the study of complex networks has recently attracted much attention in the physics community and currently represents an important area of multi - disciplinary research .network theory studies the properties of real world systems and phenomena displaying a web - like structure , usually represented mathematically as graphs .the individuals of the system under consideration are then symbolised as vertices and their relations or interactions as edges .for instance the world wide web is a network of web pages connected by hyperlinks ; the cell can be described as a network of chemicals connected by chemical reactions ; human language can be viewed as a web where words are linked if they appear adjacent or one word apart in sentences .we recall that a network is defined formally as a couple , where is the set of vertices and is a set of couples of vertices , called edges .if is an ordered set , the network is said to be `` directed '' .a network can be specified by its adjacency matrix , where if vertex is connected to vertex and otherwise .the degree of vertex is then defined as the number of nearest neighbours of vertex , that is : .the neighborhood of a vertex is the set of all vertices connected to .notice that , in contrast with the terminology usually introduced in analysis and general topology , does not contain as one of its elements .the degree function then `` counts '' the number of vertices belonging to each neighborhood and gives thus a first characterization of a network s topology . in many casesa more complete characterization of network topology is given by the weighted adjacency matrix , where the numbers represent the weight of the connection between vertex and vertex .this number can be defined in many ways . in our casewe will define it as the total number of edges connecting vertex and vertex .we then define the strength of vertex as the total number of edges belonging to vertex , that is .when multiple edges are prevalent a weighted analysis is obviously worthwhile .more refined characterizations of network topology also take account of clustering and of the average distance between two vertices .complex networks were originally studied in the context of random graph theory where it has been shown that connecting the vertices at random results in exponential degree distributions .the structure of a random graph is thus uniform , with most vertices having approximately the same degree and only few vertices of high degree .empirical results indicate however that several real - world networks display instead power - law degree distributions  , where and are constants . because of the self - similarity properties of the power law , these networks are called scale - free .it is very remarkable that both the empirical and the analytical work usually recovers scale - free networks with power law exponents ranging between and .since power laws decay much slower than exponentials , scale - free topologies are not uniform and display many highly connected vertices , called hubs .hubs work as shortcuts between apparently distant environments and thus have a crucial role in many phenomena , such as in the spreading of infective agents through a network or resilience of the network to external attacks .scale - free topologies are normally interpreted as a consequence of network self - organization , resulting in properties of the system _ as a whole _ which ( in contrast with classical physics ) can not be inferred only from properties of its _ parts _ ( that is from the interactions between the individual components considered in isolation ) .thus one of the major problems in complex network theory has been to identify the range of mechanisms by which a network can self - organise into a scale - free state .it has been shown analytically that if a network evolves by addition of new vertices at a constant rate and if the newly introduced vertices connect _ preferentially _ ( and _ _linearly__ ) to highly connected vertices , then the network displays a scale - free topology .  preferential attachment , however , is not always a natural hypothesis .for example , in many situations it is not realistic to assume that a node has the complete information about the degree distribution it would need in order to know where to attach preferentially . from the other hand this information is just what is needed to normalise properly the attachment probabilities as defined in most of the analytic models introduced in literature so far ) .alternative mechanisms generating scale - free topologies have then been introduced , as for example the static model , the varying fitness model and random walk models incorporating a copying mechanism .in addition to random networks , some interesting examples of _ deterministic _ networks have also been recently studied .a remarkable example are the apollonian networks discussed in , inspired by the ancient problem of finding a space filling packing of spheres .apollonian networks turn out to be simultaneously scale - free , small world , euclidean , space - filling and matching graphs . in this casethe scale - free topology , however , turns out to be implicitly related to a preferential attachment mechanism .it is important to recall that deterministic networks are , at least in principle , completely controllable and can thus be very useful in the design of artificial networks such as communication or economic networks , where ( depending on the particular problem under consideration ) the topology will usually have to satisfy several spacial and temporal constraints .the integer networks introduced in are another interesting example of deterministic networks where vertices represent positive integers and edges are drawn whenever there is a divisibility relation between them .these networks are directed and their topology crucially depends on the particular set of vertices under consideration .for instance , when the vertices range in the set of the prime numbers an infinite star network centered in is obtained .considering instead the set of composite numbers , the degree distribution is the sum of an in - degree distribution and an out - degree distribution , and numerical simulations indicate that the latter is well approximated by a power - law with exponent .thus our attention is drawn to abstract networks whose vertices are mathematical objects and whose edges symbolise mathematical relations .examples of this kind obviously abound .a straightforward example is the set of the infinitely differentiable functions defined in the unit interval .we identify functions ( through an equivalence relation ) when they differ by a constant and naturally define a network structure on the set of equivalence classes by connecting vertex ( for the sake of notational economy here we denote the equivalence classes by their representants ) to vertex if is the derivative of .each non - constant vertex has thus a descendent ( its derivative ) and every vertex has an antecedent ( its integral ) .we thus have defined a directed network consisting of infinite disconnected components .the degree of each vertex can only take the values or and the components can be described as infinite branches , finite branches ( such is the case when one of the vertices of the branch is a polynomial ) , cycles ( such as the one generated by ) and a loop ( generated by ) .another way to associate a network to mathematical objects is by associating networks to equations .consider any equation , where the variables range in a given set .we can then naturally associate a network to the equation by representing the points of as vertices and by forming an n - clique ( that is a completely connected subgraph ) each time that the elements of satisfy the equation .we will then say that the network thus defined is _ generated _ by the equation .cliques have been widely studied in the last few years and often provide interesting insights into network structure and function .for instance cliques naturally represent clusters , communities and groups in social networks .cliques are also very important in theoretical biology when modelling protein - protein interaction networks and gene regulatory networks .  in this paper we will study the degree distribution of networks generated by diophantine equations , that is polynomial equations whose variables range in the set of integer numbers ( originally introduced by diophantus of alexandria ) , attempting thus to establish a connection between age old unsolved problems in number theory and a very actual and practical area of science .  in particular , in model a we study the _finite _ networks generated by the pythagorean equation when the variables range in the set recovering ( at least up to ) a degree distribution that is well approximated by a power law with exponential cut - off .an ageing analysis reveals then that the hubs of the network , which are the nodes with the highest degree , are not the oldest vertices of the network as we would expect for a network grown via preferential attachment . in the pythagorean case hubs form instead in vertices whose age is between old and middle age . after a whilea freezing effect takes place and the hubs stop attracting new connections.younger vertices then take over and become the new hubs in the network .this situation looks more akin to real social networks than to the  networks .this idea has already been suggested in a recent work , in which rules are added to a stochastic network .the basic suggestion is that people in social networks agree and form hubs in response to an exact need , a deterministic rule , and not merely to an unbiased attraction to important people .the clustering coefficient analysis also displays interesting properties .in particular in pythagorean networks has a power law behaviour that can be predicted analytically .  in model bwe introduce the finite networks generated by the diophantine equation in the set .we analyse the degree distribution and clustering coefficient comparing the results to those obtained in model a. in particular we find that , although the network s topology turns out to be very different than the one obtained starting from the pythagorean equation , a power - law can still be recovered for several decades of values of .we consider the network generated by diophantine equations of the form where the variables , , and the exponent are _ positive _ integers .we study how the network generated by eq.1 evolves by introducing a new vertex at each time - step , following the natural order of the positive integers .thus at time - step the vertex is introduced and no edge forms . at each subsequent time stepanother integer is introduced and -cliques form whenever triplets of integers satisfy eq.1 .we then study the _finite _ networks obtained by limiting the variables to range within sets of the kind .the network will thus consist of vertices ( associated with the first positive integers ) and a number of edges belonging to 3-cliques .notice first of all that at each time step the degree of every vertex is expressed by an even number .this fact alone has some interesting consequences in terms of network topology .for instance , it follows from a well - known theorem of graph theory that each connected component of the finite networks we just defined has an eulerian circuit , that is a circuit joining all edges and traversing each edge only once .notice also that , according to fermat s last theorem , proven by wiles in 1994 , eq.1 has no solutions for and the corresponding networks are thus empty .we therefore need to consider only the cases and .we consider first the linear case with .  in this caseit is a simple matter to derive the degree distribution analytically . since multiple links are prevalent we performed a weighted analysis .we recall that the weight of the links is defined as the number of times two vertices are connected and the strength of a vertex as the total number of vertices it is connected to , having counted the repeated links . since we allow loops , derived by a solution of eq.[l0 ] of the type , every vertex is connected to all other vertices if , and to all other vertices but itself if , so that if and if .if is even we have . if is odd , .  to calculate the strength of the vertices we have to consider that , if , the vertex will assume links ( note that in this case the clique will give 3 new links to vertex ) , forming the following cliques : , plus links with the following cliques : if is even and if is odd . if the vertex will assume links forming the following cliques : plus links with the following cliques : if is even and if is odd .it follows that the strength of a vertex is given by if and by if .thus linearity of eq.2 results in a uniform probability distribution for the strength : if and otherwise .notice at last that the degree distribution , in the thermodynamic limit , becomes trivial as each vertex s degree diverges when .we now study the network generated by the pythagorean equation following standard mathematical terminology , we will say that is a pythagorean triplet whenever , and satisfy eq.3 .the numbers , and are usually called the `` legs '' of the triplet . when all legs are positive we say that is a positive pythagorean triplet .a pythagorean triplet is called `` primitive '' whenever , and are relatively prime .it has been known since antiquity that every pythagorean triplet is either of the form or of the form .it is also well known that every positive integer belongs to at least one pythagorean triplet and that there are infinite primitive pythagorean triplets but , despite the apparent simplicity of the problem , the topological distribution of the set of all pythagorean triplets is still largely unknown .it is known for instance that the number of pythagorean triplets with legs smaller or equal to a given positive integer can be expressed by where is an explicitly given constant and is a remainder function .it is also known that and there is a large literature aimed at giving a better estimate for the remainder .it was proved in that with some .notice however that , in order to investigate the pythagorean network just introduced analytically and derive its degree distribution we would clearly need to know much more , that is the exact number of pythagorean triplets of which is a member . as far as we are aware this is still an open problem in number theory , probably deeply connected with the factorization problem .indeed it is easy to conjecture that the properties of do nt depend only on the algebraic form of eq.3 but also on the structure of the number itself in terms of its factorization as product of prime numbers .for instance it is known that if is the number of distinct prime factors of an integer number , then the number of pythagorean triplets of which is a member is given by if all prime factors of satisfy , if is odd and some prime factor of satisfies , if and otherwise .the problem of determining the distribution of pythagorean triplets is thus reducible , at least within the set of _ primitive _ triples , to the problem of counting the prime factors of each leg , which strongly suggests that there is little hope to derive the degree distribution analytically at present .the model , however , can be described numerically , at least in the finite case .we will then consider the networks generated by allowing the variables of eq.3 to range only in the first _ positive _ integers .we performed numerical simulations of the network growth with the deterministic attachment rule given in eq.3 .we show the results for the degree distribution in the left panel of fig.[p1 ] for .what is first evident from the log - log plot is that , changing the scale of the network size , the degree distribution preserve its shape .then , with increasing values of , a wide region emerges where the probability distribution looks like to follow a power law . on the right panel of fig.[p1 ] we show the empirical distribution for after a logarithmic binning to understand better the function behind the data .we find that a power law with an exponential cut - off ,   approximates very well the empirical data .  considering that complex growing networks usually display pure power law distributions only in the thermodynamic limit , we find the finite results expressed graphically in fig.1 particularly remarkable .   in the left panelwe show the degree distribution for the pythagorean network for . in the right panelwe perform a logarithmic binning on the degree distribution for to illustrate the power law behaviour .the fitting curve is the power law with exponential cut - off in eq.[n10].,title="fig:",scaledwidth=48.0% ]  in the left panel we show the degree distribution for the pythagorean network for . in the right panelwe perform a logarithmic binning on the degree distribution for to illustrate the power law behaviour .the fitting curve is the power law with exponential cut - off in eq.[n10].,title="fig:",scaledwidth=48.0% ]  to better understand how those structures form , we performed an analysis of the network . in fig.[p2 ] we show the degree of the vertices versus their labels for different values of .small labels indicate old vertices .peak values of the degree represent the major hubs of the network . for stochastic preferential growing networks , the high degree vertices are always the old vertices of the network . in the pythagorean networkthis result does not hold . from fig.[p2 ] we can see that after a while the old vertices degree freezes and hubs form between young and middle - aged vertices .representation of the aging of the hubs for the pythagorean network for on the left and for on the right . is the degree of the vertices , while is their label .small values of correspond to old vertices.,title="fig:",scaledwidth=48.0% ]  representation of the aging of the hubs for the pythagorean network for on the left and for on the right . is the degree of the vertices , while is their label .small values of correspond to old vertices.,title="fig:",scaledwidth=48.0% ]  since the pythagorean network is made up of cliques it is interesting to look at its clustering coefficient.the clustering coefficient is defined as where is the degree of vertex and is the number of nearest neighbors of vertex that are connected to each other . we show in fig.[p3 ] our measures of on a pythagorean network for different values of .the regularity of the distribution is striking and due to the high regularity of the network . all vertices with form a triangle with their nearest neighbors , so that they have .the higher a vertex degree , the more improbable it is that its nearest neighbors are fully connected .since for the inner geometry of the network , it follows that and , obviously , .all the values of lie on well - determined levels defined by where and . to demonstrate eq.[n1 ]is a simple graphical exercise .eq.[n1 ] is compared to the numerical simulation in fig.[p3 ] . for each , the different values of equally spaced at a distance of .notice at last that , in contrast with the linear case , the network s behaviour in the thermodynamic limit is now highly not trivial and probably can not be described in detail before some open questions in number theory will be eventually answered .the numerical work we performed may however suggest some interesting conjectures .we think that the most relevant feature of our model is perhaps the ageing analysis showing that , when the network grows , the hubs are constantly rejuvenated , in sharp contrast with the behaviour of scale - free networks generated through the preferential attachment mechanism .we now study the network generated by the deterministic rule expressed by the equation this network is thus deeply connected with the classical euler and fermat problem of finding all integers that can be expressed as the sum of two squares . in a sensethis case is intermediate between the linear and the pythagorean case .notice , first of all , that in the thermodynamic limit eq.[he1 ] generates a trivial network where each vertex s degree goes to infinity . indeed given any positive integer , for any arbitrary integer ( when ) we can always find an integer satisfying eq.[he1 ] .thus we only need to consider the finite networks specified by the inequality .the network attachment mechanism , as in model a , will depend on the generating equation in the sense that every triple of integer numbers obeying eq.[he1 ] forms a clique in the network .the resulting network is very well connected since every couple of numbers at a certain time step ( when is sufficiently large ) forms a clique satisfying eq.[he1 ] .nevertheless interesting topological properties emerge . in the left panel of fig.[h1 ]we show the degree distribution for this network .the shape of the distribution is well approximated by a power law with exponent for several decades of values of . very interestingly ,after falling the distribution begins to rise again . on the right panel of the same figurewe show the degree of the vertices against their label number . this number can be considered as the birth time of the vertex so that we can observe the evolution of the hubs of the network with ageingthe degree is nearly constant and very high for small values of and it drops rapidly to small values for ( this comes from eq.7 , when then ) .notice that the power law exponent is considerably lower ( at least from the point of view of network theory where power law exponents usually range between and ) than the one we recovered in the pythagorean case .we interpret this weakening as the result of the upper thresholds we imposed to the network which , in this case , have a much stronger effect on the degree distribution than in the pythagorean case .  in fig.[h2] we show the clustering analysis for the network . on the left panelwe show the clustering coefficient , defined in eq.[n0 ] , against the degree , while on the right panel we show the average clustering coefficient against the degree. we can observe that it is very high for small values of , where the degree distribution is a power law , then it is nearly constant for several decades and finally it falls rapidly to zero for high values of the degree .this behaviour has to do with the clique structure of the network . as soon as isolated cliques form , that is for , then its value falls as new connections are added and the degree increases .left panel : degree distribution for the deterministic network generated by the rule in eq.[he1 ] for .right panel : representation of the aging of the hubs for the same network for . is the degree of the vertices , while is their label .small values of correspond to old vertices.,title="fig:",scaledwidth=48.0% ]  left panel : degree distribution for the deterministic network generated by the rule in eq.[he1 ] for .right panel : representation of the aging of the hubs for the same network for . is the degree of the vertices , while is their label .small values of correspond to old vertices.,title="fig:",scaledwidth=48.0% ]   left panel : clustering coefficient against the degree for the deterministic network generated by the rule in eq.[he1 ] for .right panel : average clustering coefficient against the degree for the deterministic network generated by the rule in eq.[he1 ] for .,title="fig:",scaledwidth=48.0% ]  left panel : clustering coefficient against the degree for the deterministic network generated by the rule in eq.[he1 ] for .right panel : average clustering coefficient against the degree for the deterministic network generated by the rule in eq.[he1 ] for .,title="fig:",scaledwidth=48.0% ]   we have introduced the networks generated by diophantine equations focusing on two classical problems of number theory widely studied since antiquity .we investigated first the pythagorean problem of finding all triples of positive integers that can be represented geometrically as legs of a right - wristed triangle .we studied then the ( closely related ) problem of determining the set of positive integers that can be represented as the sum of two squares .these two questions , despite their apparent simplicity , have not yet been answered analytically in their full generality .we noticed that these classical problems can be naturally translated into the language of network theory and studied them numerically in their finite versions ( that is by not allowing integers to be greater than a fixed upper threshold ) .it is then easy to see that the equations considered generate clique structured networks , in an interesting analogy with many relevant real world networks such as social and biological networks .  in the pythagorean case , where the network evolves through the deterministic attachment mechanism specified by eq.3, we recovered a degree distribution consistent with a power law with exponential cut - off , in accordance with the behaviour displayed by many real world random networks .we then performed an ageing analysis and showed that , in sharp contrast with stochastic preferential growing networks , hubs form between young and middle aged vertices ( the degree of each vertex , after a while , freezes ) .the power - law , in this case , is clearly not a by - product of any preferential attachment or fitness mechanism , but a consequence of the algebraic structure of eq.3 itself .this shows once more that preferential attachment is a sufficient but not a _ necessary _ condition to the generation of scale - free topologies .as far as we are aware the network generated by eq.3 is the first genuine example in literature of a _deterministic _ scale - free network ( at least in some approximation ) which does not grow preferentially .power laws are normally regarded as a sign of complexity and we found it particularly stimulating to detect them not only in natural phenomena but also at the roots of number theory itself , indicating a fascinating connection between complex systems and pure mathematics .we then studied the second classical problem mentioned above studying the network generated by eq.[he1 ] and recovered a degree distribution approaching a power - law with a relatively high exponent for several decades of values of .we believe that the study of networks generated by equations will turn out to represent an important field of interdisciplinary research and we hope that our discussion will be useful to researchers in both complex network theory and pure mathematics as a first step to recover more rigorous and general results . in particular , in future work it will be stimulating to investigate the relation between an equation s algebraic properties ( or analytic properties if , for instance , differential equations are considered ) and the associated network s topology .diophantine networks may find interesting theoretical and technological applications .first and foremost they can be used to design deterministic toy models of complex systems , allowing to find practical ways to build networks systematically without having to deal with different degrees of stochasticity in their architecture ( in this sense they could play a role similar to the one played by the euclidean grid in other traditional contexts ) .diophantine networks may also allow to overcome some limitations intrinsic in the preferential attachment method , namely the fact that the hubs are usually the oldest vertices and never `` die '' , leaving little room for network rejuvenation . here instead we have a dynamic situation where hubs are dominant , during network growth , only for a limited period of time. there may be situations where this feature is realistic ( for instance in models of technological developments ) .another interesting application of the theory may be found in the study of the behaviour of dynamically evolving agents .

Abstract: we introduce a new class of deterministic networks by associating networks with diophantine equations , thus relating network topology to algebraic properties . the network is formed by representing integers as vertices and by drawing cliques between vertices every time that distinct integers satisfy the equation . we analyse the network generated by the pythagorean equation showing that its degree distribution is well approximated by a power law with exponential cut - off . we also show that the properties of this network differ considerably from the features of scale - free networks generated through preferential attachment . remarkably we also recover a power law for the clustering coefficient .   we then study the network associated with the equation showing that the degree distribution is consistent with a power - law for several decades of values of and that , after having reached a minimum , the distribution begins rising again . the power law exponent , in this case , is given by we then analyse clustering and ageing and compare our results to the ones obtained in the pythagorean case .


the efficiency of transport systems has been of interest in various fields , including physics , biology and engineering . in transport processes ,the item being transported usually follows the steepest descent of the underlying surface , e.g. , water flowing down the slopes of a mountain .flow in networks has been modeled by toroczkai et al . by the introduction of local gradients on a substrate network. the gradient network defined on this network has provided significant insights into the dominant structures that provide transport efficiency .they have considered a fixed network of nodes with a scalar potential , at each node .the gradient of the potential at each node is a directed edge which points from to the neighbor with the minimum potential among all the neighbors of .toroczkai et al have shown several properties of gradient networks which we briefly summarize .an interesting topological property of the gradient network is that its in - degree distribution is scale - free for both scale - free ( sf ) and erds - rnyi ( er ) substrate networks  .a gradient network with a non - degenerate potential distribution , is a group of trees , hence no loops exist in this network other than self - loops .only this property makes the gradient networks very common in seemingly unrelated problems , i.e. , synchronization in oscillatory networks .the relationship between network topology and congestion has also been investigated by introducing a measure of congestion , the jamming coefficient .this measure involves the ratio of the number of nodes that receive at least one gradient link , and the number of nodes that send a link . by definition, every node sends one out link , therefore the number of senders in the network , .the jamming coefficient is    the operations and denote the statistical averaging over local potentials and networks respectively .maximal congestion occurs at , and no congestion occurs when every link receives a gradient link , corresponding to .it was also found that is independent of number of nodes for scale free substrate network , and these networks are not prone to maximal jamming .the jamming coefficient was extensively studied by park et al . where they compared it for er and sf networks with the same average degree , , for . with randomly assigned potentials on each node , below , they found that er networks are less congested than sf networks .  herewe introduce a monte carlo optimization scheme that reduces jamming significantly and introduces structural correlations into the system that are not built in .the remainder of the paper is organized as follows . in section [ sec : opt ] we introduce the algorithm for optimizing jamming coefficient that is initially calculated from randomly assigned scalar values at each node .we compare the optimized jamming values of a scale - free network and an erds - rnyi network for various values of average degrees . in section [ sec : structure ]we investigate the structure of the optimal gradient network .in particular we study the degree distribution and the correlations between the degree and the potential of each node . finally in section [ sec : conc ] we discuss the implications and the possible extensions of optimal gradient networks .previous work  has focused on random gradient networks where the potential on each node has a randomly assigned value .more generally , the potentials can be a dynamic quantity evolving in time due to perturbations , sources and sinks internal and external to the system of interest .alternatively , the potentials can evolve to become correlated to the network properties such as its degree distribution .for example , consider the networks of routers where every router has a capacity .if a router is central and highly connected , it usually has a higher capacity in order to handle the traffic en - route effectively .recently , a congestion aware routing algorithm has been introduced where the transport on the network of routers is driven by congestion - gradients .  herewe develop a monte carlo algorithm to achieve two goals : reduce jamming in the network , and observe the emerging optimal correlation between in - degree and potential of each node . for a given network and potential distribution , we redefine j in eq .[ eq : jam ] as where by definition .the initial potential distribution is chosen from a gaussian distribution , and at each iteration the potential of a random node is modified such that global congestion is reduced .we use a metropolis algorithm with the following steps :  1 .  pick a node , at random .2 .  vary by , i.e. , where is a gaussian random variable with variance .3 .  recalculate with .4 .  accept with probability  ] .for example , nodes with degrees , have a roughly gaussian potential distribution .for higher , the distribution narrows down and shifts toward larger values . for nodes with ,all nodes have large potentials , within range  ] where , is the number of neighboring links node has , is the capacity of node , is the weight of the incoming link from j to i , and the operation = 0 $ ] if .if the weights and capacity of all nodes are 1 , then this definition of reduces to the one in eq .[ eq : jam ] without the averaging . with this definition and the optimization method introduced in the paper , it is possible to study real world networks and get insights to the dominant structures of transport for these systems .the author thanks adilson e. motter for useful discussions and suggestions , gregory johnson and frank alexander for the careful reading of the manuscript .this work was carried out under the auspices of the national nuclear security administration of the u.s .department of energy at los alamos national laboratory under contract no.de-ac52-06na25396 and supported by the doe office of science ascr program in applied mathematics research .

Abstract: gradient networks can be used to model the dominant structure of complex networks . previous works have focused on random gradient networks . here we study gradient networks that minimize jamming on substrate networks with scale - free and erds - rnyi structure . we introduce structural correlations and strongly reduce congestion occurring on the network by using a monte carlo optimization scheme . this optimization alters the degree distribution and other structural properties of the resulting gradient networks . these results are expected to be relevant for transport and other dynamical processes in real network systems .   * complex networks have been shown to offer a powerful framework for the study of dynamical processes in complex systems . a fundamental question in this context is to identify how the structure of dominant connections influences the dynamical properties of the entire system . gradient networks have been introduced in references precisely to characterize the structure governing transport in complex networks . remarkably , similar structures have been identified recently in the optimization of synchronizability in oscillator networks . here , we consider the problem of optimization of transport and show that by using a monte carlo approach congestion can be reduced in complex networks . we discuss the structural correlations that emerge with optimization and its implications for real systems . *


the rapid growth of online social networks ( osns ) , such as twitter or facebook , led to over two billion accounts being active in 2014  , connecting over one quarter of the world s population and of online u.s .adults  . bridgingthe gap between social sciences , and information and communication technologies , osns constitute a crucial building block in the development of innovative approaches to the challenges society faces today .however , technological progress over the last decade has dramatically outpaced our understanding of the new systems and their impact on society .this lack of understanding of the complex dynamics of the digital world reveals the pressing need for a comprehensive and concise theory to describe and model the online world as a set of interacting networks . in this context , the activity of users has become a resource to be fought over and which drives competition in the digital world .digital services only persist if they can attract and maintain users attention . here, we describe the online world as a complex , digital ecosystem in which interacting networks constitute species in competition for survival . within this ecological context, the extinction of a network corresponds to the complete absence of activity , as an entirely passive network can not function and , more importantly , will not attract new users .in a recent study  , we demonstrated that a moderate number of identical networks can coexist in the digital ecosystem , in contrast to the principle of competitive exclusion  .in contrast to that previous work , in this paper we address the heterogeneity of networks .networks can differ in functionality , features , and most importantly they can address different peer groups . here , we show how the effect of different overlapping peer groups can be described in terms of different degrees of network fitness .we find that under certain conditions , the heterogeneity of degrees of fitness can impede coexistence which would indeed be possible for identical networks .this effect is particularly important for the competition between local networks and an international network .unlike users of local networks , users of the international network have the possibility to interact with people in other countries , providing this network an advantage over local ones , similar to a higher fitness of a certain species .a proper modeling of this effect requires taking into account the network of interactions among countries in the world , which results in a highly complex and non - linear dynamical system made of interconnected multilayer networked entities we are hence dealing with networks of multiplex networks  , in contrast to ref .  .besides , inter - country interactions induce a different type of bifurcation as the symmetry of the previous model is broken , which constitutes a fundamentally new behavior not observed in  .empirical observations have shown that facebook expanded massively in the middle of the first decade of this century , starting in the us , when local networks were the most popular services in most countries .only a few years later , facebook had become the most popular network in most countries .so , is the fate of the digital world to become dominated by a single `` big brother '' as it takes over all our digital interactions ?alternatively , is digital diversity possible from a system - level perspective ? in this paper , we show that due to the nonlinear character of our model , the answer to both questions can be positive or negative depending on a range of parameters and , quite surprisingly , depending on chance .as we will show , our model , despite its simplicity and the limited number of parameters , is able to describe surprisingly well the complex behavior of globally interacting online social networks .the digital world consists of highly connected and strongly coupled interacting subsystems .these basic building blocks are single networks , each of which obeys specific dynamics in the absence of coupling to the whole system .so the complexity of the digital world is a consequence of both the dynamics of networks in isolated environments and the interactions between many such networks .finally , not all of these building blocks are identical . instead, different networks address different peer groups or have different functionalities .hence , to reveal the fundamental mechanisms that determine the fate of the digital world , it is necessary to understand the interaction of heterogeneous networks , each driven by intrinsic dynamics .the key actors in the digital world are osns ; loosely defined as web - based platforms that enable digital social interactions over the internet .however , societies were organized as networks long before osns were even thought of . from this point of view , the growth of osns can be described through the dynamical processes by which people in the traditional offline social structure come to engage in osns .the topology of the osn is now the digital counterpart of the underlying offline social network  .  in isolation, this process of formation can be described by a set of simple dynamical mechanisms  .the system is initially given by an empty osn and the underlying social structure .individuals can be in three different states : active , passive or susceptible . while active and passive nodes exist in the online as well as the offline networks , susceptible nodes are only present in the latter .a susceptible node can join the osn via two different mechanisms : a _ viral activation _ effect which means that a susceptible node becomes active due to the influence of an active neighbor in the offline network and a _ mass media _ effect , which represents the spontaneous activation of a susceptible node .in addition , an active node can become passive spontaneously ( _ deactivation _ ) and a passive node can become active again due to the influence of an active neighbor ( _ viral reactivation _ ) .see fig .[ fig_panel1]d for a visualization of these mechanisms .notice that , in the long time limit , when the number of susceptible individuals is basically zero , this dynamics is equivalent to the susceptible - infected - susceptible model widely used in epidemiology  . as found in , this implies that online social networks can either exhibit a sustained activity ( similar to the endemic phase ) , or they can become entirely passive ( similar to the healthy phase ) .the evolution of osns rarely takes place in isolation .nevertheless , we found a perfect case study in the slovakian social network `` pokec '' , which , due to the particularities of the country , has been growing in quasi - isolation for more than ten years . by analyzing the evolution of the topology of the social contact graph of `` pokec '', we were able to rigorously validate our model .quite remarkably , with only two parameters , the model reproduced the entire topological evolution with astonishing precision ._ viral activation_ and _ viral reactivation _occur at the same rate , , and the ratio between this rate and the rate of _ mass media _ influence , , governs the topological evolution .in particular , we observed that the real system underwent a dynamical percolation transition ; that is , a phase transition between a disconnected phase and a phase in which a macroscopic fraction of the system is connected .the position of this transition is controlled by , due to the complementary roles that the _ viral activation _ and _ mass media _ effects play in the topological evolution of the network ( the former tends to connect components ; whereas the latter tends to create new components ) .finally , without loss of generality , we set the _ deactivation _rate to which is equivalent of fixing the timescale of the model .  to sum up , in our previous work , we were able to rigorously validate the dynamics ruling osns in isolation ; the fundamental building blocks of the digital world .these findings constitute the foundation for the development of a more comprehensive theory of interacting heterogeneous networks .the simultaneous existence of multiple digital services in competition for the attention of users suggests an ecological perspective from which to explain the prevalence of a given network or the coexistence of multiple networks . in ecology theory ,the principle of competitive exclusion  states that multiple species in competition for the same resource can not coexist , as even the slightest advantage of one species over the other is successively amplified ; a mechanism referred to as `` rich get richer '' or preferential attachment  .this eventually leads to the extinction of the inferior species .the key principle that drives the competition between osns is the fact that , due to the physical and cognitive limitations of users , the time they devote to online activities is limited . as a consequence , the viral parameter , , constitutes a conserved quantity that is nevertheless distributed between the competing networks as , where represents a normalized set of weights , that is , and is a vector denoting the fraction of active nodes ( activities ) in the different networks . in general, users are more likely to subscribe to and engage in networks that are more active .therefore , the viral activity of each network must be a function of the activity of the network itself .in particular , we model this by assuming that the weighting is a function , such that . in  we proposed the particular form : ^\sigma}{\sum_{j=1}^{n_l } \left[ \rho_j^\mathrm{a } \right]^\sigma } { \,,}\label{eq_weights}\ ] ] were denotes the number of networks .this choice allows us to interpolate between a set of independent networks ( ) and highly coupled ones ( ) .the activity affinity parameter , , then quantifies the tendency of users to subscribe to or engage in more active networks .interestingly , in contrast to the principle of competitive exclusion , multiple networks can coexist because the `` rich get richer '' mechanism is damped by the diminishing returns of the dynamics of network evolution . for details, we refer the reader to ref . . in the following ,we take into account the heterogeneity of networks induced by different groups of individuals that can subscribe to the different networks .these aspects are not discussed in ref . and have important implications and applications , as we will show .as mentioned above , since its official launch in 2004 , facebook has become the most popular osn in most countries ; even in countries where there was already a popular osn before facebook was launched . to mimic the real evolution of the digital ecosystem at the worldwide scale, we assume that one local network exists in each country in addition to a globally operating , international network ( see fig . [fig_panel1]a ) . in the us ,both networks are launched at the same time ; whereas the international network is launched with a delay in the remaining countries , to take into account the initial prevalence of local networks.  once launched , the international network provides the user with the possibility to connect to individuals in different countries , in contrast to local networks , making it more attractive to users . for a given country , the advantage of the international network is directly related to the abundance of social ties between that country and the rest of the world .we use passenger air travel data as a proxy for the abundance of such ties .this choice is justified by the strong correlation between air travel flows and further measures of inter - country exchange , for instance email communication  or twitter activity  .users in country experience the greater attractiveness of the international network as they perceive its activity with respect to the population of their own country and also with respect to their contacts in other countries . to account for this on a coarse grained level , in eq ., we replace the activity of the international network by an effective activity as follows    where }     \label{eqn_def_omega}\ ] ] denotes the fraction of the number of air travel passengers between countries and , ; and , the population of country . notice that , in an ecological context , this corresponds to increased fitness of the international network . in eq . , we have implicitly assumed proportionality between the number of passengers and the number of contacts in the respective countries , namely .finally , note that the arbitrary normalization in eq .  serves the sole purpose of ensuring that reasonable values for the parameter are of the order of unity .hereafter , we decompose the international network into a set of disjunct coupled subnetworks operating in each country and in competition with the respective local network ( see fig . [fig_panel1]b ) .these subnetworks are nevertheless not independent , as they are globally coupled via the effective activity defined in eq .and ultimately by the network representing the inter - country social ties . hence , our model forms a network of networks  , where each node in fig .[ fig_panel1]c represents a three - layer multiplex network  in which the bottom layer corresponds to the underlying social structure and the two upper layers denote the local and international networks operating in the respective country ( see fig . [fig_panel1]d ) .     to understand the qualitative behavior of the system , in this section we present a double meanfield approximation of the system .this reduces the system given by a network of networks to a set of evolution equations of the average activity in the international network and in local networks .as we show in the following section , the results of the full model with heterogeneous topologies exhibits similar behavior to that encountered by the double meanfield approximation .the first meanfield approximation consists of assuming a fully mixed homogeneous population in each country .let denote the fraction of active users in network in country and the fraction of nodes susceptible to joining this network .then , the fraction of passive users is given by .as explained above , in each country the virality is distributed between the local and international network via the weight functions and , as introduced in eq . . here, denotes the effective activity of the international network as defined in eq . .the evolution equations of the resulting system represent a generalization of the evolution equations for identical networks which we derived in  , where one replaces the activity of the international network with the effective activity from eq . .this procedure yields : -1 \biggl\ }  + \mu_{i , l } \rho_{i , l}^\mathrm{s}\\ \dot{\rho}_{i , l}^\mathrm{s } & = -\mu_{i , l } \rho_{i , l}^\mathrm{s } \biggr\{1+\nu{ \left < k\right>}\rho_{i , l}^\mathrm{a }  \biggl\ } { \,.}\label{eq_meanfield_full }  \end{split}\end{aligned}\ ] ] as shown in  , we further assume the the same linear relationship between virality and media influence in each country , that is : as shown in  , the value of does not affect the stability of the system . in what follows, we perform the stability analysis in the limit .this decouples the evolution of from , so that we only have to consider .plugging in the weights function defined in eq .  andthe effective activity from eq .yields the evolution equations for the activities of the local and international networks in country : ^\sigma}{[\rho_{i,\text{loc}}^{\mathrm{a}}]^\sigma + ( \rho_{i,\text{int}}^{\mathrm{a}}+\delta_i)^\sigma } [ 1-\rho_{i,\text{loc}}^{\mathrm{a}}]-1\right ] \\   \dot{\rho}_{i,\text{int}}^{\mathrm{a } } & = \rho_{i,\text{int}}^{\mathrm{a } } \left [ \frac{\lambda { \left < k\right>}(\rho_{i,\text{int}}^{\mathrm{a}}+\delta_i)^\sigma}{[\rho_{i,\text{loc}}^{\mathrm{a}}]^\sigma + ( \rho_{i,\text{int}}^{\mathrm{a}}+\delta_i)^\sigma } [ 1-\rho_{i,\text{int}}^{\mathrm{a}}]-1\right ] { \,,}\end{split }  \label{eqn_meanfield}\end{aligned}\ ] ] where .the second meanfield approximation consists of applying the hypothesis of a fully mixed homogeneous network for the inter - country social ties .we use and define the mean activity of the local networks as and the mean activity of the international network as .finally , our double meanfield approximation leads to the following system of coupled differential equations -1 \right ] \\\dot{y } & = y \left [ \lambda { \left < k\right>}\frac{(y ( 1+\bar{\omega}))^\sigma}{x^\sigma + ( y ( 1+\bar{\omega}))^\sigma } [ 1-y ]  -1 \right ] { \,,}\label{eqn_doublemf }  \end{split}\end{aligned}\ ] ] which has three relevant parameters : , , and .note that by setting to zero , we recover the equations for identical networks presented in  .in what follows , we discuss the dynamical properties of the system given by eq . .for constant , the system exhibits a saddle - node bifurcation at a critical value of the global connectivity ( see fig .[ fig_bif ] ) . above this point , coexistence is not possible and the only stable solutions correspond to the domination of either local networks or the international one . both above and below the critical value , the basin of attraction of the solution corresponding to the domination of local networks decreases with , whereas that of the international network increases ( see the rows of fig .[ fig_bif ] ) .furthermore , at the critical point , the basin of attraction of the international network is amplified discontinuously as the region of coexistence in the subcritical regime is now merged with the basin of attraction of the domination of the international network .  for constant ,the system also exhibits a saddlenode bifurcation at a critical value of the activity affinity . in  showed that the system undergoes a subcritical pitchfork bifurcation with respect to the control parameter , above which no stable coexistence is possible . breaks the symmetry of the pitchfork bifurcation and in this case the system undergoes a saddlenode bifurcation with respect to instead ( see bottom of fig .[ fig_bif ] ) .this behavior is well known in bifurcation theory and results from adding a small error term to the normal form of the pitchfork bifurcation ( see supplementary materials ) .the evolution of the basins of attraction is more complex compared to the previous case . below the critical point , both basins of attraction increase with . above the critical point , the basin of attraction of the local network increases whereas the basin of attraction of the international network decreases with ( see the columns of fig .[ fig_bif ] ) .this is particularly interesting as it implies that an intermediate value of the activity affinity just slightly above the critical point represents the worst scenario for the survival of local networks , since at this point the size of the basin of attraction of the domination of the international network is maximum .  .the white area denotes the parameters for which coexistence is possible .the blue area denotes the parameters for which only domination of either local or the international network can occur . at the blue line, the system undergoes a saddle - node bifurcation in which the stable coexistence solution disappears .the red line denotes the combination of parameters for which the system switches attractors for the initial conditions given by eq . , where we use to reflect that the us contributes about of the population .the red region shows the parameters for which the solution of domination of local networks is reached for these initial conditions . above the red line , the system approaches the domination of the international network ( solid red line ) or the coexistence solution ( dashed red line ) .[ fig_phase ] ]  in fig .[ fig_phase ] , the blue line indicates the critical line in the - plane , which separates a phase in the parameter space where coexistence is possible ( white region ) and one in which only domination can occur ( blue region ) .however , the increasing size of the basin of attraction of the domination of local networks above the critical point with respect to can dramatically alter the fate of the system for a given set of initial conditions . assume , for instance , that the international network dominates in the us and starts with a significant delay in each other country , which causes the local networks to dominate in those countries .at the time when the international network is launched globally , the state of the system can be approximated as follows \\y_0 & = \beta \left[1-\frac{1}{\lambda { \left < k\right>}}\right ] { \,,}\label{eqn_initial }  \end{split}\end{aligned}\ ] ] which we now use as initial conditions to study the further evolution .notice that if one network dominates in country , its activity is given by .hence , the initial conditions given by eq .reflect the fact that local networks dominate in the fraction of the system and the international one dominates in the remainder .the evolution of the basins of attraction makes the system approach different stationary solutions from these initial conditions for different parameters .below the red line in fig .[ fig_phase ] , the system approaches the domination of local networks starting from the initial conditions given in eq . . above this line ,the system either approaches coexistence ( white area ; crossing dashed red line ) or domination of the international network ( blue area ; crossing solid red line ) .this means that in the red region , when the international network is launched globally , it is not able to overcome the initial advantage of the local networks due to its earlier launch .  to conclude, the double meanfield approximation predicts that intermediate values of the activity affinity most favor the international network ; whereas the local networks can dominate for a high activity affinity and low global connectivity .we confirm these findings by numerical simulations in the following section .     in this section, we go beyond the meanfield approximation and study , by means of numerical simulations , the effects of the real topology of inter - country social ties and of underlying social structures . to this end , we use the air travel network ( see fig . [ fig_panel1]c and materials and methods ) as a proxy for inter - country social ties and construct 1:1000 scaled synthetic networks to model the structure of the countries with most internet users ( see tab . [ tab_countries ] ) . to generate these networks ,we make use of a model introduced in  , which produces realistic topologies of the traditional offline social networks , including heterogeneous node degrees and a high level of clustering ( see materials and methods ) .  fig .[ fig_w8 ] shows results from our model for the set of parameters that best matches empirical observations , as explained in the following section .the international network starts with a delay in all countries except the us ; so that initially in these countries the respective local network dominates .after some time , the international network obtains a significant advantage and quickly takes over in most countries .      to further study the properties of the model presented here, we define the relative prevalence of the international network compared to local networks as : where denotes the number of countries , and and are the activities of the international and local networks in country in the stationary state . with this definition, a value of implies that local networks dominate in most countries , whereas corresponds to the domination of the international network .the relative prevalence of the international network averaged over many realizations is shown in fig .[ fig_density ] for different values of as a function of the activity affinity , , and the launch time delay , . for small values of , we observe that when is small , the international and local networks coexist and we observe values around for the relative prevalence ; then , increasing favors the international network , which dominates for values of ( see fig . [ fig_tcut]a and b ) . for larger values of , this behavior smoothly translates into a more complex case , which we discuss below .we observe in fig .[ fig_density ] that for launch time delays , the actual length of the delay becomes irrelevant .this behavior corresponds to the limit of saturation of the evolution of local networks before the international osn is launched ; as discussed in the previous section .we consider this limit by averaging over regions with in fig .[ fig_tcut]c , which yields a two dimensional parameter space - .indeed , numerical simulations of the full model confirm the results from the meanfield analysis ; in particular the complex role of the activity affinity .for small and , local networks and the international osn can coexist .increasing or favors the domination of the international network , which gives rise to the blue `` v''-shaped region around .this corroborates the saddlenode bifurcation predicted by the double meanfield approximation .see supplementary video for an explicit realization . for high values of and small values of ( red region in the bottom right - hand corner of fig .[ fig_tcut ] ) , local networks dominate .note that partial states are also possible , in which the international network dominates in some countries and local networks dominate in the remaining countries .see supplementary video for an explicit realization of this case .  between the regions of domination of the international network and of local networks, there is a region in which the final fate of the system varies significantly between different realizations of the model ( `` coinflip region '' ) . in this region , if the international network wins initially in the us , it will become dominant globally ; otherwise , local networks maintain their initial prevalence .although in this region the prevalence of the international network averaged over many realizations is about , as in the coexistence region in the bottom left - hand corner of fig .[ fig_tcut ] , the behavior of the system differs dramatically from one to another . in the coexistence region, each realization of the model leads to the same final state : coexistence of local networks and the international osn .in contrast , in the coinflip region , coexistence is not possible , as this region of the parameter space corresponds to the supercritical regime ( the blue area in fig .[ fig_phase ] ) . in the coinflip region , about of the realizations end up with domination of the international network , whereas the remaining lead to the domination of local osns . as a consequence , even if we know the exact parameters , it is impossible to predict the fate of the system beforehand .we can summarize these findings as follows .a higher value of , which is a measure of the global connectivity of society , favors the prevalence of the international network and hinders the survival of the local ones .the role of the tendency of individuals to participate in more active networks ( activity affinity ) , , is particularly interesting .low values allow the networks to coexist , whereas intermediate values always lead to the prevalence of the international network and the extinction of local osns . a high activity affinity , however , enables the prevalence of local networks and thus can even lead to the extinction of the international network .   in this section, we compare the results of our model with empirical data on the recent expansion of facebook at the cost of many local networks .in particular , we consider the evolution of the number of countries in which local networks ( i.e. networks that are not facebook ) are the most popular ones , as measured in  using alexa traffic data ( see fig .[ fig_compare_empiric]b ) . we observe a significant decline of this number , which rules out the possibility that the empiric case corresponds to the domination of local networks .because the past can be considered a single realization of a stochastic process  , the empiric case can still be within the coinflip region of our model where by chance the international network was more successful .hence , we will perform the following comparison only for realizations of our model in which local networks do not dominate .the intrinsic timescale of the model is arbitrary and hence has to be mapped to real time .the optimal mapping is given such that it produces the best agreement with the empirical data .we quantify the agreement between model results and empirical data using the sum of the squared distances between the data points and model results .in particular , we use the statistic defined as : ^ 2 { \,,}\label{eqn_capital_gamma}\ ] ] where denotes the number of countries where the local network is more popular and is the corresponding result from the model .the index denotes the individual datapoints and is the estimated variance of the data ( supplementary materials ) .the real time , , is a linear function of the model time given by , where is the starting year and represents the time stretch : how many years of real time correspond one model time step . for a given set of parameters , , , and , the optimal values for and are those that minimize , as shown in fig .[ fig_compare_empiric]a .we can also use the statistic to estimate the parameters , , and which best reproduce the empirical observations . in fig .[ fig_compare_empiric]c , we plot the values of as a function of , , and , where at each point we applied the respective best time mapping , as described above .these results are averaged over several realizations of the model ; however , in the coinflip region we exclude realizations where the local networks dominate , to mimic the empirical case .interestingly , the overall best fit is achieved for at and , which lies in the coinflip region ( the optimal value of is statistically consistent with the model , given the number of degrees of freedom in the data ) with a probability for domination of the international network of .this scenario corresponds to the time mapping and , meaning the system started at the beginning of ; while the launch time delay of in the model translates to years in real time . in fig .[ fig_compare_empiric]b , we show the evolution of the number of countries where local networks are more popular for the optimal fit from the model .understanding the complex dynamics of the digital world constitutes an important challenge for interdisciplinary science . to meet this challenge , here we describe the the worldwide web as a complex , digital ecosystem in which interacting networks play the role of species in competition for survival .in particular , we study the competition between local networks operating in single countries and an international network that operates in all countries .therefore , a proper description of this system must necessarily involve the network of worldwide social interactions between different countries .we show that the effect of inter - country social ties can be mapped to the increased fitness of the international network by means of an effective activity .interestingly , there is a critical global coupling strength below which networks can coexist .however , above that threshold , only domination is possible : in general , local networks become extinct with a high probability .yet , we find that if local networks are launched earlier they can persist and dominate the international network , which happens only if local networks have accumulated a sufficiently large active userbase when the global launch of the international network takes place .the accumulation of a sufficient base depends on the parameters ; and for certain parameters on chance . for these parameters the final state of the system whether local networks dominate or become extinct can be completely unpredictable , as it varies randomly between different realizations of the model .quite remarkably , a thorough comparison of our model with empirical data from the recent takeover of facebook indicates that the most probable launch date of facebook was at the beginning of and its global launch was in late .facebook was in fact started in , but opened to the public in ; in good agreement with the estimate from our model .moreover , according to google trend data ( see supplementary materials ) , was the year when the global search volume for facebook started to increase rapidly .last but not least , our best estimation of the model parameters corresponds to the `` coinflip '' region , which means that the observed takeover of facebook only had a probability of around . with a probability , we would have been living in a world where each country had its own successful local network and a network like facebook would not exist  .our findings suggest interesting future lines of research . on the one hand , even without adjusting the parameters on a country - by - country level , our model reproduces the main features empirically observed in the takeover of facebook and the extinction of local networks in most countries for a certain parameter region .it remains an interesting task for future research to further increase the precision of the model .this could be done by improving the proxy for the similarity between countries or by adjusting parameters on a country - by - country basis . on the other hand ,the model could be extended to account for several international networks and to study their global competition . for a second international network to overcome the first , a certain minimal difference of fitness is needed ; which could be the result of different properties of the networks , such as features or functionalities .finally , random fluctuations of fitness could be incorporated to describe darwinian selection in the digital ecosystem .we use the model  to generate the synthetic networks for the underlying societies in each country .the model allows us to specify the degree distribution and the level of clustering .the model is based on a circle as a hidden metric space and works as follows :  1 .all nodes are placed on the circle with a randomly assigned variable , , which represents the polar coordinate . is uniformly distributed in . to keep the average node density on the circle constant, its radius grows linearly with the number of nodes , to satisfy .we assign each node a second hidden variable , , which represents its expected degree . is drawn from an arbitrary distribution .a pair of nodes is connected with a probability , , that depends on their hidden variables and : with . here, denotes the geodesic distance between the two nodes on the circle and the mean degree .then , the expected degree , , of a node with hidden variable can be shown to be proportional to  . as a consequence ,the degree distribution , , of the network follows the shape of the distribution .   here , we use an exponential distribution with .we set the parameters and . after generating the networks , we remove nodes with zero degree .[ fig_s1_properties ] shows the degree distribution and the clustering spectrum for the synthetic network created for the us .air travel data aggregated on a country basis were taken from http://visualising.org/datasets/global-flights-network ( date of access july 2014 ) .the original data can be accessed at http://openflights.org/data.html .the network on a country basis incorporates 230 nodes and 4600 weighted edges which correspond to the number of routes between countries , i.e. the number of total flights .the dataset contains around 60.000 of such flights .we extract the subnetwork ( see fig .[ fig_panel1]c ) by constraining to the countries listed in tab .[ tab_countries ] .this work was supported by : the european commission within the marie curie itn `` isocial '' grant no .pitn - ga-2012 - 316808 ; a james s. mcdonnell foundation scholar award in complex systems ; the icrea academia prize , funded by the _generalitat de catalunya _ ; the mineco projects nos .fis2010 - 21781-c02 - 02 and fis2013 - 47282-c2 - 1-p ; and the _ generalitat de catalunya _ grant no .furthermore , m. b. acknowledges support from the european commission lasagne project no .318132 ( strep ) .manlio de domenico , albert sol - ribalta , emanuele cozzo , mikko kivel , yamir moreno , mason a. porter , sergio gmez , and alex arenas . mathematical formulation of multilayer networks ., 3:041022 , dec 2013 .

Abstract: the overwhelming success of online social networks , the key actors in the web 2.0 cosmos , has reshaped human interactions globally . to help understand the fundamental mechanisms which determine the fate of online social networks at the system level , we describe the digital world as a complex ecosystem of interacting networks . in this paper , we study the impact of heterogeneity in network fitnesses on the competition between an international network , such as facebook , and local services . the higher fitness of international networks is induced by their ability to attract users from all over the world , which can then establish social interactions without the limitations of local networks . in other words , inter - country social ties lead to increased fitness of the international network . to study the competition between an international network and local ones , we construct a 1:1000 scale model of the digital world , consisting of the 80 countries with the most internet users . under certain conditions , this leads to the extinction of local networks ; whereas under different conditions , local networks can persist and even dominate completely . in particular , our model suggests that , with the parameters that best reproduce the empirical overtake of facebook , this overtake could have not taken place with a significant probability .


stochastic resonance ( sr ) phenomena involve optimizing a system performance measure via adding noise , and have received considerable attention in the past few decades .the prototype sr model is an overdamped bistable system , which was used to describe the earth s climatic change .since then , this generic system is a frequently used model for characterizing sr phenomena in diverse scientific fields .moreover , many experimental verifications of sr effects have been demonstrated for a bistable characteristic , such as in a schmitt trigger , bistable ring laser , paramagnetically driven bistable buckling ribbon , bistable electron paramagnetic resonance systems , bistable superconducting quantum interference devices , vertical cavity surface emitting lasers , etc .in addition , new sr - type phenomena were also observed in bistable systems subjected to suprathreshold input signals .recent research has focused on potential sr applications in signal transmission , estimation and detection .signal detection theory offers a powerful tool for analyzing linear systems , such as the matched filter with a gaussian white noise background and its impulse response matched to the input signal maximizing the filter output signal - to - noise ratio .these nonlinear systems , thereby , are potentially rich in signal detection applications , though generally more difficult to theoretically tackle .a variety of appealing nonlinear systems or models exhibiting sr effects have been explored as detection devices , such as the threshold detector , electronic circuits , biological systems , neuron models , bistable systems , etc .although many researchers stressed that sr can not do wonders in signal detection , in the sense that nonlinear systems exploiting sr can never do better than the corresponding linear systems in the presence of gaussian white noise , some promising results have been obtained for non - gaussian noise conditions , the computational and design efficiency of nonlinear systems , noise - floor limited systems , psychophysical experiments or models , and parallel arrays of nonlinear devices via the suprathreshold sr effect .  in line with this , we investigate further the performance of bistable systems versus matched filters in detecting bipolar pulse signals. some new merits of bistable systems will be emphasized in the comparison of results .section [ sec : ii ] develops a theoretical non - stationary probability density model of the bistable system .this approximate theory describes the sr phenomenon in the bistable system well , and agrees with the numerical simulation results .the comparison of the pertinent signal detection statistics , between bistable systems and matched filters , demonstrates the fact that the bistable system is a suboptimal detector against the matched filter . beyond this fact ,we evaluate in detail the detectability of bistable systems , versus matched filters , in detecting pulse signals with disturbed pulse rates and unknown arrival times in sec .[ sec : iii ] . from the power spectrum analysis of bistable system outputs ,the different pulse rates of the disturbed signal can be clearly discriminated .the matched filter , however , is more like a lowpass filter , wherein too much noise at low frequency bands weakens its resolution . in the case of detecting the unknown arrival time of received signals , the bistable system , with lower computational overhead ,is an appropriate substitute for the matched filter .the corresponding numerical simulation results also indicate that the bistable system is a more competitive detector than the matched filter , with respect to its robustness and computational efficiency .this may be of importance to bistable electronic or optical devices utilized in signal detection .we argue that the potential applicability of the bistable system deserves further study in other signal processing problems .here , we consider the detection problem of a bipolar pulse signal and introduce two hypotheses ,\nonumber \\ \textit{\textbf{h}}_{1 } : s(t)=+a,\ : [ ( n-1)t_p,\ : nt_p ] , \label{eq : one}\end{aligned}\ ] ] where .the input pulse signal , for , has the pulse duration time and pulse amplitudes ( see an example of fig . [ fig : one ] ) .the background noise is additive gaussian white noise with autocorrelation and zero - mean . here, denotes the noise intensity . the mixture of plus is then applied to a detector , such as the matched filter or the bistable system introduced below .   the bistable system under study is described as with real system parameters and  . is related to the system relaxation time .the dynamics of eq .( [ eq : two ] ) are derived from the symmetrical double - well potential , having the two minima .parameters and have the units of time and signal amplitude respectively , and define natural scales associated to the process of eq .( [ eq : two ] ) .  in each pulse duration , the system of eq .( [ eq : two ] ) is subjected to the constant amplitude or , with an additive input gaussian white noise . under these conditions , the statistically equivalent description for the corresponding probability density governed by the fokker - planck equation \rho(x , t ) , \label{eq : three}\ ] ] where and the fokker - planck operator is  . here , obeys the natural boundary condition that it vanishes at large for any .the steady - state solution of eq .( [ eq : three ] ) , for a permanent input at or , is given by , \label{eq : four}\ ] ] where is the normalization constant .this solution of eq .( [ eq : four ] ) is for global equilibrium in the double - well potential .now , we will seek the nonstationary solution of the fokker - planck equation , eq .( [ eq : three ] ) , in the case of an input transition from to , or vice versa .this calculation is carried out in appendix a. we show in appendix a that the transition from the stationary density corresponding to to the stationary density corresponding to , or vice versa , is dominated by an exponential temporal relaxation with a time constant .this allows us to deduce a response time for the bistable system , which is a measure of the time taken by the system to switch from one potential well to the other , when the input changes from to , in the presence of noise . in appendixa , the non - stationary solution for an input transition from to ( or vice versa ) is approximated with the two first terms from its asymptotic representation of eq .( [ eq : a11 ] ) , as \exp(-t / t_r ) , \label{eq : five}\ ] ] where are the steady - state solutions of eq .( [ eq : four ] ) . in eq .( [ eq : five ] ) , when , the term and starts with the initial condition of .as , the term , and tends to the stationary condition of .this theoretical model , although approximate , nicely captures the double role played by the noise , both in accelerating the switching dynamics between wells and in enhancing the fluctuations inside the wells . with this nonstationary solution of eq .( [ eq : five ] ) , as we will see , the nonmonotonic evolution of the signal detection statistics of the bistable system can be analyzed theoretically .we are interested in detecting the input pulse signal from the observation of the outputs of bistable system or matched filter .the input pulse signal is emitted at a rate of one waveform every and lasts over a pulse duration . in this detection problem , we assume that the interval is both known at the emitter and the detector .but , the waveform amplitudes and are not known at the detector . in order to detect or , we sample the system output signals at equispaced times for .the probability density of eq .( [ eq : five ] ) is then easily obtained for the bistable system . for the input bipolar pulse signals mixed with additional gaussian white noise , the impulse response of matched filter is , where is a scaled , time - reversed and shifted version of the pulse signal ( ) .if the output of the matched filter is sampled at that maximizes the output signal - to - noise ratio , the probability density has a gaussian distribution ,\label{eq : six}\ ] ] with  .  in this paper ,the signal detection statistics of the bistable system and the matched filter are measured by the false alarm probability and the detection probability where is the detection threshold , and the positive pulse signal , shown in fig .[ fig : one ] , is considered as the information carrying part of the signal in this detection problem .figure [ fig : two ] ( a ) compares the signal detection performance of the bistable system and the matched filter . as the noise intensity increases from v/hz to v/hz , the signal detection statistics of the bistable system displays a non - monotonic behavior , i.e. the sr phenomenon in signal detection .the detectability of the bistable system is optimized at v/hz .the discrete points , illustrated in fig . [ fig : two ] ( a ) , represent the corresponding numerical simulation results of the bistable system . here, we numerically integrate the stochastic differential equation of eq .( [ eq : two ] ) using a euler - maruyama discretization method with a small sampling time step  . the theoretical model of eq .( [ eq : five ] ) , although approximate , clearly describes the sr effect in the bistable system , and it agrees with the corresponding numerical results .furthermore , the output signals of the bistable system and the matched filter are also given in figs .[ fig : two ] ( b ) , ( c ) , ( d ) and ( e ) for different noise intensities .the sr - type effect in the bistable system is also visible , as shown in figs .[ fig : two ] ( b ) , ( c ) and ( d ) .  in fig .[ fig : two ] , the bistable system , even optimized at v/hz via the sr effect , can not outperform the matched filter for the case of gaussian white noise .this fact has been pointed repeatedly by many researchers in a variety of investigations .however , we have assumed complete knowledge of the probability density functions under hypotheses and , from which the conclusion of the bistable system being suboptimal by comparison with the matched filter is deduced . in next section , we will turn to more realistic problems in which some parameters of the signal are not completely known .new merits of the bistable system will be observed .in this section , we mainly study two signal detection problems : the first is detecting the pulse signal with disturbed pulse rates ; the second is a situation that the return pulse signal is delayed by the propagation time of the signal through the medium , resulting in the unknown arrival time .both the bistable system and the matched filter are employed as detectors in the two signal detection problems .the performances of the bistable system and the matched filter , with different measurements in different detection tasks , are compared thoroughly .consider the heuristic problem of detecting the received pulse signals with changed pulse rates ( i.e. multi - rate pulse signals ) , for example , which may occur due to the desynchronization of devices during the detection of moving objects .for a given noisy environment , we can deduce the response time of the bistable system in terms of the signal amplitude , system parameters and ( ref .appendix a ) .if the disturbed pulse duration is not too large ( or small ) with respect to the system response time , it will be seen in the following that the bistable system can detect different pulse rates from a frequency spectrum analysis .it is well known that intersymbol errors will appear in this situation with a matched filter .the frequency spectrum analysis of the matched filter output also shows that too many low frequency components degrade the matched filter s detectability .  in numerical simulations ,the reference pulse rate is set as hz .then , for the matched filter , the impulse response should be the `` flipped around '' version of the signal , i.e. ,  ] and .the received pulse signal  ] ( ) , but the arrival time is unknown .  for the matched filter, the test statistic can be written as }\sum_{i=\hat{i}_{0}}^{\hat{i}_{0}+m-1}h[i]x[i]\geq \gamma,\label{eq : ten}\ ] ] where =s[m-1-i] ] and the decision threshold  ] , i.e. , =s[i - i_{0}]+n[i]-a ] .after the bistable system with output  ] is the arrival time of the signal given by the detector .the matched filter or the bistable system , however , might detect a totally wrong arrival time , which is measured by the loss ratio and the loss time . if we take -t_{0 } \mid > t_{p} ] of the bistable system and matched filter in the numerical simulation .thus , the computational efficiency of both detectors can be quantified by the average computation time }{k}.\label{eq : fourteen}\ ] ]  the plots of figs . [fig : five ] ( a ) and ( b ) show the performance of bistable systems and the matched filter in detecting the unknown arrival time of input signals , this is , the average error time and the loss ratio .it is seen that the matched filter s performance depends on selecting an appropriate decision threshold  .if the decision threshold v , it will present a higher loss ratio than in bistable systems .the decision threshold v makes the matched filter optimized .two kinds of bistable systems are employed in this detection problem : one is subthreshold ( ) and the other is suprathreshold ( ) . in fig .[ fig : five ] , the average error time and the loss ratio of the suprathreshold bistable system is a monotonic increasing function of the noise intensity .this suprathreshold bistable system is suboptimal compared to the matched filter with v , but outperforms the matched filter with v . for the subthreshold bistable system shown in fig .[ fig : five ] , the average error time decreases as the noise intensity increases , and almost approaches the performance of the matched filter ( v ) before the loss ratio begins to increase .this non - monotonic influence of the noise on the performance of the subthreshold bistable system is the signature of the sr phenomenon .moreover , an important merit of the bistable system is its computational efficiency .the average computation time of the bistable system , i.e. eq . ( [ eq : fourteen ] ) , is about s for processing the data observed in the time interval  ] .the functions are eigenfunctions of the operator with the eigenvalues . multiplying both sides of eq .( [ eq : a5 ] ) by and integrating it , yields \}dy } { \int_{-\infty}^{+\infty}u^2(y)dy},\label{eq : a6}\ ] ] where eigenfunctions satisfy the boundary conditions of and . the eigenvalue problem of eq .( [ eq : a5 ] ) is then equivalent to the variational problem consisting in finding the extremal values of the right side of eq .( [ eq : a6 ] ) .the minimum of this expression is then the lowest eigenvalue , corresponding to the steady state solution of eq .( [ eq : a3 ] ) . here , we adopt eigenfunctions  ] corresponding to eigenvalues for . the elements of matrices  ] are >0,\\ k_{ij}=\int_{-\infty}^{+\infty } ij y^{i+j-2}\exp[-v(y)]dy\geq0,\end{aligned}\ ] ] where  .the matrix  ] is semi - positive definite .the minimal eigenvalue is zero .the inverse of minimal positive eigenvalue describes the main time of the system tending to the steady state solution of eq .( [ eq : a3 ] ) , what we call the system response time .note the time scale transformation in eq .( [ eq : a1 ] ) , the minimal positive eigenvalue should be and the real system response time is .   from eq .( [ eq : a9 ] ) , we can obtain the eigenfunctions  ] are normalized . because is a hermitian operator , eigenfunctions and are orthogonal where . can be expanded , according to eigenfunctions and eigenvalues , as \exp[-\lambda_i\tau ] , \label{eq : a11}\ ] ] where are normalization constants . in this paper , we take an approximate expression of \exp[-\lambda_i\tau] ] for .f. moss , d. pierson , and d. ogorman , _ stochastic resonance : tutorial and update , int .j. bifurcation and chaos _ 4 ( 1994 ) 1383 - 1397 a.r .bulsara , and l. gammaitoni , _ tuning in to noise , phys . today _ 49 ( 1996 ) 39 - 45            a. hibbs , e. jacobs , j. bekkedahl , a. bulsara , and f. moss , _ stochastic resonance in a bistable squid loop , noise in physical systems and 1/f fluctuations _ ,handel and a.l .chung , aip press ( 1993 ) 720 - 723     j.j .collins , c.c .chow , and t.t .imhoff , _ aperiodic stochastic resonance in excitable systems , phys . rev .e _ 52 ( 1995 )r3321-r3324 ; j.j .collins , c.c . chow and p. grigg , _ noise - enhanced tactile sensation , nature _ 383 ( 1996 ) 770                  d. g. luchinsky , r. mannella , p. v. e. mcclintock , and n. g. stocks , _ stochastic resonance in electrical circuits - i : conventional stochastic resonance , ieee trans .cir . and sys.- analog and digital signal processing _ 40 ( 1999 ) 1205 - 1214 ; _ stochastic resonance in electrical circuits - ii : nonconventional stochastic resonance , ieee trans .cir . and sys.- analog and digital signal processing _ 40 ( 1999 ) 1215 - 1224              h.c .papadopoulos , g.w .wornell , and a.v .oppenheim , _ low - complexity digital encoding strategies for wireless sensor networks , in proc .acoustics , speech , and signal processing _ ( 1998 ) 3273 - 3276          m.d .mcdonnell , d. abbott and c.e.m .pearce , _ characterization of suprathreshold stochastic resonance in an array of comparitors by correlation coefficient , fluctuation and noise lett ._ 2 ( 2002 ) l205-l220

Abstract: this paper presents a thorough evaluation of a bistable system versus a matched filter in detecting bipolar pulse signals . the detectability of the bistable system can be optimized by adding noise , i.e. the stochastic resonance ( sr ) phenomenon . this sr effect is also demonstrated by approximate statistical detection theory of the bistable system and corresponding numerical simulations . furthermore , the performance comparison results between the bistable system and the matched filter show that ( a ) the bistable system is more robust than the matched filter in detecting signals with disturbed pulse rates , and ( b ) the bistable system approaches the performance of the matched filter in detecting unknown arrival times of received signals , with an especially better computational efficiency . these significant results verify the potential applicability of the bistable system in signal detection field .