described in ref . , which shows that optimization can drive a population to extinction through a global bifurcation in the structure of the state space .in general , the optimization - robustness relation may depend on the type of robustness considered . in this study we focused onhow stable a state is , and hence on how resistant the network is to small changes in its dynamical state , which can be regarded as a form of robustness ( terminology used , for example , in ref .it is quite remarkable that , in seeking to optimize the network for this `` dynamical '' robustness it would lose `` structural '' robustness ( i.e. , how resistant the network is to changes in the network structure ) .but is the observed sensitive dependence on network structure really a sign of non - robustness ?the answer is both yes and no .it is `` yes '' in the sense that , because of the non - differentiability of this dependence , small parameter changes cause stability to change significantly .it is `` no '' in the sense that , because the cusps appear at valleys rather than at peaks , the stability in the vicinity of the local best parameter choices are still generally better than at locations further away ( that is , specific parameters lead to significant improvement but not to significant deterioration ) . by considering both the dynamical and structural robustness , we can interpret our results as a manifestation of the `` robust - yet - fragile '' property that has been suggested as a general feature of complex systems  .  finally , it is instructive to compare the sensitive dependence on network structure with the phenomenon of chaos , which can exhibit multiple forms of sensitive dependence  .sensitive dependence on initial conditions , where small changes in the initial state lead to large changes in the subsequent evolution of the trajectories , is a phenomenon that concerns the phase space of a fixed system .sensitive dependence on parameters may concern a similar change in the evolution of trajectories across different systems even when the initial conditions are the same , as in the case of the map when rather than is changed .but sensitive dependence on parameters may also concern a change in the nature of the dynamics , which has a qualitative rather than just quantitative impact on the trajectories ; this is the case for the logistic map , whose behavior can change from chaotic to periodic by arbitrarily small changes in and , moreover , whose lyapunov exponent exhibits a cusp - like dependence on within each periodic window .the latter concerns sensitive dependence of the stability measure of the states under consideration , and therefore is a low - dimensional analog of the sensitive dependence of network dynamics on network structural parameters investigated here . in the case of networks , however , they emerge not from bifurcations but instead from optimization .much in the same way the discovery of sensitive dependence on initial conditions in the context of ( what is now known as ) chaos set constraints on long - term predictability and on the reliability of simple models for weather forecast  , the sensitive dependence on network structure calls for a careful evaluation of the constraints it sets on predictability and model reliability in the presence of noise and uncertainties in real network systems .we thus believe that the interplay between network structure , optimization , sensitivity , and robustness is a promising topic of future research that can offer fundamental insights into the study of complex systems .this work was supported by the u.s .national science foundation under grant dms-1057128 , by the u.s .army research office under grants w911nf-15 - 1 - 027 and w911nf-12 - 1 - 0276 , and by the simons foundation under grant 318812 .in sec . [ sec : class - systems ] we made two assumptions on the coupling functions , which can be formalized as follows :  1 . , where and denote the derivatives with respect to the first and second argument , respectively , of the function .we also assume for all , which ensures that the network - homogeneous state is a valid solution of eq . .together , these assumptions are equivalent to the linear approximation of around given by . , where the scalar is independent of , and the function is independent of and .  under these assumptions , the variational equation of the system  around a given network - homogeneous state becomes where is the perturbation to the state of unit , and are the derivatives of the function with respect to the first and any of the other arguments , respectively , evaluated at , and is the laplacian matrix of the network given by eq . .an argument based on the jordan canonical form of similar to the one used in ref . then leads to a stability function , defined for given ( complex - valued ) auxiliary parameter as the maximum lyapunov exponent of the solution of \boldsymbol{\eta},\ ] ] which can be used to determine the stability of the network - homogeneous state .a key aspect of this approach is that the functional form of does not depend on the network structure , which influences the stability only through the laplacian eigenvalues  .specifically , the exponential rate of convergence or divergence is given by for the perturbation mode corresponding to the -th ( possibly complex ) eigenvalue of the laplacian matrix .thus , the perturbation mode with the slowest convergence ( or fastest divergence ) determines the stability of the network - homogeneous state through in eq . .for a system with a global coupling strength parameter , such as the networks of identical oscillators and networks of activator - inhibitor systems described in secs .s1b and s1c , respectively , the derivative in the condition ( a-2 ) above is proportional to , and can be chosen to include the factor ( thus making the stability function dependent on ) .similarly , when considering networks with equal link weights for all links ( as in the optimization problem described in sec .[ sec : opt ] ) , can be factored out of and absorbed into in the condition ( a-2 ) , so that represents a binary network .we note that the class of systems treated in ref . is a special case of our formulation in which depends linearly on the variables and the coupling function is proportional to the difference in ( some function of ) the state of the nodes ( see sec .we also note that the same stability condition is derived in ref . for a general class of systems that is different from the class of systems treated here .this general class does not include the networks of coupled second - order phase oscillators or power generators described in sec .s1a , since one of the assumptions on the coupling term is not satisfied .another advantage of our formulation is that the assumptions on the nature of pairwise interactions encoded in the coupling functions are intuitive and have clear relation to the network structure encoded in the adjacency matrix .for a given undirected network with nodes and links , suppose that the mean degree is a nonnegative integer .here we show that attains the upper bound in eq .  if and only if the network is a ucm network .suppose first that .this implies that we have .also , by the assumption we made above , the mean degree of the complement , , is an integer , and thus we have .then , eq .  becomes since this implies that the maximum and the mean degree of the complement match , i.e. , , all nodes must have the same degree in the complement .now consider a connected component of the complement .on the one hand , the maximum laplacian eigenvalue of this component is at most the maximum laplacian eigenvalue of the entire network , . on the other hand , it is at least since the maximum degree of the component is ( by applying proposition 3.9.3 of ref . to the component ) . combining these, we conclude that the maximum laplacian eigenvalue of this component equals . since the maximum laplacian eigenvalue is the maximum degree plus one , applying the second part of proposition 3.9.3 of ref . to this component , we conclude that the size of the component is .since each node in this component has degree , this component must be fully connected .note that these conclusions apply to any connected component of the complement , which implies that the number of nodes must satisfy for some positive integer .therefore , the complement consists of isolated fully - 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Abstract: the relation between network structure and dynamics is determinant for the behavior of complex systems in numerous domains . an important longstanding problem concerns the properties of the networks that optimize the dynamics with respect to a given performance measure . here we show that such optimization can lead to _ sensitive dependence _ of the dynamics on the structure of the network . specifically , we demonstrate that the stability of the dynamical state , as determined by the maximum lyapunov exponent , can exhibit a cusp - like dependence on the number of nodes and links as well as on the size of perturbations applied to the network structure . as mechanisms underlying this sensitivity , we identify discontinuous transitions occurring in the complement of optimal networks and the prevalence of eigenvector degeneracy in these networks . these findings establish a unified characterization of networks optimized for dynamical stability in diffusively coupled systems , which we illustrate using turing instability in activator - inhibitor systems , synchronization in power - grid networks , and several other examples . our results suggest that the network structure of a complex system operating near an optimum can potentially be fine - tuned for a significantly enhanced stability compared to what one might expect from simple extrapolation . on the other hand , they also suggest constraints on how close to the optimum the system can be in practice . finally , the results have potential implications for biophysical networks , which have evolved under the competing pressures of optimizing fitness while remaining robust against perturbations .


cloud radio access network ( c - ran ) is a cloud computing - based radio access network architecture in which , unlike traditional cellular systems , the protocol stack of a cluster of base stations ( bss ) is virtualized by means of a control unit ( cu ) , to which the bss are connected via a fronthaul network .this architecture enables centralized signal processing at the cu , hence allowing the implementation of effective interference management strategies , which may dramatically improve the physical - layer capacity of the system - .however , the transmission and queueing delays on the fronthaul network generally limit the gains that can be achieved when considering the performance at higher layers , particularly at layer 2 - .fronthaul latency affects most notably the operations of automatic retransmission request ( arq ) protocols in the uplink , which rely on the transmission of acknowledgement ( ack ) or negative - acknowledgement ( nak ) messages to the mobile terminals ( mts ) , indicating respectively correct or incorrect packet decoding .in fact , due to fronthaul transmission and processing , these messages incur a delay that may significantly reduce throughput and efficiency of retransmission mechanisms .for example , in long - term evolution ( lte ) systems , a latency larger than 3 - 8 ms disrupts the normal working of the hybrid arq ( harq ) protocol .  in  , in order to mitigate the effect of fronthaul latency on the performance of harq protocols , it is proposed that the bss make local decisions regarding the expected outcome of decoding at the cu without waiting for the actual feedback from the cu .the local decisions are made based on the knowledge at the bss of the modulation and coding scheme used for uplink transmission and on an estimate of the channel state information .note that this requires the bss to implement enough baseband functionalities to be able to perform resource demapping , e.g. , synchronization and fft , as well as channel estimation , hence deviating from the conventional c - ran paradigm .the drawback of this approach is that , especially for small coding blocks , the local decisions taken at the bss may be incorrect , causing inefficiencies in the use of the spectral resources and delays .the analysis in is oblivious to the magnitude of the fronthaul latency , as it is implicitly based on the assumption of instantaneous ack / nak feedback from the bss to the mts . in this letter , instead , we are interested in evaluating the performance of arq protocols as a function of the fronthaul latency .fronthaul latency translates into feedback delays , whose impact on the performance of arq , unlike in the analysis in , depends on the specific implemented arq protocol .this is because different arq protocols react in distinct ways to feedback delays .in particular , we consider the throughput and efficiency of standard arq protocols , namely stop and wait ( sw ) , go - back - n ( gbn ) and selective repeat ( sr ) ( see , e.g. , ) , as a function of the fronthaul delay .furthermore , motivated by the key role that interference mitigation plays in the evaluation of the performance benefits of c - ran , unlike prior work , we explicitly consider the impact of interference .specifically , the dynamics of the interference process , along with that of the channel gains , are modelled by means of markov processes .the analysis encompasses the standard c - ran implementation with centralized baseband processing , as well as conventional cellular systems and an hybrid approach that is akin to the solution studied in . in the hybrid approach ,a bs first attempts local decoding , and processing at the cu is invoked only if this decoding attempt is unsuccessful .it is noted that , as for the approach in , this hybrid solution requires the bss to be endowed with baseband processing functionalities .the rest of this paper is organized as follows .section [ system ] describes the system model and the arq protocols considered in this paper .section [ model ] presents the analysis based on markov modelling that is used to assess throughput and efficiency . in section [ evaluation ] ,numerical results are shown to illustrate the performance of the system .section [ conclusion ] concludes the paper .the uplink of the cellular system depicted in fig .[ fig : system ] is considered , which is composed of a cluster of bss connected to a cu through fronthaul links .slotted time is assumed , where the transmission of one packet fits the duration of one time slot .the mts store packets to be transmitted in a finite first - in first - out buffer of size packets .packet arrival in the buffer is modeled as an independent identically distributed ( i.i.d . )process , where the probability that a new packet arrives in one slot is equal to ( ) .the mts implement arq to improve packet reliability , where failed packets are retransmitted until successfully decoded .we specifically study the standard arq protocols sw , gbn and sr .we consider three different system implementations , namely : ( _ i _ ) _ conventional cellular system _ : an assigned bs performs decoding of the uplink signal of any given mt ; ( _ ii _ ) _ c - ran _ : the cu performs decoding based on the signals received by all bss on the fronthaul links ; ( _ iii _ ) _ hybrid c - ran _ : an assigned bs first attempts decoding and , if it fails , the received signal is forwarded to the cu , which attempts decoding based on all the bss signals .as mentioned , the latter solution requires the implementation of an alternative functional split between bs and cu , as compared to conventional c - rans , in which the bs implements baseband functionalities .we assume that the propagation time between mt and bs is negligible , so that packets transmitted in slot are acknowledged by means of ack / nak messages at the end of the same slot , if decoding takes place at the bs as in systems ( _ i _ ) and ( _ iii _ ) . instead , to capture the delay associated with two - way transmission over the fronthaul network, we assume that ack / nak messages relative to decoding at the cu are received at the mts after time slots in systems ( _ ii _ ) and ( _ iii _ ) .  a detailed description and discussion of the signal processing techniques used in a c - ran architecture of bss and cu can be found in , e.g. ,  . here , focusing on the performance at layer 2 , we abstract the physical layer by adopting a simple threshold model on the signal to noise ratio .specifically , a packet transmitted by an mt in a slot is successfully decoded at an assigned bs in architectures ( _ i _ ) and ( _ iii _ ) if the signal - to - interference - plus - noise ratio ( sinr ) at the bs satisfies the condition where is the decoding threshold ; is the variance of the noise ; is the channel gain between the given mt and the assigned bs in slot ; is the uplink transmitted power ; and the term captures the overall interference incurred by the packet sent by the mt in slot . instead , to account for decoding at the cu in architectures ( _ ii _ ) and ( _ iii _ ) , we assume that the cu can effectively remove the interference based on the signals received from multiple bss . as a result, a packet is decoded at the cu if the signal - to - noise ratio ( snr ) at the cu satisfies where the additional noise term captures the effect of quantization noise , see , e.g. , .the latter is generally an exponentially decreasing function of the fronthaul capacity and is proportional to the received power . here, in order to focus on the effect of the fronthaul latency , we assume this term to be small enough to be negligible as compared to the channel noise .the analysis could be easily extended to a scenario in which this assumption is violated .the temporal evolution of the interference process is modeled as an on - off process where is a binary markov chain .this simple model captures the correlation in the interference process that results from the packet arrival process at the interfering nodes , as well as from the temporal correlation of the interferers channel gains .we note that the model can be extended to include multiple levels of interference and that the transition probabilities , to be discussed below , are generally a function the transmission patterns of the interfering nodes .  to measure the performance of the network , we use the following metrics : ( _ a _ ) _ throughput _ , i.e. , the average number of packets successfully decoded per slot ; ( _ b _ ) _ efficiency _ , i.e. , the average number of packets successfully decoded per slot divided by the fraction of slots in which the mt transmits .we observe that the efficiency directly reflects the energy requirements at the mt since a low efficiency entails the need for more packet transmissions .we now briefly recall the three arq protocols under study . for a more detailed discussion, please see .   with sw ,if the buffer is non - empty , the oldest packet is transmitted by the mt in the uplink and the mt waits idle to receive an ack / nak message .therefore , in the conventional cellular architecture ( _ i _ ) and in the hybrid c - ran system ( _ iii _ ) , if the bs successfully decodes the packet , an ack is received in the same slot , and the mt can transmit the next packet in the queue in the following slot . instead , in architecture ( _ i _ ) ,if the bs fails , the packet is retransmitted in the next slot , while , in system ( _ iii _ ) , the mt waits slots and then transmits a new packet or retransmit the same packet upon reception of the ack / nack relative to decoding at the cu . finally , in the c - ran system ( _ ii _ ) , the mt always waits slots for feedback from the cu , and transmits the same or the next packet depending on whether a nak or ack message is received .   with gbn, the mt keeps a list , or window , of at most outstanding packets that have been transmitted but whose ack / nak messages have not been received or processed yet at the mt . if the window contains less than packets , a new packet is transmitted from the queue if there is any . when the window is full , the mt checks for the ack / nak messages that have been received for the packets in the window .if there are no ack / nak messages , the mt stays idle .if there are , packets are considered starting from the oldest in the window . for this packet , if an ack has been received , the window slides to the following packet , which is considered next ; instead , if a nak has been received for the packet , the mt retransmits this and all the following packets in the current window .note that gbn is meant to avoid the throughput loss associated with idle transmit periods caused in sw by delayed feedback , and is hence relevant , in our context , only for the c - ran and hybrid c - ran architectures .this is because , in these systems , an mt implementing sw must wait for feedback from the cu , whereas gbn allows the transmission of new packets , if the window is not full , even before reception of ack / nak messages for the last transmitted packet .   with sr, the mt operates in the same was as for gbn with the caveat that , when the window is full and ack / nak messages are considered for the packets in the window , only the failed packet are retransmitted . as for gbn , hence , sr is relevant only for the c - ran and hybrid c - ran architectures .in this section , we evaluate throughput and efficiency of the arq protocols discussed above by focusing on the performance of an individual mt .the dynamics of the system are modeled as a markov chain with a finite state space .the transmission process of the mt , along with the channel and the interference process are sub - chains that together compose the overall chain .the state of the system chain in slot is denoted as , where tracks the protocol state , e.g. , the state of the queue at the mt and also of the retransmission window for the gbn and sr protocols .the transition probability , for any two states , can be factored as where we have in the following , we briefly elaborate on the dynamics of channel , interference and protocol , as defined by the factors in eq .( [ factors ] ) .we adopt a correlated rayleigh fading model with correlation coefficient across two successive slots and unitary power . in order to make the model manageable within the markov chain framework at hand, we use the quantized model detailed in , in which the channel gain range is partitioned into states with equal steady state probabilities , .the corresponding transition probabilities can be calculated as detailed in  .  the threshold model in eq .( [ sinr ] ) and ( [ sinr2 ] ) determines a map between the channel and interference states , on the one hand , and the outcome of packet decoding at the bs and cu , on the other . to clarify this map, we define the thresholds and .if the channel state satisfies , then the packet transmitted in slot is decoded at the bs only if , that is , in the absence of interference , whereas decoding fails if . instead, due to interference cancellation at the cu , packet decoding at the cu is successful if irrespective of the value of .if , then the bs successfully decodes the packet transmitted by the mt in slot .[ nmarkov ] illustrates the channel sub - chain by emphasizing the thresholds and .= [ fill = white , draw = black , thick , text = black , scale=0.65 ] ( 0 ) ; ( 1)[right of=0 ] ; ( 2)[right of=1 ] ; ( 3)[above right = 1 cm of 0 ] ; ( 4)[below right = 1 cm of 0 ] ; ( q)[right of=2 ] ;  \(0 ) edge[loop left , below ] node ( 0 ) edge[bend left , above ] node ( 1 ) edge[bend left , left ] node ( 3 ) ( 1 ) edge[loop above ] node ( 1 ) edge[bend left , below ] node ( 0 ) edge[bend left , above ] node ( 2 ) ( 2 ) edge[bend left , below ] node ( 1 ) edge[bend left , above ] node ( q ) ( 4 ) edge[bend left , left ] node ( 0 ) ( q ) edge[loop above ] node ( q ) edge[bend left , below ] node ( 2 ) ;  ( 80,7 ) ( 1,5)(1,0)78 ( 1,2 ) ( 71,2 ) ( 21,2)(0,2)3(0,1)1 ( 19,0 ) ( 45,2)(0,2)3(0,1)1 ( 42,0 )   the interference process is modeled as a binary markov chain , where corresponds to a slot with interference .we parametrize the chain through the interference rate and the burstiness , where is the fraction of time spent by the chain in state , and is the average length of sequences of state . with this parametrization ,the dynamics of are defined by the transition probabilities {i , j{=}\{0,1\}}$ ] , where    the protocol sub - chain models the temporal evolution of the transmission process of the mt , as a function of the channel and interference states , according to the rules prescribed by the implemented arq protocol and by the system architecture , namely conventional cellular system , c - ran or hybrid c - ran . in general , the protocol chain tracks the number of packets in the buffer of the mt and the internal state of the arq mechanism , such as the content of the window for the gbn and sr protocols .  as an example , consider sw and hybrid c - ran . in this case, the protocol sub - chain tracks the number of packets in the buffer and the time remaining before the next transmission . to elaborate, we define the protocol state as , where is the number of packets in the mt s buffer at the beginning of slot , and is the number of slots left before the next transmission .if and , then the mt transmits a packet in slot .if the packet is decoded at the bs , then one packet is removed from the buffer and is updated by accounting for a possible packet arrival .if decoding at the bs fails , then is set to , indicating that the mt waits to receive feedback regarding decoding at the cu , and is sequentially decreased in the following slots until is reached and a new attempt is performed .  a detailed description of the protocol chain for the other configurations is not included here due to space constraints , but it follows from the same considerations outlined above . for example, a related protocol chain tracking the state of sr arq can be found in  .   by definition ,the throughput is the average number of packets successfully decoded per slot , which can be computed as , where is the steady state probability of state , namely and is equal to the number of packets delivered in state .note that is equal to the fraction of time spent by the system in state and that the limit in ( [ eq : limit ] ) exists under mild assumptions on the ergodicity of the system markov chain .the efficiency is the average number of packets successfully decoded per slot divided by the fraction of slots in which the mt transmits , which is given by where is equal to the number of packets transmitted in state .in this section , we present numerical results to assess the performance of the arq protocols sw , gbn and sr for the conventional cellular system and for the c - ran and hybrid c - ran architectures discussed above .the system parameters are set as , , db , and  db ; the channel is characterized by correlation ; the parameters of the interference chain are and ; and , unless indicated otherwise , the delay introduced by the fronthaul network is set to  slots . for the gbn and sr protocols , in the c - ran and hybrid c - ran systems, the transmission window is set to be equal the maximum delay , that is , .this choice is dictated by the observation that selecting would induce idle slots when the packet is not decoded correctly at the bs , whereas the choice would unnecessarily increase latency .slots , , db , , , , db , and ) . ]   slots , , db , , , , db , and ) . ][ thrsir ] and fig .[ effsir ] show the throughput and the efficiency , respectively , achieved by the conventional cellular system , the c - ran and the hybrid c - ran systems for the different arq protocols as a function of the . as the sir increases ,the throughput and efficiency improve due to the lower error rate for decoding at the bs , which affect both the conventional and the hybrid c - ran systems , but not the c - ran architecture .moreover , the performance of the hybrid c - ran system converges to that of the conventional and c - ran systems in the high and low sir regions , respectively .in fact , if the sir is high , decoding at the bs does not incur a significant performance degradation due to interference and hence decoding at the cu does not offer gains . as the sir decreases , instead , the decoding probability at the bs becomes smaller , and the packet is always forwarded to the cu for decoding . also , by construction , the hybrid c - ran always outperforms the c - ran system in both throughput and efficiency .considering now the impact of different arq protocols , if the basic sw is implemented , the fronthaul latency affects significantly the throughput , and the conventional approach offers the best throughput even at moderately low sir values . for time slots, the impact of the delay on gbn is limited , and both c - ran and hybrid c - ran offer performance superior to that of the conventional system . finally , the sr retransmission mechanism is the most resilient to the fronthaul delay , and hence the c - ran and hybrid c - ran are seen to achieve the best performance gain as compared to the conventional system .  in terms of efficiency , c - ran outperforms the conventional system in all the considered cases due to the higher delivery rate per transmission offered by the interference management capabilities of the cu . also , the efficiency of sw is comparable to that of sr , whereas the gbn protocol suffers due the additional transmissions of decoded packets that is due to the go - back " mechanism .[ figdelta ] depicts the throughput as a function of the fronthaul latency .the performance of both the c - ran and hybrid c - ran system degrades as increases .in particular , the sw arq mode system presents the largest sensitivity to the latency , as the latter entails an idle period , of duration , in which the channel is not used . nevertheless , c - ran or hybrid c - ran still yield significant throughput gains with gbn and sr as long as is not too large .for instance , for sr , c - ran is advantageous over the conventional system for .db , db , , , , , db , and ) . ]this letter presents a study on the impact of fronthaul latency on the performance of the standard arq protocols stop and wait ( sw ) , go - back - n ( gbn ) and selective repeat ( sr ) for c - ran systems .the analysis accounts for channel and interference temporal correlations , and offers a performance comparison in terms of throughput and efficiency with respect to traditional cellular systems and a hybrid c - ran architecture where bss attempts decoding before forwarding the received signal to the `` cloud '' .the delay introduced by the fronthaul network is observed to significantly degrade the throughput achieved by all arq protocols , although sr presents the highest resilience to fronthaul latency .moreover , the hybrid c - ran system , which can be seen as an instance of alternative functional splits currently under study for c - ran ( see , e.g. , ) , is demonstrated to provide an effective means to mitigate the throughput and efficiency degradation caused by fronthaul latency .interesting open problems include the analysis of the impact of multi - user detection in the presence of multiple mts .  20 china mobile , c - 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76 , may 2014 .a. leon - garcia and i. widjaja ._ communication networks : fundamental concepts and key architectures , 2nd ed_. mcgraw - hill , 2003 .l. badia , m. levorato and m. zorzi , `` markov analysis of selective repeat type ii hybrid arq using block code , '' _ ieee trans .1434 - 1441 , sep . 2008 .c. c. tan and n. c. beaulieu , on first - order markov modeling for the rayleigh fading channel " , _ ieee trans .12 , pp . 2032 - 2040 , dec . 2000

Abstract: in the cloud radio access network ( c - ran ) architecture , a control unit ( cu ) implements the baseband processing functionalities of a cluster of base stations ( bss ) , which are connected to it through a fronthaul network . this architecture enables centralized processing at the cu , and hence the implementation of enhanced interference mitigation strategies , but it also entails an increased decoding latency due to the transport on the fronthaul network . the fronthaul latency may offset the benefits of centralized processing when considering the performance of protocols at layer 2 and above . this letter studies the impact of fronthaul latency on the performance of standard automatic retransmission request ( arq ) protocols , namely stop and wait , go - back - n and selective repeat . the performance of the c - ran architecture in terms of throughput and efficiency is compared to the that of a conventional cellular system with local processing , as well as with that of a proposed hybrid c - ran system in which bss can perform decoding . the dynamics of the system are modeled as a multi - dimensional markov process that includes sub - chains to capture the temporal correlation of interference and channel gains . numerical results yield insights into the impact of system parameters such as fronthaul latency and signal - to - interference ratio on different arq protocols .