the genetic regulatory network ( grn ) plays an essential role in controlling cell differentiation and maintaining the stability of cell types .each cell type has its own stable gene expression profile ( the set of genes in its genome which are actively expressed ) .thus , understanding cell differentiation requires a model of the behavior of the many interacting genes that establish the expression profile associated with each cell type .gene regulatory interactions are often described as a network , with each node representing a gene and links between two genes indicating a regulatory interaction .the large - scale characteristics of genetic regulatory networks have been studied in some detail , but only in terms of network structure : the degree sequence , the prevalence of network motifs , and so forth . in terms of grn dynamics ,most efforts have been bottom - up " devoted to simulating specific regulatory interactions or analyzing models of small genetic circuits and therefore focused on specific subsections of the network .our goal is to work in a top - down " fashion to construct a minimal dynamical model which captures many important features of cell differentiation .most significantly , we capture the sequential branching differentiation of cells from the multipotent progenitor cell to more lineage - restricted cells and finally to terminal cells .  to construct our model , we first note five elementary characteristics of cell differentiation which we wish to capture : stability , directionality , branching , exclusivity , and promiscuous expression .of course , this is not meant to be an all - encompassing or fully detailed description of cell differentiation .however , these simplified characteristics prove rich enough to create an interesting model .we review here our definition of each characteristic ._ stability _ : cell types should be stable states of the dynamics. a cell which is in specific cell type should remain in that cell type for a `` long time '' ( relative to the time scale of gene activation , transcription and translation ) , unless perturbed by an external signal ._ directionality _ : progenitor cells should reliably differentiate into a variety of more specialized cell types and should only very rarely regain pluripotency by de - differentiating ._ branching _ : differentiation should occur through discrete lineage specifications , in which a multipotent cell makes a binary fate decision and commits to one of two different lineages ._ exclusivity _ : once cells in our model have committed to a specific lineage , they should lose the ability to produce cell types from the other lineage . each differentiation event both selects a specific lineage and forecloses the possibility of the other .  _promiscuous expression _ : multipotent progenitor cells should simultaneously express genes specific to all of their prospective lineages .for example , a hematopoietic stem cell expresses genes found in all of the lineages into which it can ultimately differentiate .transition of a module in our network from the multipotent cell type to one of the progenitor cell types .modules consist of two genes , each of which activates its own transcription and inhibits the transcription of the other gene . in the top panels , both genes in module a are active at a moderate level , represented by the gray coloration .this intermediate steady state is poised between the two subordinate lineages .modules b and c are also both expressing their genes at a moderate level , as a result of the influence of module a. in the lower panels , the cell has differentiated from the multipotent cell type to progenitor type 1 .module a has switched from the intermediate stable state to the stable state in which gene is highly active ( as represented by the black coloration ) , whereas gene has become inactive ( as represented by the white coloration ) . as a result, module c has become completely inactive due to the signals from module a , whereas module b remains in the intermediate steady state , and is thus prepared to make further cell fate decisions . ]  to create a model which exhibits these characteristics , we take our inspiration from the reported network properties of real grns . biological networks exhibit modularity , i.e. they can be divided into largely independent subnetworks , called modules .the nodes in a given module interact strongly with other nodes in the module , but only weakly with nodes in other modules . in our model ,the network is composed of many modules , each responsible for a specific cell fate decision .these modules influence each other in a hierarchically organized fashion , as shown in fig .[ biftree ] , which ensures that differentiation events occur in a branching , sequential fashion .the dynamical state associated with each cell type is a result of the dynamics of the associated module .arrows pointing from one module to another indicate influence of the former on the latter .influence between modules acts as follows : a parameter in the regulated module depends on the expression level of a gene in the regulating module . in the top pairs of panelsthe system is in the multi " ( multipotent ) state , and in the bottom pair , the system is in one of the two available progenitor cell types . on the dynamical side , the module a undergoes a change in its state ( the specifics of which we will discuss shortly ) . in making the change ,it exerts an influence on the two modules downstream , in this case turning off module c and allowing module b to enter a state which corresponds to pro 1 " ( progenitor 1 ) .notice that like module a , modules b and c are also linked to down - stream modules on which they exert influence , and that module a is itself influenced by a module higher up " on the differentiation cascade .this allows the process to proceed in series through as many levels as are required .our model suggests an important role for modularity : the topology of the regulatory network sequentially restricts the trajectory of the dynamical system to ever smaller subspaces of the state space .these restrictions ensure that a given cell type can only take the appropriate pathways of subsequent differentiation .the dynamics of each module of the network imposes the branching structure on cell differentiations .previous work has studied the behavior of grns using a variety of different dynamical system models : boolean networks , differential equations , and chemical master equations .although each model has its merits , for our current purposes differential equations provide the best combination of rich behavior and analytical tractability .   to model a set of interacting genes with differential equations , we associate the expression level ( number of proteins produced by the gene which are present in the cell ) of each gene with a variable .the state of the system is given by the vector  ] , which points in the direction of the future time evolution of the state under the dynamics .these vectors trace out the trajectory of the system , i.e. the time series of states the system will visit .a fixed point is any point in state space in which the state of the system is trapped " under the deterministic dynamics ( ignoring noise ) .formally , this occurs for any point such that for all .the fixed point is stable to small perturbations if the eigenvalues of the jacobian at are all negative . in our model , each cell type is represented by a stable fixed point of the dynamics .our goal is to construct the simplest plausible set of genetic regulatory interactions for which the state of the system moves , either by random fluctuations or external signals , through a sequence of branching , exclusive steps from the fixed point which represents a multipotent cell type , to states which represent a series of intermediary cell types , and finally to a terminal cell type .huang _ et al ._ provide experimental evidence that the genetic circuit controlling hematopoietic ( blood cell ) stem cells consists of a pair of mutually repressing genes , each of which also acts as a self - activator .such a feedback loop is reported to be an over - represented motif in real genetic networks .we have chosen the huang switch for our cell - fate controlling module , as shown in fig .[ biftree ] .the differential equations describing the huang switch are : these equations model the dynamics of self - activation and mutual inhibition with hill functions .the parameters give the maximum transcription rate due to self - activation , and control the concentration of protein or at which the inflection point in the hill function occurs .parameters give the basal rate of transcription , which is suppressed by a downward sloping hill function with controlling the concentration of protein at which its inflection point occurs .the cooperativity of the interactions is given by , which governs the steepness of the hill function .the rate of protein degradation is controlled by .  to simplify our analysis we impose certain symmetries on the equations , and select specific values for certain parameters to produce the desired behavior .we make the maximum contributions of the basal expression rate and the self - activation the same ( ) and thus have a single parameter which controls the overall maximum transcription rate of the gene . for simplicity, we also set the inflection points of the two hill functions ( for the self - activation and inhibition ) to the same value ( ) .finally , we choose to make all parameter values , , and identical between the two equations ( so , etc . ) , thus making the equations symmetric between and .we can write the new set of parameter values as : where we have chosen to set the inflection point and the cooperativity .these parameter choices are not unique , and do not require careful tuning : as long as a wide range of values of , , and produce the desired tri - stable behavior .this leaves us with the new simplified equations : these equations have only two free parameters : and .these are the parameters which are influenced by other modules , following the schematic shown in fig [ biftree ] , and as detailed below . in order to better understand the effect of these parameters , we explore the bifurcation that occurs as they are changed .a bifurcation is a dynamical event in which a small change of a parameter leads to a qualitative change in the dynamics of the system .bifurcations can destabilize fixed points , or cause points which were previously unstable to become stable .thus bifurcations that change the stability of a fixed point are the dynamical system analogues of cell fate decisions in differentiating cells , where we represent cell types as fixed points for simplicity .for example , a parameter change may destabilize the fixed point representing a multipotent cell type , thus driving the state to a different fixed point , which represents the new progenitor cell type .note that while a fixed point may be stable under the deterministic dynamics of the system , in the presence of noise even a stable fixed point may not trap the system indefinitely ; see the next subsection .each panel shows the state space of the two gene toggle switch for the parameter values chosen ( and ) . for are three stable fixed points in the dynamics ( shown as black dots ) and two unstable fixed points ( shown as gray diamonds ) .the arrows give the trajectory of the system at each point in phase space , with longer arrows indicating faster movement and shorter arrows slower movement .as changes from to the module undergoes a subcritical pitchfork bifurcation ( shown in more detail in fig . [ hyster ] ) .the two unstable fixed points converge on the interior stable fixed point , eventually destabilizing it when and leaving only the two exterior stable fixed points . ]  for and , the two variable system is tri - stable " , meaning that there are three stable fixed points in the dynamcs .this state space is shown in the top left panel of fig [ trans2 ] .the three stable fixed points are given by black dots : there is an intermediate fixed point , characterized by moderate amounts of and , and two exterior fixed points , characterized by large amounts of one variable , and small amounts of the other .the gray diamonds represent the two unstable fixed points . leaving the parameter fixed and tuning from to drives a bifurcation from tri - stablity to bi - stabliltiy , as shown in the subsequent three panels of fig . [ trans2 ] .the intermediate fixed point is destabilized , leaving only the two exterior stable fixed points .the bifurcation also occurs if the overall transcription rate , , is increased while is kept constant only the ratio of these parameters has a real effect on the qualitative dynamics .their absolute values merely set the time scale .the position of the fixed points as the parameter changes from to .the black circles represent stable fixed points while the gray diamonds represent unstable fixed points .as the module undergoes a subcritical pitchfork bifurcation , a cell at the interior stable fixed point moves along the trajectory shown by the arrow * a*. when the fixed point is destabilized and the cell is pushed by any random fluctuations from the now unstable fixed point to one of the exterior fixed points following either trajectory * b * or the dotted trajectory .once the cell has been drawn to one of the exterior fixed points , if the module reverses the bifurcation and once again becomes tri - stable , the cell will remain at its new stable fixed point , following trajectory * c*. so once the state of the system has undergone the bifurcation , it can not easily be driven back to its starting position , thus making the differentiation event modeled by this bifurcation one - way . ]  in their analysis of the tri - stable switch , huang _ et al . _ argue that the interior " stable fixed point , characterized by intermediate expression levels of both genes , represents the hematopoietic stem cell , and the two exterior " stable fixed points represent the two committed lineages . they provide evidence that the differentiation from hematopoiesis to the erythroid or myelomonocytic lineages is caused by the destabilization of the intermediate fixed point via a subcritical pitchfork bifurcation .this in turn forces the cell to shift to one of the two other fixed points . in fig .[ hyster ] we show the bifurcation diagram corresponding to this event .the black dots indicate the stable fixed points , and the gray , the unstable fixed points .the -axis shows the value of the bifurcation parameter , .when the circuit is tri - stable : there are three stable fixed points . when , the interior stable fixed point is destabilized by the approach of the two unstable fixed points .thus , the state of the system will eventually migrate to one of the two remaining stable fixed points under perturbation .however , once the system sits at an exterior fixed points , it will not return to the interior fixed point if we reverse the process and decrease the bifurcation parameter the exterior fixed points remain stable to small perturbations throughout .thus , to reverse the cell differentiation , not only would the interior fixed point need to be re - stabilized ; the state of the system would also need to be driven from the exterior fixed point .this ensures that cell differentiation is directional : a simple , transient , one parameter perturbation ( increase in ) can drive the system toward specificity , but there is no similarly simple way to de - differentiate back to multipotency .we chose the tri - stable switch proposed by huang _et al _ for its simplicity and empirical support .several other models have been proposed for the genetic control circuit in hematopoiesis .each of these represents a reasonable model for a genetic switch , and would most likely function as desired if used as modules in the hierarchical , modular network proposed here .this underscores a key strength of the top - down approach : important properties of the dynamics are driven by the hierarchical architecture and may be insensitive to many of the specifics of each module .noise is ubiquitous in biological systems and important for many functions .the grn as a whole , but especially the genes responsible for the process of development ( and cell differentiation ) , must be organized for reliable performance .thus we require that our model behaves robustly in the presence of noise .feedback loops and modularity have been shown to contribute to robustness , and because our model makes use of these design characteristics , we find that robustness is easily attained .we simulate noise in the dynamics using a langevin equation : here , is the same function as in the deterministic dynamics . in the second term , is a wiener process ( an independent white noise process ) .this process introduces additive noise with a magnitude given by the parameter and with no dependence on the state .such noise was proposed by kepler and elston for models of genetic regulatory interactions . to simulate the forward evolution of the system for a length of time , we must integrate the wiener process over the interval which gives us , where is a gaussian random variable with zero mean and unit variance .intuitively , the magnitude of the integral grows like because it behaves like a one dimensional random walk : for a -d random walk , the expected displacement after steps is proportional to .thus we can compute the new state of the system after a time step using the forward euler integration scheme : although the forward euler scheme is only first order accurate , the equations are quite simple and simulating them is very fast .thus , we can easily choose to be small enough to ensure accuracy .naturally , higher order schemes could be applied as needed . note that since the variables in our system describe the concentration of protein products , we require that no variable drop below zero : a negative concentration is non - physical .   the behavior of the module in the presence of noise as it approaches the bifurcation .each panel shows the final positions of 500 simulations ( cells " ) starting from the interior stable fixed point simulated for units of ( in time steps ) .the noise magnitude is and the overall activity for various values of .black circles show the positions of the stable fixed points and gray squares show the end points of each simulation . as increases , the variance of expression levels of and increases near the interior stable fixed point . for none of the simulated cells escape " from this fixed point . for approximately 50 cells ( or ) escape , divided evenly between the two exterior fixed points .when approximately escape . in similar simulations at the same noise strength , ,none of the cells " starting near one of the exterior fixed points transitioned to the interior fixed point for any parameter values tested . ]  in the presence of noise , we find that the transition between stability and instability for the interior stable fixed point is no longer abrupt . as the module approaches the bifurcation point , the restoring force which pushes the state of the system back towards this interior fixed point weakens .we simulated 500 cells starting at each of the three stable fixed points for values of with and for time steps ( units of , a fairly lengthy period in simulation time relative to the timescale of transcription and translation ) .the distribution of the final states for cells starting at the interior stable fixed point spreads as increases ( shown in fig .[ bifurcation - noise ] ) , i.e. as the value of increases , the variance of the gene expression increases .our model hence makes a sharp prediction that may soon be testable _ in vivo _ , using flow cytometry to measure the approximate expression level of a gene for each cell in a population of cells approaching differentiation .if our model is correct , the expression level of significant genes should become more varied across the population as the cells approach a cell fate decision .thus , observing the variance of genes as cells differentiate would provide a clearer picture of which genes are controlling cell fate decisions : those with higher variance .before the actual bifurcation point is reached , we see non - trivial numbers of cells have already moved to one of the exterior , ` differentiated ' fixed points .this indicates that for values of the interior stable fixed point becomes metastable ( weakly stable ) .thus the state of the system will not remain near this fixed point indefinitely in the presence of noise . as we discuss later, progenitor cell types may be metastable fixed points in which the state of the grn rests for a short time before continuing its differentiation .this would explain the difficulty of culturing and maintaining such cell types _ in vitro_.  in closing this section , we note that our model provides a potential dynamical explanation of the observed prevalence of binary ( as opposed to trinary , etc . )branches in cell differentiation .the subcritical pitchfork bifurcation found in our modules arises from the topology of the regulatory interactions and is present for many choices of parameter values without careful tuning .this bifurcation leads naturally to binary cell differentiations .in general , only one of the eigenvalues of the interior fixed point becomes positive at the bifurcation .this implies that flows along the other eigenvectors near the interior fixed point still push the state of the system toward that fixed point .this mechanism concentrates flows along the unstable eigenvector . in fig .[ bifurcation - noise ] the distribution of final states does not spread evenly through the state space ; it spreads preferentially along trajectories following the two sides of the unstable manifold of the interior fixed point . near the fixed point, the unstable manifold is tangent to the unstable eigenvector ; farther from the fixed point , the unstable manifold need not be a straight line .these trajectories , in turn , lead to one of the two exterior fixed points .trinary branchings , since they can not depend on such simple properties of the state space , are comparatively much more fragile and require correspondingly careful selection of parameters .we have found in our model that binary branching occurs ` for free ' from a robustly occurring pitchfork bifurcation , with the simulated cells flowing away from their starting state along one of the two ends of the single , destabilized eigenvector .the modular network and consequences of a single cell fate decision , in detail .the and parameters in all three modules are functionally dependent on one of the genes in the module directly above them . in the case of modulea , we denote the stimulus as a generic external signal to reflect the fact that it could come from another gene , or some external differentiation signal . in the top panel ,the state of the system is poised in the intermediate , meta - stable state of module a ( ) .modules b and c are both in the intermediate stable state , expressing , , , and at a moderate level . in the lower panel ,module a has undergone differentiation , driven by noise , switching to one of the two exterior stable states , high , low .this in turn has caused module b to become meta - stable by increasing the value of from to .module c is inactivated by the change in parameter from to , essentially silencing the genes in the module . ]now we are ready to build the full hierarchical model . in fig .[ celldiff ] , we show a differentiation event in more detail : three modules are shown , and . module influences and . in the top panel ,module is in the intermediate fixed point , but this fixed point is only meta - stable .this meta - stability is caused by the influence of the signal .if module is part of a larger cascade , then the signal will originate from a gene in the module above in the hierarchy ; if is at the top of the hierarchy then the signal may originate from outside the cell . module expresses moderate levels of and , which in turn influence the parameters and in modules and , through the functional relationships : for simplicity , the same functions and are used throughout the network . in fig .[ gandh ] we show how the piecewise linear functions and respond to changes in the variable on which they depend .the specific functional form chosen is not important ; sequential differentiation will occur as long as the functions satisfy two basic requirements .first , if the value of increases substantially , then the interior fixed point in the influenced module should be destabilized , preparing it to undergo differentiation .thus , the function , which controls , should increase slightly in value in response to higher values of .this higher value of will drive the bifurcation above .second , if the value of decreases dramatically , indicating that the gene has been shut off , then the transcription rate , , of the downstream module , which is under the influence of , should be dramatically decreased . to achieve this , for small values of , should be small , effectively silencing the genes in the influenced module .   the piecewise linear response functions and . in the top panelwe show the function , which controls the decay rate , and in the bottom panel we show the function , which controls the transcription rate . specifically , for and for , interpolating between these values for .similarly for and for , interpolating between these values for . ]returning to fig .[ celldiff ] , moderate levels of and result in parameters in modules and which maintain stable , moderate expression levels of their genes , , and .note that any modules downstream of and would also express their genes at an intermediate level .since module is in a meta - stable state , the state of the system is driven by noise after some time to one of the two exterior fixed points ; in the figure , high , low .the result is shown in the lower panel : module has been silenced by the decrease in expression of ( thus also shutting off all the genes in modules downstream from it , under the control of or ) .module , under the influence of increased , has become meta - stable , and is thus waiting to drive a differentiation event as the system chooses " between expressing and .the proposed links integrating the modules correspond to the simplest possible regulatory connections that generate the observed branching behavior . in most models , regulatory control acts on the process of transcription. however , the decay rate can also act as a control parameter , if one protein promotes the degradation of another protein . in the preceding discussion , we chose to use both and as control parameters to simplify the model . since the qualitative behavior of the modules depends only on the relative values of the decay and the activation ,the desired behavior can be accomplished using only the activation rate as a parameter .this requires that the activation rate in a regulated module respond non - monotonically to the relevant gene in the controlling module : for increased levels of the relevant gene , the activation rate decreases slightly ( which has the same effect as increasing ) , whereas for decreased levels of the relevant gene , the activation rate drops dramatically , as it does for , to silence the module .precisely what regulatory connections appear in real networks remains to be seen ; however , modules of mutually regulating transcription factors clearly are linked in real grns .  thus our hierarchical network model allows a single , multipotent cell type to access a multitude of distinct , terminal states in a biologically sensible way .cinquin and demongeot present an alternative mathematical model : a single high - dimensional genetic switch with more than two possible outcomes .their model produces promiscuous expression of antagonistic factors ( which represents the undifferentiated state ) , whereupon a symmetry breaking bifurcation results in increased expression of one of the genes in the circuit , and decreased expression of all the others .while cinquin and demongeot argue that such multi - switches are necessary to capture particular biologically observed differentiations , as a generic mechanism for generating a diversity of terminal cell types , the modular , hierarchical approach taken in our model is more robust : it is not reliant on the highly symmetric , all - to - all coupling present in the cinquin model .our model should work with a diversity of different modules , so long as they conform to certain , loose criteria .although we imposed an , symmetry on the huang circuits used in this paper , we did so only to reduce the number of parameters and simplify the discussion : the huang switch maintains tristability , and all its other important properties , when the parameters are asymmetric .there has been much interesting mathematical work on symmetry breaking in biological systems . coupled cell systems " present a potentially interesting mathematical toolkit for analyzing the dynamics of modular networks , such as the network proposed in this paper : dynamical components called cells " interact with one another according to a given network of connections . studyingthe hierarchical network architecture proposed in this paper using the tools of coupled cell systems may provide useful dynamical insights , and is a fruitful avenue for future work .a simulation of three interacting modules with noise . at interior fixed point of module is changed from stable to meta - stable by an external stimulus . for the next units of time top module remains meta - stable , then finally falls into the high exterior fixed point , as shown in the top panel .this silences module ( and thus genes and , which are rendered inactive ) as shown in the bottom panel .finally , in the middle panel we see module , which became meta - stable after module made its cell fate decision and entered the exterior fixed point .module remains meta - stable for units of until it transitions to an exterior fixed point , completing the differentiation of the cell through this segment of the network . ][ stochres ] shows the result of a simple three module simulation , with the noise parameter .the differentiation tree functions properly in the presence of noise : at the outset , all genes are expressed at a moderate level . at time , a signal is introduced which destabilizes the top module . after some time , the state of the the top module is decided in favor of .as decreases , we see that genes and are also silenced .the genes and continue to be expressed at moderate levels for some time , until through random fluctuations , the system exits the meta - stable state , resulting in increased expression and decreased expression .the individual cell fate decisions are strongly stochastic in the model as presented . once the interior fixed point is destabilized , random fluctuations determine into which exterior fixed point the state of the system falls .this is consistent with the stochastic component of cell fate decisions observed in previous work . as noted in huang _ , the perturbation from the interior fixed point can be done asymmetrically , resulting in intrinsic " as opposed to stochastic cell fate decisions .  at the outset , when any outcome was possible and the cell multipotent ,all of the genes are expressed .as the cell differentiates and loses its pluripotency , the genes which correspond to the lineages which are no longer accessible to the cell are silenced .this sequence of silencing events provides a clear explanation of the exclusivity of cell differentiation , as well as a mechanism for ensuring an orderly progression of cell types .consider a differentiation tree with levels and assume the tree is complete and symmetric , so that binary decisions occur .[ biftree ] is an example of such a tree for .a general tree will contain modules and genes . at the beginning of the differentiation processall genes will be expressed promiscuously at intermediate levels . when the module at the top of the tree transitions to one of its two possible exterior fixed points , the cascade of silencing turns off an entire -level sub - tree .thus the effective dimension of the state space is reduced from to .the silenced genes are fixed near and subsequent time evolution of the system can only take place in the remaining state space , which is associated exclusively with the available lineages .exclusivity and orderly cell type progression follow immediately from this successive restriction of the dynamics to smaller and smaller subspaces .this picture also yields an experimental prediction .if the dynamical mechanism proposed here is correct , then differentiation from multipotent to more specific cell types should be characterized by a sequence of gene silencing events , with many more genes silenced ( on average ) early in the differentiation process and fewer genes silenced as the terminal type is achieved .as seen in the simulation described above , individual cells can remain in the meta - stable state for a long time before continuing their differentiation .we simulated iterations of the model , and measured the length of time the system spent in meta - stable states .we found an exponential distribution of waiting times , with some of our simulated cells spending as many as units of in the meta - stable state .for the purposes of comparison , we note that once the state escapes the region of state space close to the meta - stable fixed point and the transition to an exterior fixed points begins , the transition often occurs in less than units of .so while individual cells can not be maintained in these meta - stable states indefinitely , such states can be quite long - lasting .we have not considered the proliferation of cells ( i.e. division maintaining cell type ) , as described by hoffmann _et al _ .the proliferation rate is a function of the state of the grn .thus if the proliferation rate is high in the meta - stable states of intermediate gene expression , a population of cells in the meta - stable state could be maintained indefinitely .the number of progenitor cells found _ in vivo _ is the result of a balance between : cells differentiating into the meta - stable progenitor state from more multipotent cell types ; progenitor cells proliferating ; and progenitor cells differentiating out of the progenitor cell type to a more lineage - committed state through the stochastic dynamics described above .thus the meta - stability of the dynamical fixed point may provide an explanation for the limited capacity for self - renewal of some progenitor cell types _ in vitro_.our model reproduces all of the desired behaviors of cell differentiation by using established network properties of real genetic networks .cell differentiation occurs between dynamically distinct , stable cell types .cell fate decisions are directional : small parameter changes can easily drive differentiation down the tree , but not back up again .differentiation occurs in a branching , exclusive fashion , due to the properties of the bifurcation and the progressive restriction of available state space . and finally , a cell type higher in the differentiation tree exhibits promiscuous expression of genes specific to all its accessible lineages , by construction .it is easy to add additional control mechanisms to this model ; for example , we could require external signals , in concert with the influence of higher " modules , to drive each bifurcation .our goal , however , is merely to outline a minimal dynamical system capturing certain generic features of cell differentiation .in addition , our model suggests a new role for modularity and an explanation for the prevalence of binary cell fate decisions .modules allow for the easy implementation of repeated behaviors , in this case facilitating the repeated branchings observed in cell fate decisions .the regulatory structure of the hierarchical , modular network topology serves to channel trajectories in the state space , by progressively restricting them to lower dimensional subspaces .this guides the state of the system through an orderly progression of fixed points , representing the various cell types .further , binary cell fate decisions emerge effortlessly and robustly from the topology of our network modules due to the behavior of flows near the interior fixed point of intermediate gene expression .the dynamical simplicity and robustness of binary cell fate decisions could explain their prevalence in cell differentiation .our model thus illustrates that many of the important properties of cell differentiation can be captured quite naturally by a system of loosely coupled genetic modules .building progressively more accurate whole - system models that recapitulate key functionalities provides an important complement to the traditional bottom up " approach , which seeks maximal integration of all available details .top down " approach , focused on less detailed minimal models , allows us to focus on understanding the influence of the topology of the genetic regulatory network on the resulting dynamics .the details of our individual modules and their interconnections are of course oversimplified and incomplete . however, emerging evidence from the topology of real genetic networks suggests that the overall modular architecture is likely to be correct .we hope that future experimental results will show to what degree the genetic regulatory network in general , and the cell differentiation network in particular , can be treated as a collection of loosely coupled modules .our model makes two clear predictions regarding the cell differentiation network . on the one hand ,the variance in expression level of genes involved with the differentiation process should increase across a population of cells as the population approaches differentiation .on the other , the progressive restriction of available state space should be reflected in a sequence of gene silencing events as cells proceed from more multipotent , promiscuously expressing types down a particular lineage , with fewer genes ( on average ) going silent as the terminal cell type is approached .as large - scale models improve through interaction with experimental evidence , we believe that the top down and bottom up approaches can eventually meet in the middle " , fitting carefully specified subsections of the genetic network into a larger network framework .danos , v. , e. a. ( 2008 ) , in _ proceedings of the first international workshop , formal methods in systems biology , fmsb. ( j. fisher ed . ) , vol .5054 of _ lecture notes in bioinformatics _ , pp . 103122 , cambridge , uk : springer , berlin , germany

Abstract: multi - potent stem or progenitor cells undergo a sequential series of binary fate decisions , which ultimately generate the diversity of differentiated cells . efforts to understand cell fate control have focused on simple gene regulatory circuits that predict the presence of multiple stable states , bifurcations and switch - like transitions . however , existing gene network models do not explain more complex properties of cell fate dynamics such as the hierarchical branching of developmental paths . here , we construct a generic minimal model of the genetic regulatory network controlling cell fate determination , which exhibits five elementary characteristics of cell differentiation : stability , directionality , branching , exclusivity , and promiscuous expression . we argue that a modular architecture comprising repeated network elements reproduces these features of differentiation by sequentially repressing selected modules and hence restricting the dynamics to lower dimensional subspaces of the high - dimensional state space . we implement our model both with ordinary differential equations ( odes ) , to explore the role of bifurcations in producing the one - way character of differentiation , and with stochastic differential equations ( sdes ) , to demonstrate the effect of noise on the system . we further argue that binary cell fate decisions are prevalent in cell differentiation due to general features of the underlying dynamical system . this minimal model makes testable predictions about the structural basis for directional , discrete and diversifying cell phenotype development and thus can guide the evaluation of real gene regulatory networks that govern differentiation .   cell differentiation , genetic regulatory network , dynamics , modularity


one of the solutions currently discussed in the field of quantum communication , , is a quantum repeater .its construction is not so simple as building classic amplifier because of non - cloning theorem which is one of the most basic and important foundations of quantum mechanics .however , the lack of ability of making perfect copies of quantum information may be compensated with use of teleportation and the phenomenon of entanglement .the protocol of quantum teleportation may be utilized to amplify the signal i.e. a quantum state during the transfer process .this approach may be used , e.g. in a fiber , to improve the quality of transferred information . applying the teleportation protocol and the entanglement to amplify the quantum information results with building of so - called quantum repeater ( qr ) .it should be emphasized that the notion of qr is currently in the center of interest of many researchers , .apart from the theoretical analysis of this subject there are also physical experiments carried out with use of qrs as the elements of quantum networks .these experiments are being accomplished for the transmission of quantum states in an optical fiber and also in the air .therefore , the progress in the field of quantum computing , , , , is strongly connected with the progress of quantum communication because of the need to send information .the notion of qr is currently discussed as a potential solution to the problems of quantum communication and cryptography ( especially for quantum key distribution ) .the three generations of qr were presented do far .the first type utilizes the phenomenon of entanglement and its purification , , .the second generation is based on the near - perfectly entangled pairs generation ( however , in this approach still the entanglement purification is used ) .the third type , of qr utilizes the protocol of teleportation .the quantum information is transmitted from one point to another , so the communication is organized as a one - way scheme .the solution presented in this chapter belongs to the third generation of qr and it uses so - called one - dit teleportation protocol to correct the errors which may appear during the transfer process . the novelty of described solution consists on using only pauli - z gate for error correction .more precisely : the result of measurement performed during the teleportation protocol determines the number of pauli - z operations which have to realized as an error correction .the reminder of this chapter is organized as follows . in section ( [ lbl :sec : single : dit : teleportation : ms : jw : cn:2016 ] ) there is a quantum teleportation protocol presented and the error correction is performed with use of pauli - z gate .section ( [ lbl : sec : protocol : quantum : repeater : ms : jw : cn:2016 ] ) describes the realization of qr with use of previously mentioned teleportation protocol .the interpretation of repeater protocol as a quantum circuit is shown in section ( [ lbl : sec : qc : for : qr : network : ms : jw : cn:2016 ] ) .the summary and the final conclusions are presented in section ( [ lbl : sec : conclusions : ms : jw : cn:2016 ] ) .the protocols presented in this section are defined with notions of dits and qudits .a concept of dit is a generalization of classic bit .for classic bit there can be distinguished only two states : zero and one . in the case of ditmore states are possible .the number of these values is symbolized by the letter .for example , the classic bit is a dit with .a unit of quantum information is so - called qubit .a definition of qubit may be presented as : where vectors , stand for the computational base ( in this case the standard base is used ) : , \enspace \vert 1 \rangle = \left [ \begin{array}{c } 0\\ 1\\ \end{array } \right].\ ] ] these vectors represent the classic states zero and one .  by analogy ,a generalization of qubit is a qudit . in this casemore base states are admissible .the state of unknown qudit with base states is represented as : the number of base states for a given qudit , expressed as , will be also called a qudit s freedom level .standard base for qudits requires more vectors e.g. for so - called qutrits ( qudits with ) standard basis vectors are : ,\enspace { | 1 \rangle } = \left [ \begin{array}{c } 0 \\ 1 \\ 0 \end{array } \right],\enspace { | 2 \rangle } = \left [ \begin{array}{c } 0 \\ 0 \\ 1 \end{array } \right].\ ] ]  the presented repeater protocol is realized as a single - dit teleportation protocol .the described protocol differs from other one - dit protocols , in this way that only pauli - z gate is used as a correction gate .the pauli - z ( or just ) gate for qudits may be defined as :    where stands for the root of unity : more precisely , is a -th root number of unity .the symbol represents the imaginary unit .there are also hadamard gate and cnot gate used .definitions of these gates for qudits are : where . in the case of cnot gatethe gate may be given as well - this gate is obtained as the hermitian conjugate of cnot operation .the gate is useful for a quantum teleportation protocol described in this section .the exemplary matrix form of cnot gate for qutrits ( ) is : where the zeros were replaced by dots to make the notation more legible . in general , the cnot gate for qudits is constructed in a following way : where the symbol stands for matrix direct sum and means the negation operation for qudit with freedom level .the gates , mentioned above , allow to present the teleportation protocol of an unknown quantum state : the 3-qudit state , where belongs to alice and to bob , after performing the hadamard , cnot and operations on it , due to fig .[ lbl : fig : circuit : teleporation : ms : jw : cn:2016 ] , may be described as : where .the gates cnot and realize the entanglement .function is expressed as : a correction gate s form depends on the result of measurement performed on the first qudit by alice ( however , both qudits are measured in the standard base what is a consequence of hadamard gate s use in a quantum cirucit see fig .[ lbl : fig : one - way - quantum - repeater : ms : jw : cn:2016 ] ) .if alice , after the measurement , obtains one of given states then the bob s qudit is characterized by the proper values of probability amplitudes and there is no additional operations needed to correct the obtained state .this also stands with the accordance to the values of function : , - they equal zero .if alice , also after the operation of measurement , obtains one of following states then the state of bobs qudit is the values of function in this case are following : when the gate is used only once , then the correct values of some amplitudes may be recovered . in general, the result of measurement determines the number of operations which should be performed .it can be expressed as if stands for the result of measurement performed on the first qudit .the above description presents the teleportation protocol for an unknown qudit , where gate is needed to correct the final state and only one measurement decides about the way how the correction is done after the teleportation process .  [ cols="^ " , ]   although the suggested number of qudits needed to carry a correct simulation is , it should be emphasized that the phenomenon of entanglement occurs inside of repeaters .it means that the entanglement is present in a teleportation process and it occurs between three qudits which consist the repeater itself .there is no entanglement between particular repeaters .this assumption makes easier the implementation of mentioned quantum phenomena with use of classic computers .it is also important to know that the repeaters are connected physically ( e.g. optical fiber ) with suitable quantum channel .if the numeric or symbolic simulation is to be carried out on a classic machine , then only the three - qudit repeater needs to be simulated .the features of communication channel , which transmits a quantum state , may be simulated apart from the simulation of qr .  ....-- n : numer of nodes -- i : index of node -- : initial state -- h : change history of the transmitted state  h append(h , , 0 ) [ , r ] qrepeater ( ) h append(h , , r )  while i < n do begin   [ , r ] qrepeater ( , r )   ...   other operationse.g. noise generation   entanglement level measuring and etc .   ...   h append(h , , r )         i i + 1 end  ....  fig .[ lbl : fig : classical - sim - of - quantum - repeater : ms : jw : cn:2016 ] presents the pseudo - code illustrating the simulation s algorithm for a given state .the changes are saved in a classic variable which collects the quantum states obtained after the calculations in every node .the initial state is assigned to variable and the final state to .the one - dit teleportation protocol with error correction performed by pauli - z gate allows the realization of qr for transmitting the unknown qudit state .two presented strategies for z - type error correction provide the choice to the user who can run the protocol with the error correction at the end of the whole process or the correction may be performed locally on every qr .the approach with the correction at the end of transmisssion reduces the use of z gate application .we use this gate only at the end of the process to complete the communication protocol by correcting the received information .this solution excludes the need of using pauli - x and hadamard gates , which are utilized in other known protocols for qr realization .the advantages connected with presented approach are : the influence of possible errors , caused by the imperfect realization of quantum gates , is lower and the construction of qr is easier because only the operation of measurement have to be performed ( the error correction is redundant ) .the another interesting problem for further research is using quantum error correcting codes on transmitted state .the presented protocol concerns the pure states , so it would be also important to analyze the quality of transmitted information taking into account the quality of utilized quantum gates and the influence of environment .we would like to thank for useful discussions with the _ q - info _ group at the institute of control and computation engineering ( issi ) of the university of zielona gra , poland .we would like also to thank to anonymous referees for useful comments on the preliminary version of this paper .the numerical results were done using the hardware and software available at the gpu -lab located at the institute of control and computation engineering of the university of zielona gra , poland .klamka , j. , wgrzyn , s. , znamirowski , l. , winiarczyk , r. , nowak , s. : nano and quantum systems of informatics .nano and quantum systems of informatics .bulletin of the polish academy of sciences .technical sciences 52(1 ) , 110 ( 2004 )         ritter , s. , nolleke , c. , hahn , c. , reiserer , a. , neuzner , a. , uphoff , m. , mucke , m. , figueroa , e. , bochmann , j. , rempe g. : an elementary quantum network of single atoms in optical cavities .naturem vol .195 200 ( 2012 )   d. stucki , m. legr , f. buntschu , b. clausen , n. felber , n. gisin , l. henzen , p. junod , g. litzistorf , p. monbaron , l. monat , j .- b .page , d. perroud , g. ribordy , a. rochas , s. robyr , j. tavares , r. thew , p. trinkler , s. ventura , r. voirol , n. walenta , and h. zbinden : long - term performance of the swissquantum quantum key distribution network in a field environment .new journal of physics , vol . 13 , no . 12 , pp . 123001 ( 2011 )  ursin , r. , tiefenbacher , f. , schmitt - manderbach , t. , weier , h. , scheidl , t. , lindenthal , m. , blauensteiner , b. , jennewein , t. , perdigues , j. , trojek , p. , mer , b. , frst , m. , meyenburg , m. , rarity , j. , sodnik , z. , barbieri , c. , weinfurter , h. , zeilinger , a. : entanglement - based quantum communication over 144 km .nature physics 3 , 481 - 486 ( 2007 )

Abstract: in this chapter a quantum communication protocol with use of repeaters is presented . the protocol is constructed for qudits i.e. the generalized quantum information units . one - dit teleportation is based on the generalized pauli - z ( phase - flip ) gate s correction . this approach avoids using pauli - x and hadamard gates unlike in other known protocols based on quantum repeaters which were constructed for qubits and qudits . it is also important to mention that the repeaters based on teleportation protocol , described in this chapter , allow a measurement in the standard base ( what simplifies the measurement process ) and the use of teleportation causes only pauli - z operational errors .   information transfer , quantum repreater , qudit teleportation protocol


ieee 802.15.4  specifies a _ physical _ ( phy ) and a _ medium access control _( mac ) layer for use in _ low rate wireless personal area networks _( lr - wpans ) .802.15.4 itself is used as a stand - alone communication solution for resource - constrained wireless sensor nodes or as a basis for additional standards like wirelesshart or zigbee .it facilitates the so - called _ wireless embedded internet _ and _ internet of things _ ( iot ) scenarios in combination with the ipv6 adaptation protocol 6lowpan  .layers and interfaces of 802.15.4 are depicted in figure [ fig : ieee802154_stack ] .the phy is responsible for the transceiver management ( incl .frequency selection ) , receiving and transmitting data , as well as _ energy detection _ ( ed ) and _ clear channel assessment _ ( cca ) operations , both required for the _ carrier sense multiple access collision avoidance _ ( csma - ca ) functionality of the mac layer .the mac provides lr - wpan management operations , csma - ca , network beacon operations , security functions , a guaranteed time slot ( gts ) mechanism with a superframe structure , as well as the provision of a reliable link between mac instances .both layers define header and frame structures and a number of command messages .in addition , a _ service specific convergence sublayer _( sscs ) is specified to adapt ieee 802.2 llc frames for use with the 802.15.4 mac .additional descriptions are provided by .802.15.4 simulation models are already included in the omnet`++`extension frameworks inetmanet , mixim and castalia .our reasons to create a new implementation for inet _2.x _ are manifold .mixim s 802.15.4 implementation focuses on the phy layer , with an implementation of the 802.15.4a impulse radio ultra wide - band ( ir - uwb ) standard in addition to the narrow - band version .the mac model only covers the csma - ca functionality of 802.15.4 ; other functions are not included .castalia s implementation includes an extensive phy model and many mac functions .castalia s main drawbacks are the lack of higher layer protocols for internet applications ( i.e. , ipv6 , tcp and udp for use in iot scenarios ) as well as support for beacon management and newer omnet`++`releases .inetmanet includes a port of an older ieee 802.15.4 model  , originally developed for inet-2006/omnet`++`3.4 .inetmanet s implementation is missing support for newer 802.15.4 revisions as well as several mac layer service primitives , channel scan operations , and various beacon and wpan management functions .as the wireless embedded internet requires protocols from the tcp / ip family , we favor using the inet framework and implementing a new 802.15.4 simulation model from scratch , to enable feature completeness as well as further combinations with other iot standards like 6lowpan , coap and rpl .the composition of implemented ( compound ) models is shown in figure [ fig : ieee802154_host ] .we modeled each layer , their connecting interfaces as well as the service primitives according to the standard specifications ( refer to figure [ fig : ieee802154_stack ] and ) and general modeling guidelines for ieee 802.15.4 ( * ? ? ?12.3 ) .the 802.15.4 model itself consists of the following parts :  * ` ieee802154radio ` : this base class is an extension of inet s abstract _ radio class_. it includes the reception and the radio model with adjustments for 802.15.4 ( e.g. , _ energy detection _( ed ) , _ clear channel assessment _ ( cca ) functions , link layer service primitives ) . * ` ieee802154phy ` : this class represents the phy layer .message exchange between the mac layer and the radio interface is handled here and service primitives for cca and ed functions are generated in this class . * ` ieee802154mac ` : the mac layer completes the _ network interface card _( nic ) of a ieee 802.15.4 host .this layer manages mac functions such as csma - ca , gts , beacon and wpan management , amongst others . * ` sscs`/`stdllc ` : both modules connect applications to 802.15.4 s mac layer .sscs is used for ieee 802.2-compatible _ logical link control _( llc ) instances ( i.e. , applications ) , whereas the ` standardllc ` provides means for conversions of messages from non-802.2-compatible applications like inet s ` ipvxtrafgen ` .we combined above described layers and modules into a so - called ` ieee802154host ` , similar to inet s ` standardhost ` .    the following classes / headers are not included in fig . [ fig : ieee802154_host ] , yet they are crucial for the simulation model s functionality :  * ` ieee802154enum ` : header that describes 802.15.4 s command , frame and timer types , amongst other enumerations from the standard specification . *` ieee802154fields ` : header that defines _ structs _ for complex 802.15.4 data structures like the pan descriptor or the superframe specification . * ` macaddressext ` : extends inet s ` macaddress ` class with support for eui-64 ( 64-bit _ extended unique identifier _ ) mac addresses and 16-bit ( ieee 802.15.4 ) short addresses . * ` macframecontrolfield ` : class that generates the mac frame control field ( refer to ( * ? ? ?7.2.1.1 ) ) .* ` macpib ` / ` phypib ` : classes that represent mac and phy pan information bases ( pib ) databases that hold attributes required for the layer management .the modules are initialized according to inet s multi - stage initialization procedure .parameters are set to their default or user - specified values ( via ` omnetpp.ini ` ) .necessary timers for the mac and phy layer ( i.e. , the radio module ) are started and the 802.15.4 node goes into receive state if nothing else is specified and no traffic generator is started .packet reception and transmission is modeled with the use of according service primitives as specified in the standard .packets and command frames are transferred over their corresponding interfaces .different use cases ( e.g. , direct data transfer , indirect data transfer , gts handling ) are modeled and available as example configurations in the ` omnetpp.ini ` .we are currently in the process of adapting the 802.15.4 model to the latest inet version and update its specifics to recent revisions of the 802.15.4 standard . at this stage , the model is used and developed as a standalone project .we plan to integrate it completely into inet , thus simplifying the use of 802.15.4 in inet - based simulations .for this , we plan to add ` ilivecycle ` support and integrate the 802.15.4 modules in inet s layer structure and its ` standardhost ` . providing an integration in omnet`++\:/\:`inet s project feature dialog is also outstanding work .  certain modulation schemes ( e.g. , ask and qpsk ) are not used in the 802.15.4 model nor provided in inet yet .we need to tackle this issue along with the parametrization , inclusion of newer phys ( from current 802.15.4 revisions ) and the practical validation of the 802.15.4 model against real - life sensor nodes .choosing phy and energy consumption parameters is another open question , as 802.15.4 hardware platforms from different vendors provide different characteristic numbers .last but not least , we plan to combine and integrate the 802.15.4 model with our 6lowpan simulation model  for inet to expand the use of omnet`++\:/\:`inetin the area of sensor networks and iot simulation .     f. chen and f. dressler , `` a simulation model of ieee 802.15.4 in omnet++ , '' in _ 6 .gi / itg kuvs fachgespraech drahtlose sensornetze , poster session _ , aachen , germany , july 2007 , pp[ online ] .available : http://www7.informatik.uni-erlangen.de/~dressler/publications/fg-sensornetze-2005.pdf   m. kirsche and j. hartwig , `` a 6lowpan model for omnet++ , '' in _ proc .of the 6th omnet++ workshop , co - located with the 6th icst conference on simulation tools and techniques ( simutools 2013 ) _ , march 2013 .[ online ] .available : http://www.omnet-workshop.org/2013/general/technicalprogram

Abstract: this paper introduces a new ieee 802.15.4 simulation model for omnet`++\:/\:`inet . 802.15.4 is an important underlying standard for wireless sensor networks and internet of things scenarios . the presented implementation is designed to be compatible with omnet`++`__4.x _ _ and inet _ 2.x _ and laid - out to be expandable for newer revisions of the 802.15.4 standard .   ieee 802.15.4 , simulation model , omnet`++ `


due to the rapid expanding of the internet , we are overloaded by the unlimited information on the world wide web  . for instance , one has to choose among millions of candidate commodities to shop online .comprehensive exploration is infeasible  . as a result , various recommendation approaches have been proposed to help filtering the relevant information  .for instance , the popularity - based recommendations ( pr ) , which recommend the most popular items to users , are commonly adopted in online recommender systems . however , such recommendations are not personalized such that identical items are recommended for individuals with far different taste . by comparison ,the collaborative filtering ( cf ) makes use of collective data of individual preference and provides personalized recommendations  .so far , cf has been successfully applied to many online applications .recently , recommendation algorithms have been proposed from a physics perspective  .for instance , diffusion is applied on the user - item bipartite networks to explore items of potential interest for a user .this mass diffusion ( md ) algorithm is shown to outperform cf in the recommendation accuracy  . however, a similar problem as observed in pr is found in md : diffusion - based recommendations are biased to popular items even individual preferences are considered . in fact , a good recommendation algorithm should recommend items of personal interest and at the same time maximize the diversity of choices .  an alternative approach based on the heat conduction ( hc ) on the user - item graphsis thus introduced  .this method provides users with many novel items and leads to diverse recommendation results among users .however , hc has low accuracy compared with md .the paradox is eventually solved by combining md with hc in a hybrid algorithm  , which can be well - tuned to obtain significant improvement in both recommendation accuracy and item diversity .though they are helpful in filtering information , recommendation algorithms may impose reinforcing influence on the system , by guidance to one s choices which influence subsequent recommendations and hence choices of others .the influence is amplified with successive recommendations .we note that such perspective is employed to explain the evolution movie popularity  , which yields consistent predictions compared with observed data .it is thus interesting to examine such influence on recommender systems .unlike most existing works which are devoted to improving recommendation accuracy  , our present study presents a physics perspective and utilizes microscopic interactions to explain and predict macroscopic behaviors of recommender systems  .  in this paper , we use the gini coefficient to measure the dispersion in item popularity  .we note that a small dispersion implies similar popularity among items , and hence diverse recommendations for users .we consider various conventional algorithms including the popularity - based , the collaborative filtering , the mass diffusion and the heat conduction algorithms .we focus on the physical aspects and study numerically and theoretically the reinforcing influence of recommendations on the dispersion of item popularity .the result indicates that md and cf reinforce the popularity of popular items , as similar to pr .on the other hand , the heat conduction algorithm increases the popularity of the niche items and generates smaller dispersion in item popularity .our results suggest that recommender systems have reinforcing influence on global diversification .we quantify the influence of recommender systems by measuring the changes in the dispersion of item popularity after successive recommendations .if the dispersion is large , some items dominate in popularity and users have limited choices . on the other hand ,if the dispersion is small , items have similar popularity and users enjoy diverse recommendations .to quantify such global diversity , we make use of gini coefficient  to measure the dispersion of item popularity , as in the case of individual wealth . in addition to wealth, it has been used to measure dispersion in sociology , science and engineering .mathematically , it is given by where is the normalized cumulative popularity when items are ranked in ascending order of popularity , with being the normalized rank .specifically , corresponds to uniform popularity among items , while corresponds to maximal dispersion .  to see how gini coefficient quantifies the changes in popularity dispersion we study as examples the scientific citation data and the baby name data .the scientific citation data is based on the citation relation in the aps ( american physics society ) journals from 1893 to 2009  , and the baby name data is based on the first names taken from us social security administration , and contain the top 1000 boy and girl names every year from 1880 to 2009  .what we are interested most is how the dispersion changes with time in these two systems .the results are reported in fig . 1 from which we can see the gini coefficient keeps increasing in aps citation system while decreasing in baby name system . due to the technological advances , ones get access to far more information than before .good papers can thus have wider spread and are cited more which leads to a larger dispersion and hence lower global diversity .similarly , parents know more candidate names for babies , and the system shows increasing diversification .the above examples show that the changes in global diversity are well captured by the gini coefficient .we thus make use of gini coefficient to examine the influence of recommender systems on popularity dispersion .we investigate in this section the influence of recommender systems on the global diversity by examining dispersion in item popularity . here , we consider four recommendation algorithms including mass diffusion ( md ) , heat conduction ( hc ) , user - based collaborative filtering ( ucf ) , item - based collaborative filtering ( icf ) . in addition , we consider two benchmark algorithms including popularity - based recommendation ( pr ) and random recommendation ( rr ) , corresponding to the recommendations of respectively most popular and random items .we first give brief descriptions of the md algorithm .consider a system of users and items represented by a bipartite network with adjacency matrix , where the element if user has collected object , and otherwise ( throughout this paper we use greek and latin letters , respectively , for object- and user - related indices ) .for a target user , the algorithm starts by assigning one unit of resources to objects collected by , and redistributes the resource through the user - item network .we denote the vector as the initial resources on items where is the resource possessed by object .the redistribution is represented by , where is the diffusion matrix , with and denoting the degree of object and user respectively .technically , recommendations for a given user are obtained by setting the initial resource vector in accordance with the objects the user has already collected , that is , by setting . the resulting recommendation list of uncollected objectsis then sorted according to in descending order .physically , the diffusion is equivalent to a three - step random walk starting with units of resources on the target user .the _ recommendation score _ of an item is taken to be the resources on the item after the diffusion .the scores for objects that user have already collected are set to .the recommendation list for user is generated by ranking all his / her uncollected objects in descending order of their final resources .the hc algorithm works similar to the md algorithm , but instead of a diffusion process , the scores are evaluated by a conduction process as represented by physically , the temperature of an object is considered to be the average temperature of its nearest neighborhood , i.e. its connected objects. the higher the temperature of an item is , the higher its recommendation score .the cf algorithms provide recommendations based on user or item similarities .it is divided into two main categories : the user - based cf and the item - based cf . in ucf ,the recommendation score of an item is evaluated based on the similarity between the target user and the users who collected the item .the final recommendation score for each item can be written as where is the similarity between user and .  in icf ,the recommendation score of an item is evaluated based on its similarity with the collected items of the target user .similarly , the final recommendation score for each item can be written as where is the similarity between item and .the measure of similarities used in cf is subject to definition .here we define the measure of similarity as the number of common neighbors  in the bipartite networks .  with the above mentioned algorithms , we consider a scenario of recommender systems as follows . at every stepa random user is selected as the active user , based on whom the recommendation scores of all items are evaluated . for simplicity , we assume that the active user would accept the recommendation results and select the uncollected item with the highest recommendation score , i.e. adding a link between the active user and the item in the bipartite network .an illustrative example is shown in fig .the red node corresponds to the active user , and the red link corresponds to the choice made by the user according to recommendation results .  in one _ marco - step _ of our simulation , we randomly choose 10 percent of users as active users .after each macro - step , we evaluate the dispersion the item popularity by gini coefficient . note that we do not consider the growth of the system since introducing new users or items may involve the cold start problem for them  .the datasets we will examine are the subsets of data obtained from four online systems : movielens , netflix , delicious and amazon .these data are random samplings of the whole records of user activities in these websites , the descriptions of data are given in table i.  = 0.4em  .description of the data [ cols="<,^,^,^,<,^,^,^",options="header " , ]   we show in fig .3 the evolution of gini coefficient in simulations as a function of macro - step . as we can see , the gini coefficient increases in the presence of md , ucf and icf algorithms .this corresponds to their reinforcing influences on the system , leading to a wider dispersion of item popularity after successive recommendations .a further evidence can be seen in fig .4 , which shows that popular items become more popular , while neglecting the rest of the items .this corresponds to an undesired influence , as choices and visions for users become more limited in the presence of these recommendation algorithms .we can further understand the reinforcing influences of the md , ucf and icf recommender systems by comparing their gini coefficients with the unpersonalized popularity - based algorithm . as shown in fig .3 , similar trends are observed between the four algorithms .we can examine the underlying reasons in fig .4 , which shows that the md , ucf , icf and pr algorithms only recommend to users the most popular items .these results imply that the changes in the distribution of item popularity are similar in these four algorithms .therefore , personalized elements in the md , ucf and icf algorithms do not increases the global diversity as compared to the unpersonalized pr algorithm .      on the other hand , the hc algorithm behaves quite differently from the other algorithms . as we can see in fig .3 , it generally decreases the gini coefficient in movielens and netflix , where density of links is high . in sparse systems ,the three - step conduction process can only reach some items with large degree , and inevitably add links to hot items .this leads to an increasing gini coefficient .moreover , the hc method is also different from the random recommendations as we can see from fig .4 . instead of uniform addition of links ,it inclined to add links to items with small degree .it implies that the hc algorithm does not reinforce the popularity of hot items as the md and cf algorithms .to better understand their influences of recommender systems , we derive analytically the distribution of item scores after the recommendation processes .the major difficulty in analysis comes from the particular network topology of each dataset , which embeds the non - trivial correlations between users and items  . herewe focus on the recommendation influences , and assume a simple topology where users and items are randomly connected  .this corresponds to a crude mean - field approximation , but such assumption facilitates the analysis and the illustration of physical behaviors underlying the recommendation algorithms .  to begin our analysis, we derive the probability that a user and an item are connected in a random graph .suppose we start with cavities on user and on item , which are respectively the degree of and .if one cavity is picked randomly among the items , the probability that being picked is .it implies that , where is the probability that is not connected to . as , expansion to the first order of leads to , where is the total number of links in the bipartite network .we then derive the mean - field expression of recommendation scores in the md recommender system .as mentioned above , the md method is based on the three - step diffusion .the resource vector for items in the first step and last step are denoted respectively by and . in the second step ,the resources are in users side and the corresponding vector is denoted as . by considering the last step of the diffusion process , the score of from user is given by .substitution of leads to as .next we derive the scores for the hc algorithms by again considering the last step of the conduction process. however , the total resources " does not conserve in heat conduction but instead the temperature of user is given by , where corresponds to the random choices of initial collected item for .therefore ,   in the user - based cf , scores of items are evaluated by the similarity between the target user and the users who have collected it .the user similarity is given by the number of common neighbors .therefore , where in the mean - field approximation . the score for object then approximated by where is a constant for a given network .  as similar to user - based cf ,the item similarity in item - based cf can be approximated by in the mean - field approximation . the score for object then approximated by where is a constant for a given network .  in order to compare the simulated results and the mean - field predictions , we evaluate the corresponding total scores that a item receives from all the users .as shown in fig .5 , the mean - field approximation captures both the magnitude and the trend of the recommendation scores .further insights are drawn by noting in most systems , which implies in eqs .( 4 ) , ( 5 ) and ( 7 ) .since we assume that users always accept the item with highest recommendation scores , the recommendation scores in the md , ucf and icf cases are thus similar to the pr algorithm which recommends the most popular items .this again shows the reinforcing influence of these recommendation algorithms . on the other hand , eq .( 5 ) suggests in the hc algorithm , which is item independent as in the case of random recommendations .  though the approximated scores of hc agree well with rr , their behaviors are difference in terms of choices of items . according to ,users select the reachable items with lowest degree after three - step conduction , compared to the random choice .therefore , the hc and rr algorithms show different influence on the dispersion of item popularity , as we can see in fig .3(c ) and ( d ) .as we have seen from the previous sections , the md algorithm reinforces the popularity of hot items and limits available choices , while the hc algorithm recommends items with low popularity and increases global diversity .it is thus interesting to examine the influence on diversity if these two algorithms with opposite influences are combined .we thus adopt the hybrid algorithm of md and hc proposed in  , with the new recommendation score given by the parameter adjust the relative weight between the two algorithms . when increases from to , the hybrid algorithm change gradually from hc to md .we remark that though eq .( [ eq_hybrid ] ) corresponds to a linear combination of scores , the hybrid algorithm is a non - linear combination of hc and md as users select only items with highest scores .   in the hybrid recommendation algorithm in real systems .the results are averaged on 100 independent realizations.,width=340 ]  the influence of the hybrid algorithm on gini coefficient is shown in fig .6 as a function of .the lines with different symbols correspond to gini coefficient measured after increasing macro - step . as we can see from fig .6 ( a ) and( b ) , the gini coefficient increases with , corresponds to a transition from hc to md recommender systems .it is interesting to note that gini coefficient shows a significant increase in a short range of on the netflix and movielens datasets , and becomes saturated afterwards .the saturated gini coefficient corresponds to dominance of the md algorithm such that only popular items are recommended , despite the presence of hc algorithm .similar behaviors are not observed in fig. 6 ( c ) and ( d ) in the delicious and amazon datasets , which are sparse compared to the netflix and movielens datasets .another interesting behavior is noted in fig . 6 ( a ) and ( b ) when we compare the gini coefficient after different macro - steps of recommendations .as we can see , the lines with different symbols intersect at a particular value of , suggesting a steady gini coefficient after the reinforcement of recommendations .the corresponding value of thus corresponds to the balance between the hc and md algorithms , leading to steady dispersion in item popularity .this is desirable when one considers the reinforcing influence on global diversity as undesired side - effect of recommender systems .these values of and the corresponding gini coefficient are compared respectively to the values of with optimal recommendation accuracy and the gini coefficient before recommendation algorithms are implemented .these results show that high recommendation accuracy does not always guarantee a global diversity , leading to a paradox in recommendations .recommendation is an effective way to solve the problem of excess information .however , it is unclear how they allocate popularity among items . in this paper, we simulate successive recommendations and measure their influence on the dispersion of item popularity by gini coefficient .our result indicates that local diffusion and collaborative filtering reinforce the popularity of hot items , widening the popularity dispersion .on the other hand , the heat conduction algorithm increases the popularity of the niche items and generates smaller dispersion of item popularity .simulations are compared to mean - field approximation .our results indicate that there is reinforcing influence of recommender systems on global diversification .this work provides a deeper understanding of these recommendation methods , highlights the importance of the global diversity and may shine some light for developing a new recommendation method that can directly controls the global diversity .this work was partially supported by the swiss national science foundation under grant no .( 200020 - 132253 ) , the national natural science foundation of china(grant no .60973069 ) and the sichuan provincial science and technology department(grant no.2010hh0002 ).  99 a. broder , r. kumar , f. moghoul , p. raghavan , s. rajagopalan , r. stata , a. tomkins , j. wiener , comput .netw . * 33 * , 309 ( 2000 ) .s. maslov and y .- c .zhang , phys .lett . * 87 * , 248701 ( 2001 ) .g. adomavicius and a. tuzhilin , ieee trans .* 17 * , 734 ( 2005 ) .f. cacheda , v. carneiro , d. fernndez , v .formoso , acm trans .web * 5*,1 ( 2011 ) .j. a. konstan , b. n. miller , d. maltz , j. l. herlocker , l. r. gordon and j. riedl , commun .acm * 40 * , 77 ( 1997 ) .j. l. herlocker , j. a. konstan , k. terveen , and j. t. riedl , acm trans .* 22 * , 5 ( 2004 ) .t. zhou , j. ren , m. medo and y .- c .zhang , phys .e * 76 * , 046115 ( 2007 ) .c . zhang , m. blattner and y .- k .yu , phys .lett . * 99 * , 154301 ( 2007 ) .t. zhou , z. kuscsik , j .-liu , m. medo , j. r. wakeling , and y .- c .zhang , proc .sci . * 107 * , 4511 ( 2010 ) .m. b. diaz , m. a. porter and j .- p .onnela , chaos * 20 * , 043101 ( 2010 ) . c. h. yeung , g. cimini , and cjin , phys .e * 83 * , 016105 ( 2011 ) .shang , l .- y .l , w. zeng , t. zhou , and y .- c .zhang , europhys .lett . * 88 * , 48006 ( 2009 ) .r. lambiotte , m. ausloos , phys .e * 72 * , 066107 ( 2005 ) . c. gini , ( 1912 ) ( italian : variabilita e mutabilita ( variability and mutability ) , c. cuppini , bologna , 156 pages . reprinted in memorie dimetodologica statistica ( ed .pizetti e , salvemini , t ) .rome : libreria eredi virgilio veschi ( 1955 ) .http://www.ssa.gov/oact/babynames/ f. lorrain and h. c. white , j. math .* 1 * , 49 ( 1971 ) .k . zhang , c. liu , y .- c .zhang and t. zhou , europhys .lett . * 92 * , 28002 ( 2010 ) .m. e. j. newman , phys .lett . * 89 * , 208701 ( 2002 ) .m. peltomaki , m. alava , j. stat .p01010 , ( 2006 ) .m. e. j. newman , s. h. strogatz and d. j. watts , phys .e * 64 * , 026118 ( 2001 ) .

Abstract: recommender systems are promising ways to filter the overabundant information in modern society . their algorithms help individuals to explore decent items , but it is unclear how they allocate popularity among items . in this paper , we simulate successive recommendations and measure their influence on the dispersion of item popularity by gini coefficient . our result indicates that local diffusion and collaborative filtering reinforce the popularity of hot items , widening the popularity dispersion . on the other hand , the heat conduction algorithm increases the popularity of the niche items and generates smaller dispersion of item popularity . simulations are compared to mean - field predictions . our results suggest that recommender systems have reinforcing influence on global diversification .