many real - world complex systems ranging from society to critical infrastructure operate through multiple layers of distinct interactions among constituents as well as the interplay between these interaction layers .multiplex network is a class of networks introduced to model such systems , in which the same set of nodes ( say , individuals in a society ) are connected via more than one type of links ( say friendships , kinships , co - workerships , _ etc _ ) .each type of links in multiplex networks forms the network layer .the study on multiplex networks is one of the major contemporary topics of network theory , together with closely related topics such as interacting networks , interdependent networks , and interconnected networks , _ etc . _introduction of distinct layers in multiplex networks necessitates new conceptual as well as computational developments beyond the well - established single - network framework , as one has to properly deal with the coupling between layers as well .one such factor is the pattern of multiplexity , that is , how the layers are coupled structurally . in real - world multiplex systems ,the coupling structure between layers should by no means be random but correlated . the term correlated multiplexity was introduced to refer to this correlation property .correlation properties of the layer coupling have been assessed empirically in term of the interlayer degree correlation and the link overlap , and indeed both were found to be significant .effect of the interlayer degree correlation has been theoretically investigated for percolation and network robustness in multiplex networks ( see also related studies on the interdependent networks ) . the multiplex network model based on the coevolution of layers has been proposed to model the emergence of interlayer degree correlation in evolving multiplex networks .a more comprehensive framework for characterization of the correlation properties of multiplex networks has recently been proposed .meanwhile , in social network literature the existence and consequences of link overlaps across social network layers have been the central issues of multiplexity studies , proving to play instrumental roles in social structure and dynamics in diverse ways .however , most theoretical studies on multiplex networks thus far have focused on the case of randomly coupled sparse layers , thereby the effect of the link overlap across layers has largely been ignored . only recently two groups of researchers have investigated explicitly the effect of link overlap for the problem of mutual percolation using different theoretical approaches and obtained different conclusions about the existence of tricritical point .  mutual percolation is a percolation model originally introduced for interdependent networks yet applicable to multiplex networks straightforwardly .a pair of nodes in multiplex network are said to be mutually connected if the pair is connected through each and every layer in the network .a set of nodes within which every pair is mutually connected form the mutually - connected component , or the mutual component for short .if there exists an extensive mutual component , called the giant mutual component , we say that mutual percolation has occurred and the system is mutual - percolated .the mutual percolation is a minimal model of cooperative coupling between layers in multiplex systems .recently the so - called viability of multiplex networks has been introduced ( see also closely related weak percolation models proposed in ) .the viability of multiplex networks is based on the concept of multiple resource demands for proper functioning in multiplex systems .for such kind of systems , a node can be viable only when it is supplied with each and every kind of resources successfully through corresponding network layer . as we will see in the following sections , the problem of viability of multiplex networks turns out to include the mutual percolation as a particular limit , thus provides unifying framework for existing different approaches for mutual percolation with link overlap , thereby helps clarify how and why the two studies produce different results .thus we study in this article the effect of link overlap on the viability of multiplex networks with two - fold objectives : i ) to present generalized theoretical formalism and understanding of the problem of viability of multiplex networks with link overlap ; and ii ) to clarify the discrepancy of existing results for the mutual percolation with link overlap by way of the limiting case of the multiplex viability problem .the rest of the article is organized as follows . in section [ sec : viability ] , we introduce the model of viability in multiplex networks and summarize main results from .effect of the link overlap on viability is investigated in section [ sec : viability - overlap ] . as a limiting case of the viability problem, we revisit the mutual percolation with link overlap and discuss about the two existing approaches  in section [ sec : mutual ] .implications to the network robustness under random link failures in networks with link overlap are investigated in section [ sec : robustness ] .finally summary and discussions in section [ sec : conclusion ] will follow .the viability of multiplex networks was introduced based on the concept of multiple resource demands for proper functioning in multiplex systems such as the civil society  and biological systems  .for example , in the case of society , our life in modern society relies on interrelated infrastructure networks including water supply networks , gas supply networks , and power grid systems . in such systems , simultaneous connectivities of a node with the nodes producing resources ( thereafter called the resource nodes ) , such as power plants in the power grid and water sources in the water supply networks , through a series of functioning nodesare essential for the proper functioning , _i.e. , _ to be viable . in this sectionwe briefly summarize the previous work to outline the viability model and its solution properties , before generalizing it to cases with overlapping links in the following sections .for more details and additional results regarding the multiplex viability model the reader is referred to .   let us consider a network with -multiple layers , where each layer of the multiplex network corresponds to a certain infrastructural network .a given fraction of resource nodes generates and distributes resources essential to be viable .a key assumption of the model is that only viable nodes can function properly and transmit resources further to their connected neighbors .then , a node is viable only if it can reach , via the viable nodes , to a resource node in each and every layer .one is interested primarily in what fraction of the nodes would be viable for given network parameters and distribution of resource nodes .specifically , we define the viability to be the fraction of viable nodes in the limit .  to identify clusters of viable nodes ( or viable clusters for short ) in a given network , two algorithms were proposed , called the cascade of activations ( ca ) and deactivations ( cd ) , respectively .how the two algorithms proceed when applied to a small multiplex network of nodes with two layers ( without overlapping links ) is illustrated in fig . 1 .   at each step .reproduced from ref . . ][ fig : algorithm - viability ]    [ fig3 ]  the ca algorithm runs as follows :  * initially all nodes except resource nodes are labeled unviable .* at each step , unviable nodes that are linked to viable nodes through each and every layer [ colored green in fig .1(a ) ] are activated to become viable . *this cascade of activations continues until no nodes newly become viable .the cd algorithm runs as follows :  * initially all nodes including resource nodes are labeled viable .* at each step , the nodes that do not reach the resource nodes in any of the layers [ colored green in fig .1(b ) ] are deactivated to become unviable . *this cascade of deactivations continues until there are no nodes to deactivate .note that the viable clusters obtained from the ca and cd algorithms can be in general different from each other , as exemplified in fig . 1 . while the viable nodes for the ca are always viable for the cd as well , it is not always so _ vice versa_. this already suggests that the problem of multiplex viability can have multiple solutions , depending on the system s history .to gain more insight , we compute the viability analytically .this can be done for the cases of multiplex networks with layers that are locally - tree - like ; we further assume that the layers are coupled randomly , so that the link overlaps among layers are negligible for sparse networks we consider ; finally , resource nodes are assumed to be distributed randomly over the network .  in -layer networks ,each node has degrees over different layers drawn from the joint degree distribution . to calculate viability , we first consider the probability that the node reached by following a randomly chosen -type link ( is not viable . given the initial fraction of randomly distributed resource nodes on locally - tree like structures , s satisfy the following self - consistency equations , where is the mean degree of the -layer network .the first term is the probability that the chosen node is a resource node .the second term is the probability that it is connected with viable nodes through each type of links .the viability is obtained as the probability that a randomly chosen node is viable , which is given in terms of s by by solving eqs .( 12 ) for given and , one obtains the viability . one may note that when eqs .( 12 ) reduce to those of mutual percolation with the viability identified with the size of the giant mutual component .we illustrate basic features of the multiplex viability model with a specific example of the multiplex networks with two randomly - coupled erds - rnyi ( er ) layers ( duplex er graphs ) . for simplicity the mean degrees of two layersare chosen to be the same , denoted as .then eqs .( 1 ) and ( 2 ) reduce to a single equation for as by solving eq .( 3 ) , can be obtained for given and .( 3 ) is found to have two stable solutions for a range of parameters .one can construct the phase diagram [ fig .2(a ) ] , in which the low and high viability phases are separated by the two lines of loci of the saddle node bifurcations .the solid line corresponds to the points at which the viability for the ca algorithm undergoes a discontinuous change , and the dashed line indicates that for the cd algorithm . between the two lines ( hysteresis region ) , there are two possible stable solutions , and is determined by its initial value .for example , for the ca algorithm , keeps to be in low viable state with increasing until the abrupt jump at on the solid line . for the cd case, high viability sustains until abrupt collapse at on the dashed line [ fig .two lines merge at the critical point located at . above ,the viability changes gradually without discontinuity as increases and the distinction between the two phases disappear .such a bistability leads to a hysteresis in viability of multiplex networks [ fig .in order to demonstrate the hysteresis explicitly , let us suppose the following scenario of a sequence of systemic collapse and subsequent recovery of viability .initially the system is in high viability state with well - established networks ( that is , sufficiently large ) .as decreases due to random failures of links , abruptly collapses at .after collapse , if one tries to restore viability to the level before the collapse , one needs to increase the link density up to , which is significantly larger than the point of collapse .this suggests that the multiple resource demands lead to not only a potential danger of abrupt collapse but also an excessively severe complication in recovery . note that the bistability and hysteresis are absent in the single networks , [ figin this section , we study the viability of multiplex networks with link overlap . with link overlap , one can still apply the ca and cd algorithms in a straightforward manner ( fig . 3 ) .however , one should note different roles of overlapping and non - overlapping links for mutual connectivity and viability . for ordinary connectivity , making link overlap over different layersdoes not extend the existing connected clusters but merely provides redundancy .for mutual connectivity , by contrast , the overlapping links can play distinctive role : a cluster formed by overlapping links ( hereafter called the overlap cluster for short ) is by itself a mutually - connected component .consequently the whole overlap cluster becomes viable once any of its member nodes becomes viable .similarly , the overlap cluster becomes deactivated also as a whole .  in what followswe will focus on the duplex systems .generalizations for cases will be discussed later . given the distinct roles of non - overlapping and overlapping links , it is useful to represent the degree of a node by distinguishing the two classes .therefore we use the vectorial degree for each node where  is the degree for non - overlapping link of type and is for overlapping links , all of which are drawn from the joint degree distribution .we further assume that the layers are randomly coupled as well as the overlapping and the non - overlapping degrees are uncorrelated .we provide analytic treatments for the ca and the cd algorithms , respectively , for viability in the presence of link overlap .it is shown that the two algorithms are described by different sets of equations , displaying a rich variety of behaviors in viability .[ fig : algorithm - overlap ]   to become viable in the ca algorithm , a node must be connected individually with viable nodes either through both types of links simultaneously or through overlapping links .accounting for this , the self - consistency equations for the probabilities ( ) that the node reached by following a randomly chosen type- nonoverlapping ( overlapping ) link is not viable in the ca algorithm thus can be written as , \nonumber \\1-u_o=\rho+(1-\rho ) \sum_{\vec{k } } & \frac{k_o p(\vec{k})}{z_o } \big [ ( 1-u_o^{k_o-1})\nonumber\\ & + u_o^{k_o-1 }  ( 1-u_1)^{k_1 } ( 1-u_2)^{k_2 } \big ] .\end{aligned}\ ] ] the first term is the probability that the chosen node is a resource node ; the second term is the probability that it is connected with viable nodes through overlapping links ; and the last term is the probability that it is connected with viable nodes through each type of non - overlapping links but not through overlapping links .similarly , the final fraction of viable nodes can be obtained as .\end{aligned}\ ] ]  like eqs .( 12 ) , eqs .( 45 ) can have two stable solutions for a range of parameters . as illustrated in fig . 4, the outcome of ca algorithm corresponds to the lower branch solution for of eqs .( 45 ) . in passing , it is worthwhile to note that for , eqs .( 45 ) reduce to those for the mutual percolation with link overlap proposed in ref . .   for ca ( red square ) and cd ( blue circle ) are presented . herethe each layer of the network is an er graph with and . ][ fig : solution ]   as noted before , overlapping links play a more intricate role in the cd algorithm . since an overlap cluster deactivates as a whole ,they behave effectively as a single `` supernode . ''then the condition for a node to be viable becomes whether the supernode to which it belongs is connected with other viable supernodes through each and every type of ( non - overlapping ) links attached to some nodes within the supernode . to take this into account ,it is convenient to `` renomalize '' the network by regarding each overlap cluster as a single supernode , . ] which are connected via non - overlapping links connecting nodes in different supernodes .this approach was first proposed for the percolation of interdependent networks with link overlap in ref . .  to proceed with computing the viability for the cd algorithm with link overlap using the renormalized networkone further needs two additional quantities .first is the distribution of the mass ( number of nodes ) of a supernode by following a randomly chosen non - overlapping link ; and the second is the distribution of ( non - overlapping ) degrees of a supernode of mass in the renormalized network , where denotes the ( non - overlapping ) degrees of a supernode in the two layers .once and are at hand , the viability can be obtained as follows : the probability that a mass- supernode including a randomly chosen node does not contain resource nodes is .thus the probabilities that the node reached by following a randomly chosen type- ( non - overlapping ) link is not viable satisfy \nonumber\\ & + ( 1-\rho)^m \sum_{\vec{q } } \frac{q_i p_m(\vec{q})}{z_i } ( 1-u_i^{q_i-1 } ) ( 1-u_j^{q_j } ) \big\}.\end{aligned}\ ] ] here is the probability that the node belongs to the infinite mass supernode ( containing the resource node with probability one ) . the first term in the summationgives the probability that the mass- supernode contains the resource node so that it is viable by itself ; the second term gives the probability that it does not contain the resource node but is connected to other viable nodes ( belonging to different supernodes ) through each and every layer of non - overlapping links .similarly , the final fraction of viable nodes can be obtained as \nonumber\\ & + ( 1-\rho)^m \sum_{\vec{q } } p_m(\vec{q } ) \prod_{i=1}^2 ( 1-u_i^{q_i } ) \big\}. \end{aligned}\ ] ] note that when only term contributes in eqs .( 67 ) , thereby eqs .( 67 ) and eqs .( 45 ) become identical .-phase diagram of viability with link overlap .( b ) projections of the phase boundary surface edges ( thick lines ) for the ca ( red ) and the cd ( blue ) on plane .( c ) same projection plots on the plane . ]the auxiliary quantities and can be obtained as follows .first , assuming that the overlapping and non - overlapping degrees of a node are uncorrelated , the supernode mass distribution can be obtained from the component size distribution of the ( original ) network of overlapping links using the marginal distribution for overlapping degrees from .the possibility of non - overlapping links connecting nodes within the same supernode ( self - links ) and those connecting nodes between two supernodes more than once ( multi - links ) complicates the computation of .fortunately , however , there exist some simplifying conditions .one can show that unless the renormalized network has diverging second moments in the supernode degree distribution both can be neglected . under such circumstances and with additional assumption of uncorrelated network layers, one can obtain by the -th order convolutions of the marginal distribution from   as in the case of eqs .( 45 ) , eqs .( 67 ) can have two stable solutions . between the two branches ,the outcome of cd algorithm corresponds to the upper branch solution for of eqs .( 67 ) ( fig . 4 ) .the lower branch solution corresponds to the activation process in terms of the supernode , rather than that of the individual nodes as in the ca algorithm . finally , we note that eqs .( 67 ) for are equivalent to the formulation for the mutual percolation with link overlap proposed in ref . , completed with the additional term which was missing in that study .   for explicit illustration, we again consider the duplex random graphs .each layer of the network is an er graph with the mean degrees and for non - overlapping and overlapping links , respectively .one may construct such a network by randomly placing non - overlapping links on each layer of nodes , after which additional random overlapping links are placed simultaneously in both layers .overlaps between non - overlapping and overlapping links can be neglected in large networks .( ) , combinations of hysteretic behaviors with discontinuous ( red ) , continuous ( blue ) , and mixed ( green ) changes are observed as the non - overlapping link mean degree is varied .the parameters and color codes follow those indicated as three dots in fig .( b ) more examples but with varying the overlapping link mean degree , combinations of hysteretic behaviors with discontinuous ( red ) and mixed ( black and magenta ) changes are observed .the parameters and color codes follow those indicated as three dots in fig .]  eqs .( 45 ) for the viability in the ca algorithm can be written straightforwardly as .\end{aligned}\ ] ] solution for the viability in the cd algorithm can also be obtained by noting that duplex er graphs satisfy the simplifying conditions for and discussed in the previous subsection ( with possible exception at ) .( 67 ) becomes \nonumber\\ & + ( 1-\rho)^m ( 1-e^{-zmv})^2 \big\ } , \end{aligned}\ ] ] where and with being lambert -function .using the lower branch solution of eq .( 8) and the upper branch solution of eq .( 9 ) , one can construct the three dimensional phase diagram in the parameter space ( fig .the phase diagram is rather complicated , which should read as follows : the system is in high ( low ) viability phase on the right ( left ) side of red ( blue ) surface .the two phase boundary surfaces correspond to the loci at which lower branch solution loses stability in eq .( 8) and discontinuous jump of viability in ca algorithm occurs ( red ) and those at which upper branch solution in eq .( 9 ) and cd algorithm do so ( blue ) . asthe parameters are varied , discontinuous changes in the viability occur when the parameter trajectory penetrates the phase boundary surface . on the other hand ,if the parameter trajectory does not penetrate the boundary surface , the viability changes smoothly. therefore one can observe diverse sets of hysteretic behaviors , depending on the way the system parameters are varied ( fig .the facilitating effect of the overlapping links for viability can also be seen from the phase diagram .as noted , mutual percolation corresponds formally to the viability problem with .let us look at this correspondence more closely . without link overlap , one can obtain the equations for mutual percolation by simply setting in eqs . ( 12 ) . for , the lower branch solution of eqs .( 12 ) becomes a trivial one . above , where denotes the mutual percolation transition point without link overlap, the nontrivial solution appears in the upper branch .therefore the size of giant mutual component in mutual percolation can be identified with the upper branch solution for of eqs .( 12 ) with .  in the presence of link overlap, one should take into account the distinctive role of the overlapping links forming the mutual component by themselves .this can only be implemented by the cd - type eqs .( 67 ) , and can not be properly accounted for with the ca - type eqs .note that the upper branches of the two approaches do not converge when .hence we obtain the analytic solution for the mutual percolation with link overlap as follows .the probability that a node reached by following a randomly chosen link in -layer does not belong to the giant mutual component is given by setting of eq .( 6 ) as using the same auxiliary quantities and .then the size of the giant mutual component is obtained in terms of the probability that a randomly chosen node belongs to the giant mutual component by which is the case of eq .( 7 ) with and s in place of and s , respectively .we note once again that these eqs .( 1011 ) are equivalent to the formulation proposed in ref . , except for the additional term which was missing in that study .   for fixed ( red ) .symbols ( circles ) are numerical results with .comparison with the predictions from eq .( 8) ( line ) and the message - passing algorithm proposed in ref . ( squares ) is made ( purple ) .( b ) the phase diagram for the mutual percolation of duplex er graphs .phase boundary obtained from eq .( 12 ) ( red ) is displayed , together with that from ref . ( black lines ) , shown for comparison .solid lines represent lines of discontinuous transitions which terminate at the filled circle .dotted line is the continuous transition line proposed in .( c , d ) two example configurations of mutual - percolated clusters . ]   applying eqs .( 1011 ) to duplex random graphs with non - overlapping link mean degree and overlapping link mean degree , one obtains from eq .( 9 ) the following equation for the giant mutual component size , with and given in sec . 3.3 .solving eq .( 12 ) for , one can analyze the mutual percolation behaviors , showing good agreement with numerical simulations [ fig .one can also obtain the -phase diagram for the mutual percolation [ fig .7(b ) ] , which predicts discontinuous mutual percolation transitions except at , which is nothing but the continuous ordinary percolation transition point .this prediction is in agreement with the claims in but in contrast with those of , which predicted the existence of tricritical point [ fig . 7(b ) ] . as noted in sec .3.1 , the approach of corresponds to the ca - type eqs .( 45 ) with ( upper branch ) , which has different solution than that of eqs . ( 1011 ) unless [ fig .being based on local update , this approach is able to detect mutual connectivity in configurations like fig .7(c ) , but can fail for configurations like fig .7(d ) .   for various overlap fractions ( from right to left ) .lines are obtained by eq .( 12 ) and symbols are numerical results with .( b , c ) plots of the total mean degree for mutual percolation transition ( b ) and the size of discontinuous jump in the giant mutual component size at the transitions ( c ) as functions of the overlap fraction . ]another view at the facilitating role of link overlap for mutual connectivity is provided by examining how the transition points for mutual percolation of the total mean degree , where , changes with the overlap fraction ( fig .for the same total mean degree , the giant mutual component size increases with the overlap fraction [ fig .as the overlap fraction increases , the mutual percolation transition occurs at lower , from to [ fig .the size of discontinuous jump in the giant mutual component size at the transition decreases with the overlap fraction , from to ( that is , continuous ) [ fig .given the distinctive role of overlapping links for multiplex viability , it is of interest to examine how differently the network would respond to failures of the two classes of links . having established the analytic approaches for the viability and mutual percolation in previous sections , one can readily employ the existing method for examining the effect of random link failures .  herewe will present analytic formulae for the case of mutual percolation as they are similar to yet simpler than those for the viability .suppose fraction of overlap links is removed , the degree distribution of overlapping links becomes , and consequently the mass distribution of supernodes changes accordingly .then the giant mutual component size can be obtained by solving eqs .( 1011 ) with the supernode mass distribution corresponding to . on the other hand ,when the non - overlapping links are removed , eqs .( 1011 ) for and should be modified slightly , while remains unchanged .suppose fraction of -type non - overlapping links is removed , the probability that a node reached by following a randomly chosen non - overlapping link in -layer does not belong to the giant mutual component becomes ^{q_i-1 } \ }  \ { 1-[1+(v_j-1)f_j]^{q_j } \},\end{aligned}\ ] ] for .then the giant mutual component size is obtained by ^{q_i } \ } . \ ] ]     for er layers , the supernode mass distribution after the random removal of fraction of overlapping links of unperturbed mean degree is given by then the giant mutual component size is obtained by with and from eq .( 15 ) . for the random removal of non - overlapping links ,( 1314 ) reduces into a single equation for the giant mutual component size as   we consider a variety of random link removal strategies : i ) removing the overlapping link simultaneously ; ii ) removing only the type-1 link among the overlapping link ; iii ) removing any one link among the overlapping link ; iv ) removing only the type-1 non - overlapping link ; v ) removing overlapping link of any type ( fig .we found as expected that overlapping link removals are significantly more detrimental than non - overlapping link removal in both system s viability [ fig .9(a ) ] and mutual connectivity [ fig .surprisingly , removals of only one of the overlapping links are quite closely as efficient as the removals of the entire overlapping links , reflecting the critical role of the link overlap .such a critical role of link overlap may provide mechanistic basis for understanding of the importance of multiplexity in social context . finally , it is also observed that removing the non - overlapping links in a particular layer is more efficient to break the network down than the removals of the same number of non - overlapping links from any layers .in this article , we have studied the effect of link overlap in the viability of multiplex networks , both analytically and numerically .distinctive role of overlapping links in viability and mutual connectivity is emphasized and exploited for setting up analytic framework properly .especially , it was shown that with link overlap the ca and the cd algorithms for the viability are described by different equations . as a result , a rich phase diagram for viability is obtained and the hysteretic patterns of viability become greatly diversified in the presence of link overlap . mutual percolation with link overlapis revisited as the limit of the cd case of multiplex viability problem , and controversy between existing results was clarified . for duplexrandom graphs , the transition is obtained to be discontinuous as long as at the transition , thus the tricritical point is absent .the distinctive role of overlapping links was demonstrated explicitly by the different responses of the network under random removals of overlapping and non - overlapping links , respectively , as well as under several removal strategies .our results show that the link overlap strongly facilitate viability and mutual percolation as the overlapping links form mutually - connected cluster by themselves . at the same time , by forming cluster the effect of link overlap can become long - ranged , posing challenge in analytical approach to the problem .the analytical approach for cd algorithm based on eqs .( 67 ) and eqs .( 1011 ) can be extended easily for general -layers in case of the complete link overlap , _i.e. _ , when the overlapping links span across all the layers .in case of the partial overlap , for instance when some links overlap across only two of three layers , the present approach can not be readily applied , although the numerical algorithm can .it is left an open problem to establish theory for the cd algorithm with general overlaps . the theory for ca algorithm , eqs .( 45 ) , however , would be extended to -layers straightforwardly by generalizing the degree vector to include the partially overlapping link degrees as well .speaking in clich , the link overlap is more than sum of its part links .the impact of link overlap on multiplex network dynamics is expected be seen widespread given its strong empirical evidences across diverse real - 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Abstract: many real - world complex systems are best modeled by multiplex networks . the multiplexity has proved to have broad impact on the system s structure and function . most theoretical studies on multiplex networks to date , however , have largely ignored the effect of link overlap across layers despite strong empirical evidences for its significance . in this article , we investigate the effect of link overlap in the viability of multiplex networks , both analytically and numerically . distinctive role of overlapping links in viability and mutual connectivity is emphasized and exploited for setting up proper analytic framework . a rich phase diagram for viability is obtained and greatly diversified patterns of hysteretic behavior in viability are observed in the presence of link overlap . mutual percolation with link overlap is revisited as a limit of multiplex viability problem , and controversy between existing results is clarified . the distinctive role of overlapping links is further demonstrated by the different responses of networks under random removals of overlapping and non - overlapping links , respectively , as well as under several removal strategies . our results show that the link overlap strongly facilitates viability and mutual percolation ; at the same time , the presence of link overlap poses challenge in analytical approach to the problem .   multiplex networks , link overlap , viability , mutual percolation , network robustness


since the development of the electronic computer in the 1940s , the serial processing computational paradigm has successfully held sway .it has developed to the point where it is now ubiquitous .however , there are many tasks which are yet to be successfully tackled computationally .a case in point is the multifarious activities that the human brain performs regularly , including pattern recognition , associative recall , etc . which is extremely difficult , if not impossible to do using traditional computation .this problem has led to the development of non - standard techniques to tackle situations at which biological information processing systems excel .one of the more successful of such developments aims at `` reverse - engineering '' the biological apparatus itself to find out why and how it works .the field of neural network models has grown up on the premise that the massively parallel distributed processing and connectionist structure observed in the brain is the key behind its superior performance . by implementing these features in the design of a new class of architectures and algorithms , it is hoped that machines will approach human - like ability in handling real - world situations .extremely simplified models of neurons connected in a network via suitable connection weights were known to implement various logical functions since the 1940s .the subject received fresh impetus a decade and half ago due to some breakthroughs .these were however restricted to networks subject to convergent dynamics .such systems tend to a time - independent solution ( a `` fixed - point '' attractor ) after starting off from some initial condition .on the other hand , the brain never settles down to a steady state but appears to exhibit a rich variety of non - periodic behavior .the development of nonlinear dynamical systems theory - in particular , the discovery of `` deterministic chaos '' in extremely simple systems - has furnished the theoretical tools necessary for analyzing non - convergent network dynamics .several neurobiological studies have in fact indicated the presence of chaotic dynamics in the brain and its possible role in biological information processing .thus , the ability to design networks with aperiodic behavior promises to add a new dimension to our understanding of how the brain works .several efforts in designing and applying chaotic neural networks have already appeared - .( for further references see ) . in the present work ,a particularly simple model whose state is updated after discrete time intervals is studied .the piecewise linear nature of the model neuron used , not only makes detailed theoretical analysis possible , but also enables an intuitive understanding of the dynamics , at least for a small number of connected elements .this makes it easier to extrapolate to larger networks and suggest possible applications .the proposed model is also particularly suitable for hardware implementation using operational amplifiers ( owing to their piecewise linear characteristics ) .the rest of the paper is organized as follows .the basic features of the piecewise linear neural model used is described in section 2 , along with the biological motivation for such a model .the next section is devoted to analyzing the dynamics of a pair of excitatory and inhibitory neurons , with self- and inter - connections .this simple system shows a wide range of behavior including periodic oscillations , chaos and border - collision bifurcations .section 4 extends the model to larger networks under certain restrictive conditions .this is followed by a discussion of the possible application of the model to various information processing tasks , such as associative memory and nonlinear function approximation .the rich dynamics of the system allows it to respond to specific inputs with periodic or aperiodic responses ( in contrast to a time - independent constant output , as in convergent networks ) and also to act as a _ central pattern generator_. we conclude with a short discussion on possible directions for further studies .let denote the activation state of a model neuron at the time interval . if , the neuron is considered to be active ( firing ) , and if , it is quiescent .then , if is the input to the neuron at the instant , the discrete - time neural dynamics is described by the equation assuming there to be no effects of delay .the input is the weighted sum of the activation states at the -th instant of all other neurons connected to the neuron under consideration , together with external stimulus ( if any ) .the form of is decided by the input - output behavior of the neuron .usually , it is taken to be the _ heaviside step function _ , i.e. , where is known as the _ threshold_.  if the _mean firing rate _, i.e. , the activation state averaged over a time interval , is taken as the dynamical variable , then a continuous state space is available to the system . if be the mean firing rate at the time interval , then here , is known as the _ activation function _ and is the parameter associated with it .the first term of the argument represents the weighted sum of inputs from all neurons connected with the one under study . is the weightage for the connection to the neuron . represents the external stimulus at the instant .considering the detailed biology , there are two transforms occurring at the threshold element . at the input end , the impulse frequency coded information is transformed into the amplitude modulation of the neural current . for single neurons , this pulse - wave transfer function is linear over a small region , with nonlinear saturation at both extremities . at the output end , the current amplitude is converted back to impulse frequency .the wave - pulse transfer - function for single neurons is zero below a threshold , then rises linearly upto a maximum value . beyond this maximum, the output falls to zero due to `` cathodal block '' .these relations are time - dependent .for example , the slope of the wave - pulse transfer function decreases with time when subjected to sustained activation ( `` adaptation '' ) .  the net transformation of a input by a neuron is therefore given by the combined action of the two transfer - functions .let us approximate the nonlinear pulse - wave transfer function with a piecewise linear function , such that the wave - pulse transfer function is represented as where are the slopes of respectively , is the inhibitory saturation value and represents a threshold value .it is easily seen that the combined effect of the two gives rise to the resultant transfer function , , defined as in the present work we will assume that .this condition ensures that the operating region of the neuron does not go into the `` cathodal block '' zone .this allows us to work with the following simplified neural activation function ( upon rescaling ) throughout the rest of the paper : where is called the _gain parameter _ of the function ( fig .1 ) . for infinite gain ( ), the activation function reverts to the hard - limiting heaviside step function .note that , if neural populations had been considered , instead of single neurons , then sigmoidal activation functions , e.g. , would have been the appropriate choice . by varying , transfer functions with different slopes would be obtained .although , in the present study , the gain parameter , , of the transfer function is considered constant , in general it will be a time - varying function of the activation state , decreasing under constant external stimulation until the neuron goes into a quiescent state .the threshold is also a dynamic parameter , changing as a result of external stimulation .we have also assumed that the neuron state at the instant is a function of the state value at the previous instant only . introducing delay effects into the model , such that , might lead to novel behavior .having established the response properties of single neurons , we can now study the dynamics when they are connected .it is observed that , even connecting only an excitatory and an inhibitory neuron with each other leads to a rich variety of behavior , including high period oscillations and chaos .the continuous - time dynamics of pairwise connected excitatory - inhibitory neural populations ( with sigmoidal nonlinearity ) have been studied before . however , in the present case , the resultant system is updated in discrete - time intervals and the dynamics is governed by piecewise linear mappings .this makes chaotic behavior possible in the proposed neural network model .chaotic activity has been previously observed in piecewise linear systems , for both continuous - time as well as discrete - time evolution of the system .if and be the mean firing rates of the excitatory and inhibitory neurons , respectively , then their time evolution is given by the coupled difference equations :   the network connections are shown in fig .2 . without loss of generality ,the connection weightages and can be absorbed into the gain parameters and and the correspondingly rescaled remaining connection weightages , and , are labeled and respectively . for convenience , a transformed set of variables , and , is used .the dynamics is now given by  note that , if , the two - dimensional dynamics is reduced to an effective one - dimensional one , simplifying the analysis .we will now examine the cases : , , and , in detail .unless mentioned otherwise , the threshold will be taken as 0 ( a non - zero value of introduces an affine transformation to ) .this represents the condition when the connection weights and , .the dynamics is that of an _ asymmetric tent map _3(a ) ) :   the fixed points of this system are , and . is stable for , whereas exists only when , and is stable for . beyond this, chaotic behavior is observed unless the maximum output value , i.e. , , iterates to the region where .the parameter space diagram is shown in fig .4 .  along the line , we get the symmetric tent map scenario .so the lyapunov exponent along this curve grows as for .this is one of the two special cases where an analytical expression for can be obtained .the other instance is when the map s invariant probability distribution , .this occurs when along the curve defined by the above relation , the lyapunov exponent evolves with the parameter according to   in general , has to be obtained computationally .5 shows plotted against for , when the map is in the chaotic region . a sharp drop to zerois observed in both the terminal points , indicating sharp transition between chaotic and fixed - point behavior at and 0.75 . at , the entire interval ] is divided into a chaotic region of measure zero , defined on a non - uniform cantor set ( in general ) and an `` escape set '' which maps to .this is because , for , f(z)=0 .any time an iterate of falls in this region , in the next iterate the trajectory will converge to .the points left invariant after one iteration , will be in the two intervals  ] . the phase space is thus fragmented into two invariant regions .after iterations , there will be fragments of the chaotic invariant set , with ( ) intervals of length .the fragmentation of the phase space , therefore , has a _ multifractal _ nature .the presence of multiple length scales is due to the fact that the slope magnitude of the map is not constant throughout the interval $ ] .it is to be noted that , even for not belonging to the fractal invariant set , the trajectory might show long _ chaotic transients _ until at some iterate it maps to . for ,the map has a constant slope .as a result , the cantor set is uniform , having exact geometrical self - similarity and a fractal dimension , .so , the phase space of the coupled system has a fractal structure in this parameter region , i.e. , where .6 shows the bifurcation structure of the map for . for ,the fixed point is stable . at becomes unstable , leading to bands of chaotic behavior .the chaotic bands collide with the unstable fixed point at and merge into a single chaotic band .this band - merging transition is an example of _ crisis _ and has been studied in detail for the symmetric tent map .the -value at which the band - merging occurs for a given value of , can be obtained analytically by solving the quartic equation : for , all the roots are complex , implying that band - merging does not occur over this range of -values .uniform chaotic behavior occurs at ( the entire interval [ 0,1/b ] is uniformly visited by the chaotic trajectory ) .the chaotic band collides with the unstable again at .this boundary crisis destroys chaos and stabilizes the fixed point .this represents the condition when the connection weightages are such that , , ( ) .the dynamics is given by the following map ( fig .3(b ) ) key difference with the earlier case is that now the dynamics supports superstable period- orbits ( ) .this is a result of the existence of a region of zero slope ( giving a non - zero output .there are two fixed points of the map , ( as before ) , and ,  , if it exists , is superstable , as the local slope is zero . on the other hand, is stable , only if .if the fixed points are unstable , but iterates of fall in the region , superstable periodic cycles occur .the fixed point , , becomes stable when .chaotic behavior occurs if none of the fixed points are stable , and no iterate of falls in the region .the vs. parameter space diagram in fig . 7( for ) shows the different dynamical regimes that are observed .  the bifurcation diagram for ( fig .8) shows how the dynamics changes with . for , ,is the stable fixed point . at , becomes unstable , giving rise to a superstable period-2 cycle .a periodic regime is now observed , which was absent in the previous case .the periodic orbits initially follow a _ period - doubling _ sequence until a period-32 ( ) orbit gives rise to a period-48 ( ) one .this occurs as a result of border - collision bifurcations by which `` period-2 to period-3 '' bifurcations have been seen to occur . in the above instance ,each of the sixteen period-2 orbits give rise to a period-3 orbit .the structure of the superstable periodic orbits is quite complex .the length of the cycles is plotted against in fig .the remarkable self - similar structure of the intervals is to be noted .numerical studies indicate that cycles of all periods exist having the following ordering : between any superstable period- and period- cycle , there exists an interval of for which a period- orbit is superstable . at periodic orbits become unstable , leading to onset of chaos .the chaotic behavior persists till , when becomes stable .let us now study the effect of introducing a _threshold_. a positive threshold , , applied to both and , has the effect of shifting the map to the right by .this allows the segmentation of activation state -space , according to dynamical behavior . for initial conditions lying in the region bounded by the two straight lines , and , the trajectories are chaotic ,provided the maximum point of the map , , does not iterate into the region . for the region , , any iterate will map to the fixed point , .initial conditions from will map to the chaotic region , if the maximum point of the map does not iterate into . otherwise , a fractal set of initial conditions will give rise to bounded chaotic motion , the remaining region falling in the `` escape set '' , eventually leading to periodic orbits .  a negative threshold shifts the map to the left .this has the consequence of destabilizing the fixed point .if the other fixed point is also unstable then this will imply that all possible values will give rise to periodic or chaotic trajectories .this indicates the possible existence of a global chaotic attractor .this corresponds to the condition when all the connection weights are different .the dynamics is irreducible to 1-dimension .we need to consider only the positive region , as otherwise , is the stable fixed point . in the non - zero region ,different dynamical behavior may occur depending on the region where the fixed point occurs and on its stability .one of the fixed points is , whose stability is determined by obtaining the eigenvalues of the corresponding jacobian , evaluating the above matrix , gives the following condition ^{1/2 } < 2,\ ] ] for stability of the fixed point .the other fixed point may occur in any one of the four following regions of the -space :  * region i * : , .+ is the only fixed point .* region ii * : , . +the fixed point is , which is stable if .  * region iii * : , . +the fixed point is , which is stable if .* region iv * : , .+ the fixed point is .this is a superstable root , as the local slope is zero under all conditions .the abundance of tunable parameters in this case makes detailed simulation study extremely difficult .however , some preliminary studies in the parameter space ( keeping the other parameters fixed ) gives indication of dynamics similar to that seen in cases and .the parameter space is shown in fig .[ fuinfig10 ] for , .a variety of dynamical behaviors is observed - from fixed points to periodic cycles to chaos , as indicated by the different regions .in addition , there are regions exhibiting periodic behavior which have fractal intervals of chaotic activity embedded within them .in the previous section , the behavior of a pair of excitatory - inhibitory neurons ( number of neurons , ) was shown to have sufficient complexity .the dynamics of a n - neuron network ( ) described by where is the set of activation state values ( both excitatory and inhibitory neurons ) , and is the matrix of connection weights between different neurons. the full range of behavior shown by such a system will be impossible to study in detail , as the number of available tunable parameters are too large to handle .however , under certain restrictions , the dynamics of such large networks can be inferred .let denote the connection weight from to the neuron .then , under the condition the n - neuron network dynamics is reducible to that of a 1-dimensional map with linear segments ( for ) .the occurrence of `` folds '' in a map have already been shown to be responsible for creation and persistence of localized coherent structures within a chaotic flow . as in this case, the resultant map will have a number of such folds , the system might show coexistence of multiple chaotic attractors ( isolated from each other ) . a simple example to illustratethis point is a fully connected network of four neurons : two excitatory ( , ) and two inhibitory ( , ) .let and represent the slope of the transfer functions for the excitatory and inhibitory neurons , respectively .the 4-dimensional dynamics is reducible to the 1-dimensional dynamics of .simulations were carried out for the set of parameter values : and .furthermore , , have a threshold equal to .11(a ) shows the return map and time evolution of in the absence of any bias .there is only a single global chaotic attractor in this case . when a small negative bias is applied to the whole network , the previous attractor splits into two coexisting isolated attractors having localized chaotic activity .which attractor the system will be in , depends upon the initial value it starts from .11(b ) shows the return map for a bias value of -0.15 and the superposed time evolutions of starting from initial conditions belonging to two different attractors .so , an increase in bias , can cause transition from global chaos to localized chaotic regions .this property can be used to simulate a proposed mechanism of olfactory information processing .it has been suggested that the olfactory system maintains a global attractor with multiple `` wings '' , each corresponding to a specific class of odorant . during each inhalation ,the system moves from the central chaotic repeller to one of the wings , if the input contains a known stimulus .the continual shift from one wing to another via the central repelling zone has been termed as chaotic `` itinerancy '' .this forms the basis of several chaotic associative memory models .the above picture can be observed in the present model by noting that , if the external input has the effect of momentarily increasing the bias from a negative value to zero , then the isolated chaotic regions merge together into a single global attractor . in this condition ,the entire region is accessible to any input state .however , as the bias goes back to a small negative value , the different isolated chaotic attractors re - emerge , and the system dynamics is constrained into one of these .sustained external stimuli will cause the gain parameters to decrease ( adaptation ) , thereby decreasing the local slope of the map .if the stimulus is maintained , the unstable fixed point in the isolated region will become stable leading to a fixed - point or periodic behavior .the above scenario , in fact , is the basis of using the proposed model as an associative memory network .chaotic dynamics enables the microscopic sensory input received by the brain to control the macroscopic activity that constitutes its output .this occurs as a result of the selective sensitivity of chaotic systems to small fluctuations in the environment and their capacity for rapid state transitions . on the other hand ,chaotic attractors are globally extremely robust .these properties indicate that the utilization of chaos by biological systems for information processing can indeed be advantageous .it has been suggested , based on investigations into cellular automata , that complex computational capabilities emerge at the `` edge of chaos '' .  based on this notion ,efforts are on to use chaos in neural network models to achieve human - like information processing capabilities .chaotic neural networks have been already been applied in designing associative memory networks and solving combinatorial optimization problems , using chaos to carry out an effective stochastic search .the superposition of chaotic maps for information processing has also been suggested before .the model presented in this paper can be used for a variety of purposes , classified as follows :  * associative memory * : a set of patterns ( i.e. , specific network state configurations ) are stored in the network as attractors of the system dynamics , such that , whenever a distorted version of one of the patterns is presented to the network as input , the original is retrieved upon iteration .the distortion has to be small enough so that the input pattern is not outside the basin of attraction of the desired attractor . in networks using convergent dynamics, the stored patterns necessarily have to be time - invariant or at most , periodic .chaos provides rapid and unbiased access to all attractors , any of which may be selected on presentation of a stimulus , depending upon the network state and external environment .it also acts as a `` novelty detector '' , classifying a stimulus as being previously unknown , by not converging to any of the existing attractors .this suggests the use of chaotic networks for _ auto association_.  in the previous section , the basic mechanism for constructing an associative memory network has been described . in this proposed model, both constant and periodic sequences can be stored .this is made possible by introducing `` folds '' in the return map of the network , so that a large number of isolated regions are produced .the nature of the dynamics in a region can be controlled by altering the gain parameters of individual neurons .accessibility to a given attractor depends upon the initial condition of the network and the input stimulus .so , regions with fixed - point or periodic attractors may be embedded within regions having chaotic behavior .in addition , chaotic trajectories confined within a specific region can also be generated when presented with a short - duration input stimulus belonging to that region .`` novelty detection '' is implemented in the above model by making the basins of attractors ( corresponding to the stored patterns ) of some pre - specified size .input belonging outside the region , therefore , can not enter the basin and will not be able to converge to the stored pattern .  * pattern classification * :in this information processing task , different input sets need to be classified into a fixed number of categories .decision boundaries , i.e. , boundaries between the different classes are constructed by a `` training session '' where the network is presented with a series of inputs and the corresponding class to which they belong . in the proposed model, classification can be on the basis of dynamical behavior .for example , input sets belonging to different input classes may give rise to different periodic sequences .otherwise , the distinction can be made between categories of inputs which give rise to chaotic and non - chaotic trajectories . for a pair of neurons ( ) , under the condition , linear separation of the -spacecan be done ( as shown above ) . by varying the parameters and , the orientation and size of the class regionscan be controlled .if and , nonlinear decision boundaries between different classes can be generated . by using suitably adjusted weights ,any arbitrary classification can be achieved .an example of nonlinear decision boundary generation is shown in fig .the network used for this purpose consists of 4 neurons - 2 excitatory ( ) and 2 inhibitory ( ) ( fig .the gain parameters are and .all the neurons have a threshold , .the network is fully connected with all weightages equal to unity .the input stimulus is taken to be the initial value of the excitatory neurons and the inhibitory neuronal states are initially taken to be zero .  as shown in fig .12(b ) , for = 0.25 , the phase space is segmented into basins leading to either fixed point or chaotic attractors . by increasing ,the width of the chaotic band can be reduced .changing the initial value of the inhibitory neurons will cause translation of the band and manipulating the connection weights gives a rotation to the band .thus , any general transformation can be applied to the segment .more complex network connections might permit segmenting isolated box - like regions in the phase space .this possibility is currently under investigation .  * system dynamics approximation * : a system may be described by a nonlinear input - output relation , where the mapping function , , is unknown . by having access to a limited set of input - output pairs ,the function has to be approximated - in effect , building a system simulator . in the present model, a sufficient number of coupled neurons can be used to construct any arbitrary piecewise linear input - output relation . by use of a suitable learning rule, the available data set can be used to determine the gain parameters , thresholds and connection weights of the network .a close approximation of the system dynamics will enable prediction and control of its behavior .the approximation s accuracy is not restricted to systems with piecewise linear functions - but can also give good qualitative reconstruction of smooth nonlinear systems .* periodic sequence generation * : capability for periodic sequence generation can be exploited for modeling _central pattern generators_. these are a class of biological neural ensembles which control well - defined rhythmic muscle movements such as swimming , running , walking , breathing , etc .usually they are found in the spinal cord , producing periodic sequences without feedback from the motor system or higher - level control .the ability to generate multiple sequences from the same neural assembly is another interesting feature .postulating the existence of single pacemaker neurons acting as the ` system clock ' to initiate periodic activity can not explain all the observed phenomena .the existing network models for simulating this behavior mostly suffer from the drawback that they can not generate multiple non - overlapping sequences .this shortcoming is overcome in the model presented here . for and , a rich variety of periodic sequencescan be chosen from the same network , simply by altering the gain parameters by a very small amount . as mentioned above ,numerical investigations indicate that cycles of any period can be generated by suitably altering the value of .the network has been studied in detail only for .nonetheless it shows capability for supporting extremely complex behavior . under certain restrictions ,the dynamics for networks can also be understood .relaxation of these restrictions will provide a challenging task for the future .one important point not touched here is the issue of _ learning_. the connection weights \{ } have been assumed constant , as they change at a much slower time scale compared to that of the activation state values . however , modification of the weights due to learning will also cause changes in the dynamics . such bifurcation behavior , induced by weight changes , will have to be taken into account when devising learning rules for specific purposes . the interaction of chaotic activation dynamics at a fast time scale and learning dynamics on a slower time scale might yield richer behavior than that seen in the present model .an interesting study for the future will be to introduce time - varying hebbian synaptic weights , evolving according to where and denote the neuron state and connection weight at the instant , and is related to the time - scale of the synaptic dynamics .the combined effect of such synaptic dynamics with the chaotic network dynamics ( for small ) described above might lead to significant departure from the overall features described here .real biological systems reside in an extremely noisy environment .this is incorporated into neural models by using stochastic updating rule and/or explicit introduction of a term representing external noise .we plan to introduce similar features in the proposed model . in dissipative chaotic systems ,the effect of external noise seems to be limited to destroying the fine structure of the bifurcation sequence .the interaction of deterministic chaos and stochastic noise in the network will be interesting to study .another possible modification to the present model will be to incorporate time - varying operating conditions .in time - dependence of a suitable system parameter was shown to give rise to interesting dynamical behaviors , e.g. , transition between periodic oscillations and chaos .this suggests that varying the environment can facilitate memory retrieval if dynamic states are used for storing information in a neural network . in the proposed network, periodic variations can be incorporated into the available system parameters : gain , threshold and connection weights . in nonlinear systems ,time - varying dynamics often leads to the phenomenon of `` stochastic resonance '' where a subthreshold signal is amplified by the background noise .it has been observed that replacing noise by chaos also gives rise to this resonance behavior in a bimodal piecewise linear map .it will be of interest to see whether such phenomena can occur in the present model , on the introduction of temporal variation in the parameters .the occurrence of `` resonance '' might enhance the capability of the network to detect subthreshold signals , through signal amplification by deterministic ` noise ' .

Abstract: the paper examines the discrete - time dynamics of neuron models ( of excitatory and inhibitory types ) with piecewise linear activation functions , which are connected in a network . the properties of a pair of neurons ( one excitatory and the other inhibitory ) connected with each other , is studied in detail . even such a simple system shows a rich variety of behavior , including high - period oscillations and chaos . border - collision bifurcations and multifractal fragmentation of the phase space is also observed for a range of parameter values . extension of the model to a larger number of neurons is suggested under certain restrictive assumptions , which makes the resultant network dynamics effectively one - dimensional . possible applications of the network for information processing are outlined . these include using the network for auto - association , pattern classification , nonlinear function approximation and periodic sequence generation .   excitatory - inhibitory neural networks , chaos , nonlinear dynamics .