* keywords * : model uncertainty ; stochastic differential equations ( sdes ) ; probability measures ; hellinger distance ; stationary probability density ; parameter estimation ; numerical simulation     mathematical models for scientific and engineering systems often involve with some uncertainties .we may roughly classify such uncertainties into two kinds .the first kind of uncertainties may be called _ modeluncertainty_. they are due to physical processes that are not well understood or not well - observed , and thus are not or not well represented in the mathematical models .the second kind of uncertainties may be called _ simulation uncertainty_. this arises in numerical simulations of multiscale systems that display a wide range of spatial and temporal scales , with no clear scale separation . due to the limitations of computer power , not all scales of variabilitycan be explicitly simulated or resolved .these uncertainties are sometimes also called _ unresolved scales _ , as they are not represented ( i.e. , not _ resolved _ ) in modeling or simulation .although these unresolved scales may be very small or very fast , their long time impact on the resolved simulation may be delicate ( i.e. , may be negligible or may have significant effects , or in other words , uncertain ) .thus , to take the effects of unresolved scales on the resolved scales into account , representations or parameterizations of these effects are desirable .model uncertainties have been considered in , for example , .research works relevant for parameterizing unresolved scales include , among others .stochastically representing unresolved scales in fluid dynamics has considered as well .  in this chapter , we only consider model uncertainties .specifically , we consider dynamical systems containing uncertain parameters or unknown system functions , and examine how to estimate these parameters , using observed probability distributions of the system evolution .after briefly comment on estimating uncertain parameters based on observed sample paths for the system evolution in [ paths ] , we then , in [ measures ] , propose a method of estimating uncertain parameters based on observed probability distributions ( i.e. , probability measures ) and present a few examples to demonstrate this method , analytically or numerically .since random fluctuations are common in the real world , mathematical models for complex systems are often subject to uncertainties , such as fluctuating forces , uncertain parameters , or random boundary conditions .stochastic differential equations ( sdes ) such as are appropriate models for many of these systems . here is a brownian motion or wiener process , the drift and diffusion contain uncertain parameters ( or and are unknown ) , to be estimated based on observations .for example , the langevin type models are stochastic differential equations describing various phenomena in physics , biology , and other fields .sdes are used to model various price processes , exchange rates , and interest rates , among others , in finance .noises in these sdes may be modeled as a generalized time derivative of some distinguished stochastic processes , such as brownian motion ( bm ) or other processes .theoretical results on parameter estimations for sdes driven by brownian motion are relatively well developed , and various numerical simulations for these parameter estimations are also implemented .see for a more recent review about estimating and computing uncertain parameters , when dynamical systems are submit to colored or non - gaussian noises.  these research works on estimating uncertain parameters in dynamical systems are based on observations of sample paths . in the next section ,we devise a method to estimate uncertain parameters based on observations of probability distributions of the system evolution .consider a dynamical system with model uncertainty , modeled by a scalar sde where the drift and diffusion contain uncertain parameters , to be estimated based on observations of probability distributions ( i.e. , probability measures ) of the system paths .  to this end , we need to introduce the hellinger distance between two probability measures .it is used to quantify the similarity between two probability distributions .this is a metric in the space of probability measures .  for our purpose here, we define the hellinger distance between two probability density functions and as follows the hellinger distance satisfies the property : .we estimate uncertain parameters by minimizing the hellinger distance between the true probability density for the solution process and its observed probability density . in reality , the probability density has to be numerically formulated or discretized .but in order to demonstrate the method , we consider two examples for which the true probability density can be analytically formulated . in the first example , we minimize the hellinger distance between the true stationary probability density for the solution process and its observed stationary probability density , while in the second example , we do this for time - dependent probability densities .   under appropriate conditions on and ( see , p.170 ) , such as , and as well as some smoothness requirements , there exists a stationary probability density for the sde , as a solution of the steady fokker - planck equation , where the positive normalization constant is chosen so that and , i.e. , note that here may be an arbitrary point so that the integral exists ( say , take if that is possible ) .\(i ) a special case : langevin equation + with parameter . given an observation " of the stationary probability density . find a so that the hellinger distance ^ 2 dx ] is minimized .insert into the hellinger distance , which is now an algebraic function of parameter .thus we use deterministic calculus to find a minimizer ( possibly by hand , or matlab if needed ) .note : .insert into the hellinger distance , which is now a functional of and thus we use calculus of variations ( on ) to find a minimizer .we then derive the euler - lagrange equation to be satisfied by , together with appropriate boundary conditions ( needed for and ) .consider a scalar sde the fokker - planck equation for the probability density for the solution is with an observation of , we can estimate parameters , or , or , by examining the inverse problem of the fokker - planck equation . for more information about inverse problems of partial differential equations ,see .* solution : * + the fokker - planck equation for is in this case , we define the hellinger distance : }\int_{-\infty}^{\infty } ( \sqrt{q_i(x , t)}-\sqrt{p(x , t , b)})^2 dx\ ] ] where , and is the time period when are observed .we numerical find by minimizing .    by the definition }\int_{-\infty}^{\infty } ( \sqrt{q_1(x , t)}-\sqrt{p(x , t , b)})^2 dx ] , we have the plot of in figure [ pq2 ] .so we see that if , is minimized when and if , is minimized when .   and the observation : ( top ) and ( top).,title="fig:",width=245,height=188 ]  and the observation : ( top ) and ( top).,title="fig:",width=245,height=188 ]         l. arnold , hasselmann s program visited : the analysis of stochasticity in deterministic climate models . in j .- s .von storch and p. imkeller , editors , _ stochastic climate models_. pages 141158 , boston , 2001 .birkhuser .s. cha , comprehensive survey on distance / similarity measures between probability density functions ._ international journal of mathematical models and methods in applied sciences _ , vol .1(4 ) , 2007 pp.300 - 307 .                                f. moss and p. v. e. mcclintock ( eds . ) , _ noise in nonlinear dynamical systems_. volume 1 : theory of continuous fokker - planck systems ( 2007 ) ; volume 2 : theory of noise induced processes in special applications ( 2009 ) ; volume 3 : experiments and simulations ( 2009 ) . cambridge university press .t. n. palmer , g. j. shutts , r. hagedorn , f. j. doblas - reyes , t. jung and m. leutbecher .representing model uncertainty in weather and claimte prediction .earth planet .sci . _ * 33 * ( 2005 ) , 163 - 193 .c. penland and p. sura , sensitivity of an ocean model to details " of stochastic forcing . in _ proc .ecmwf workshop on represenation of subscale processes using stochastic - dynamic models_. reading , england , 6 - 8 june 2005 .j. yang and j. duan , quantifying model uncertainties in complex systems ._ progress in prob .65 , p.4980 , kohatsu - higa , arturo ; privault , nicolas ; sheu , shuenn - jyi ( eds . ) , 2011 .arxiv : 0912.0280

Abstract: due to lack of scientific understanding , some mechanisms may be missing in mathematical modeling of complex phenomena in science and engineering . these mathematical models thus contain some uncertainties such as uncertain parameters . one method to estimate these parameters is based on pathwise observations , i.e. , quantifying model uncertainty in the space of sample paths for system evolution . another method is devised here to estimate uncertain parameters , or unknown system functions , based on experimental observations of probability distributions for system evolution . this is called the quantification of model uncertainties in the space of probability measures . a few examples are presented to demonstrate this method , analytically or numerically .


the emergence of cooperation in different real systems has been puzzling researchers in several areas devoted to the study of systems involving social , economic or biological organization . even though each of these systems is conformed by single units with natural competitive tendencies ,the emergence of collective behaviours is undeniable .while natural selection operates through competition , cooperation is essential to the evolution and emergence of higher degrees of complexity .the struggle between competition and cooperation is then one of the keys in understanding the self organization of complex systems conformed by interacting units .still , many questions arise regarding how such opposites forces can coexist .the survival of the cooperative behavior is a classical problem of game theoretical approaches . in this context , the paradigmatic prisoner s dilemma game has been widely studied in different versions .it is usually formulated as a standard model for the confrontation between cooperative and selfish behaviors . for many yearsit was implemented in zero dimensional systems , where every player can interact with any other , until the crucial effects of spatial distribution were finally noticed . since then , several mechanisms for the evolution of cooperation have been proposed .some of them are summarized in : kin selection , direct reciprocity , indirect reciprocity , group selection and network reciprocity .here we have chosen to focus on this last mechanism , which is associated to the fact that a cooperative individual can take advantage of the topology of the network to form clusters of cooperators that are often resilient to the invasion of cooperators .studies about the effect of network reciprocity have dominated the literature on spatial distributed games in the last years .they were the response to the need of studying the evolution of the strategies of players of a game beyond the simplifying assumption of a well - mixed population , where everybody interacts equally likely with everybody else .the observation that real populations are not well mixed and the fact that spatial structures could affect the evolution of a game and the strategies of the players demanded a new approach .a natural step was to consider complex networks as models for the underlying topology characterizing the spatial or social structures . in the case of a game played on top of a network or graph ,the individuals of a population are located on the vertices of the graph .the edges of the graph determine the links through which individuals can interact . in a spatial model for the prisoner dilemma ,the players are classified either as cooperators or defectors , and it is assumed that every agent can only play with his / her neighbours .it has been shown that extremely simple rules determine whether network reciprocity can favor cooperation .but it is the concept that cooperators can prevail by forming clusters what we want to analyze here .this idea has been discussed and analyzed in many works . in is found that cooperators can prevail by forming network clusters , where they help each other . in one of the pioneering works on spatially extended games authors have analyzed several shapes for a cluster of cooperators and test the stability of each one against the invasion by defectors .they found that cooperators can only survive and grow if they form clusters .another work pointing out the clustering effect is where it was stated that cooperators can survive by forming clusters within which they benefit from mutual cooperation , that in turn , allows them screening the exploitation by defectors throughout the borders of the cluster .it must be mentioned that some authors found an inverse relationship between the formation of clusters and the success of cooperation .  in this workwe intend to show that the survival of cooperators involves much more than the conformation of clusters .if it depended only on that , cooperation success would increase monotonically with the clustering coefficient of the underlying network . by focusing on the analysis of networkswhich only differ in their clustering coefficient , in the next sections we show not only that this does not happen ( i.e. the equilibrium fraction of cooperators is a non monotonic function of ) but also that there is a strong dependence on the composition of the initial population .this hints at the existence of several mechanisms responsible for the expansion or extinction of cluster of cooperators . to find these mechanisms , in the last section we focus on what happens for populations incompletely ordered and completely disordered networks .a detailed analysis allows us to understand the important of the initial fraction of cooperators for the evolution of the different systems .the prisoner dilemma is a caricature of a real situation in which selfish and altruist tendencies compete .it has been the subject of study of game theory for the last 60 years .its name and formal elaboration is attributed to a. tucker , who mentioned it in a classroom in 1950 , but it was not until 1952 that the first results about it were published .the formulation of the prisoner dilemma as a game is rather simple .it is played by two players who must choose their moves between two strategies : to cooperate ( c ) or to defect ( d ) .the reward , or _ payoff _, obtained by each player after one round of the game is given by table [ tabla1 ] :  .payoff table for the prisoner s dilemma : the strategy in each row gets the payoff given by the table when playing again the strategies in the columns .[ cols="^,^,^",options="header " , ]   in order to study the possibility that the players can change their strategies as a result of their previous interactions , thus generating an evolutionary dynamics of strategies , many authors started to work with the iterated prisoner dilemma , in which players interact by iteratively playing the game several times .the history of successes or failures of each player is recorded in what is called his cumulative payoff .how the players use the information accumulated in their own and others cumulative payoffs is what defines the rules of evolution .operationally , the evolutionary dynamics acts at a certain instance of the game , for example after everybody has played against everybody else , when players decide whether to change strategies or not , following certain update rules .before all the players start again playing the game , all the cumulative payoffs are set to .the spectra of rules of evolution is wide and ranges from purely deterministic to stochastic dynamics .complementary to the evolutionary aspects mentioned above , many authors started to analyze spatial games in order to cope with the limitations associated with the assumption that players were always part of a well mixed population . .the evolutionary behaviour of the populations of surviving strategies of spatial games on networks can be affected by several features of the underlying topology as , for example , the degree distribution of the graph , the average distance between nodes , or the clustering coefficient .the concept that cooperators can survive by grouping in clusters has been discussed and analyzed in many works .intuitively , the reasoning goes as follows .the effect of the cluster would be to screen the nodes at the interior from the presence of defectors .as defectors can only get an advantage from their interaction with cooperators , only those located next to the border of a cluster of cooperators should collect any benefits . in turn , although the cooperators at the border of the cluster should have lower payoffs because of their interaction with defectors , their cooperator neighbours at the interior of the cluster should perform better than the defectors at the border . thus , imitating the internal cooperators should be always more convenient than imitating the bordering defectors , which should lead to the survival , and even expansion , of the cluster of cooperators .the problem is that all these arguments , as well as the very definition of ` cluster ' , depend crucially on the structure of the network .the most important feature in this regard is the clustering coefficient , which measures how connected is the neighbourhood of each node , on average .the existence of local transitive relationships , closely related to the clustering , is what defines the possibility of survival of small clusters of cooperators .paradoxically , it will be also responsible for the negative effect that an isolated cooperator may have on incipient cooperative clusters .here we use the definition of global clustering coefficient of watts and strogatz  . for each node , its local clustering coefficient is defined as the quotient between the number of links joining nodes of the neighbourhood of divided by the total number of possible links ( ) . is then defined as the average over of all local clustering coefficients .we study the influence of on the evolutionary dynamics of the iterated prisoner s dilemma , but keeping the degree distribution constant , to disentangle both contributions .for this we analyze regular networks ( i.e. with the same number of neighbours for every node ) with different values of , generated with the following algorithm .starting from an ordered network ( defined below ) we select at random two pairs of connected nodes .then we ` cut ' both connections and connect each individual to one of the individuals it had not been connected before . in other words , the connections are swapped . if this change gives a network with larger , it is accepted and the network is updated . if it does not increase , the change is only accepted with a fixed ( and typically small ) probability .this process goes on until the clustering coefficient has reached the desired value .notice that this procedure leaves the degree distribution of the original network unchanged .when the desired clustering coefficient is very low , it is to be expected that the resulting networks is very close to a regular random network , independently of the starting one . on the other hand , for larger values of is to be expected that the effect of the starting network is much larger .for this reason we use two different starting networks : ring networks where each node is connected symmetrically to the closest nodes , and 2-dimensional lattice networks .the networks generated from these two classes are called , respectively , random ring networks or random lattice networks .three different starting lattice networks are used : regular square lattices ( ) , triangular lattices ( ) and square lattices where each node is connected to its moore neighbourhood ( ) .for all values of ring networks can be considered as one - dimensional because for a given cluster of nodes the size of the surface is independent of the volume whereas for lattice network the relationship is .  throughout our simulations , we have considered two types of evolutionary dynamics , one deterministic and the other stochastic  . in both cases ,each player either copies the strategy of one of its neighbours or sticks to the same strategy used in the previous round . in the deterministic dynamics each player copies the strategy of its most successful neighbour , if the payoff of that neighbour is larger than its own . in the probabilistic dynamics ,previously used in , it copies the strategy of a neighbour chosen at random , with a probability proportional to its relative payoff . his own strategyis also included in the pool of eligible strategies .as the results we have obtained are qualitatively the same for both types of dynamics , in the following we focus on the deterministic dynamics , and comment briefly on the small differences obtained when using the stochastic dynamicsas mentioned in the previous section , we consider two different dynamics , though explicit results corresponding to only one of them will be shown in the following paragraphs . in all the cases we consider regular networks with 1000 to 10000 nodes with even degrees between 4 and 8 .we observe no dependence on the size but different regimes associated to the degree .the state of the nodes is synchronically updated and the payoff of each player is not cumulative in time . even though we observe that different initial concentrations of cooperators , , lead to qualitatively the same results , when properly scaled , there are some important differences . to show this we use two different initial concentrations of cooperators , and for every network analyzed in this paper .if the equilibrium value of is plotted as a function of , leaving all the other parameters constant , a piecewise constant function is obtained as is shown in fig.1 .this has also been previously noticed , but with a different payoff table ( in the case considered in a cooperator gets payoff when playing against a defector ) . to understand the origin , and quantify the limits , of these steps, we must consider the necessary conditions for the propagation of the cooperating behavior .for a cooperator to have a chance to turn a defecting neighbour into a cooperating one , its payoff should be at least larger than that of the defecting neighbour .this leads to the condition , where is the number of cooperator neighbors of the cooperator and is the number of cooperator neighbors of the defector .the condition on can be written as where .note that , as and , is a natural number that must satisfy .this gives a maximum of possible steps .note however that in some networks the range of possible values for is smaller , and therefore the number of steps of is at most . in general , for networks with the same number of the number of possible steps will be smaller for the networks with smaller clustering coefficients .as an example , consider the two extreme cases of a tree and a lattice network with : whereas the tree has the maximum possible of steps , the lattice network can have at most steps . in all cases the last step corresponds to because for those values of a cooperator , regardless of the composition of its neighbourhood , is not able to turn a defecting neighbour into a cooperating one .furthermore , it is also possible that , because of geometrical constraints , also vanishes for other steps . for the networks analyzed in this paper ,we have confirmed that only the height of the steps depends on .furthermore , simulations show that only for the first two steps the final number of cooperators is non vanishing ( see fig.[figsteps ] ) . for these reasonswe have only analyzed the dependence of in these first two steps , i.e. we have used only two values of , and , that satisfy and .   as a function of the parameter for three different networks : ( lattice ) , random lattice with starting from a lattice with , and idem with . ]  in figs .[ k4 ] to [ k8 ] we plot the numerical results obtained from computational simulations with to agents .each curve corresponds to the average fraction of cooperators in the steady state , as a function of the clustering of the networks .the highest clustering value corresponds to the ordered network ( lattice or ring ) , and networks get increasingly disordered as is decreased .   as a function of the clustering coefficient , for and .( l ) and ( r ) in the caption refer to lattice and ring networks respectively . ]   as a function of the clustering coefficient , for and .( l ) and ( r ) in the caption refer to lattice and ring networks respectively . ]we begin by analyzing what happens for evolutions whose initial state consists of the same number of cooperators and defectors . in other words ,the initial probability that a given agent is a cooperator is . in this case , the steady state is always composed by a finite fraction of cooperators .as can be seen in the figures , there are some features that are common to all the families of networks analyzed .the first is that , for each class of network , the behaviors of the curves is qualitatively the same for the two values of used .the only difference is that , as is to be expected , curves for are below curves for .another important feature is that the final fraction of cooperators for ordered networks is always larger than what is obtained in completely random networks .even though this seems to confirm the impression that clustering is beneficial to cooperators , it must be noticed that many curves are not monotonic with , as for example , all curves corresponding to random ring networks .another interesting feature to notice is that , for ring networks the addition of a very small amount of disorder causes an abrupt decrease in the steady fraction of cooperators .this happens because of the one - dimensional nature of the ring : rewiring very few links at each side of a cooperation cluster can be very effective in stopping its expansion .when more links are rewired the dimensionality of the system begins to increase and cooperators clusters find new directions to expand .for all values of curves for random ring networks are always below those for random lattice networks , for the same values of .this is probably related to the lower dimensionality of the substrate of the random ring network that may have an influence even for high values of the disorder .notice that the curves only overlap for very small values of the clustering coefficient .this means that a large amount of disorder is needed for the network to ` forget ' the starting substrate .   as a function of the clustering coefficient , for and ( l ) and ( r ) in the caption refer to lattice and ring networks respectively . ]  in fig .[ k4 ] only one point is shown for random lattice networks because both the square lattice and the completely random network with have a vanishing clustering coefficient .the large difference seen in fig .[ k4 ] between the steady state fraction of cooperators could be attributed to the much shorter minimal distances between nodes in random regular networks ( which have a diameter  ) or to the presence of short loops in the square lattice ( see next section ) .  when the initial state has less collaborators , the situation is more complex to analyze because for some systems the population evolves to an equilibrium state where all the cooperators have been eliminated .if , however , we consider only those systems that have a steady state with a non vanishing fraction of cooperators , the picture is very similar to what is found for .an example of this for is shown in fig .[ k6c ] where the initial fraction of cooperators was ( compare figs .[ k6b ] and [ k6c ] ) .when , instead , the fraction of realizations that converge is considered , the picture that emerges is rather different , as figs .[ k8d ] and [ k6c ] show . in this case ,ordered networks are less favourable for the preservation ( and eventual expansion ) of cooperation than completely random networks .as before , the behaviour between these two extremes is not monotonic .a feature of these curves that stands out is that , for the same values of , they seem to depend very weakly on the type of substrate used to generate them .   as a function of the clustering coefficient , for and .( l ) and ( r ) in the caption refer to lattice and ring networks respectively .only those realizations that showed to the survival of cooperators were considered . ]  . ]the difference of the evolutionary dynamics of populations starting from many , or few , cooperators can be shown even more clearly using the same variable for both cases : the average fraction of steady state cooperators , with the average taken over the whole population .but it must be recalled that for populations with small numbers of initial cooperators the variable does not give atypical value of final cooperators because the steady state cooperators distribution has at least two modes clearly separated , one with zero cooperators and other with many cooperators .we have used this variable to show that for some stochastic dynamics the results are very similar to what has been described above for a deterministic dynamics . in the stochastic dynamicswe have used , the agents choose the strategy of a neighbor with a probability proportional to the corresponding cumulative payoff , but only if it is larger than his / her own cumulative payoff  .[ figstoch ] shows some results for this dynamics .curves for several different values of are shown because in this case , the dependence on is not as simple as in the deterministic case . in any case , several qualitative similarities with the deterministic case are apparent . for ordered networks are more favourable to cooperation than completely disordered ones , at least for .for the situation is reversed , and now the most favorable networks in terms of cooperation are completely random ones .there are even some values of for which the dependence with is not monotonic .   for random ring networks with , and for ( panel a ) and ( panel b ) . ]  . ]so far we have shown that the clustering coefficient of the networks has an important influence on the steady state .but whether or not this influence is beneficial depends strongly on the initial fraction of cooperators , both for deterministic and stochastic evolutionary dynamics .the possible causes for this are addressed in the next section .to understand the conflicting ways in which network clustering can affect cooperation we study in some detail the deterministic dynamics , and hope that some conclusions apply also to the stochastic case .furthermore , in terms of the range of possible clustering coefficients we limit ourselves to analyze what happens for networks at the two extremes : completely ordered ( large ) and completely disordered ( small ) networks . for this last classwe concentrate on lattice networks with and which have clustering coefficients ( ) and ( ) ,  first , we analyze the fate of a cluster of 3 cooperators . in the case of the lattice networks with and , there are two and three possible configurations , respectively ( see fig.[k8 t ] ) .however , all of them are unstable because , thanks to the large clustering , some neighbors of the cluster can be connected to 2 or 3 cooperators in the cluster , having thus a larger payoff than any of them . on the other hand , in a random networkthere is a finite probability ( ) that all the neighbors of a 3-cluster are not neighbors of more than 1 cooperator , and thus the cluster is stable .furthermore , if this cluster does not disappear it will grow to become a cluster of cooperators ( a central cooperator surrounded by cooperators ) .this cluster , in turn , has a non vanishing probability of continuing its expansion .for example , if there is a link joining two of the new surrounding cooperators ( which happens with probability , the cluster grows by turning into cooperators the non - cooperating neighbors of the nodes that share the link ( see fig .[ figer]a ) .but there is now a non vanishing probability that there is also a link joining the nodes of the cluster surface which would lead to an increase of its size of . and, in general , at any step of its growth it could keep growing with a probability roughly proportional to , where is the size of the cluster surface . considering that most of the nodes of the clusters lie in its surface , implies that once the cluster has reached a size of order it will keep growing until it spans the whole lattice .thus , if we consider the evolution of all possible 3-clusters , the distribution of final cluster sizes should be non - vanishing only for sizes and for sizes smaller than .  in the case of 4-clusters , it is easy to see that the situation is qualitatively the same for regular random networks . on the other hand , for lattice networksthe picture is completely different : square 4-clusters expand until they occupy the whole lattice because every cooperator is connected to 2 or 3 others whereas non cooperating neighbours can only have at most 2 cooperating neighbours .taking these ideas into account , and assuming that cluster expansion or death is not influenced by the presence of cooperators outside the cluster , we can try to predict what happens when cooperators are placed at random in a network . in the case of a lattice network ,the probability of an initial set of cooperators taking over the network is simply the probability that there is at least one square cluster of cooperators : . for a random regular networkthe probability is , where is the probability that a given node is the center of a 3-cluster and that it expands during at least two steps :    where the first term in the product is the probability that a given node has at least two cooperating neighbors , the second is the probability that the 3-cluster expands in the first step , and the third is the probability that it continues expanding in the second step .we assume that after the second step the expansion goes on until all the lattice is occupied by cooperators .[ k8 t ] shows that these functions overestimate the fraction of populations that are able to take over the whole network .this shows that , somewhat paradoxically , the presence of other cooperators can sometimes hinder the expansion of a cluster . for example , if a square has 2 defecting neighbors , connected to opposite sides of the square , and in turn connected to at least one cooperator outside of the cluster , the square disappears .note that a 4-cluster in a random network is harder to destroy because a neighbour of the cluster is a neighbour only to one cooperator of the cluster and thus it needs two other cooperating neighbors to be able to destabilize the cluster .evidently , 3-clusters are much easier to destabilize , as the partial failure of the estimate shows .[ figer]b shows one way a 3-cluster can be destabilized by a close cooperator .interestingly , there is also a mechanism by which different clusters can collaborate in each other s expansion , given that the average distance between nodes is small enough .consider for instance a couple of stable stars of cooperators in a random network .if they are connected , i.e. if a cooperator of one star is connected to a cooperator in the other , these cooperators would be able to turn their non - cooperating neighbors into cooperators , thus increasing the size of the two clusters by . in a random network ,the probability that two clusters of size and are connected is .  .black circles represent cooperators and white circles represent defectors.,width=377 ]  taking all these features into account it is possible to give a better estimate for the fraction of systems that converge to a final state dominated by cooperators or with only a few ( ) stable cooperators .the probability that the cooperator population dies out is where is the probability that a cooperator survives the first time step :   .black circles represent cooperators and white circles represent defectors .a ) expansion of a 3-cluster .b ) disappearance of a 3-cluster.,width=377 ]  which is simply the sum of the probabilities of having cooperator neighbours multiplied by the probability that none of the defecting neighbors has more than cooperating neighbors .the probability of the final state being dominated by cooperators can be approximated by :   each addend gives the probability of having stars and that they are not connected .  .the lines show the theoretical estimates , assuming independence ( full lines ) or dependence ( dashed lines ) among all clusters of cooperators . ]it is interesting to see what happens for networks with because both lattice and random regular networks have vanishing clustering coefficients .[ k4 t ] shows that the probability of taking over the whole population is rather similar for both networks .interestingly , in this case there is also the possibility of having a final state with a small number of stable cooperators , for the lattice network .the reason is very similar to the case of random regular networks and is a consequence of having a vanishing cluster coefficient : a lineal cluster of cooperators can not be destabilized because no neighbor can be a neighbor to more than cooperator of the cluster .  .the lines show the theoretical estimates assuming dependence among all clusters of cooperators . ]it has been sometimes suggested that one of the possible reasons for the success of cooperating strategies in spatially structured populations is the possibility of forming globular clusters . in this way , cooperators inside the cluster are ` protected ' by the ones on the border . if the populations is placed on a graph , the ` globularity ' of the possible clusters is proportional to the clustering coefficient . therefore , cooperating strategies should be more successful in networks with large than in networks with small . for the evolutionary dynamics studied herewe confirm that this is indeed the case when the fraction of cooperators in the steady state is compared in random regular networks ( low ) and lattice networks ( large ) having the same degree distributions .however , by analyzing populations in graphs with intermediate values of , we find that the equilibrium fraction of cooperators is not a monotonic function of .the results commented above were obtained using initial populations with many cooperators ( half of the population , on average ) .but if the initial population has much less cooperators , the situation becomes more complex . onthe one hand one finds that some evolutions lead to the elimination of all cooperators . on the other hand , when cooperators do not disappear , their final fraction , as a function of has a similar behaviour to that observed when there are many initial cooperators .the problem is that the number of such evolutions is much smaller for ordered networks than for disordered ones , the behaviour for intermediate values of also being non monotonic .the situation is then very different from that obtained from initial populations with more cooperators .the same difference appears when stochastic evolutionary dynamics are analyzed .the non monotonicity of the curves , together with the dependence on the initial condition suggest that there might be several mechanisms that influence the success or failure of cooperation . in the last sectionwe have shown that this is indeed the case , at least for completely ordered or completely disordered networks . in ordered networksthe evolution is isotropic and deterministic : wherever it is placed , a square cluster of 4 cooperators is always able to expand .but in disordered networks the fate of the cluster depends on where in the network it is located : given a large enough network there are positions from where a cluster of 3 cooperators will be able to grow to a very large size . in other words , in disordered networkssmaller clusters are able to expand than in the case of ordered networks , but only if they are placed in the right places .additionally , we have shown that it is not uncommon that cooperators that are outside , but not very far , from a cluster of cooperators , can actually hinder its expansion and even lead to its disappearance .

Abstract: in the last years the prisoner dilemma ( pd ) has become a paradigm for the study of the emergence of cooperation in spatially structured populations . such structure is usually assumed to be given by a graph . in general , the success of cooperative strategies is associated with the possibility of forming globular clusters , which in turn depends on a feature of the network that is measured by its clustering coefficient . in this work we test the dependence of the success of cooperation with the clustering coefficient of the network , for several different families of networks . we have found that this dependence is far from trivial . additionally , for both stochastic and deterministic dynamics we have also found that there is a strong dependence on the initial composition of the population . this hints at the existence of several different mechanisms that could promote or hinder cluster expansion . we have studied in detail some of these mechanisms by concentrating on completely ordered networks ( large clustering coefficient ) or completely random networks ( vanishing clustering coefficient ) .


in this work we aim at _ characterizing _ outliers ._ outliers _ are the _ exceptional _ objects in the input dataset , that is to say objects that significantly differ from the rest of the data .approaches to outlier detection introduced in the literature can be classified in supervised , which exploit a training set of normal and abnormal objects , semi - supervised , which assume that only normal examples are given , and unsupervised , which search for outliers in an unlabelled data set .it is worth to notice that the above mentioned methods focus only on _ identification _ , and they do not concentrate on providing a _ description _ or an _ explanation _ of why an identified outlier is exceptional , which is vice versa the problem we are intended to face here . while outlier detection in datasets has been one of the most widely investigated problems in data mining , the related problem of outlier explanation received less attention in the literature .it must be noticed that the outlier explanation problem is completely different from supervised and semi - supervised outlier detection , and , moreover , is to be considered orthogonal to the unsupervised outlier detection task .  as an example of outlier explanation task ,assume you are analyzing health parameters of a sick patient , which include several features such as body temperature , blood pressure measurements and others .if an history of healthy patients is available , then it is relevant to single out those parameters that mostly differentiate the sick patient from the healthy population .it is important to highlight here that the abnormal individual , whose peculiar characteristics we want to detect , is provided as an input to the outlier explanation problem , that is , this individual has been recognized as anomalous in advance by the virtue of some external information , mean or procedure .the focus of this paper is the discovery of _ outlying properties _ : we are interested in unveiling the hidden structures that make an input outlier object special w.r.t . an input population .this can be accomplished ( ) by detecting the subsets of the input population that represent an homogeneous sub - population , intuitively a set of objects sharing similar features ( which we will refer to as _ explanations _ ) , including , and ( ) by identifying attributes ( also referred to as _ properties _ ) where substantially differentiates from the other objects in .  with this aim , subspace outlier mining techniques , like the one presented in ,could in principle be used to extract information about outlier properties . however , the originary task considered in is different from the task investigated here , since subspaces therein highlight the outlierness , whereas in our approach they represent an homogeneous subpopulation upon which to compare a given property . in ,the authors focus on the identification of the intensional knowledge associated with distance - based outliers .however , this setting models outliers which are exceptional with respect to the whole population , but it does not capture objects that are exceptional only if compared to homogeneous subpopulations . in outlier subpopulation is given in input and compared with the inlier population in order to simultaneously characterize the whole exceptional subpopulation . despite the latter approach shares with ours a common rationale , we note that the framework considered in is different and the solutions there proposed can not be applied to the special case in which the outlier sub - population consists of just one single individual , which is precisely the scenario considered here .a viable solution to the outlying property detection problem has been devised in .specifically , a set of attributes witnesses the abnormality of an object if the combination of values the object assumes on these attributes is very infrequent with respect to the overall distribution of the attribute values in the dataset , and this is measured my means of the so called _ outlierness _ score . a major problem with the outlierness score presented in that it was specifically designed and shown effective for categorical attributes .hence the question is how to adapt that idea to a more general setting with both categorical and numerical attributes .we point out that discretizing numerical attributes and applying the technique of to the discretized attributes is not a suitable solution , for several reasons .first of all , the result of the analysis will strongly depend on the results of the discretization process .this drawback is further exacerbated by the peculiarities of the outlierness measure , which assigns higher scores to very unbalanced distributions , and by contrast provides low scores to uniform frequency distributions . in a sense, the discretization process should be supervised by the outlierness score , in order to detect in the first place the bins capable of magnifying the score itself .the appropriate treatment of numerical attributes is indeed one of the main problems we deal with in this paper .specifically , the main contribution of this work amounts to provide an _ outlierness _ measure representing a refined generalization of that proposed in and which is able to quantify the exceptionality of a given numerical or categorical _ property _ featured by the given input anomalous object with respect to a reference data population .in particular , in order to quantify the degree of unbalanceness between the frequency of the value under consideration and the frequencies of the rest of the database values , our measure analyzes the curve of the cumulative distribution function ( _ cdf _ ) associated with the occurrence probability of the domain values .it is worth noting that relying on the _ cdf _ allows to correctly recognize exceptional properties independently of the form of the underlying probability density function ( _ pdf _ ) , since the former compares the occurrence probabilities of the domain values rather than directly comparing the domain values themselves .this enables us to build a general methodology for uniformly mining exceptional properties in the presence of both categorical and numerical attributes , so that a fully automated support is provided to decode those properties determining the abnormality of the given object within the reference data context .the rest of the paper is organized as follows .section [ sect : outlierness0 ] introduces the outlierness measure and the concept of explanation .section [ sect : algo ] describes the method for computing outlierness and determining associated explanations .section [ sect : experim ] discusses experimental results . finally , section [ sect : concl ] presents conclusions and discusses future workto begin with , we fix some notation to be used throughout the paper . in the following , denotes an _ attribute _ , that is an identifier with an associated domain , and denotes a set of attributes .the _ value _ associated with the attribute in the object will be denoted by  ] , where ( ) , ( ) , and ( ) , if is numeric , and , if is categorical .if , the interval  ] on .an object of satisfies the condition , if and only if  ] , if is numerical .moreover , satisfies a set of conditions if and only if satisfies each condition .given a set of conditions on .the _ selection _ of the database w.r.t . is the database consisting of the objects satisfying .next , the definitions of outlierness and explanation are introduced .this measure is used to quantify the exceptionality of a property .the intuition underlying this measure is that an attribute makes an object exceptional if the relative likelihood of the value assumed by that object on the attribute is rare if compared to the relative likelihood associated with the other values assumed on the same attribute by the other objects of the database .let be an attribute of .we assume that a random variable is associated with the attribute , which models the domain of .then , with we denote the pdf associated with .the pdf provides a first indication on the outlierness degree of a given value , as usually we would expect low pdf values associated to outliers .however , the sole pdf value is not enoughs .a given pdf value represents a hypothetical `` frequency '' for that value in the sample under consideration .how typical is that `` frequency '' provides a better insight on the outlierness degree : a low pdf value in a population exhibiting low values only is not an indicator of an outlier , whereas an anomalous low pdf value in a population of significantly higher values denotes that the value under observation represents an outlier .thus , analyzing how the values distribute on a pdf is the key for measuring the degree of outlierness .let denote the random variable whose pdf represents the relative likelihood for the pdf to assume a certain value .the cdf of is :   assume that the height of the individuals of a population is normally distributed with mean and standard deviation .then , let be the attribute representing the height , is a random variable following the same distribution of the domain and is the associated pdf , reported in the first graph of fig .[ fig : example_prelim ] .the pdf assumes value in the domain \subset \mathbb{r} ] such that ( ) for , and ( ) is monotone increasing for . in the followingwe employ the mapping the first integral measures the _ area above _ the cdf for ) ] .intuitively , the larger the first term , the larger the degree of unbalanceness between the occurrence probability of  ] . as for the second term , the smaller it is , the more likely the value  ] and in particular it is close to zero for usual properties .by contrast , values closer to one denote exceptional properties .  consider fig .[ fig : outlierness ] , reporting on the left a gaussian distribution ( with mean and standard deviation ) .consider the values and , for which and hold .assume that an outlier object exhibits value on .the associated outlierness corresponds to the whole area ( filled with horizontal lines ) above the cdf curve , that is . for an object exhibiting value on ,instead , the associated outlierness corresponds to the difference between two areas ( filled with vertical lines ) detected at frequency , that is .    for the sake of clarity , in the above example we considered a pdf having a simple form .however , we wish to point out that our measure is able to correctly recognize exceptional properties irrespectively of the form of the underlying pdf , since it compares the occurrence probabilities of the domain values rather than directly comparing the original domain values .  given an object and a dataset on a set of attributes , an attribute showing a large ( i.e. exceeding a given threshold ) value of outlierness will be called a ( _ outlying _ ) _ property _ of in ._ explanations _ are useful in our framework to provide a justification of the anomalous value characterizing an outlier. intuitively , an attribute of that behaves normally with respect to the database as a whole , may be unexpected when the attention is restricted to a portion of the database .relevant subsets of the database upon which to investigate outlierness can be hence obtained by selecting the database objects satisfying a condition , and such that a property is exceptional for .  a condition ( set of conditions , resp . )is , intuitively , an _ explanation _ of the property , if ( , resp . ) and is an outlying property of in ( , resp . ) . finally , the _ outlierness _ of the attribute in w.r.t . with _ explanation _ is defined as .it is worth noticing that , according to the relative size of , not all the explanations should be considered equally relevant . in the following , we concentrate on -explanations , i.e. , conditions such that , where  ] and  ] , each term is equal to , with a global bandwidth ( as a rule of thumb , the global is set to ) and .the rationale underlying this choice is that we want that each value at hand ( ) contributes in equal manner ( ) to the estimation of the underlying pdf .moreover , we employ the _ parzen window _ kernel function , that is , for , and otherwise , since this kernel represents a good trade off between simplicity of computation and accuracy . indeed , the above density estimate can be computed in time by means of a sort of the attribute domain .we also notice that , since the outlierness depends on the cdf of the pdf values , this greatly mitigates the impact of the non - smoothness of the estimate of the pdf through parzen windows , other than making the measure robust w.r.t. deviations of the estimate from the real distribution .proper conditions are the basic building blocks for the explanations .to single them out , our strategy consists in finding , for each attribute , the `` natural '' interval including  ] is assigned to .the interval is then uniquely identified by  ] ,  ] and the second one in the range  ] holds . in the following , we concentrate the comparison on the analysis of the behavior of the two methods on the attribute , in order to demonstrate that while is naturally perceived as an outlying property by the technique hereby introduced , it is very unlikely to obtain the same goal when the technique in is employed .         in order to apply the latter method to the _dataset , we discretize the attributes by grouping attribute values in equi - width bins .figure [ fig : unif2a ] reports on the left the value of the outlierness ( as defined in ) on the value  ] the outlierness measure fluctuates between 0.3 and 1 .this means that even small changes in the number of bins produce results which can dramatically change .this is a very undesirable property , since determining the right number of bins for the analysis at hand is a very challenging task .figure [ fig : unif2b ] , showing different frequency histograms associated with the attribute , should further clarify things .the histogram associated with the best outlierness value , namely outlierness , is the one using bins ( at the center of the figure ) . in this case , the central bin ( centered in zero ) scores a low value of absolute frequency .differently , for both bins ( reported on the left in the same figure ) or bins ( reported on the right ) , the fact that the outlierness of in is sensibly smaller can be explained by looking at the displayed histograms . in both cases ,the value of is grouped with some more frequent values and , hence , the corresponding outlierness value gets sensibly smaller .providing a larger number of bins does not solve the problem : as already pointed out , the scoring functions assigns a score close to to very unbalanced distributions , while its value rapidly decreases when frequencies spread . and ,indeed , with a large bin size the number of different categorical values ( each associated with a different bin ) becomes large , and these values score about the same absolute frequency .the consequence is that the outlierness values get small as well .we can conclude that in order to enable the method to discover meaningful knowledge , the bins that maximize the score should be detected in the first place .however , the interaction with explanations ( which select subsets of the overall population ) makes it difficult to provide optimal a - priori intervals , since the distribution of the property attribute are likely to change when switching from one explanation to another .this is clearly not the case with the technique proposed in this paper .since the outlierness measure defined here directly exploits the density estimate of the object value , it is completely adaptive to numerical data and does not suffer of the aforementioned drawbacks .the outlierness computed by our method is .figure [ fig : unif2b ] on the right shows the density estimate of attribute , together with the value associated to ( notice the circle on the curve ) , which is exploited in order to compute the outlierness associated with .the purpose of this paper has been that of devising techniques by which the outlying properties detection problem can be solved in the presence of both categorical and numerical attributes , which represents a step forward with respect to available literature .the core of our approach has been the definition of a sensible outlierness measure , representing a refined generalization of that proposed in , which is able to quantify the exceptionality of a given property featured by the given input anomalous object with respect to a reference data population .also , we have developed algorithms to detect properties characterizing the anomalous object provided in input .the experimental results we have obtained confirm that the presented approach is more than promising . as a matter of fact, there are several application scenarios where the proposed technique can be profitably applied .further scenarios include rank learning problems like in : there , the problem of detecting rules for characterizing individuals who are scored as exceptional according to a specific scoring function ( like , e.g. , the amount of fraud they commit in a fraud detection scenario ) is investigated .it is clear that if exceptional objects are reputed as outliers , then the outlier explanation technique described in this paper could be exploited as a basic building block for rule learning in that domain .m. m. breunig , h. kriegel , r.t .ng , and j. sander .lof : identifying density - based local outliers . in _ proc .international conference on managment of data ( sigmod ) _ , pages 93104 , dallas ,tx , usa , 2000 .s. papadimitriou , h. kitagawa , p.b .gibbons , and c. faloutsos .loci : fast outlier detection using the local correlation integral . in _ proc . of the international conference on data enginnering ( icde ) _ , pages 315326 , bangalore , india , 2003 .b. schlkopf , c. burges , and v. vapnik .extracting support data for a given task . in _ proc .of the acm sigkdd international conference on knowledge discovery and data mining ( kdd ) _ , pages 252257 ,montreal , canada , 1995 .

Abstract: the _ outlying property detection problem _ is the problem of discovering the properties distinguishing a given object , known in advance to be an outlier in a database , from the other database objects . in this paper , we analyze the problem within a context where numerical attributes are taken into account , which represents a relevant case left open in the literature . we introduce a measure to quantify the degree the outlierness of an object , which is associated with the relative likelihood of the value , compared to the to the relative likelihood of other objects in the database . as a major contribution , we present an efficient algorithm to compute the outlierness relative to significant subsets of the data . the latter subsets are characterized in a `` rule - based '' fashion , and hence the basis for the underlying explanation of the outlierness .