since the seminal work of chamberlin , the economics literature on monopolistically competitive markets has been rooted on the idea of product differentiation : goods are provided by many producers in several versions and models , despite they aim at satisfying the very same need .such a market framework can be easily experienced everyday in real - life markets , where several products compete with each other in offering very similar services to the final consumer .product differentiation has been studied from two perspectives : ( i ) _ horizontal _ , often referred to as spatial differentiation , and ( ii ) _ vertical _ .if a good is defined as a bundle of characteristics ( in the sense of lancaster ) , the distinction between horizontal and vertical differentiation is that the former refers to goods with different features and the same qualitative level , whereas the latter takes into account differences in quality .however , as cremer and thisse have shown , results of hotelling - type models and vertical differentiation models are basically equivalent . in this paperwe assume that the differentiation has a comprehensive characterization : in fact , different features and qualitative levels will be described in a unique multidimensional environment .consumer s behavioral choice is studied in the literature from several , yet complementary , points of view . from a buyer s perspective , differentiation among goodsis advantageous insofar as it helps in seeking a feasible good offering the highest satisfaction . in this sense ,an increase in differentiation is usually associated to a positive value for buyers , because it increases the possibility of a making a selection that is close to an ideal target .monopolistic competition and oligopoly characterize the largest part of the real market experience .these two market structures share many common characteristics ; however , they differ from each other in the way the total market output is looked at .in fact , oligopolistic firms explicitly consider the effect of their role in determining the total market output , whereas monopolistically competitive firms consider the aggregate market output as exogenously given in the process of setting their individual production , .we focus our attention on the demand side of a _ monopolistically competitive market _ , where firms supply composition is taken as an exogenous environmental configuration .our goal is to study a consumer s individual choice process , that is , to analyze the way in which she selects a final product among several version of it supplied by different brands . to this aim, we make the ( non - simplifying ) assumption that the consumer considers quality , features , and prices as a unified multi - criteria discrimination concept .market failure due to asymmetric information occurs when the buyer and the seller have two different informative sets and the more aware of them tries to extract a supplementary benefit at the expenses of the other .indeed , the case of a consumer who does not know the complete characterization of the goods she needs must be considered : that consumer chooses blinded by her ignorance .the result is a lower level of satisfaction than the potential one .this may require a policy intervention to solve the resulting inefficiency . to further support this point of view ,consider the perverse effect of differentiation : the tyranny of choice " a term coined by schwartz  represents the situation when the abundance of available choices on the market may even become undesirable if the consumer can not count on a valid consciousness to guide her selection . in this respect ,some authors stress the role of advertisement in informing consumers .however , adverts may barely provide a neutral source of knowledge .thus , the model here proposed will show effects of advertising .the relevance of _ information _ and _ knowledge _ in consumption is widely recognized , because of their ability to influence the consumer s attitude in perception and searching .indeed , some analysis supports the point of view that a lack of knowledge may truly generate a reduction of the consumer s judgement capacity . in our opinionthis conclusion can not be fully ascribed to the well - known bounded rationality " concept , which appears more linked to the completeness of the informative set , as advocated in and in . in this paperwe explicitly emphasize the semantic difference between the two concepts of information " and knowledge " .our interpretation of these two terms relies on a suggested distinction between ( i ) what the consumer knows with regards to the current transaction being concluded ( which should be considered as `` information '' , as generally accepted in literature ) , and ( ii ) what the consumer knows in a broader sense , in the form of cultural background and capability to understand and to evaluate ( which is a notion of `` knowledge '' ) . the interested reader may refer to existing surveys .a possible rationale for the reduction of the consumer s judgement capacity can be ascribed to the complexity of choice environment .in fact , it is known that consumers decide to make purchases for many reasons , which range from very basic needs to utmost volatile instincts . as a consequence , even their perceived desires / needs and the way they are expressed respond to a very wide set of stimuli , as documented in an ample literature . also , there is evidence that choices made in consumption may affect the development dynamics of the economy .we approach the problem of individual consumer s choice by describing the different phases of the informative process that leads the buyer to her final selection . in this sense , we assume that the consumer is able to build an `` experiential cognitive map '' , where all the variables are taken into account in a spatial sense .our approach basically belongs to the well - known framework of consumer decision - making research , which considers consumption as a process that generally involves the following phases : ( 1 ) problem recognition ( i.e. , market selection according to the need being satiated ) ; ( 2 ) information search ; ( 3 ) evaluation of alternatives ; ( 4 ) final choice ( i.e. , purchase ) ; ( 5 ) post - purchase evaluation .we assume that the consumer has chosen the market where to enter , and focus on phases ( 2 ) , ( 3 ) , and ( 4 ) .we will neglect phase ( 5 ) .  in order to build a spatial configuration , here we use a hotelling - type approach , to qualify the satisfaction in terms of the distance from the perfect choice " .our model informally follows a multi - attribute approach : we assume that the consumer s exploration will reveal her behavioral attitude in recognizing and appreciating desired characteristics .thus , the set of available options is analyzed , and , after comparisons , the most satisfying alternative is chosen .market complexity affects many aspects of the decision process .in fact , motivational elements and topological configurations of attributes space unavoidably link the maximizing choice to the effort put in the search .this is the reason why our model is based on two distinct features : ( i ) the consumer s ability in discriminating similar goods , and ( ii ) her knowledge of the market structure .furthermore , a very strong influence on present choices and decisions is exerted by similar previous experiences . as a consequence, we assume that a purchase decision passes through an analysis that takes into account every cognitive element of the consumer s activity , either present or stored in memory from past experiences .the theoretical context employed here allows individuals to adaptively select strategies in order to manage the actual situation at their best  .  in this paper, we investigate the existence of some safety guidelines " to help consumers with different degrees of information to make their choices .inspired by previous studies on the beneficial role of randomness in socio - economic systems , we actually test the effectiveness of several strategies to reach a final decision , ranging from the hypothetical scenario of perfect information , to a completely random walk .as it was found for financial markets , for career advancements in managerial organizations , and for efficiency of political institutions , we anticipate that also in this case randomness gives positive results .the paper is organized as follows .section 2 presents the model , in its static and dynamic features . in section 3we run various types of simulations , and present numerical results .conclusions and future directions of research are addressed in section 4 .in this section we describe our model .we first deal with its static features , which are related to the exogenous market structure ( [ subsect : model_for_market ] ) , the endogenous consumer s characteristics ( [ subsect : model_for_consumer ] ) , and the interaction between market and consumer ( [ subsect : gravitational field ] ) .successively , we describe the dynamical procedure employed by the consumer to explore the market , gathering information in an attempt to reach her target ( [ subsect : dynamics_of_model ] ) .   the topological structure of the market is represented by a graph with three types of nodes , identified by different shapes , see figure 1(a ) .formally , the graph has the theoretic structure of a _forest _ , that is , a free union of _ trees _ ( connected acyclic graphs ) . is a pair , where is a nonempty set of _ nodes _ , and is a ( possibly empty ) set of ( undirected ) _ edges _ , with an edge being a subset of having size two .two distinct nodes are _ adjacent _ if is an edge in .a _ path _ in a graph is a sequence of adjacent nodes , in the sense that the set is an edge in for each .a graph is _ connected _ whenever any two distinct nodes are joined by a path , and is _ acyclic _ if there are no paths beginning and ending at the same node . for further details ,refer to . ]each tree is star - shaped , and represents a `` cluster '' : for instance , the graph in figure 1(a ) has three clusters .the meaning of the three types of nodes is described below .these nodes are located at the center of the trees composing the forest , and are denoted by pentagons .they are the _ hubs _ ( i.e. , highly connected nodes ) of the graph , and may represent either a `` brand '' or a `` category '' of products .for example , in the market of cameras , we can look at each central node as a specific producer ( canon , minolta , nikon , etc . ) or as a type of camera ( bridge , mirrorless , reflex , etc . ) .these are the _ leaves _ of the forest ( i.e. , the nodes with degree 1 ) , and are denoted by circles . is the number of nodes that are adjacent to .the nodes adjacent to are also called the _ first neighbors _ of . ]the terminal nodes represent all final products that are sold on the specific market at hand .for instance , in dealing with the market of cellular phones , a terminal node may be a 4-tuple of the type , iphone6,32,silver .these are the nodes that belong to paths connecting the hub of a cluster to the terminal nodes of the same cluster , and are denoted by squares .intermediate nodes represent the distinct phases of a consumer s _ informative journey _ , intended as a cognitive walk that guides her from a brand / category to a final product present in the market . to illustrate the role of intermediate nodes , consider the following example in a computer market .a consumer decides to buy a final product represented by the -tuple  here the central node is the brand , the terminal node is a laptop with some specified features , and the intermediate nodes are all restrictions of the -tuple to the first components , apple , macbook air , , macbook air, , , macbook air,,i7 - 2.2 , and , macbook air,,i7 - 2.2,8 . ] with .thus , each product can be identified with an -tuple of features specified by the sequence of informative steps needed to reach it , starting from the central node ( which represents its first feature ) . in this way , a product actually sold in the marketis completely determined by the topology of the cluster ( brand / category ) to which it belongs .it is important to emphasize that in our approach the market structure is totally exogenous .therefore , at this initial stage , the distinct clusters constituting the market are not connected to each other , and live in a space that is _ not _ endowed with any metric structure .   once that both the number and the topological structure of the clusters are determined , the market has to be evaluated by consumers .in fact , each ( type of ) consumer has different features , preferences and perceptions , which constitute the `` lens '' through which the market is viewed by her .below we describe in detail how the consumer s point of view may affect the representation of the market , in terms of her : _ satisfaction _ , _ knowledge _ , and _discrimination ability_.  the presence of a personal point of view naturally induces a deformation of the graph structure , which however leaves unchanged the original topology of the market .in fact , the market assumes a shape that reflects the existence of an underlying _ satisfaction metric _ , defined in an appropriate space and intrinsic to each ( type of ) consumer : see figure 1(b ) , where , for the sake of graphical representation , we embed the graph into the two - dimensional space , with denoting the standard euclidean distance .  in order to better specify such an intrinsic satisfaction metric , we assume that each consumer has in mind an `` ideal goal '' , a hypothetical target product with several well - specified characteristics .this target , which may or may not exist in the market , occupies a specific position in the euclidean space , say , . in particular , if the target is a real product sold in the market , then it is located on a terminal node of the graph .a natural assumption of our model is that a consumer purchasing her ideal product will retrieve from it the maximum _ ex - post _ satisfaction , that is , . on the other hand , purchasing a product different from her target and corresponding to a terminal node will provide her with a non - perfect satisfaction , that is , in this case .note that the satisfaction retrieved from a non - ideal product shall naturally depend on the euclidean distance of the chosen product from the target .more generally , we define the satisfaction associated to any point by where is the maximum possible distance in the bounded part of the considered space ( which is a square , in our case ) . due to its generality , this definition of satisfaction works for any pair of nodes ( not necessarily terminal ) .it is important to note that in our model the distance between two terminal nodes is independent from the set of features of the corresponding products .for instance , the top blue node and the lower red node in figure 1(b ) which are very far from each other in the underlying metric space might well have very similar features ; nevertheless , it is possible that the satisfaction deriving from buying is very different from that deriving from buying , even just because they belong to different clusters . on the other hand, the three nodes of different colors placed at the center of the metric space ( , and ) might present very different features despite being very close to each other ( hence giving a similar satisfaction to the consumer ) .  in reality ,consumers knowledge of the exogenous market structure may vary quite a lot .we measure knowledge in terms of perception of the existence of clusters ; those clusters that are known to the consumer are labeled as _ active_. figure 1(b ) graphically describes a deformed topological structure of a simple market with three clusters , as viewed by a perfectly informed consumer , i.e. , a consumer with a maximum knowledge . in the represented case , the three clusters are all active , in fact their central nodes are connected to each other by red edges ( whereas the _ intra - clusters _ links are colored in yellow ) .note such a perfectly informed consumer could theoretically visit all the nodes of the graph during her informative journey , using both red and yellow edges .the situation is quite different for a consumer with an imperfect knowledge . in figure 2we describe a market with seven clusters , as it is viewed by a consumer with a knowledge : in this case only three of the seven clusters are active , hence the informative journey of this type of consumer will be limited to the three corresponding brands / categories . in figure 2we also represent the following additional features :  * a possible position of the consumer at the beginning of her journey over the graph , indicated by a red human shape located on a node belonging to the top - right ( active ) cluster . in what follows, we refer to this initial node as the _ source _ of her psychological journey .( for example , in the process of searching for a new cellular phone to buy , the consumer might start her exploration from the node of the graph that represents or is very close to her old phone . ) * a possible position of her target product , indicated by a shape with concentric red circles . in the represented case ,the target is located on a terminal node of the middle - right ( inactive ) cluster .( however , as already emphasized , the location of the target may well fail to coincide with a terminal node of a cluster , thus indicating that the consumer s ideal product is not present in the market . )  in the described situation , despite the target coincides with a terminal node of the graph , the consumer will never be able to reach it , due to her imperfect knowledge : in fact , the target belongs to an inactive cluster .therefore , the consumer s greatest ambition can only be to get as close as possible to her target , while remaining within the subset of the market that is reachable by his partial knowledge .  ) and a quite high number of indifference levels ( ) , represented by annuli with different shades of gray .the indifference areas are centered at the target , which is located on a node , and is denoted by concentric red circles .the random initial position of the buyer is located on a node of the top - right cluster , and is denoted by a human shape.,scaledwidth=50.0% ]  the last feature that influences the market evaluation by the consumer is her preference structure . in our model , we assume that the buyer has only a sort of `` fuzzy '' perception of the level of satisfaction associated to each point of the graph , until he actually reaches a node and physically explores it . specifically , we distinguish ( 1 ) an _ ex - ante _ perception " , which is the individual presumption of the satisfaction that a good can provide ; and ( 2 ) an _ ex - post _ evaluation " , which is the satisfaction that a good existing on the market actually provides . in this respect , the level of satisfaction associated to a given node of the graph represents what we call its `` ex - ante utility '' , which is just an estimate of its `` effective utility '' ( measurable only after the consumer physically explore the node ) .we model this fuzzy preference structure by a _total preorder _ on the set of nodes . is a binary relation on , which is _ reflexive _( for each ) , _ transitive _( and imply for each ) , and _ complete _ ( or for each such that ) . in this case , the _ indifference relation _ associated to , defined by if and , is an equivalence relation on . ]this is a classical hypothesis in individual preference theory .since the number of nodes is finite , the advantage of this framework is that it admits a _utility representation _ with `` thick '' indifference classes .on is _ representable _ if there exists an _order - embedding _ from into the set of real numbers , i.e. , a map such that for each , we have if and only if . in this case, the function is called a _utility representation _ of .it is well - known that a total preorder on a countable set ( hence , in particular , on a finite set ) is always representable : see , e.g. , for a general discussion about utility representations and some technical results . ]figure 2 provides a graphical description of such a consumer s preference structure . the space endowed with the satisfaction metric is partitioned into equivalence classes , called _ indifference levels _ and represented by concentric annuli centered at the target .all the nodes in an annulus are _ ex - ante _ equally preferred by the consumer , i.e. , they display the same _ presumed _ utility : is the total preorder on representing the preference structure of a consumer , is the indifference associated to , and is an utility representation of , then we have if and only if .so each annulus can be seen as a fuzzy multidimensional representation of indifference curves . ]the further from the target an annulus is , the lower the presumed utility associated to the corresponding indifference level is , and vice versa .clearly , a low value of is typical of a consumer with a rather low ex - ante discrimination power , whereas a large value of indicates a consumer with a high ability in discriminating among similar goods .  as a conclusive remark ,note that we do not assume _ a priori _ the existence of a relationship between knowledge and discrimination ability .therefore , we can identify several `` categories of buyers '' , which display various combinations of these two features . in the next subsectionwe illustrate how consumers interact with the market structure , and how this interaction along with some additional individual features determines different categories of consumers .each central node ( hub ) of the graph can be considered as a `` pole of attraction '' for the consumer . in the case that hubs represent brands , the strength of such attraction depends on many factors , such as marketing suggestions , variety of offered products , advertising and promotional campaigns , communications and brand - management policies , etc . to simplify our analysis , in this paperwe assume that the attractive power of a hub is only a function of the amount of products having that brand .in fact in analogy with the gravitational field of a body in physics we introduce the _ mass _ of a hub , which is by definition equal to the number of leaves of the corresponding cluster .note that the attraction effect generated by each hub becomes effective only if the consumer is aware of the presence of the corresponding cluster : said differently , `` knowledge determines attraction '' .  in figure 3we graphically represent the _ attraction field _ induced by the three active clusters of figure 2 ( the four inactive clusters have no effect whatsoever on the consumer , since she can not even perceive their existence ) .for the sake of visualization , only the central nodes are reported in the figure .the integer number next to each hub is its mass , which provides a discrete measure of the strength of directional attraction .the arrows of the attraction field may possibly push the consumer to choose in a certain direction .however , not every consumer will be affected in the same way by the field : in fact , as we shall argue later on , the more a consumer is well aware of her personal tastes / goals , the less she is influenced by market suggestions . in order to take into account this kind of behavior , next we introduce the last feature that characterizes a consumer in our model .this is the capability of the consumer to discern the features of a product without being subject to the influence of the market : the higher her awareness , the less likely that she makes her choice according to the market s attraction field .in fact , a highly aware consumer will always take advantage of both her knowledge and her ex - ante discriminating power to effectively explore the market while moving towards the target . in our modelwe assume that awareness is a parameter that varies in the closed interval $ ] .knowledge ( ) , discriminating ability ( ) and awareness ( ) are the three parameters that allow us to distinguish among several categories of consumers , characterized by suitable combinations of these features . in table we list a taxonomy of cases , in which the extreme values of the parameters better emphasize the differences among different types of buyers .in particular , consumers from type # 1 to type # 5 ( with ) display an imperfect knowledge of the market , while consumers from type # 6 to type # 10 ( with ) know the market perfectly . within each of these two general categories, we have the extreme cases of consumers with either minimum ( ) or maximum ( ) awareness , as well as a special type of consumer who as we shall describe later on walks along her informative path completely _ at random_. finally , for non - random cases , we consider consumers with low and high discriminating ability ( respectively , and ) .these ten combinations of the main individual parameters describe a fictitious community made by several distinct ( non - interacting ) consumers , who aim at maximizing their own utility according to their subjective awareness , discrimination capacity and knowledge . as we explain in detail in the next section, each type of consumer will navigate the graph in search of her ( existing or non - existing ) target , following step by step ( i.e. , node by node ) a personal informative path starting at the source ( where she is located at time ) .._ten types of consumers characterized by different values of individual parameters . _only rather extreme values of knowledge ( ) , awareness ( ) and discrimination ability ( ) are used . the special type of random consumer also appears . [ cols="^,^,^,^",options="header " , ]   in particular ,comparing the performances of consumer , which is naturally the best type in both the tables , one notice that in table 4 the efficiency drops from down to , while the utility does not reach ( against a of table 2 ) ; at the same time , the social welfare passes from to , and the average stopping time increases from to time steps . on the other hand , comparing the random consumer in the two tables , one can see that the collapse of performance is less dramatic : this means that the random strategy confirms its superiority with respect to the completely uninformed one ( visible by comparing consumers and in table 4 ) , and seems also to be unaffected by the position of the target ( on or off product ) .  in figure 11 ,the behavior of and ( for buyers with and ) is reported as a function of consumer awareness , as already done in figure 8 .compared with the latter , it immediately appears that the final utility trend shown in figure 11(b ) remains almost unchanged ( it rapidly saturates to a maximum value that is just slightly lower than in the target - on - product case ) , but the shape of the efficiency behavior looks now very different : indeed , due to the impossibility to reach the target , the efficiency score never exceeds , staying also below the one of the random consumer for any value of awareness .thus , considering both efficiency and utility , the random consumer s performance results again quite effective , not only with respect to the totally unaware consumers , but also with respect those with : again , this is particularly relevant in view of the fact that gathering information has a cost , and so randomness appears even more convenient than what one can derive from the results of figure 11(b ) .  finally , let us to focus the attention on a strange effect visible in both tables 2 and 4 .some reader may have noticed that , when consumers with can not reach the target ( because either or the target is off product ) , the percentage of events with efficiency greater than seems to _ decrease _ when the discriminating ability of consumers increases ( from to ) , whereas ( as expected ) the final utility increases .this happens , in particular , for consumer types # 3 and # 4 in table 2 , and types # 3 , # 4 , # 8 and # 9 in table 4 .although counterintuitive at a first sight , such a behavior has a natural explanation if one deeper analyzes how the efficiency of a consumer is affected by the presence of a small number of indifference levels around the target ( which is a symptom of a low discriminating ability ) .   of table 4 )remains trapped in a cul - de - sac , yet her efficiency stays quite high.,scaledwidth=45.0% ]  in figure 12 we show an example of informative journey for a consumer with , and , in the target - off - product case ( that is , we are considering consumer # 8 of table 4 ) . due to her low discriminating ability , the buyer considers all the nodes within the first indifference level ( colored in black ) as equivalent : therefore , guided only by the degree of nodes , she wanders among the three hubs , and , with only occasional raids down some branches of the respective clusters , until she finally ends up in the branch where is depicted in the figure .as one can see , she has reached ( at , say , time ) the product placed at the minimum possible distance from the target , and so her satisfaction and , in turn , her utility jumps to its ( relative ) maximum value ( see equations ( 1 ) and ( 2 ) ) .however , her journey is bound to end soon : in fact , due to the high number of previous visits , hub has been switched - off , hence the consumer s fate is to remain trapped in a cul - de - sac . in particular , she will stop , after four more steps , on node at time , thus producing an efficiency score very close to ( see equation ( 3 ) ) ; on the other hand , her utility score may be quite lower than .translated into reality , this effect captures the behavior of an undecided ( and possibly well informed ) consumer who , due to her scarce discriminating ability , oscillates for a while among different brands , and suddenly decides to buy a certain product , even if it does not perfectly matches her target .in this paper we have presented a graph - based model of consumer choice , which describes the hypothetical cognitive journey that each individual experiences in the process of buying a product .the role of the causes that influence the decision is measured by means of behavioral differentiation in several parametric simulations .our results explicitly show the relevance of information and knowledge , in the form of individual awareness , discriminating ability , and perception of market structure .  we have focused our attention on some prototypical categories of consumers , taking in account their subjectively distorted visions of the market , as it seems to happen in everyday consumption experiences .many of our results confirm what one would naturally expect : for example , a perfect knowledge of the market structure paired with a high discrimination ability and a good individual awareness usually determines a very satisfactory choice . on the other hand ,a few results of our simulations look rather surprising and maybe intriguing in terms of individual satisfaction , efficient strategies , and decision procedures , and therefore call for a deeper analysis of their explanation and consequences .first of all , our model shows that consumers provided with a minimal level of knowledge and information may unexpectedly reach very high levels of utility .this appears to be in sharp contrast with the classical paradigm that `` perfect information is mandatory to obtain optimal results '' . however , considered that actual markets are far from being characterized by perfect information , our results provide some justification for more realistic approaches , which do not rely on perfect information as an unquestionable tenet of optimality .second , the results of our simulations consistently suggest that whenever consumers fail to have a minimal level of knowledge and information , random decisions will make them better off .this translates into a new type of behavioral strategy , which may operate as a sort of protective shield for the unaware category of consumers .said differently , a consumer , who wants to avoid that social - economic forces advertising , bandwagon effects , persuasive market power may well defeat market attraction by employing a random approach , especially since the latter is free of charge .it is worth noting that a random behavioral model could be a source of inspiration for alternative strategies of both firms and policy makers , for example concerning new anti - trust and competition laws .  a last consideration arising from the results of our simulations concerns the emerging category ofthe `` informed - but - undecided '' consumer .in fact , for this peculiar type of buyers , it turns out that the higher their discrimination ability is , the worse their efficiency in consumption becomes .this seeming counterintuitive effect has however an explanation : with a great capacity to distinguish differentiated characteristics of goods , the final efficiency can be very high just in case of few indifference levels .lucie sirieix , marion delanchy , herv remaud , lydia zepeda , and patricia gurviez .consumers perceptions of individual and combined sustainable food labels : a uk pilot investigation . , 37(2):143151 , 2013 .joffre swait and wiktor adamowicz . choice environment , market complexity , and consumer behavior : a theoretical and empirical approach for incorporating decision complexity into models of consumer choice ., 86(2):141167 , 2001 .subramanian sivaramakrishnan , fang wan , and zaiyong tang . giving an `` e - human touch '' to e - tailing : the moderating roles of static information quantity and consumption motive in the effectiveness of an anthropomorphic information agent ., 21(1):6075 , 2007 .alessandro pluchino , cesare garofalo , andrea rapisarda , salvatore spagano , and maurizio caserta .accidental politicians : how randomly selected legislators can improve parliament efficiency . , 390(21):39443954 , 2011 .

Abstract: we present a graph - theoretic model of consumer choice , where final decisions are shown to be influenced by information and knowledge , in the form of individual awareness , discriminating ability , and perception of market structure . building upon the distance - based hotelling s differentiation idea , we describe the behavioral experience of several prototypes of consu.pdfmers , who walk a hypothetical cognitive path in an attempt to maximize their satisfaction . our simulations show that even consumers endowed with a small amount of information and knowledge may reach a very high level of utility . on the other hand , complete ignorance negatively affects the whole consumption process . in addition , rather unexpectedly , a random walk on the graph reveals to be a winning strategy , below a minimal threshold of information and knowledge .   * keywords : * product differentiation ; hotelling ; consumer behavior ; information ; knowledge ; random choice .


large networks of coupled oscillators are pervasive in science and nature and serve as an important model for studying emergent collective behavior .some examples include synchronized flashing of fireflies , cardiac pacemaker cells , walker - induced oscillations of some pedestrian bridges , josephson junction circuits , and circadian rhythms in mammals .a paradigmatic model of the emergence of synchrony in systems of coupled oscillators is the kuramoto model , in which each oscillator is described by a phase angle that evolves as where is the intrinsic frequency of oscillator , represents the strength of the coupling from oscillator to oscillator , and .the classical all - to - all kuramoto model corresponds to .the study of generalizations of the kuramoto model has become an important area of research .some examples of such generalizations include systems with time - delays , network structure , non - local coupling , external forcing , non - sinusoidal coupling , cluster synchrony , coupled excitable oscillators , bimodal distributions of oscillator frequencies , phase resetting , time - dependent connectivity , noise , and communities of coupled oscillators .in this paper we study the case where the coupling strength is not uniform , but rather defines a network that has strong modular , or community , structure .synchrony on heterogeneous networks has been studied in the past , both for phase oscillator systems and other dynamical systems .much recent work has focused on the synchronization of phase oscillators on networks with modular structure . while the link between community topology and synchronization is well established  , there are few analytical results that describe synchronization in modular networks .reference  developed a framework to study a general number of communities , assuming that oscillators within communities are identical .reference analyzed the linear stability of the incoherent state for a system of coupled communities of heterogeneous phase oscillators .the same system was considered in ref . , where a set of coupled low - dimensional equations governing the dynamics of the community order parameters was formulated . here , we study this system of equations , finding for some important cases analytical expressions for local and global order parameters describing synchronization within communities and on the whole network , respectively .we find that , in the limit of a large number of communities , the ott - antonsen ansatz introduced in ref . can be used to obtain a low dimensional description of community synchrony . using this description , we characterize the phase space of the system where the parameters are the local and global coupling .one of our results is that , depending on the relative strength of local and global coupling , the transition to synchrony in the network can be mediated by local or global effects .this paper is organized as follows . in sec .ii we describe the model . in secs .iii and iv we present in detail the local and global dimensionality reductions , respectively . in sec .v we discuss the effect of community structure of the network on the dynamics and how it promotes hierarchical synchrony . in sec .vi we discuss how our results generalize when certain heterogeneities are introduced into the network . in sec .vii we conclude this paper by discussing our results .we are interested in studying coupled oscillators on a network with strong community structure such that ( i ) the coupling strength between oscillators within the same community is much larger than the coupling strength between oscillators in different communities and ( ii ) the intrinsic frequency for an oscillator is drawn from a distribution specific to the community to which that oscillator belongs .condition ( i ) serves as a model of situations where all the coupling strengths have similar magnitude , but the density of connections between communities is less than the density of connections within a community .the motivation for condition ( ii ) is that oscillators in different communities could have different frequency distributions due to different functional needs ( e.g. , as in cardiac myocytes in different regions of the heart ) , or as an approximation to fluctuations inherent to large but finite communities .thus , for a network with communities labeled where community contains oscillators , we assume that the coupling matrix in eq .( [ eqh ] ) can be written in block form as , where and , respectively , denote the communities to which oscillators and belong .furthermore , we assume that the intrinsic frequencies for oscillators in community are drawn from a distribution particular to that community , denoted by .we denote the fraction of oscillators in community by , where is the total number of oscillators in the whole network .  with this notation , eq .( [ eqh ] ) results in the following system , considered in refs . : where denotes the phase of an oscillator in community , , , and the intrinsic frequency is randomly drawn from the distribution .next , in order to measure synchrony within and between communities we define the local and global order parameters respectively , such that measures the degree of local synchrony in community and measures the degree of global synchrony over the entire network .we note that the linear stability of the incoherent state in this model was studied in ref . ( see also ) .in this section , we will study local synchrony by assuming there are a large number of oscillators in each community . using the definition of in eq .( [ eqordr ] ) , we simplify eq .( [ eqmodel ] ) to where denotes complex conjugate .we now move to a continuum description by taking the limit in such a way that all remain constant .accordingly , we introduce the density function that represents the density of oscillators in community with phase and natural frequency at time .since the number of oscillators in each community is conserved , satisfies the local continuity equation , , or \right\}=0.\end{aligned}\ ] ]  following ott and antonsen , we expand in a fourier series , , and make the ansatz , namely which , when introduced in eq .( [ eqcontintra ] ) , yields a single ordinary differential equation ( ode ) where in the continuum limit is given by   finally , by letting the distribution of frequencies be a lorentzian with spread and mean , i.e. \}$ ] , we can calculate by closing the contour of integration with the lower - half semicircle of infinite radius in the complex plane and evaluating at the enclosed pole of : thus , by evaluating eq .( [ eqodea ] ) at , we close the dynamics for : which defines complex odes , or equivalently real odes , given by   equation ( [ eqsimpintra ] ) was formulated originally in ref . , but its consequences for hierarchical synchrony have not been studied in detail .equations ( [ eqrho1 ] ) and ( [ eqpsi1 ] ) describe the dynamics of local synchrony .the synchrony of community is described by the magnitude of its order parameter and phase .the phase variable obeys an equation similar to that of the network - coupled kuramoto model , eq .( [ eqh ] ) , but the effect of community on community is modulated by the degree of synchrony of community , , and its relative size .in contrast to the kuramoto model , each community has an additional variable which evolves in conjunction with the phase variable . in this sense , the dynamics of the community order parameters resembles a network of coupled complex ginzburg - landau oscillators ..summary of local and global state variables and parameters of the system .[ cols="^,^",options="header " , ]   in what follows , we will consider the illustrative case in which all communities have the same size and spread in natural frequencies , i.e. and .furthermore , we let the coupling strength within each community be the same , as well as the coupling strength between oscillators in different communities .we assume the coupling strength within communities is much larger than that between communities , namely where and are of the same order .we clarify that the local coupling strength is chosen so that the local coupling within a community is of the same order as the sum of the coupling to every other community .more generally , a local coupling strength of the form with can be analyzed from our results by rescaling by a factor of . in sectionvi we relax these assumptions and discuss the case where community sizes , spread in frequency distributions , and coupling strengths vary from community to community .we now use the definition of in eq .( [ eqordr ] ) to rewrite the system in eqs .( [ eqrho1 ] ) and ( [ eqpsi1 ] ) as we note that although we will let in the next section , eqs .( [ eqrho ] ) and ( [ eqpsi ] ) are valid when is any positive integer and can be used to study synchrony on networks with a small number of communities .finally , we assume that the mean frequencies are drawn from a distribution , which we assume to be lorentzian with spread and mean .however , by entering a rotating frame , we can set without any loss of generality .for the sake of convenience we summarize all local and global state variables and parameters of the system in table [ table ] .we note that choosing a lorentzian distribution for is a natural choice if the heterogeneity in the distributions is assumed to originate from fluctuations arising from the random sampling of frequencies from the same lorentzian distribution . in this case, since a sum of lorentzian random variables has a lorentzian distribution , the distribution of the average frequencies in finite communities is lorentzian .  before analyzing eqs .( [ eqrho ] ) and ( [ eqpsi ] ) , we illustrate the behavior of the local and global order parameters over a range of values for and .we define as a measure of local synchrony and show the behavior of and in fig .[ bifd ] . while this behavior will be deduced from the analysis that follows , we find it convenient to present the phase space now to provide a framework for our subsequent analyses .we note that although the diagram above is theoretical , we present plots of and following various paths in the diagram , and all show excellent agreement with the theory . in the parameter space , we find the following four regions : region a where ( bottom left red ) , region b where ( top left yellow ) , and regions c and d where ( bottom right green and top right blue , respectively ) . in regiona there is neither local nor global synchrony , in region b there is local synchrony but no global synchrony , and in both regions c and d there is both local and global synchrony .we note that although both in both regions c and d , the nature of solutions for are qualitatively different , as will be discussed later . finally , solid and dashed curves indicate bifurcations between these regions and will be discussed as we proceed with the analysis . in the rest of this section, we will study local synchrony , characterized by the community order parameters .we will do this by assuming a given value of the global synchrony order parameter . in the next section , we will study the dynamics of using a dimensionality reduction on the global scale .we note here that in the rest of the figures in this paper , since we are interested in networks with a large number of communities and a large number of oscillators per communities , we will compare the results from direct numerical simulation of eq .( [ eqmodel ] ) on networks with large and with the theoretical curves obtained from our analysis of the continuum limit .first we study local synchrony when . in this case , from eqs .( [ eqrho ] ) and ( [ eqpsi ] ) we see that each community decouples from all others and evolves independently .the phase of community moves with velocity , and the stable fixed points of eq .( [ eqrho ] ) are so that all are equal .bifurcation ( i ) , indicated as a solid black line in fig .[ bifd ] , is described by this analysis , and occurs at .to illustrate this bifurcation , we plot in fig .[ intra1 ] the results of simulating the system as is varied from zero to six with , , and fixed and plot the resulting from simulation ( blue circles ) against the theoretical prediction of eq .( [ eqrsigma ] ) ( dashed red ) .the interpretation of this result is that the oscillators in each community synchronize as in the all - to - all kuramoto model , but with an effective coupling strength , which shows that the weak coupling to other independently evolving communities slightly inhibits synchrony .the analysis above assumes .now , we will analyze local synchrony when . in this casesome of the communities become synchronized with each other .given a value of ( which can be obtained using another dimensionality reduction , as we will show in the next section ) , community synchronizes with the mean field [ i.e. , a solution for eqs .( [ eqrho ] ) and ( [ eqpsi ] ) exists ] if ,\ ] ] and otherwise the community drifts indefinitely .the degree of local synchrony for locked communities can be found by setting in eq .( [ eqrho ] ) to zero and using eq .( [ eqpsisync ] ) , which gives the implicit equation   eq .( [ eqrhoimp ] ) determines the steady - state value of for locked communities and yields two possible kinds of solutions for : either eq .( [ eqrhoimp ] ) has a real solution for every , or it has a real solution for only some .it can be shown that when , eq .( [ eqrhoimp ] ) has a real solution for all , and thus each community becomes phase - locked and each reaches a fixed point as . on the other hand ,if , there is a real solution for only some with magnitude less than a critical locking frequency , which we denote as . in this case communities with phase - lock and given by the solution of eq .( [ eqrhoimp ] ) , while other communities continue drifting indefinitely .the phase angle of a drifting community increases or decreases monotonically and therefore its order parameter might be time dependent , according to eq .( [ eqrho ] ) .however , assuming a stationary global order parameter with constant and ( as will be discussed in the next section ) , the solution of the two - dimensional autonomous system in eqs .( [ eqrho ] ) and ( [ eqpsi ] ) must approach a limit cycle ( this can be shown , for example , using the poincare - bendixson theorem ) . to estimate the time averaged value of in this limit cycle, we neglect the effect of the cosine term in eq .( [ eqrho ] ) over one period and find that the time averaged order parameter for drifting communities is approximated by .this value agrees with the solution of eq .( [ eqrhoimp ] ) when is the locking frequency in eq .( [ eqcommlock ] ) .therefore , the community locking frequency can be determined by inserting the expression for above into eq .( [ eqcommlock ] ) , obtaining that communities lock when their frequency satisfies , which defines a new bifurcation .when ( region d ) , the locking frequency is finite and only some communities phase - lock .as approaches from above , the locking frequency diverges . for ( region c ) , all communities phase lockthe boundary between these two regions for larger is denoted as bifurcation ( ii ) , and is indicated as a solid black line in fig .a heuristic interpretation of this transition is that when is increased through bifurcation ( ii ) , communities with large desynchronize because the local coupling strength causes them to prefer an angular velocity much closer to their own mean frequency than the mean frequency of the entire network .  to test eqs .( [ eqrsigma ] ) , ( [ eqrhoimp ] ) , and ( [ y ] ) , we simulate the system with and with , , and ( parameters from regions b , c , and d , respectively ) and plot time - averaged as a function of in figs .[ intra2](a ) , ( b ) , and ( c ) , respectively . results from direct simulation are plotted in blue circles and compared to theoretical predictions , which are plotted as dashed red curves .[ intra2](a ) corresponds to region b , where and is given by eq .( [ eqrsigma ] ) and is therefore independent of .[ intra2](b ) corresponds to region c , where all communities lock and their order parameter is a solution of eq .( [ eqrhoimp ] ) .[ intra2](c ) corresponds to region d , where some communities lock and their order parameter is a solution of eq .( [ eqrhoimp ] ) , and other communities drift and their order parameter is independent of and given by .the vertical arrows indicate the theoretical value for the locking frequency obtained from eq .( [ y ] ) .theoretical results match very well with the numerical simulations .in the previous section , we studied local synchrony by assuming a steady - state value for the global synchrony order parameter .we now discuss how the global order parameter can be found by making a second dimensionality reduction on a global scale .as we previously let tend to infinity in order to enter a continuum description within each community , we now consider the limit and introduce the density function that describes the density of communities with average phase , mean natural frequency , and degree of local synchrony at time . in analogy with individual oscillators ,the number of communities is conserved and must satisfy the continuity equation . however , we find that the degrees of local synchrony quickly reach a stationary distribution , so we seek solutions where . in analogy to the classical kuramoto model , we find that approaches a fixed point if community phase - locks , or otherwise forms a stationary distribution with other drifting s . with eq .( [ eqpsi ] ) the continuity equation becomes \right\}=0,\ ] ] where is the steady - state value of given by eq .( [ eqrsigma ] ) or implicitly by eq .( [ eqrhoimp ] ) .  like eq .( [ eqcontintra ] ) , eq .( [ eqcont ] ) is of the form studied by ott and antonsen in refs . , and can be solved with a similar ansatz .thus , we make the ansatz inserting eq .( [ eqansinter ] ) into eq .( [ eqcont ] ) , we find that   we calculate as : since is defined implicitly by eq .( [ eqrhoimp ] ) for locked communities and by eq .( [ eqrsigma ] ) for any drifting communities , it is potentially piecewise - defined and not smooth . however , to a very good approximation we can do this integral using residues by considering the solution of eq .( [ eqrhoimp ] ) for which is real and positive for as a function of complex .the function is analytic when , and its real part converges to as with , while its imaginary part converges to an odd function . as for , the real part of differs from eq .( [ eqrsigma ] ) by a bounded amount .if decays so quickly that the error in approximating by for can be neglected when computing the integral , we can approximate the integral above by the integral which has instead of ( due to the symmetry of , the imaginary part of does not contribute to the integral ) .the integral with on the real line can be done by deforming the contour of integration to the line connecting to , where , , and closing the contour with the semi - circle in the negative complex plane connecting to . using the residue theorem , and taking and , we obtain where we have defined . for the lorenzian distribution with , we expect this approximation to be excellent when , but the agreement between the direct numerical simulation of eqs .( [ eqmodel ] ) and the theoretical predictions is very good even for situations in which is smaller [ e.g. , fig . [ hier ] ( a ) close to the transition for ] . we note that if is in region c [ see fig .[ bifd ] ] using to evaluate the integral in eq .( [ eqz1 ] ) is exact since all communities lock and are described by eq .( [ eqrhoimp ] ) .evaluating eq .( [ eqadot ] ) at and closes the complex dynamics for : the evolution of and are given by we note that these equations are valid provided that ( a ) is in the manifold of poisson kernels [ i.e. is of the form in eq .( [ eqansinter ] ) ] and ( b ) the distribution of degrees of local synchrony remains stationary as the system evolves .regarding assumption ( a ) , ref . shows that in the kuramoto model all solutions approach this manifold as .the stable fixed points of eq .( [ eqrho ] ) are   to eliminate assume nonzero ( and thus ) , and insert eq .( [ eqrss ] ) into eq .( [ eqrhoimp ] ) with .we choose the real , positive solution given by which we insert back into eq .( [ eqrss ] ) to obtain .we note that other solutions for are purely imaginary or negative . from the top line of eq .( [ eqrss ] ) , the imaginary solutions for result in a critical value for larger than , while real solutions result in a critical value smaller than , and thus we choose the positive real solution ( the negative solution results in ) .finally , to calculate the bifurcation curve for the onset of global synchrony , we let which yields the curve .this curve is indicated as a dashed black curve in fig .[ bifd ] and gives bifurcation ( iii ) from region a to c and bifurcation ( iv ) from region b to d.  we now seek to compute the mean degree of local synchrony . in the large limit we consider here , is given by an integral equation .if is in region c , i.e. , then since each community becomes phase - locked , we simply have however , if is in region d , i.e. , then because some communities phase lock and some do not , we have that where is the locking frequency given by eq .( [ y ] ) .  to illustrate these results ,we simulate the system with , , , and let vary between zero and six . in fig .[ inter ] we plot ( main ) and ( inset ) from simulation in blue circles and the theoretical predictions from eqs .( [ eqrss ] ) , ( [ rhat ] ) , ( [ eqsolve1 ] ) and ( [ eqsolve2 ] ) in dashed red .theoretical predictions agree well with simulations .+  with a complete understanding of both local and global synchrony in the system studied above , we now discuss hierarchical synchrony .we consider moving slowly ( compared with ) along some path in parameter space , restricting paths to lines starting at for simplicity . from our analysiswe find that bifurcations intersect at .thus , for lines , if the onset of local synchrony occurs before the onset of global synchrony . on the other hand , if the onset of local and global synchrony occur simultaneously .choosing and , in figs .[ hier ] ( a ) and ( b ) we plot the steady - state values of and resulting from moving along the lines and , respectively , for and .we note that is used in these simulations rather than as in the previous simulations because we find that finite - size effects are more prevalent near bifurcation ( iii ) .this is most likely due to the fact that at this bifurcation the onset of local and global synchrony occurs simultaneously .the values of and from simulation are plotted in blue circles and red triangles , respectively , with theoretical predictions plotted in black dashed and dot - dashed , respectively .note that for these parameters , so , and accordingly we see a separation of local and global onset in fig .[ hier](a ) , but not in fig .[ hier](b ) .we interpret these results as follows . along paths where local coupling effects dominate global coupling effects . in this casethe community structure is strong enough to yield a hierarchical ordering of synchrony , i.e. a separation in the onset of local and global synchrony .however , when global coupling effects dominate local coupling effects . in this casethe community structure is weak enough to yield a simultaneous onset of local and global synchrony .we now discuss how the results above generalize when some of the assumptions previously used are relaxed .we allow for heterogeneities in both the sizes of communities and spread in frequency distributions , i.e. we allow and to vary from community to community. we also allow the local and global coupling strengths to vary , letting for and for .beginning with local synchrony , we carry out a dimensionality reduction on the local scale and obtain the following odes :   thus , when , we have that the onset of local synchrony in community occurs at , i.e. in general synchrony occurs at different values for different communities . when , community becomes phase - locked if in which case satisfies otherwise community will drift .note that for a given value of , the behavior of depends not only on , but also , , , and , so in general there is no single locking frequency that separates locked and drifting communities at .  to study global synchrony, we again perform a dimensionality reduction on the global scale .since , , , and vary from community to community , after sending we introduce the density function that represents the fraction of communities with phase , mean natural frequency , degree of local synchrony , size , frequency distribution spread , and local and global coupling strengths and at time . noting that depends on , , , and and again looking for solutions with stationary , satisfies the continuity equation \right\}=0,\ ] ] where now depends on , , , and in addition to and .we assume that for each community the mean frequency , size , frequency distribution spread , and local and global coupling strengths and are all chosen independently and make the ansatz which yields the ode   finally in the continuum limit can be calculated by the integral   equations .( [ heteroglobal1 ] ) and ( [ heteroglobal2 ] ) govern the global synchrony of the system and must be solved self - consistently with the local dynamics , governed by eqs .( [ heterolocal1 ] ) and ( [ heterolocal2 ] ) . for arbitrary distribution functions , , , and the integral in eq .( [ heteroglobal2 ] ) might need to be evaluated numerically , but for certain choices , e.g. exponentials or linear combinations of dirac delta functions , further analytical results are attainable but not presented here .we have described and solved fully the steady - state dynamics of coupled phase oscillators on a modular network with a large number of oscillators in each community and a large number of communities . in particular , we have studied local and global synchrony , i.e. synchrony within and between communities , respectively .first we assumed a large number of oscillators in each community and used a local dimensionality reduction to study local synchrony .next , when the number of communities is large , we showed that a global dimensionality reduction can be done to study global synchrony .our analytical results shed light on the phenomenon of hierarchical synchrony , characterized by synchronization on a local scale before it occurs on a global scale , which occurs when the community structure of the network is strong enough .the system analyzed in this paper modeled synchrony on a network with two community levels , but synchrony on networks with more levels , e.g. communities with subcommunities , can be modeled in a similar way and analogous analytical results can be obtained .although we have assumed strong uniform coupling within communities and weak uniform coupling between communities , we conjecture , based on preliminary numerical experiments , that the system studied in this paper is in some cases a good quantitative model for networks where links between oscillators in the same community are dense and links between oscillators in different communities are sparse .an interesting result is that the system of planar oscillators representing community interactions [ eqs .( [ eqrho ] ) and ( [ eqpsi ] ) ] admits an approximate low dimensional description .the analysis of community synchrony in sec .iv is , to the best of our knowledge , the first low - 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Abstract: we study synchronization of sinusoidally coupled phase oscillators on networks with modular structure and a large number of oscillators in each community . of particular interest is the hierarchy of local and global synchrony , i.e. , synchrony within and between communities , respectively . using the recent ansatz of ott and antonsen , we find that the degree of local synchrony can be determined from a set of coupled low - dimensional equations . if the number of communities in the network is large , a low - dimensional description of global synchrony can be also found . using these results , we study bifurcations between different types of synchrony . we find that , depending on the relative strength of local and global coupling , the transition to synchrony in the network can be mediated by local or global effects .


chaos synchronization appears in networks of interacting non - linear units . due to the finite velocity of signal transmission ,the couplings might be delayed .time delay may generate instability , therefore such a network tends to become chaotic . on the other side, interaction enforces synchronization . under certain circumstances ,even when the time delay is much larger than the internal time scale , the units lock to a common chaotic trajectory without any time shift .the phenomenon has been demonstrated both numerically and experimentally in small networks of nonlinear oscillators and coupled semiconductor lasers , and is of natural interest in the fields of neuroscience and secure communication . for any network of identical non - linear units with a single time delay ,the stability of chaos synchronization is determined by the maximum lyapunov exponent of a single unit with delayed feedback , and the second largest eigenvalue of the adjacency matrix .firstly , two chaotic regimes are possible depending on the scaling of the single unit lyapunov exponent , namely strong and weak chaos .synchronization is only possible in the regime of weak chaos , where the exponent is positive and scales inversely with the delay time .secondly , the stability of the synchronized trajectory is determined by the difference in magnitude between the largest and the second largest eigenvalue .if there is a gap , stable chaos synchronization is possible , otherwise it is ruled out .for example , a ring of nonlinear units with unidirectional bonds has no eigenvalue gap , hence it can not synchronize .more specifically , the number of synchronized groups , is determined by the greatest common divisor ( gcd ) of the network loop lengths .complete synchronization is linked with a non - zero eigenvalue gap and is possible if .additionally , for , the network shows a pattern of synchronized groups , where units belonging to the same group are not connected to each other , only to units from other synchronization groups .this is called sublattice synchronization .the gcd condition is exact for networks with a single large delay time and is related to mixing of information between the units .however , it has been argued that it is also true for networks with multiple large delay times with a fixed ratio , as resonances between the delayed signals influence the stability of synchronization . while the problem of 2 coupled units is solved ,an analytic proof is still not available for general networks .nontheless , the extended gcd condition has been demonstrated in numerical simulations and in experiments on semiconductor lasers : in , two lasers interacting by transmitting their laser beams with a single delay time , become chaotic but can not synchronize at zero lag . when a second , twice as long delay time is added with beam splitters, the two lasers could synchronize to a common chaotic intensity .hence , adding the second delay time produced synchronization .more recently , the gcd condition correctly predicted the number of phase synchronization clusters in networks of up to 16 coupled lasers .our work extends the previous results to small networks with unidirectional bonds and two large delay times .our main contributions are two .firstly , complete and sublattice synchronization triggered by time delay resonance , as predicted by the gcd condition , are demonstrated analytically in networks of iterated maps .secondly , the scope of the gcd condition is investigated beyond chaos synchronization showing that time delay resonances also trigger high correlations among units in non - synchronizable networks .gcd - induced high correlations are found in two cases , in an asymmetrical chaotic network and in doubly connected rings of noisy linear oscillators .the paper is structured as follows . in section[ s : formalism ] we generalize the formalism of master stability function to networks with double time delay .we provide analytic results for doubly connected rings of bernouilli maps .complete and sublattice synchronization are explained by means of the master stability function symmetries and we give special time delay ratios for which complete synchronization is not possible . in section [ s : phasemaps ] we study complete and sublattice synchronization induced by different time delay resonances in doubly connected rings of tent and bernoulli maps .we also discuss sensitivity to detuning and compare with equivalent single - delay networks with virtual units .section [ s : corrs ] investigates the validity of the gcd argument in non - synchronizable networks .we present an asymmetric chaotic network showing high correlations for .we also study doubly connected rings of noisy linear oscillators , for which we find gcd - induced correlation peaks .finally , the results are summarized in section [ s : summ ] .generally , a network of iterated maps with two time delays can be modeled as follows : , \label{eq : msf1}\end{gathered}\ ] ] where \to[0,1] ] with eigenvalues .this mode is parallel to the sm and preserves synchronization .every other mode is perpendicular to the sm .if the lyapunov exponent along the parallel mode is positive , , the synchronized trajectory is chaotic .aditionally , along the transverse modes , the msf should be negative , for all , to guarantee the stability of the synchronized state .  in this manuscriptwe consider networks of two different kind of maps .bernouilli maps , modeled by are chaotic for .since the derivative is constant , their msf can be calculated analytically . moreover ,the analytic results are known to reproduce qualitatively several features of more complex chaotic delay systems .for some properties , however , the fluctuations of the derivative play a role .therefore , we compare our results to tent maps , modeled by for bernouilli maps , eq . has constant coefficients and we can assume an exponentially evolving perturbation .then , we find the characteristic polynomial \label{eq : ypoly}\,\ ] ] which has complex roots , with for each set of eigenvalues .the spectrum of lyapunov exponents along the eigenmode is then given by .a perturbation mode is stable if all the roots lie inside the unit circle .if holds , one immediately finds an unstable solution irrespective of .this corresponds to the strong chaos regime where the two delay terms vanish in the limit  .the condition for weak chaos is thus given by assuming both delays to be large , we write and and consider the limit .we introduce a weak chaos ansatz ,  ] . in the limit of large delay for weak chaos ,we obtain .then , for each eigenmode , eq . becomes \ , .\label{eq : pseudospec}\ ] ] since for , the phases of the multipliers are uniformly distributed between and , we can solve for , to obtain the so - called pseudo - continuous spectrum , on which the multipliers are densely located : . the generalized master stability functionis then given by eq . andits symmetries are our main tools to study the network synchronization properties .we will now apply the previous formalism to study directed rings with two time delays ( fig .[ fig : ringmaps ] ) .this is a paradigmatical system that can not synchronize with a single time - delay , but shows synchronization by time - delay resonance .the simplest possible ring , consisting on two mutually coupled units , was addressed in .there , it was shown that synchronization with multiple time delays was possible under certain conditions for time delays values .it is immediate to check that the provided , values leading to zero lag synchronization corresponded to network configurations with , thus being coherent with our results . in this sectionwe generalize to the case of rings of units with two time delays and no self - feedback .we will study the symmetries of the msf in order to re - derive the time - delay ratios allowing either complete or sublattice synchronization as predicted by the gcd condition .the polynomial allows us to study synchronization stability analytically .-unit doubly coupled unidirectional rings translate to identical adjacency matrices with eigenvalues , where .thus , the pseudo - continuous spectrum eq . becomes \label{eq : pseudospecring}\,.\ ] ] considering the case of single delay , , we can easily solve eq ., and find a msf for a single delay , the stability of a perturbation mode only depends on the magnitude of its corresponding eigenvalue , i.e. the master stability function is spherically symmetric in the complex plane with respect to .consequently , the stability of all the eigenmodes , transverse and parallel , is the same . since for chaotic dynamics , both complete and sublattice synchronization are unstable in unidirectional rings of any size with a single delay .if we consider networks with two time delays , i.e. , this spherical symmetry can be broken , and the transversal modes can be stabilized , depending on the ratio of the delays . however , depending on the the msf , , still can have some symmetry .it is straightforward to check that the gcd of doubly connected rings is .consider and relatively prime and . if the number of elements in the ring is a multiple of , we find .then , the spectrum contains eigenvalues of the form , with , where the mode is the parallel mode .it can be shown that the master stability function is invariant under a transformation . hence all the modes have the same lyapunov exponent , .since the parallel mode is unstable , all of them are unstable as well .  to demonstrate this point , we can write and , for some integers and . since and are relatively prime , we find that and are relatively prime as well .then , the pseudo - continuous spectrum eq . along the eigenmode with eigenvalue is . \leftrightarrow\\    1 & = & a(1-\epsilon)e^{-i\phi } + a\epsilon\left[(1-\kappa)(e^{\frac{i2\pi}{k}}y)^p+\kappa ( e^{\frac{i2\pi}{k}}y)^q \right ] \nonumber \,.\end{aligned}\ ] ] since the msf only depends on the magnitude , we conclude that .the corresponding eigenvectors have distinct entries , with a phase difference of .these unstable modes give thus rise to different sublattices corresponding to the distinct entries of .moreover , it is possible to rule out beforehand some time delay ratios that do not allow complete synchronization .the limit to stability is given by for the roots of the characteristic polynomial eq . .hence , with uniformly distributed along the unit circle if is sufficiently large .choosing , we get where we have considered .for the parallel mode , this reduces to but eq . also holds for a perpendicular mode whenever both hold at the same time . by substituting and taking the quotient of the phases of eqs ., we find that mode will be unstable for a delay ratio where , are integers and .once an unstable mode is found , the periodicity of its eigenvector determines the number of synchronized sublattices .we proved this result analytically for bernouilli maps , however this is a rather general phenomenon .the symmetry arguments of the master stability function and eq .apply also to steady states and periodic orbits in general since the derivatives along the trajectory are also constant or periodic in this case .it can be argued that these symmetry arguments can be extended to chaotic attractors , as these consist of unstable periodic orbits , but rigorous analytic proofs for chaotic systems other than bernouilli maps are difficult .we provide numerical evidence for the generality of our analytic results in section [ s : phasemaps ] .in this section , we demonstrate both complete and sublattice synchronization in rings with double delay , for the cases and , respectively .moreover , we compare the synchronization regions with those of equivalent single delay networks with the same loop lengths .we investigate the sensitivity to a small detuning of the two delays as well .if we choose , a ring of nonlinear units contains loops of all lengths , with .thus , the gcd is always equal to one .is a second degree polynomial and can be solved for each mode .the complete synchronization region in - space is then the intersection of all the transverse modes stability regions .we show the master stability function for a ring of bernouilli maps in fig . [fig : msf ] .the spherical symmetry is clearly broken ; the closer the phase of the eigenvalues to , the smaller the corresponding lyapunov exponent .consequently , the stable parameter region for is largest .it is hence easiest to stabilize zero - lag synchronization for only two coupled elements , where this is the only transverse eigenvalue . for our choice of parameters ,zero lag synchronization is stable for , as , as indicated by the square .also for , we find in the stable region of the msf , both eigenvalues are indicated by triangles . for msf is unstable for the modes , and for we find unstable transverse modes for .zero lag synchronization is hence unstable in both cases for the chosen parameters .  for a different parameter choice, the phase maps showing the parameter region for which zero lag synchronization is stable in rings of bernouilli maps , are depicted on fig .[ fig : ringmaps1 ] .the size of the synchronization regions shrinks with increasing number of units , in agreement with the shape of the master stability function .the analytic phase diagrams have been confirmed by numerical iterations of the chaotic network .  in order to demonstrate the gcd condition for other chaotic units , for which analytic results are not available ,we simulated analogous networks of tent maps . in order to compare both dynamics, we picked the tent and bernouilli maps parameters , and , laying the same lyapunov exponent for the single isolated maps without delay , .the resulting synchronization regions are shown in fig .[ fig : ringmapstents ] . we find a similar structure of nested smaller regions for increasing number of units , but due to the fluctuations of the derivative of the map , the corresponding coefficients of eq . become time dependent , shrinking the regions of stable chaos synchronization .   for a double delayed network as obtained by solving eq ., with ( left ) and ( right ) .parameters are and .the solid line marks the contour .thus , the eigenmodes whose corresponding eigenvalue lies inside the contour are stable and those staying outside are unstable .the eigenvalues are indicated for ( triangles ) , ( squares ) and ( inverted triangles ) .the eigenvalue along the synchronization manifold is indicated with a circle.,title="fig : " ] [ fig : msfa ]  for a double delayed network as obtained by solving eq ., with ( left ) and ( right ) .parameters are and .the solid line marks the contour .thus , the eigenmodes whose corresponding eigenvalue lies inside the contour are stable and those staying outside are unstable .the eigenvalues are indicated for ( triangles ) , ( squares ) and ( inverted triangles ) .the eigenvalue along the synchronization manifold is indicated with a circle.,title="fig : " ] [ fig : msfb ]  for a double delayed network as obtained by solving eq ., with ( left ) and ( right ) .parameters are and .the solid line marks the contour .thus , the eigenmodes whose corresponding eigenvalue lies inside the contour are stable and those staying outside are unstable .the eigenvalues are indicated for ( triangles ) , ( squares ) and ( inverted triangles ) .the eigenvalue along the synchronization manifold is indicated with a circle.,title="fig : " ]   for a delay ratio of the resulting network contains loops of length .the is equal to 2 for rings with an even number of units , and to 1 if the number of nodes is odd .hence , for even , this time delay ratio produces sublattice synchronization . as shown in fig .[ fig : msf](b ) , the master stability function is symmetric under a transformation .a transverse mode with eigenvalue is thus always unstable for chaotic dynamics .we provide an example by solving the specific case of a 4 unit doubly coupled ring ( fig . [ fig:4ring ] ) . here , eigenmode $ ] with eigenvalue is unstable , generating two groups formed by units 1 and 3 , and 2 and 4 respectively .the modes , which do not allow sublattice synchronization , are both stable .units belonging to the same group develop identical chaotic trajectories , although they are not directly coupled , but receive input from units of the other group .the resulting parameter region showing stable sublattice synchronization for bernouill maps is shown in fig .[ fig:4ringmap ] . in orderto show that the phenomenon is not limited to constant slope maps , we also provide the synchronization region for an equivalent network of tent maps . we observe again a shrinking of the synchronization region due to time derivative fluctuations .the effect of time delay resonances can not be explained by the gcd condition alone . in reference , the effect of multiple time delay is studied by transforming the network to an equivalent network with homogeneous delay times .this is done by inserting imaginary units coupled with a single time delay along the longer connections . here, we demonstrate how the resulting phase maps are different , despite both networks being equivalent from the gcd point of view .take for instance a directed ring of three units with .with respect to the gcd condition , this network is completely analogous to a single delay triangle where the longer links have been substituted by a 2 link chain of simple delays mediated by an auxiliary unit ( see fig . [fig : auxiliaryunits ] ) .the corresponding synchronization region turns out to be smaller than that of a directed triangle with double delay .moreover , a single delay network like this one does not suffer from detuning effects .the synchronization properties of chaotic networks with multiple time - delay depend on a precise ratio between the time delays .the synchronization phase maps presented above are found to be very sensitive to detuning . as it is shown in fig .[ fig : ringmaps2 ] , the parameter regions showing synchronization stability shrink drastically after a small mismatch of the delays , and .it is more natural to study the effect of detuning between the two delay times in continuous chaotic systems with larger internal correlation time . therefore , we consider a doubly connected unidirectional ring of stuart - landau oscillators , modeled by where the last unit , , is coupled to the first , . here is the amplitude phase - coupling , and are the coupling strengths and and are the coupling delays . without coupling , the oscillators are in a stable periodic orbit . in fig .[ fig : sl](a ) we show the unit - to - unit zero lag crosscorrelation , given by , as a function of the ratio between the delay times in a ring of two oscillators .the unit - to - unit zero lag crosscorrelation in a three - unit ring is shown in fig .[ fig : sl](b ) .both ring configurations show high correlations around delay ratios of and , as predicted by the gcd - argument .we observe , firstly , that the correlations are higher for two than for three oscillators .for the ratio , this corresponds to the smaller transverse eigenvalue found for two bernoulli maps , represented in fig . [fig : msf ] .secondly , we find higher correlations for simpler ratios .the width of the delay resonance peaks depends on the internal decay time of the oscillators : while the bernouilli maps are found to be very sensitive to detuning of the two delay times , we find a considerable width of the resonances for stuart - landau oscillators .moreover , the crosscorrelation between the oscillators is not always positive , but has an oscillatory shape as the delay ratio varies .this can be explained by phase effects : for two coupled oscillators we find anti - synchrony for delay ratios , with being the dominant frequency of the chaotic motion .indeed , having one finds that for these delay ratios the synchronization manifold is destabilized , as the two delayed signals interfere destructively with each other .the manifold is however not suppressed .therefore we observe anti - correlation between the oscillators .consequently , we find a frequency of at the delay resonance , i.e. the first order resonance at has an oscillation frequency of , the second order resonance at has a frequency of approximately , etc .similar results can be found in the ring of three doubly connected oscillators .the gcd argument provides information about the possible number of synchronization groups .this holds for networks where complete synchronization is a solution of the dynamical equations .but it was argued in reference that the gcd condition is applicable to other networks as well . even if chaos synchronization is not a solution ,the gcd determines how the information about the trajectories mixes according to the network topology .we demonstrate this in two different systems .first we show how the gcd affects correlations among the non - linear units in an asymmetrical network .secondly , we study correlations in directed rings of coupled linear oscillators with noise , identifying correlation peaks for the time delay ratios predicted by the gcd condition .when the network gcd is equal to , at each time - step each unit is driven just by the units belonging to one of the synchronized sublattices .on the other hand , when the gcd is equal to 1 , the graph is aperiodic and after enough iterations of the dynamics each unit is being driven by the initial state of every other unit .this topological effect is able to induce correlations among the network trajectories even when the system does not have a synchronized solution .consider the network depicted in fig .[ fig : detour]a . here, we have a single coupling detour between two units with variable time delay embedded in a 3 unit ring with single a time delay . this network can not synchronize by construction .we have simulated the dynamics of bernouilli maps coupled with this topology and computed time - correlations among each unit trajectories .the results are shown in fig .[ fig : detour]b . for integer values of , we can distinguish two situations . for values of , we have . however , for we have and the correlations are practically zero .it is useful to compare chaotic systems to linear systems with noise . by replacing the chaotic dynamics by a linear system with white noise ,one recovers properties that relate solely to network structure , and not to the specific chaotic system .however , a major difference between deterministic chaotic systems and stochastic linear systems is that the latter can not synchronize , as synchronization is a nonlinear phenomenon .nevertheless , linear stochastic systems have been shown to mimic several qualitative features of the auto- and cross - correlation functions of delay - coupled chaotic elements . in some casesone can even quantitatively model the auto - correlation function of a chaotic delay - system with a linear model .we demonstrate here that also these delay resonances can be explained by a stochastic linear delay model .we consider a ring of oscillators , where each oscillator is characterized by a natural frequency , an internal decay rate , and an internal white gaussian noise source , with zero mean ( the variance is irrelevant , as the whole system can be rescaled ) .we thus approximate the chaotic signal by a linear response , which is captured by the correlation functions , and a component which effectively acts as a source of noise .each node is coupled to its neighbor with a strength over a first connection with a delay , and a second connection with a delay and a strength .this system is modeled as with .we can decompose the system into its eigenmodes , given by with .the dynamics is then modeled by with .we can easily solve the system in fourier space , and find for  the spectrum has maxima at and at ; the peaks are most pronounced when those two conditions are both hold .clearly , the variances are maximal when the resonances due to the two delays maximally overlap , and minimal when none of the resonances overlap .consequently , the in - phase mode has a maximal variance for being rational ; the simpler this ratio , the larger the variance .the out - of - phase modes have a larger variance for and a minimal variance for rational ratios for which this conditions does not hold .we show the variances of the different modes in a ring of elements in fig . [fig : linearsystems]a .the in - phase mode has large maxima at , and less pronounced maxima at other rational ratios as .the out - of - phase mode shows maxima if , we see indeed extrema at and minima for and .the crosscorrelation between two nodes in the ring , can be computed as a sum of the eigenmodes : if all the eigenmodes have the same variance the sum cancels out and there will not be any zero lag correlation between the two elements .this is the case if is irrational and if multiple eigenmodes have overlapping resonances for the same delay ratio .this happens whenever for a given mode , effectively recovering eq .[ eq : forbidden_ratio ] . for sublattice synchronization, it should hold for all , except for the distance between the nodes of the group . for a ring of elements ,we show the magnitude of the zero - lag correlation in fig .[ fig : linearsystems]b .we have provided a formalism for studying the stability of synchronization in networks with two or more time delays .some networks that do not synchronize with a single time delay can be brought to synchrony with an appropiate choice of the second time delay .ultimately , the relationship between both time delays determines the available synchronization regime , as predicted by the gcd condition .we exemplified this comprehensively in the case of directed rings with two delay times .we were able to provide certain time delay ratios that do not allow complete synchronization , as well as to identify the unstable modes giving rise to synchronized sublattices .the resulting synchronization stability regions are different from those in gcd equivalent networks with single time delay. also , the synchronization properties can be very sensitive to a detuning between the two time delays : as one of the time delays varies , we observe positive or negative correlations depending on the internal correlation time and frequency of the chaotic units .  the global influence of network topology and time delay resonance via the gcd argumentis made evident not only in synchronization phenomena .the network structure also induces high correlations among the units trajectories in non - synchronizable systems .moreover , the gcd - induced time delay resonances observed in rings of chaotic map are reproduced by networks of noisy linear oscillators . yet , a mathematically rigorous explanation of the gcd argument in general chaotic networks remains to be found .m. jimnez and e. korutcheva warmly thank the hospitality and the financial support of dep . of theoretical physics iii at the university of wrzburg , as well asthe financial support of dep . de fsica fundamental , uned .e. korutcheva also thanks the sponsorship by the alexander von humboldt foundation within the renewed research stay program .

Abstract: chaos synchronization may arise in networks of nonlinear units with delayed couplings . we study complete and sublattice synchronization generated by resonance of two large time delays with a specific ratio . as it is known for single delay networks , the number of synchronized sublattices is determined by the greatest common divisor ( gcd ) of the network loops lengths . we demonstrate analytically the gcd condition in networks of iterated bernouilli maps with multiple delay times and complement our analytic results by numerical phase diagrams , providing parameter regions showing complete and sublattice synchronization by resonance for tent and bernouilli maps . we compare networks with the same gcd with single and multiple delays , and we investigate the sensitivity of the correlation to a detuning between the delays in a network of coupled stuart - landau oscillators . moreover , the gcd condition also allows to detect time delay resonances leading to high correlations in non - synchronizable networks . specifically , gcd - induced resonances are observed both in a chaotic asymmetric network and in doubly connected rings of delay - coupled noisy linear oscillators .   = 1