recent advances in additive manufacturing allow the precise placement of one or multiple materials at micrometer resolution with essentially no restrictions on the geometric complexity of the spatial arrangement .complex three dimensional solids can be created with highly non - regular material distributions in a near optimal fashion , enabling the fabrication of structures with enhanced performance .topology optimization has emerged as a promising approach to utilize the benefits of additive manufacturing .structural topology optimization seeks to find the optimal geometry and/or the material layout of a body within a given design domain .the geometry is represented by the spatial distribution of two or more material phases ; in structural problems , one of these material phases may represent void .originally topology optimization methods were developed primarily to create conceptual designs in the early stage of the design process .later , topology optimization was applied to directly design micro - electro - mechanical systems ( mems ) , utilizing the ability of thin - film fabrication techniques , such as photolithography in combination with chemical etching , to create geometrically complex devices at low cost . as the mems design problem is essentially two - dimensional , traditional approaches were sufficient to achieve the necessary geometric resolution at acceptable computational cost .motivated by the availability of affordable additive fabrication methods for three dimensional structures , this paper focuses on topology optimization of three dimensional structures and introduces a new approach for finding optimized designs with high geometric resolution on rather coarse computational meshes .most approaches for structural topology optimization are density methods .for a two - phase problem , the density is considered a design variable and can assume intermediate values between the density of the material phase `` a '' and the density of the material phase `` b '' .the most popular density method is the solid isotropic material with penalization method introduced by and .it features great versatility , robustness , efficiency , and ease of use and implementation for a broad range of applications .density methods typically describe the boundaries between the material phases either via intermediate density values or by discrete material distributions leading to jagged boundary geometries . in both cases ,the enforcement of boundary and interface conditions at the material interface is hampered and may result in non - physical responses , such as premature yielding .often this issue can be mitigated by mesh refinement or adaptive re - meshing .however , for problems that require an accurate description of the boundaries , such as boundary layer problems in fluids and skin - depth issues in electromagnetics , it was reported that density methods like simp fail .the shortcomings of density methods have promoted the development of the level set method ( lsm ) for topology optimization .the lsm allows a crisp representation of the phase boundaries and the accurate enforcement of boundary conditions on fixed meshes .the material interface in the lsm is described implicitly by the iso - contours of a level set function ( lsf ) , usually the zero level - set contour .the lsf is typically discretized by the same mesh used for the physical field and is updated in the optimization process via the solution of the hamilton - jacobi equations .alternatively , the parameters of the discretized lsf are defined as explicit functions of the optimization variables , and the resulting parameter optimization problem is solved by standard nonlinear programming methods .the key challenges for the lsm include ( a ) controlling the spatial gradients of the lsf in the vicinity of the zero - level set contour to avoid ill - conditioning of the optimization problem , ( b ) controlling local feature sizes , and ( c ) accelerating the convergence of the geometry in the optimization process . for a detailed discussion of the lsm, the reader is referred to the comprehensive review by and .  in lsms, the structural geometry can be represented in the mechanical model via an ersatz material approach , immersed boundary techniques , or by adaptive geometry conforming meshes .the first two approaches allow the use of fixed , design independent meshes while the last approach requires local or global re - meshing as the structural geometry evolves in the design process .  using an ersatz material approach ,the void phase is modeled by a soft material and the material properties in elements intersected by the zero - level set contour are interpolated between the `` void '' and solid phase , proportional to the volume ratio of the individual phases .however , this approach faces the same issues as density methods in regards to enforcing boundary conditions across the material interface .other approaches to model the mechanical response include generalized and adaptive finite element schemes such as the super - imposed finite element method ( sfem ) , the extended finite element method ( xfem ) , and local re - meshing schemes .  in this paper , we focus on the lsm in combination with the xfem . the xfem does not require a mesh that conforms to the material interfaces and reduces the complexity of mesh construction .spatial discontinuities in the structural response are captured by augmenting the standard finite element interpolations with additional shape functions .this approach is similar to the sfem as the solution is obtained by super - imposing the standard and enriched shape functions .however , unlike the sfem , the xfem can combine multiple types of shape functions and thus allows for greater flexibility .the xfem builds upon the partition of unity concept developed by .the idea of xfem was originally proposed by to model crack propagation .the reader is referred to for an overview of the application of xfem to problems in fracture mechanics .the xfem has been used for a variety of problems in computational mechanics , such as fluid - structure interaction , multi - phase flows , and nano - scale heat transfer . a general overview of the method is presented by . originally introduced the xfem into shape optimization using a simplified xfem formulation .this formulation does not use additional enrichment functions and is limited to problems where one of the material phases represents void and the geometric configuration is `` simple '' , i.e. does not contain geometric features that are smaller than the size of two elements .in this instance , the weak form of the governing equations is only integrated over the solid material in each element . in addition , if the interface between the material phases is traction free , this simplified version of the xfem only differs from the traditional finite element method with respect to the domain of integration .the simplified xfem version was applied to structural shape optimization , for example , by , , and , and to topology optimization of three dimensional structures by .an xfem approach based on a standard enrichment strategy allows to model two - phase problems with a simple intersection pattern .this formulation is considered , for example , by to solve `` solid - void '' structural topology optimization problems , modeling the `` void '' as a soft material . used a standard enrichment strategy to discretize the phonon boltzmann transport equation and optimize the thermal conductivity of nano - composites .however , this enrichment strategy is not guaranteed to consistently approximate the state variable fields for configurations with complex intersection patterns .  an enhanced version of the xfem was proposed by , who presented a generalized enrichment based on the step enrichment of and applied it to two dimensional structural topology optimization .this formulation captures consistently the mechanical response for complex geometries and intersection patterns for general multi - phase problems .this paper will expand the combination of the lsm and the xfem onto general two - phase , three dimensional problems .we will compare results of the proposed lsm - xfem framework with simp results for structural topology optimization examples .the numerical results will show that the lsm - xfem combination is a promising approach for three dimensional problems and allows for the use of coarse meshes to represent the structural geometry and to describe the structural response with acceptable accuracy .the main challenges of expanding the previous lsm with xfem approaches to three dimensions stem from the increased complexity of possible intersections patterns .such patterns include elements that are intersected multiple times and elements containing only a small volume of a particular phase .in contrast to , we will adopt the generalized enrichment strategy of to ( a ) accurately model the structural response on complex three dimensional patterns and ( b ) solve solid - solid material distribution problems .further , we will expand the preconditioning scheme of onto three dimensions to mitigate ill - conditioning issues in the xfem analysis problems due to elements with small volume fractions .the main contributions of the paper are : ( a ) we present a numerically robust and computationally viable approach for solving general two - phase , three dimensional topology optimization problems , and ( b ) we provide a direct comparison of lsm - xfem and simp results for three dimensional problems , highlighting key features of the two methods .the remainder of this paper is structured as follows : the geometry models of the lsm - xfem and simp methods are described in section [ sec : geometry - modeling ] .the mechanical model and the xfem formulation are summarized in section [ sec : structural - analysis ] .details of the lsm - xfem and simp optimization approaches are presented in section [ sec : optimization - model ] .section [ sec : computational - considerations ] highlights specific computational challenges of the xfem approach for three dimensional problems .section [ sec : numerical - examples ] presents structural topology optimization examples , comparing the lsm - xfem and simp approaches .section [ sec : conclusions ] summarizes the main conclusions drawn from this study .in topology optimization , the geometry of a body is defined via its material distribution . in density methods , such as simp, the material distribution is discretized by finite elements , with either elemental or nodal parameters defining the distribution within the element .most often the same mesh is used to approximate the density distribution and the structural response .alternatively , the state and density fields can be discretized by different meshes with different refinement levels ; see for example the multi - resolution topology method ( mtop ) by .the optimization variables define analytically or by means of auxiliary partial differential equations the nodal or element density parameters . for two - phase problems, the density is continuously varied between `` 0 '' ( phase `` a '' ) and `` 1 '' ( phase `` b '' ) .implicit or explicit penalization schemes , optionally combined with projection methods , are used to encourage `` 0 - 1 '' solutions .the crispness of the interface geometry , as described via the optimized material distribution , depends on ( a ) the formulation of the optimization problem , i.e. the objective and constraints , ( b ) regularization techniques , such as density or sensitivity filters , and ( c ) the optimization algorithm . in general, the resolution of the phase boundaries increases as the mesh is refined . for three dimensional problems ,often coarse meshes are used to limit the computational costs . in this case, the boundary geometry either lacks crispness due to the presence of elements with intermediate densities or is approximated by a spatially discontinuous material distribution , leading to jagged interfaces .alternatively , the material distribution can be described via a level set function .typically the zero level set contour defines the phase boundaries .considering a two - phase problem , the interface is defined as follows :    where the vector collects the spatial coordinates , is the domain of material phase `` a '' , is the domain of material phase `` b '' , and defines the material interface between phase `` a '' and phase `` b '' . for example , to model a sphere in a three dimensional mesh centered at ( ,, ) , the value of the level set function at a grid point ( ,, ) is :    where represents the radius of the sphere , and the sign value of at each node determines if the node is inside or outside the circle .  the level set field is typically discretized by shape functions with either local or global support ; the reader is referred to the review paper by . in the simplest and most common approach, the level - set field is approximated on the same mesh used for discretizing the governing equations . in this study, we follow this approach and define the nodal level - set values explicitly as functions of the optimization variables ( see [ sec : optimization - xfem ] ) .the resulting parameter optimization problem is solved by a standard nonlinear programming method .  while lsms provide a crisp definition of the phase boundaries , they require seeding the initial design with inclusions and/or introducing inclusions in the course of the optimization process , for example via topological derivatives .an example of an initial design with a regular pattern of spherical inclusions is shown in fig .[ fig : long - cantilever - beam - swiss - cheese ] .the images in the upper row and the image in the lower left corner show only phase `` a '' .the material layout of both phases is depicted in the lower right image .this layout can be generated by superposing eq .[ eqn : circle ] for spheres at uniformly spaced center locations .the optimization results of the lsm are typically dependent on the initial layout .furthermore , it is non - trivial to generate an initial design that satisfies geometric design constraints , such as volume or perimeter constraints . in our experience , severe constraint violations may cause the optimization process to converge to a feasible design with otherwise poor performance .in this section , we briefly discuss the structural model and the xfem analysis used in this paper .the governing equations are presented first , followed by a summary of the xfem discretization and analysis .this paper considers the topology optimization of structures using the lsm approach described above and the xfem to predict the structural response , assuming infinitesimal strains , a linear elastic material behavior , and static conditions .we consider the two - phase problem depicted in fig .[ fig : xfem - problem - setup ] , where denotes the surface where traction forces are applied , denotes the surface with prescribed displacements , and is the normal at the material interface pointing from phase `` a '' to phase `` b '' .the weak form of the governing equations can be decomposed into the following terms :    where collects the contributions from the static equilibrium , including body forces and surface tractions , models the interface conditions along the phase boundaries for `` solid - solid '' problems , and is due to a fictitious spring model to pin free floating material in `` solid - void problems '' .the weak form of the structural equilibrium equations is :    where is the kinematically admissible test function , is the strain tensor associated with the test function , is the displacement vector , is the stress tensor , is the applied body force , and is the external traction applied along .  to enforce continuity of the displacements along the phase boundaries, the static equilibrium equations are typically augmented by either an enhanced lagrange multiplier or penalty formulations , such as the stabilized lagrange multiplier and the nitsche method .note that the standard lagrange multiplier approach is not suitable for the xfem as it suffers from numerical instabilities .the reader is referred to , , and for more details .  in this paper, we enforce displacement continuity along phase boundaries for `` solid - solid '' problems via the following stabilized lagrange multiplier method :   \cdot \pmb{\lambda } \d \gamma_{a , b }      + \gamma \int_{\gamma_{a , b } } \pmb{\mu } \cdot \left [ \mathbf{u } \right ] d \gamma_{a , b } \\      + \int_{\gamma_{a ,b } } \mu \cdot \left ( \pmb{\lambda } - \bar{\pmb{\sigma } } \cdot \mathbf{n}_{a , b } \right ) d \gamma_{a , b},\end{gathered}\ ] ]   = \mathbf{u}^{(a ) } - \mathbf{u}^{(b ) } \ ,   \left [ \mathbf{v } \right ] = \mathbf{v}^{(a ) } - \mathbf{v}^{(b ) } \ , \ ] ]    where is the lagrange multiplier , and is the associated test function . the higher the weight is , the better the interface condition is satisfied , at the cost of numerical stability .the formation of free floating solid particles surrounded by void material is possible in `` solid - void '' topology optimization , leading to a singular analysis problem .this issue does not exist in an ersatz material approach as the void phase is modeled via a soft material .a similar approach can be applied to the xfem to suppress singularities by modeling the `` void '' phase via a soft material .however , this approach requires accounting for the interface contributions ( [ eq : stabilized_residual ] ) and integrating the governing equations over the void phase . to avoid the associated complexity and computational costs ,we extend the approach of onto three dimensions and assume that the solid phase is supported by fictitious springs .this model leads to the following contribution to the governing equations , assuming that phase `` a '' is the solid phase :    where denotes the stiffness of the distributed system of springs .  for a more detailed explanation of this xfem formulation ,the reader is referred to the paper by .   to capture the discontinuities in the strain and stress fields along the phase boundaries, we enrich the standard finite element approximation with additional shape functions .we adopt the generalized enrichment strategy of which resolves consistently the displacement fields in the presence of small features and does not suffer from artificially coupling disconnected phases . considering a two - phase problem ,the displacement field is approximated as follows :    where is the enrichment level , is the maximum number of enrichment levels used for each phase , are the shape functions , is the vector of nodal displacement components at node for phase  ] .both phases have a poisson ratio of .the maximum volume of the stiff phase is limited to of the total volume .the design domain is discretized by elements .because of the symmetry condition , only one half of the cuboid is numerically analyzed .we compare the simp and lsm - xfem results .the optimization problem is solved by a simp approach with a penalization factor of , a smoothing radius of ( the element edge ) and the projection parameter of .the design domain is initialized with a uniform material distribution of .the optimized material distributions are shown in fig .[ fig : long - cantilever - beam - simp - xfem ] where material with a density lower than is transparent .the strain energies are reported in tab .[ tab : long - cantilever - beam - two - phase - optimization ] .the lsm - xfem results for a smoothing radius of ( the element edge ) are shown in fig .[ fig : long - cantilever - beam - simp - xfem ] and the strain energies are given in tab . [ tab : long - cantilever - beam - two - phase - optimization ] . considering the full design domain ,the level set field is initialized with a array of equally spaced holes with radius of .the initial design is shown in fig .[ fig : long - cantilever - beam - swiss - cheese ] and satisfies the volume constraint for the stiff phase .note that the interface condition is enforced via the stabilized lagrange multiplier method ( [ eq : stabilized_residual ] ) with an element wise constant lagrange multiplier , , and a consistency factor of .comparing the simp and lsm - xfem results , the same trends can be observed for this `` solid - solid '' problem as for the `` solid - void '' ones studied earlier .the lsm - xfem approach leads to three dimensional structures with thinner walls and higher stiffness .in contrast , the simp method generates truss - type structures , in particular if the discretization is too coarse and the optimum wall thickness is less than the size of an element .  for illustration purposes only , we show a realization of the lsm - xfem optimized design for in fig . [fig : manufacture - long - cantilever - beam ] .the structure was fabricated with a polyjet 3d printing process on a connex objet 260 printer .white material represents phase `` a '' , black represents phase `` b '' .the left and center pieces show the individual phases printed separately , the printed two - phase design is shown on the right .cc|c|c|c|c|c & & + & & & + & & & + & & & + & & & + & & & + & & & + & & & + & & & +in this paper we presented an optimization approach combining a level set method ( lsm ) for describing the geometry and an extended finite element method ( xfem ) for predicting the structural response . building upon generalized enrichment and preconditioning schemes , previously developed for two - dimensional problems ,the proposed optimization scheme was applied to two - phase `` solid - void '' and `` solid - solid '' problems in three dimensions . in all examples ,the strain energy was minimized subject to a volume constraint on the stiff phase .the results of the lsm - xfem approach with and without perimeter constraints were compared with the ones of a simp method which employs density filtering and projection .the numerical studies suggest that the lsm - xfem method features an improved convergence as the mesh is refined and is able to represent thin - walled structures on coarse meshes .the simp approach may require a strong projection to achieve clear `` 0 - 1 '' results with comparable strain energies .while density filtering is an efficient and intuitive method to control the local feature size , neither level set smoothing nor imposing a perimeter constraint achieve a similar effect on lsm - xfem results .the current lack of a feature size control and the significant improved complexity of the lsm - xfem formulation limit the attractiveness of this scheme .however , for problems where a high mesh resolution is not tolerable and/or interface conditions need to be enforced with high accuracy , the lsm - xfem approach might be an interesting alternative to simp - type methods .the advantages of the lsm - xfem problem have been shown by for fluid problems at high reynolds numbers in two dimensions .the authors plan to study three dimensional flow problems with the lsm - xfem approach in the future .deaton j , grandhi r ( 2013 ) a survey of structural and multidisciplinary continuum topology optimization : post 2000 .structural and multidisciplinary optimization pp 138 , http://dx.doi.org/10.1007/s00158-013-0956-z    duysinx p , miegroet l , jacobs t , fleury c ( 2006 ) generalized shape optimization using x - fem and level set methods . in : iutam symposium on topological design optimization of structures , machines and materials ,springer , pp 2332       gerstenberger a , wall wa ( 2008 ) an extended finite element method / lagrange multiplier based approach for fluid - structure interaction .computer methods in applied mechanics and engineering 197:16991714  guest j , prvost j , belytschko t ( 2004 ) achieving minimum length scale in topology optimization using nodal design variables and projection functions . international journal for numerical methods in engineering 61(2):238254    heroux m , bartlett r , hoekstra vhr , hu j , kolda t , lehoucq r , long k , pawlowski r , phipps e , salinger a , thornquist h , tuminaro r, willenbring j , williams a ( 2003 ) an overview of trilinos .tech . rep .sand2003 - 2927 , sandia national laboratories      lee p , yang r , maute k ( 2011 ) an extended finite element method for the analysis of submicron heat transfer phenomena , multiscale methods in computational mechanics , lecture notes in applied and computational mechanics , vol 55 , springer , pp 195212       maute k , kreissl s , makhija d , yang r ( 2011 ) topology optimization of heat conduction in nano - composites . in : 9 congress on structural and multidisciplinary optimization , shizuoka , japan  meisel n , gaynor a , williams c , guest j ( 2013 ) multiple - material topology optimization of compliant mechanisms created via polyjet 3d printing . in : twenty forth annual international solid freeform fabrication symposium an additive manufacturing conference , august 12 - 14 , 2013 , austin , tx                    yamasaki s , nomura t , kawamoto a , sato k , nishiwaki s ( 2011 ) a level set - based topology optimization method targeting metallic waveguide design problems .international journal for numerical methods in engineering 87(9):844868

Abstract: as the capabilities of additive manufacturing techniques increase , topology optimization provides a promising approach to design geometrically sophisticated structures which can be directly manufactured . traditional topology optimization methods aim at finding the conceptual design but often lack a sufficient resolution of the geometry and structural response , needed to directly use the optimized design for manufacturing . to overcome these limitations , this paper studies the viability and characteristics of the extended finite element method ( xfem ) in combination with the level - set method ( lsm ) for topology optimization of three dimensional structural design problems . the lsm describes the geometry by defining the nodal level set values via explicit functions of the optimization variables . the structural response is predicted by a generalized version of the xfem . the lsm - xfem approach is compared against results from a traditional solid isotropic material with penalization ( simp ) method for two - phase `` solid - void '' and `` solid - solid '' problems . the numerical results demonstrate that the lsm - xfem approach can describe crisply the geometry and predict the structural response of complex three - dimensional structures with acceptable accuracy even on coarse meshes . however , the lsm - xfem studied here lacks a robust and intuitive formulation to control the minimum feature size , and the optimization results may depend on the initial design .


vs. agent difference in size , for different scaling similarity parameter ( ) values.,title="fig:",width=529 ] +  multiagent systems and complex networks are very active and growing areas of research with applications found in a diverse variety of problems found in e.g. engineering systems complexdyncomm , biological systems biological , arrowsmith , bagni , amaral2 , neural networks , socioeconomic systems caiafacontrarians , huberman , huberman2,netmisk , gligorma , redelico .let it be recalled that growth , innovation pace of life in cities are scaled using the population size as urban metrics .similarly , biological metrics such as number , size and metabolic rate of cells are also scaled using body sizes . in multimarket economies , size is also relevant beside the number of competitors , entry barriers , etc . .thus many phenomena , outside well known condensed matter physics , related to size , appear in areas that , at first sight , do not have something in common ; like sociology , biology , economy , etc . , but for which similar mathematical models can be used in fact economo , proto .  in complex networks , individuals can behave in two opposite ways : as cooperators or defectors as stated in the n - person prisoner dilemma hauert .recently much attention has been given to the emergence of `` cooperation '' in public goods games , and more generally in `` evolution theory '' .as has been used for conflict resolution according to game theory , as well as reference marchandsimon , simon2 in the so - called theory of the organization . there , they treat the conflict between groups within an organization and conflict between organizations , these conflicts , which could be interpreted as a situation of competition for example , are produced by : conflicting objectives and have different perceptions of reality .so the conflict occurs when an agent or agent group experiences a decision problem .negotiation and politics are the procedures used by the organization to manage these conflicts and lead to tensions in status and power systems . if those who are formally more powerful prevail , they give rise to a stronger perception of differences in status and power , if not sustained its position is weakened . to address the conflict between organizations has used the theory of negotiation , albeit with a gap that empirical and negotiation situations are so complex that it is not possible to develop a theory general .within this framework we can also quote . which defines five forces that determine the competitive strategy of an organization , within them the bargaining power and market threats result in the rivalry of existing competitors .finally one of the latest nobel laureate in economics , coase , coasewilliamson , williamson referred to the formation of firms , companies , rather than individual goods and services marketed by itself , this is given by the structure of transaction costs .in the present paper , we focus on a competitive scenario based on a set of differential equations as it will be explained in section 2 . in short ,we start from a lotka - volterra set of equations for describing prey - predator situations , i.e. a system of agents competing for some common resource .an original constraint is imposed on the agent dynamics , i.e. an increase in size is favored in order to obtain resources , while an agent decreasing size implies the loss or lack of getting such resources . by sizewe mean something like the fraction of a given market that an agent has , i.e. its market share in economy .other intuitive notions of size are easily imagined in many other systems .let us mention that in , a different generalization of the lotka - volterra model to the agent ( competitor ) case was used for describing the information electronic web s dynamics .two behavior states were found depending on a global interaction parameter : on one hand , a stationary state exists in which only one agent gets all the available resources ( `` winner takes all '' ) , while the rest of agents gets nothing ; on the other hand , there is another stationary state where the resource is fairly shared among all agents ( `` sharing the market '' ) . in several works , spatial configurations of agents were also considered allowing them to interact only with a limited set of neighbors , thereby resulting in spatially distributed patterns as stationary states of the system .let us also finally cite in which the interaction coefficient of the model of has a linear form , giving rise to the effect `` riches get richer '' .practically we introduce an interaction function ( if ) in the lotka - volterra model which depends on an agent size in order to let the agents compete among themselves in a self - adjusting way .this contrasts with assuming a fixed interaction coefficient like in the previously cited references .that means that the competition process and outcome are now intertwined variables since they are related to changes in the size of agents .a parameter that governs the spread in competition strength , i.e. for scaling the difference between the size of agents , is also introduced .the if modification will be shown to lead to a dynamics of the system characterized by a time dependent competition .as is kept fix along each simulation , we are able to see how different agent groups compete among themselves . in short, this means that the if is varying with time , accordingly with the size of agents is changing , implies an intrinsically novel dynamics of competition .this paper is organized as follows : in section 2 , the mathematical model is outlined ; in section 3 , the system fixed points are theoretically analyzed and an illustrative example for a system with a small number ( five ) of agents is provided ; in section 4 , simulation results are presented for the case of a larger number ( ten ) of agents , in order to establish the qualitative validity of the quantitative findings obtained for the small size systems .this allows us in section 5 , to outline the novelty of this work and emphasize some conclusion .let us consider a system with agents competing for some common resource , as in a generalized predator - prey model or lotka - volterra model studied in . on this model , a competition parameter , ,is introduced , which allows the agents to compete among themselves in a self - adjusting way .the system is governed by the following set of differential equations: where is the size of agent in the range ; is its time derivative ; is the growth rate of agent if no interaction is present ;  is the maximum capacity of agent and is the if hereby defined by:  \label{gamma}\ ] ] in which is a global positive parameter which controls / scales the degree of similarity in the competition .notice that the analytical form in eq.([gamma ] ) , resembling the gauss function used in statistics , has been chosen because it allows both analytical and numerical approaches ; it has many attractive mathematical properties , i.e. , it is a continuously differentiable function allowing us to make a proper theoretical analysis of the system dynamics .in fact , one could simply require any positive and even function of the absolute difference of agent sizes with the property that its maximum is located at and it is a decreasing function for positive .we suggest that another case could be the kac potential .other forms can be imagined but to compare them is not the primary purpose of this paper .  from eq.([gamma ] ) , we see that the interaction is symmetric and always positive representing a competitive and fair scenario ; the competition factor is maximum and equal to one when . additionally , we observe that , as the absolute difference of agent sizes becomes large the competition factor tends to be small .plots of the if versus the absolute difference of two agent sizes for different values of are shown in fig.[figure 1 ] in order to suggest quantitatively reasonable values .as we let the interaction coefficient depends on the difference in agent sizes , a peer - to - peer competition is imposed , in the sense that agents with a similar size compete more aggressively than agents with different sizes .the name of peer - to - peer competition model makes sense in this context , as only agents with similar sizes are able to compete reciprocally .we emphasize that this kind of competition is a phenomenon that can be observed in many socioeconomic systems , i.e. the competition is strong among big companies while the competition is weak between a big company and a medium or small one .similarly for banks , hedge funds , pension funds , countries , etc . even in poker game, it is the money the player has at his / her disposal which allows him / her to compete and influence the game .as it will be shown below , different stationary states emerge as a function of , which are interestingly characterized by agent grouping .self - organized clustering as well as different possible scenarios appear , ranging from chaotic to non - chaotic motion , depending not only on the value of the size similarity measure but also on the agent size initial condition .  it should be further noticed that , as tends to be larger ( ) the similarity of competition tends to be static , i.e. constant , approaching to the particular case of `` winner takes the maximum , or almost all '' .in this paper we consider that all agents have the same dynamics properties restricting our study to the case where and for which the equations are: , the existence of fixed points and a study of their stability are presented . by definition ,a fixed point is a point in the phase space where all the time derivatives are zero , i.e.,   from eq.([condicionpf ] ) , we detect at least three trivial fixed points which are :  \(i )  for ( all agents with zero size ) ;\(ii ) and for every ;  \(iii )  for ( all agents with the same size )  in the latter case ( iii ) , we can directly calculate the corresponding constant by using eq.([final model ] ) as follows : whence we necessarily have .  as usual , for the analysis of a system fixed points stability , one needs to look at the eigenvalues of the jacobian matrix evaluated at the corresponding fixed points .it is shown that the elements of the jacobian matrix for the system are: _ { \left( i , k\right ) } = \frac{\partial\dot{s}_{i}}{\partial s_{k}}=\left\ {  \begin{array}{ll } 1 - 2s_{i}-{\displaystyle\sum\limits_{i\neq j } } s_{j}\gamma\left ( s_{i},s_{j}\right ) \left [ 1-\frac{2}{\sigma^{2}}s_{i}\left ( s_{i}-s_{j}\right ) \right ] & \mbox{\ \ for \  } k = i \\  -s_{i}\gamma\left ( s_{i},s_{k}\right ) \left [ 1+\frac{2}{\sigma^{2}}s_{k}\left ( s_{i}-s_{k}\right ) \right ] & \mbox{\ \ for \  } k\neq i.\end{array } \right .\label{jacobian}\ ] ]  if we evaluate the jacobian matrix at the fixed points ( i ) , from eq.(jacobian ) we obtain the identity matrix ; all its eigenvalues are equal to one and therefore it is an unstable fixed point .  evaluating eq.([jacobian ] ) at the second type of fixed point ( ii ) for the case with and , we obtain that the jacobian matrix is :  ] whose eigenvalues are: reveals that it is not a stable fixed point .  additionally to the fixed points of type ( i ) , ( ii ) and ( iii ) there are many other points that verify the conditions in eq.([condicionpf ] ) .these points are found by seeking the roots of a set of non - linear equations .   in order to illustrate the analysis we restrict ourselves to examine the case of agents .considering the degeneracy of several solutions , seven possible states appear in which a different combination in the number of agents can be involved . in this senseit can be said that such final states can be called `` size levels '' or in short `` levels '' .moreover since several agents can be on the same level , it can be said that `` clusters '' of agents are obtained .table 1 enumerates these cases and the type of stability ; the latter has been determined by numerically evaluating the jacobian matrix eq.([jacobian ] ) at the fixed point through a newton - raphson ( nr ) algorithm .let us examine the seven cases obtained through the numerical solution of eq.([jacobian ] ) .they are summarized in table 1 .* case i : only one state final size , level , is found to be occupied by the five agents ( 5 ) .* case ii and iii two levels appear ; with two distributions : called ( 1 - 4 ) and ( 2 - 3 ) i.e. made either of 1 agent and a group of 4 agents or a group of 2 agents and a group of 3 agents , respectively . * likewise cases iv and v are two distributions with three levels ( 1 - 1 - 3 ) and ( 1 - 2 - 2 ) respectively .* case vi has only one distribution with four levels , ( 1 - 1 - 1 - 2 ) .* finally in case vii again only one distribution is found , now with five levels , ( 1 - 1 - 1 - 1 - 1 ) .notice that from eq.([jacobian ] ) the upper levels are always occupied than the lower one .this fact reflects what seems to happen in real life , as `` big players '' are less numerous than small ones .observe also that the number of levels and the fixed point stability depend on .it is also found that the result depends on the initial conditions .indeed , observe in fig.[figure 2 ] , the time evolution of the set illustrated when , and for two different sets of initial conditions . as it can be seen , for these cases , after 1000 iteration ( time ) steps different states ( `` levels '' ) values are asymptotically reached . for arbitrary , random ,ic we obtain the solution corresponding to case vi , i.e. ( 1 - 1 - 1 - 2 ) .however when the size initial conditions are imposed to be taken from a narrow range , the resulting solution corresponds to the case iv , i.e. ( 1 - 1 - 3 ) . since , as in fig.[figure 2 ] , two different stable solutions can exist for the  value , but for two different sets of initial conditions ( ic ) , this sort of local degeneracy must be stressed : we call it an overlap region .we emphasize that according to the initial conditions the population of the final states can be markedly different .this observation is reinforced when we analyze the agent size  , here below ..the seven cases emerging from the numerical solution of eq .( 5 ) by the newton - raphson algorithm are summarized .cases are ordered according with the number of final levels and their characteristic stability are enumerated in the last column .further explanations are in the text . [cols="^,^,^,^",options="header " , ]   in fig.[figure 3 ] , the corresponding agent sizes are shown at the stable fixed points as a function of the parameter . in fixed pointsearching , we have used the nr method , departing from 100000 random initial conditions looking for the stable fixed points . the numerical evaluation of the jacobian matrix at different values has been done through the nr procedure .any solution that is not a stable fixed point was discarded .so all the stable fixed point solutions are considered to be found .it can be observed that there are overlap regions in space where more than one fixed point exists . in the overlap region , cases vi and iv coexist . in the overlap region , cases iv and ii coexist . in simple words , agents with different asymptotic in time value , or final level , can be simultaneously found , for a given , thus for a given if .  given that absolute eigenvalues determine the strength of the attractor , we can search for the value of the maximum absolute of eigenvalue real part of the jacobian matrix for each fixed point ; this result is shown in fig.figure 4 .a similar feature , i.e. overlap regions , as in fig.[figure 3 ] is observed as a function of , depending on the ic .thus it is found that the dynamics of the system is dominated by the fixed point of the maximum absolute of eigenvalue real part ; the precise dynamics _ also _ depends on the initial conditions .recall that cases i , iii , v and vii are not stable. however cases ii , iv and vi show an interesting type of degenerate stability ( two possible solutions coexist within specific ranges of ) as reported in the previous paragraph .while unique solutions are found in : ( i ) case vi for , ( ii ) case iv for and ( iii ) case ii for .many numerical investigations lead us to conclude that  * for random initial conditions , i.e. for a widely spread range of initial sizes , the solution corresponding to the maximum absolute of eigenvalue real part appears .* when the initial conditions are taken from a non uniform , i.e. narrow , distribution , the solution corresponding to the other eigenvalue emerges .in this section we present some simulation results for a system with agents , all of them varying between and in size .we have made simulations where the parameter has been varied in a logarithmic way , from up to . in summary, one can observe different dynamics , ranging from the trivial result with almost no competition , when is small in relation to the difference of agent sizes , passing through oscillatory regimes , cluster formation regimes , up to the state where only one agent is the biggest and the rest of them remain at a very low level state , when is large .  to solve our differential equations system ,we have used a typical algorithm for numerical integration .the initial size of each agent in each simulation was randomly chosen from a uniform distribution .the system evolution for ten agents for and is shown in fig.[figure 5 ] .when grows , a change in the system behavior occurs . for ,four size levels having 1.0 , 0.5 , 0.33 and 0.25 values are obtained populated by 1 , 2 , 3 and 4 agents respectively .it is found again that small size agents form more populated clusters .the clusters are formed after a transitory regime .a similar highly complex behavior can be seen , up to around 500 iterations for . for higher time iterationssome erratic , oscillations appear .no asymptotically strictly stable in time solution exists . for very large timeone can consider that a non stable stationary state is reached ; it can be conjectured that the corresponding fixed points are non stable either .the evolution goes toward a band clustering rather chaotic situation .however if is further increased above ( fig.[figure 6 ] ) , the system asymptotically reaches a stable stationary state , so that a sort of condensation of agents occurs and a cluster type situation appears , with a densely populated level at the lowest size . for , it can be observed that there are 5 clusters ( 1 - 1 - 1 - 2 - 5 ) ; for there are 3 clusters ( 1 - 1 - 8 ) and finally for only two clusters are present ( 1 - 9 ) .it can be seen that agents abandon the oscillatory behavior , seen at intermediate values , in order to form different groups or clusters with strong internal competition ; this case corresponds e.g. already to the 1 - 1 - 1 - 2 - 5 set , and is huge in the last case ( ) when the lowest size cluster is made of 9 agents ; recall that the competition factor is maximum and equal to one when ( fig.figure 2 ) .it can be also observed that further increasing induces a systematic decrease in the number of levels , - the lowest level being always the more crowded .our numerical simulation has shown that the situation is found for two levels only , i.e. , it emerges when is around 0.5 .this persists for higher values .this behavior in which one agent obtains the highest level while the rest of agents reaches the lowest common level illustrates that some sort of monopoly has been configured as the final state of the competition .the main input contribution of this paper consists in the introduction of a self - organized competition scheme related to agent sizes .the agent size acts into our context , through the if , as a limitation of the number of possibilities for agent interaction or as a constraint .if the difference in sizes is out of the scale they are not able to interact anymore .so they are `` constrained '' in some sense .this constraint allows that only `` players '' of similar sizes are able to or truly compete with each other .a quite simple example is given by a poker game when , at the end of the game , only those with the same capacity to bet can be continuing the game .other cases occur in many finite size markets , like pharmaceutical companies through drugstores , fuel distribution through gasoline stations , clothing and specialty goods in supermarket chains .  in the present analysis of the clustering of agentsno use is made of techniques like those used physics literature on synchronization ermentrout , strogatz , liu , zheng1,hu , zheng2 , which have been around for a couple of decades , because we do not use chaos synchronization in our contribution .such extensions should nevertheless be suggested .our model leads to describe cases when a strong self - organized feedback scheme essentially occurs , i.e. the number of interacting agents being a somewhat secondary aspect .as we are dealing with a nonlinear system , the initial conditions impose severe constraints leading to specific dynamics and a variety of final stationary states .indeed several behaviors emerge , going from one extreme in which a few agents compete with each other , passing through oscillations with clustering , up to the case of a `` winner takes the top '' state , and all others drop out .this latter state reflects the fact that the if tends to a constant value , as the size scale parameter increases .it can be said in simple words that this corresponds to when `` the competition is at its maximum '' .we have stressed the case of a small number of agents which seems the most reasonable practical case . the stability analysis for five agents as presented exemplifies the different dynamics and final state possibilities .here it was demonstrated that there exists a critical of the system , and , where the solution is no longer unique .it can be extrapolated the present findings to other finite number of agents cases .  as shown in several figures ,a clustering phenomenon can be also obtained in which the competition is in fact between clusters of agents . in market language , this situation represents the natural segmentation into big , medium , and small players .we have shown that the segmentation can be extreme , even of the binary type . from a socioeconomic point of view, this means that a monopolistic situation is sometimes likely .notice also that even when clusters ( in size ) appear there still exists competition among the agents the cluster ; this competition is stronger at the lowest size levels , which are the most densely populated .this fact well reflects the complexity of markets : not only agents with equal sizes are in competition with those with bigger or smaller sizes , but also they are still in strong competition with each other at their own level . in some sense , the only final state will be , sooner or later , only one agent at the top , because of the attrition of the small ones .  finally and in addition we would like to re - emphasize that the competition scheme presented here eq.([gamma ] ) gives rise to a very complex dynamics , although based on a simple idea , and on having only one quantity , as the regulation parameter of the different behaviors .this can be put in line with the observed complexity of evolution of true systems , though is clearly a reductionist view of the world .99 r. albert , a.l .barabsi , rev .74 ( 2002 ) 47 - 97amaral , j.m .ottino , eur .j. b 38 ( 2004 ) 147 - 162 .knyazeva , s.p .kurdyumov , evolution and self - organization laws of complex systems , moscow , nauka publisher in russian 1994 p 236 .l. kocarev , g. vattay , complex dynamics in communication networks , springer verlag 2005 .f. liljeros , c.r .edling , l. a. n. amaral , microbes infect . 5 ( 2003 ) 189 - 196 .zhang yi , k.k .tan , ieee trans .circuits syst .52 ( 2005 ) 2482 - 2489 .caiafa , a.n .proto , intern .c 17 ( 2006 ) 385 - 394 .maurer , b.a .huberman , j. econom .control 27 ( 2003 ) 2195 - 2206 .adamic , b.a .huberman , quart .j. electron .commerce 1 ( 2000 ) 5 - 12 .j. miskiewicz , m. ausloos , in proc . of the 3rd nikkei econophysics symposium : practical fruits of econophysics , tokyo , ed h. takayasu , springer 2006 , p. 312 .m. gligor , m. ausloos , eur . phys .j. b 57 ( 2007 ) 139 - 146 .redelico , a.n .proto , m. ausloos , physica a 388 ( 2009 ) 3527 - 3535 .bettencourt , j. lobo , d. helbing , c. kuhnert , g.b .west , proc .104 ( 2007 ) 7301 - 7306 .savage , a.p .allen , j.h .brown , j.f .gillooly , a.b .herman , w.h .woodruff , g.b .west , proc .104 ( 2007 ) 4718 - 4723 .moran , j.d .allen , biol .bull . 212 ( 2007 ) 143 - 150 .y. sawada , physica a 204 ( 1994 ) 543 - 554 .pilloff , rev .( 1999 ) 163 - 182 .economo , a.j .kerkhoff , b.j .enquist , ecol.lett . 8( 2005 ) 353 - 360 .n. olivera , a.n .proto , f. redelico , rc332008 7 th international conference on social science methodology , campus di monte santangelo , naples , italy ( 2008 ) . c. hauert , s. de monte , j. hofbauer , k. sigmund , science 296 ( 2002 ) 1129 - 1132 .santos , m.d .santos , j.m .pacheco , nature 454 ( 2008 ) 213 - 216 .n.s . glance , b.a .huberman , sci am . 270 ( 1994 ) 76 - 81 .j. von neumann , o. morgenstern , theory of games and economics behavior , princenton , 1944 .caram , c.f .caiafa , a.n .proto , intern .c 17 ( 2006 ) 435 - 445 .li yanhui , zhu siming , appl .modelling 31 ( 2007 ) 912 - 919 .m. kac , phys .fluids 2 ( 1959 ) 8 - 12 .press , b.p .flannery , s.a teukolsky , w.t .vetterling , numerical recipes in fortran : the art of scientific computing , 2nd edn , cambridge : cambridge university press , 1992 , p.364 .

Abstract: the dynamic behavior of a multiagent system in which the agent size is variable it is studied along a lotka - volterra approach . the agent size has hereby for meaning the fraction of a given market that an agent is able to capture ( market share ) . a lotka - volterra system of equations for prey - predator problems is considered , the competition factor being related to the difference in size between the agents in a one - on - one competition . this mechanism introduces a natural self - organized dynamic competition among agents . in the competition factor , a parameter is introduced for scaling the intensity of agent size similarity , which varies in each iteration cycle . the fixed points of this system are analytically found and their stability analyzed for small systems ( with agents ) . we have found that different scenarios are possible , from chaotic to non - chaotic motion with cluster formation as function of the parameter and depending on the initial conditions imposed to the system . the present contribution aim is to show how a realistic though minimalist nonlinear dynamics model can be used to describe market competition ( companies , brokers , decision makers ) among other opinion maker communities .


a keynesian beauty contest is a popular concept used to explain price fluctuations in equity markets. keynes described the action of rational agents in a market using an analogy based on a fictional newspaper contest . in the contest , entrants are asked to choose a set of the six most beautiful faces from among photographs of different women . those entrants who would select the most popular facewould be then eligible for a prize .a naive strategy would be to choose the most beautiful face according to the opinion of the entrant .entrants are known as employing such a strategy independent voters .a more sophisticated entrant , aiming to maximize his / her chances of winning a prize , would try to deduce the majority s perception of beauty .this implies that the entrant would make a selection on the basis of some inference from his / her knowledge of public perception .such voters are known as copycats . to estimate public perception , people observe the actions of other individuals ; then , they make a choice similar to that of others . because it is usually sensible to do what other people are doing , the phenomenon is assumed to be the result of a rational choice .nevertheless , this approach can sometimes lead to arbitrary or even erroneous decisions .this phenomenon is called an information cascade .  collective herding phenomena in general pose quite interesting problems in statistical physics . to name a few examples , anomalous fluctuations in the financial market , and opinion dynamics been related to percolation and random field ising model . a recent agent - based model proposed by curty and marsili focused on the limitations that herding imposed on the efficiency of information aggregation .specifically , it was shown that when the fraction of herders in a population of agents increases , the probability that herding produces the correct forecast ( i.e. that individual information bits are correctly aggregated ) undergoes a transition to a state in which either all herders forecast rightly or no herder does .we can observe super - diffusive behaviour in the sense that variance grows asymptotically faster than ( where is the long memory ) in several fields.,,, , it is characterized by the variance when .when , the diffusion of the variance becomes a standard brownian motion .for example , in the case of daily financial data , represents the time series of data .the past price affects the present price , and the diffusion becomes faster than brownian motion .such phenomena can be attributed to long - range positive correlations .we may observe dynamical phase transition ( from normal to super - diffusive behaviour). in such a phase transition , correlation plays an important role .further our voting model shows a similar transition .the herders make long - range correlations and display super - diffusive behaviour .therefore , a majority of voters reach the wrong conclusion .  in this paper, we discuss a voting model with two candidates and .as mentioned above , we set two types of voters - independent and copycat .independent voters voting is based on their fundamental values ; on the other hand , copycat voters voting is based on the number of votes . in our previous paper , we investigated the case wherein all the voters are copycats. in such a case , the process is a plya process , and the voting rate converges to a beta distribution in a large time limit. our present model exhibits a scale - invariant behaviour .this behaviour is observed in the mixing of the binary candidates .furthermore , the power law holds over the entire range in a double scaling limit .this paper is an extension of our previous works .although our model is very simple , it contains three phases .we believe that it is as adequate as the percolation and random field ising models , and that it is useful for understanding phase transition in several fields .we discuss two specific issues : one is the distribution in votes that appears for a mixture of independent and copycat voters and the other is the change in the vote distributions over time . on the basis of these above mentioned points , we discuss phase transition for information cascade .the organization of this paper is as follows . in section 2, we introduce our voting model and define the two types of voters - independent and copycat - mathematically . in section 3 , we calculate the distribution functions strictly for the special cases - independent voters always vote for either of the two candidates ; their behavior is not probabilistic .then , we obtain a solution that is an extension of the solution given in ; in this case , there is no phase transition . in section 4 , we discuss more general cases .we use a stochastic differential equation , the fokker - planck equation , and a numerical simulation . in this model , we can observe phase transition at the ratio of copycats to independents through the variance of the distributions .there are three phases .if copycats constitute a majority or number half of the total number of voters , the voting rate converges more slowly than it would in a binomial distribution . if independents constitute the majority of voters , the voting converges at the same rate as it would in a binomial distribution .this implies that the proportion of copycats influences the results of the voting .the last section presents the conclusions .we model the voting of two candidates , and . at time , each candidate has and votes . at the beginning ( ) , the two candidates , and , have and votes , respectively .hereafter , we omit the time for the initial votes ( and ) and define . at each time step , one voter votes for one candidate .voters are allowed see the number of votes for each candidate when they vote so that they have knowledge of public perception .there are two types of voters - independent and copycat .independent voters vote for and with probabilities and , respectively .their votes are independent of others vote and depend on what they think their fundamental value is .copycat voters vote for each candidate with the probabilities that are proportional to the candidates votes . if the number of votes are and at time , a copycat voter votes with probability for and for .copycat voters votes are based on the number of votes .  here , we set the ratio of independent voters to copycat voters as and , respectively . if we set , this system becomes a plya model with starting elements of type and starting elements of type . in this case, it is well known that the distribution of the voting rate is a beta distribution . as such ,this system is a plya process doped with a binomial distribution .  the evolution equation for a candidate is  is the distribution of the number of votes at time for candidate .the first and second terms of ( [ m1 ] ) denote the votes of the copycat voters ; the third and fourth terms denote the votes of the independent voters .if we set as the distribution of the number of votes at time for candidate , we have the relation the initial condition is .this is the relation between the back and front .in this section , we study the exact solution of ( [ m1 ] ) for a special case . for ,we obtain the following master equation : at this limit , independent voters always vote for only one candidate , ( if we set , independent voters vote only for ) .the master equation has a simpler form : when we substitute in the above equation , the last term of the rhs vanishes , and thus , the probability can be calculated easily : for , we can prove ( see appendix a ) that the following equality holds : this is the distribution of the votes for the special case wherein the independent voters always vote for only one candidate , .if we set , all voters are copycats , and we obtain the following reduction : this is a beta binomial distribution . at the limit ,the above equation becomes a beta distribution .note that to obtain ( [ 13 ] ) , we use the identity in , we discussed the physical characteristic of this model . in the limit and with fixed , the scale invariance holds over the entire range .  here , we can calculate the momentum of these distributions to analyze them .the momentum is given by we introduce quasi - momentum as   we can prove ( see appendix b ) that the quasi - momentum can have the following form : if we set , we get the average vote   we study .the coefficients of master equation ( [ process ] ) do not contain the initial votes for ; given by .if we set , the master equation does not depend on the initial conditions .therefore , for a large limit , the behaviour of the moment does not depend on the initial conditions , and .we can also observe this in ( [ fp ] ) and ( [ ito ] ) in the next section . here, we set for the representative case .direct calculation using ( [ moment ] ) is difficult .hence , we study as . in the above case , distribution ( [ distribution ] ) becomes a constant distribution with cut - off .the cut - off implies a fast decay for larger values ( ) .if we set , we get a constant distribution . using ( [ distribution ] ) , we get for large time values , the only time dependent term is . in the case of , we can assume that only the first term of the summation is non - negligible .therefore , cut - off is the inflection point of .using ( [ limit ] ) , we can obtain .then , the momentum is ( [ 3 ] ) is a continuous function of .hence , there is no phase transition throughout . in the next section ,we study the general case in the continuous limit .to investigate long - ranged correlations , we analyze in the limit . we can rewrite ( [ process ] ) as we define with the initial condition .we change the notation from to for convenience .then , we have .the support for the law of is thus .given , we obtain a random walk model let .we now consider } , \nonumber \\p(x,\tau)&=&\epsilon p(\delta_t/\epsilon , t/\epsilon),\end{aligned}\ ] ] where and .approaching the continuous limit , we can obtain the fokker - plank diffusion equation for this process ( see appendix c ) : we can also obtain such that it obeys a diffusion equation with small additive noise : \textrm{d}\tau+\sqrt{\epsilon } \textrm{d}b_\tau,\hspace{2 cm } x_0=\frac{c_0-c_1}{c}. \label{ito}\ ] ] though ( [ fp ] ) and ( [ ito ] ) are equivalent , hereafter , we only deal with ( [ ito ] ) for simplicity .assume is random or deterministic . let be the variance of . if is gaussian or deterministic , the law of ensures that the gaussian is in accordance with density where is the expected value of and is its variance .if is the logarithm of the characteristic function of the law of , we have and identifying the real and imaginary parts of , we obtain the dynamics of the mean of as the solution for is since we are interested in the voting rate obtained , we introduce a new scaled variable : the solution for is when , .this implies that the average percentage of s votes againt the total poll is .when , .this agrees with our assertion that the scaled distribution of votes becomes a beta distribution when is large . in this case , the mean value does not change .  from the above discussion, we can infer that the distribution becomes similar to a delta function .the question of how this distribution converges to a delta function constitutes the next problem . to investigate this, we analyze the dynamics of the variance .the dynamics of are given by the riccati equation if , we get if , we get now , we can summarize the temporal behaviour of the variance as  here , we introduce rescaled variables the solution for is  if , becomes .this agrees with our assertion that the distribution of votes becomes a beta distribution .if or , candidate gathers of all the votes in the scaled distributions , but the voting rate converges more slowly than that in a binomial distribution .if , the voting rate becomes , and the distribution converges as it would in a binomial distribution .hence , if independent voters form a majority , the distribution of votes becomes similar to a delta function and the convergence is at the same rate as that in a binomial distribution .if copycat voters form the majority , the distribution remains the same but the convergence is at a rate slower than that in a binomial distribution . in this phase , it is difficult to ascertain the causes for the delay of the convergence .similar phenomena can be seen in several fields . in daily financial data ,the motion of the price does not represent a markov process , and it is difficult to forecast the future price . in fact , it has been pointed out that the motion of price is super - diffusive behaviour and the stochastic differential equation for price is similar to ( [ ito]). when all voters are copycats , the distribution becomes a beta distribution and does not converges .curty and marsili recently introduced a model about information cascade .their model is based on game theory .they showed that when the fraction of herders in a population of agents increases , the probability that herding produces the correct forecast undergoes a transition to a state in which either all herders forecast rightly or no herder does .their model is similar to the limitation of our model in the case wherein voters are unable to see the votes of all the voters but can only see the votes of previous voters .however , there is a significant difference between our model and their model with respect to the behaviour of copy cats . in their model, copycats always select the majority of votes , which is visible to them .thus , the behaviour becomes digital ( discontinuous ) .we aim to carry out an analysis of the influence of this behaviour in the future .we now consider the correlation . for a beta binominal distribution, we can define the parameter . this parameter represents the strength of following a decision .if we set , everyone votes for the candidate who received the first voter s vote . on the other hand , when , copycats become independent .it should be noted that our conclusion does not depend on except when . for large , convergence is not related to , but is related to , the appearance probability of independent voters .  herewe discuss the solution in the previous section .if we set ( is the same as relation([re ] ) ) , ( [ pd ] ) becomes for large .this is because independent voters votes become deterministic .hence , in ( [ fp ] ) , the diffusion term disappears . in ( [ gp ] ) ,the noise term disappears , and the dynamics of are given by then , phase transition disappears , and the behaviour of is continuous this result is acceptable , following the discussion in the previous section and that ( [ moment ] ) is continuous with respect to .( see ( [ 3 ] ) . )if there is a consensus about the fundamental value , copycat voters affect the convergence in proportion to their ratio .further , in this case , the convergence does not depend on correlation .  in order to confirm the analytical results pertaining to the asymptotic behaviour , we perform numerical simulations .we use the master equation ( [ 1 ] ) directly .figures 2 and 3 display the deformation of the distribution of votes for over time .figure 2 is the case wherein the independent voters vote for with the probability , and figure 3 is the case wherein the independent voters always vote for .we can see that the distribution converges to a delta function for the votes of the independent voters .if all voters are copycats ( ) , the distribution becomes a beta binomial distribution .because of the doped binomial distribution ( independent voters ) , the distribution is deformed . for the case , we can obtain an exact solution in section 3 .   at for and .,title="fig : " ] [ fig3 ]   at for and .,title="fig : " ] [ fig4 ]  figures 4 and 5 display the resulting scaled variance for different .the distribution converges to a delta function over time . however , there are differences between figures 2 and 3 at .the difference is in the speed of the convergence that is characterized by the scaled variance . in the general case ( ), we can recognize the phase transition ( figure 4 ) .if , the variance converges at the same rate as that in a binomial distribution ( slope ) . if or , super - diffusive behaviour is exhibited , and convergence is slower than that in a binomial distribution ( slope )the cases and represent two different phases .   for and .,title="fig : " ] [ 1 ]   for and .,title="fig : " ] [ 2 ]  for other cases ( ) , the slope changes continuously ( figure .we investigated a voting model that is similar to a keynesian beauty contest .mathematically , our model is a binomial distribution ( independent voters ) doped in a beta binomial distribution ( copycat voters ) .we calculated the exact solution for special cases and analyzed the general case using a stochastic differential equation .in the special cases , there is no phase transition .we will extend this function to the general in the future .we believe that the obtained solution is a useful clue to understand phase transitions clearly .in general , , the correlation structure , exhibits a dramatic change at a critical value of the doping .if copycats constitutes a majority or number half of the total number of voters , the variance converges slower than it would in a binomial distributions .this implies that our conclusion is extremely volatile because the fundamental value becomes irrelevant , and it is difficult to estimate the conclusion of the vote .we observed phase transition in the limit .however , the long memory is finite ; therefore , this gives rise to the question of whether we can observe the phase transition .this question arises because in this case , voters are unable to see the votes of all the voters but can only see the votes of previous voters .this model is useful to understand the model introduced by curty and marsili. we intend to address this issue in the future .this work was supported by grant - in - aid for challenging exploratory research 21654054 ( sm ) .we prove the assumption ( [ rec ] ) . multiplying ( [ process ] ) by and summing over we get ] , \nonumber \\\ \label{6}\end{aligned}\ ] ] where we call the second and third terms of the rhs without the coefficient  and , respectively .we can rewrite as we can rewrite as then , is given by substituting ( [ 9 ] ) in ( [ 6 ] ) , we can obtain the time evolution of the summation : for , we can prove that the following equality holds : the analytic form can be obtained by multiplying both sides with = ( -1)^{l-1}(c_0)_{k'-c_0 - 1}/[(1)_{k'-l - c-0 + 1}(1)_{l-1}]$ ] and summing over       we use and , a standard iid gaussian sequence ; our objective is to identify the drift and variance such that given , using the transition probabilities of , we get + 1}-\delta_{[\tau/\epsilon ] } ) = \epsilon(2p_{[\frac{l/\epsilon+c+\tau/\epsilon-1}{2}],\tau/\epsilon}-1 ) = \epsilon[(1-p)(1 - 2q)+\frac{px}{\tau+1}].\ ] ] then , the drift term is . moreover , ,\tau/\epsilon } + ( -1)^2(1-p_{[\frac{l/\epsilon+c+\tau/\epsilon-1}{2}],\tau/\epsilon } ) ] = \epsilon^2,\ ] ] such that in the continuous limit , we can obtain the fokker - plank diffusion equation for this process :      eguluz v and zimmermann m 2003 _ phys .lett . _ * 85 * 5659 stauffer d 2002 _ adv.complex syst . _ * 7 * 55 curty p and marsili m 2006 _ jstat _ p03013 kanter i and kessler d f 1995 _ phys .lett . _ * 74 * 4559 schenkel a , zhang j and zhang y c 1993 _ fractals _ * 1 * 47 hod s and keshet u 2004 _ phys . rev. e. _ * 70 * 11006 usatenko o v and yampolskii 2003 _ phys .lett _ * 90 * 110601 huillet t 2008 _ j. phys .a _ * 41 * 505005 mori s and hiskado m 2010 _ j.phys.soc.jpn _ * 79 * 034001 hisakado m , kitsukawa k and mori s 2006 _ j. phys .a. _ * 39 * 15365 kullmann l and kertesz j 2001 _ phys .e. _ * 63 * 051112

Abstract: we introduce a voting model that is similar to a keynesian beauty contest and analyze it from a mathematical point of view . there are two types of voters - copycat and independent - and two candidates . our voting model is a binomial distribution ( independent voters ) doped in a beta binomial distribution ( copycat voters ) . we find that the phase transition in this system is at the upper limit of , where is the time ( or the number of the votes ) . our model contains three phases . if copycats constitute a majority or even half of the total voters , the voting rate converges more slowly than it would in a binomial distribution . if independents constitute the majority of voters , the voting rate converges at the same rate as it would in a binomial distribution . we also study why it is difficult to estimate the conclusion of a keynesian beauty contest when there is an information cascade .   * standard and poor s , marunouchi 1 - 6 - 5 , chiyoda - ku , tokyo 100 - 0005 , japan  department of physics , school of science , kitasato university , kitasato 1 - 15 - 1 + sagamihara , kanagawa 252 - 0373 , japan


sensor networks  consist of a set of small , autonomous devices which may be used , for example , for environmental monitoring , patient monitoring in health care , and industrial machinery surveillance . as sensor nodes may be distributed in areas without power supply , or they may be mobile , they are generally equipped with batteries , which makes energy a scarce resource .a basic idea for saving energy is to periodically switch off the sensor nodes . the mechanism that establishes the alternation between the active and inactive statesis generally called _ duty - cycling _ ( see , for example ,  ) . in some cases ,duty - cycling is energy - aware ( see , for example ,  ) .the main disadvantage of these approaches is that they require a quite regular pattern for the availability of energy from the environment .  in previous work we introduced and studied a possible technique for energy - aware duty - cycling in ( mobile ) sensor networks with energy harvesting capabilities .this system is inspired by self - synchronized sleeping patterns of natural ant colonies  .the focus of these first studies was purely on the swarm intelligence aspects of the proposed system .the experiments were performed without considering , for example , packet loss , collisions and network failures .before we outline the contributions of this work , we introduce already a glimpse of the basic behavior of this previously introduced system ; see figure [ fig : generic - behavior ] .the solid line shows the fraction of active nodes over time , whereas the slashed line shows the average battery level of the nodes over time .finally , the dotted line represents the sun power that is used to establish the amount of energy which can be harvested by the sensor nodes at each time step .note that all three measures are scaled to  ] .hereby , corresponds to a full battery . in the following we first outline the determination of a so - called _ ideal transmission power level _ , which then leads to the choice of the real transmission power level .the ideal transmission power level ( ) of a sensor depends on the current battery level : , where , respectively , are parameters that fix the minimum , respectively maximum , transmission power levels .only when batteries are fully charged the ideal transmission power level may reach .the ideal transmission power level is then translated into the _ real transmission power level _ as follows : such that \ ] ] at this point it is important to realize that the transmission power level is used only for sending the duty - cycling message . for other messages during the second phase of each period ,the user application is responsible for choosing transmission power levels .the duty - cycling event described above is summarized in algorithm [ algo : basic ] . as mentioned above , the battery level of the sensor nodes is responsible for their choice of a transmission power level for sending the duty - cycling message. therefore , the battery level of course affects the communication topology in the context of the duty - cycling mechanism .calculate ( see eq .[ eqn : calculate - activity ] ) draw a random number  ] .hereby , corresponds to absolute darkness . in  we described a model for the evolution of the sun light intensities , that is , for the evolution of over time .here we consider exactly the same model .additionally , we assume a variable cloud density  ] range that is used by the description of duty - cycling given in section [ sec : concept ] .   assuming that is , the length of one period corresponds to 60 seconds , the simulations that we conducted span days ( each day consists of periods ) . the first phase of each period , which is reserved for the duty - cycling events ,was given seconds .information about the state of the sensor nodes ( active versus inactive ) is collected at the start of each period .the most important measure taken is the _mean activity _ of the sensor network , which is measured at any time as follows : \ ] ] note that , the greater the more sensors are active at the specific time at which was determined .self - synchronization behavior is characterized by an oscillating value of over time .this was shown already for a mobile sensor network with sensor nodes in figure [ fig : generic - behavior ] of the introduction ( see solid line ) .however , the results from that figure were obtained in a _ perfect environment _ with no collisions or transmission failures and no propagation times . moreover , the energy model that was used had no relation to real sensor node hardware .  the experiments that we present in this section aim at proving the applicability of the proposed mechanism in real sensor networks .all experiments are done on the basis of a static network of isense nodes as simulated by the shawn sensor network simulator .for the first experiment that we conducted we assumed a zero probability for packet collisions .moreover , we assume a cloud density of zero , that is , for .figure [ fig : behavior ] shows the obtained duty - cycling behavior .again , the solid line shows the fraction of active sensor nodes over time , whereas the slashed line shows the average battery level of the sensor nodes over time . finally , the dotted line represents the sun power that is used to establish the amount of energy which can be harvested by the sensor nodes at each time step .the graphic shows the behavior for one day of simulation , that is , 1440 periods .self - synchronized duty - cycling is indicated by the appearance of activity peaks over time .it is remarkable how the system adapts to the available energy resources , reducing the height of the peaks when the battery level of the nodes is reduced .note that when a lot of energy is available the system can even prescind from switching off sensor nodes .this can be seen by the existence of a large activity peak of about 200 periods of length located around period 14000 .note that for this experiment the average fraction of nodes that are active at each period is approximately .this measure will henceforth be called the _mean system activity_.  concerning the energy spent by the duty - cycling protocol with respect to the user application , we took measures over the whole simulation of periods ( that is , 30 days ) .  the energy spent by duty - cyclingis hereby split into the `` idle '' and `` active '' states as well as the energy spent for transmitting the duty - cycling messages ( tx ) and receiving duty - cycling messages ( rx ) .note that message reception is the task which consumes most of the energy . in total, the duty - cycling mechanism consumes approximately of the total amount of spent energy .this may seem quite high at first .however , consider that this percentage strongly depends on the value of , which we have set to a very moderate value of 0.001 . increasing this valuewill obviously cause the decrease of the percentage of energy spent by duty - cycling .  after these initial studies we will now test the duty - cycling mechanism in two adversarial scenarios . in first place , it is shown how the system responds to situations with communication failures . in second place , the behavior of the system is studied in scenarios where energy harvesting is limited , for example , due to cloudy weather . finally , we present a mechanism for the automatic parameter adaptation of the system for what concerns different network sizes .   with the next experiment we aim at studying the robustness of the system with respect to communication failures .the experiment consists in simulating the duty - cycling protocol under different packet loss rates. a packet loss rate of $ ] means that the probability of correctly receiving a message is .we repeated the initial experiment as outlined above for all packet loss rates between 0 and 1 , in steps of 0.01 .the results are shown in terms of the obtained _ mean system activity _ for each considered packet loss rate in figure [ fig : duty - experiments ] ( top ) .it can be observed that the behavior of the system does not visibly change until a packet loss rate of about .this means that the proposed system is quite robust against packet loss . only for packet loss rates greater than about the system behavior degrades .another interesting question concerns the possible change in system behavior when energy harvesting is restricted .considering the example of solar panels , this is the case , for example , with bad weather conditions .remember that the proposed model is able to simulate bad weather conditions by means of cloud densities greater than zero .we repeated the initial experiment ( that is , without considering packet loss ) for a range of different cloud densities between and .figure [ fig : duty - experiments ] ( middle ) shows the evolution of the obtained mean system activity when moving from low to high cloud densities . as expected , with increasing cloud density the mean system activity decreases .interestingly , the relation between cloud density and the mean system activity is linear .   when changing to differently sized networks ( we only considered 120 node networks so far ) , it is intuitively clear that some parameter values must be adjusted in order to maintain a functional system .note that when changing the network size , the node density changes .hence , it is reasonable to assume that for maintaining the shape of the activity peaks , the choice of the transmission power level and the probability of spontaneous activation should be adapted to the new network size . a way of obtaining the new system parameters is described in the following .with , and we refer to the new number of sensor nodes , the probability of spontaneous activation and the transmission power level of the new , differently sized , network .first , in order to obtain the same wake - up rates as in the case of a 120-node network , the following rule can be applied : , where and are the parameters from the original network .note that this rule increases the probability of spontaneous activation of the nodes when the network size is decreased , and vice - versa when the number of nodes increases .moreover , the average number of nodes spontaneous activations per time unit is maintained .next we introduce a rule for adapting the transmission power level .the basic idea is to have a constant average number of sensors being reached by a transmission . due to the fact that the sensor nodesform a random topology , the following reasoning was used . in general, the number of nodes that can be reached by the ideal transmission of a sensor can be estimated as follows : , where is the number of sensor nodes and is the space in which the sensor nodes reside . in our caseit holds that .therefore , this term reduces to .as is known for the case of 120-node networks , an adjusted transmission power level can be calculated for networks of different sizes as follows : where is the size , respectively is the transmission power level , of the new network .however , note that the transmission power level is not directly modifiable as an algorithm parameter .the only parameters of our algorithm for what concerns to the transmission power level are and .these values are used as the boundaries of the region for the ideal transmission power levels . therefore , our scaling method consists in using eq .[ transmission - power - scaling ] for obtaining and for delimiting the value of the new networks ideal transmission power level .  with this scaling methodology we repeated the initial experiment ( that is , without packet loss ) for a range of different network sizes graphic in figure [ fig : duty - experiments ] ( bottom ) shows the evolution of the resulting mean system activity .ideally we would have expected a more or less straight line at about .this would have meant that the introduced parameter scaling method leads to a system that behaves equally for all network sizes .however , as can be seen , the scaling method only works well for networks with more than 100 nodes . for smaller networks , the mean system activity decreases .however , this can be explained by the decreasing connectivity and communication ability .in this paper we introduced a protocol for self - synchronized duty - cycling in wireless sensor networks with energy harvesting capabilities .this protocol , inspired by real ant colonies , has been implemented in the generic algorithm library wiselib .moreover , experiments have been performed with the network simulator shawn configured to simulate isense hardware .the proposed technique adapts to changing energy conditions in a self - organized way .moreover , it is very robust for what concerns packet loss and limitations of energy harvesting . in the future we plan to combine this protocol with real user applications such as monitoring or tracking .moreover , we plan to verify the experiments on real hardware .finally , as our technique does not depend on nodes being static , the results obtained in this paper can be reproduced for mobile sensor networks .this work was supported by the eu project fronts ( fp7-ict-2007 - 1 ) . in addition ,christian blum acknowledges support from the spanish _ ramn y cajal _ program , and hugo hernndez acknowledges support from a catalan _ fi _ grant .t. baumgartner , i. chatzigiannakis , s. fekete , c. koninis , a. krller , and a. pyrgelis .wiselib : a generic algorithm library for heterogeneous sensor networks . in j.sa silva et al . , editors , _ proc . of ewsn 2010 _ , volume 5970 of _ lncs _ ,pages 162177 .springer , 2010 .i. chatzigiannakis , s. fischer , c. koninis , g. mylonas , and d. pfisterer . : an open large - scale wireless sensor network testbed . in n.komninos , editor , _ proc . of sensappeal 2009 _ , volume 29 of _ lecture notes of the institute for computer sciences _ , pages 6887 .springer , 2009 .h. hernndez and c. blum .self - synchronized duty - cycling in sensor networks with energy harvesting capabilities : the static network case . in f. rothlauf et al . , editor , _ proc .of gecco 2009 _ , pages 3340 .acm press , 2009 .h. hernndez , c. blum , m. middendorf , k. ramsch , and a. scheidler .self - synchronized duty - cycling for mobile sensor networks with energy harvesting capabilities : a swarm intelligence study . in _ proc . of sis 2009_ , pages 153159 .ieee press , 2009 .

Abstract: in this work we present a protocol for self - synchronized duty - cycling in wireless sensor networks with energy harvesting capabilities . the protocol is implemented in wiselib , a library of generic algorithms for sensor networks . simulations are conducted with the sensor network simulator shawn . they are based on the specifications of real hardware known as isense sensor nodes . the experimental results show that the proposed mechanism is able to adapt to changing energy availabilities . moreover , it is shown that the system is very robust against packet loss .