recently h5n1 avian influenza and sars have raised the profile of emerging infectious diseases . both can infect humans , but have a primary animal host .typically such zoonotic diseases emerge periodically into the human population and disappear ( _ e.g. _ , ebola , hanta virus , and rabies ) , but sometimes ( _ e.g. _ , hiv ) the disease achieves sustained person - to - person spread . with the advent of modern transportation networks ,diseases that formerly emerged in isolated villages and died out without further spread may now spread worldwide .a number of interventions are available to control emerging diseases , each with distinct costs and benefits . to design optimal policies, we must address several related , but nevertheless distinct , questions .how fast would an epidemic spread ?how likely is a single introduced infection to result in an epidemic ?how many people would an epidemic infect ?we quantify these using , the _ basic reproductive ratio _ , which measures the average number of new cases each infection causes early in the outbreak ; , the probability that a single infection sparks an epidemic ; and , the _ attack rate _ or fraction of the population infected in an epidemic .understanding these different quantities and what affects them helps us to select policies with maximal impact for given cost .many different models are used to study disease spread .perhaps the most important decision in developing a model is how the interactions of the population are represented .because of the complexity of the population , it is invariably necessary to make simplifying assumptions . the errors ( and therefore , the conclusions ) resulting from many of these approximations are not well - quantified . in this paperwe will focus on quantifying the impact of clustering ( the tendency to interact in small groups ) and individual - scale heterogeneity on the spread of an epidemic .based on how they handle clustering , models for population structure fit into a hierarchy of three classes ( which in turn may be subdivided ) . at the simplest levelthe population is assumed to mix without any clustering .most existing models fall into this category . at the most complex level ,agent - based models are used : the movements of each individual are tracked , and people who are in the same location are able to infect one another .these models typically require significant resources to develop , and the clustering is explicitly included . an intermediate level of complexity attempts to introduce the clustering as a parameter ( or several parameters ) .usually these models only consider clustering in terms of the number of triangles in a network , but as we shall see , other structures may play a role .  before introducing the details of our model , we review some previous work .all the models we consider are susceptible - infected - recovered ( sir ) epidemic models  , in which individuals begin susceptible , become infected by contacting infected individuals , and finally recover with immunity .  for unclustered populations , ordinary differential equation ( ode ) modelswere among the earliest models used  and remain the most common .they are deterministic , and so can not directly calculate , but they give insight into the factors controlling and . because they assume mass - action mixing , it is difficult to incorporate individual heterogeneity in the number of contacts. more recently some network - based models have been introduced for unclustered populations  .these models represent the population as nodes with edges between nodes representing contacts , along which disease spreads stochastically .heterogeneity in the number of contacts is introduced by modifying the degree ( number of edges ) of each node . by neglecting clustering ,these studies are able to make analytic predictions through branching process arguments . a recent sociological study  used surveys with participants recording the length and nature of their contacts .this data is valuable for providing the contact distribution needed for the above network models , and allows us to apply network results to real populations .however , this data does not directly tell us anything about the clustering of the population resulting from family / work / other groups .other recent work by  analytically addresses the impact of heterogeneity in infectiousness and susceptibility in unclustered networks .using agent - based simulations  allows us to directly incorporate clustering . in these simulations ,the population is a collection of individuals who move and contact one another .the modeller has complete control over the parameters governing interactions and how the disease spreads .this allows us to study many effects , but also introduces many parameters .it is difficult to test the accuracy of the assumptions used to generate these models and to extract which parameters are essential to the disease dynamics .the expense of developing these simulations is frequently prohibitive .  in this paperwe introduce a systematic approach for calculating the impact of clustering , and quantifying the error . because our model investigates disease spread in clustered networks ,we provide a more detailed review of previous work on clustering and disease .a few investigations have been made into the interaction of clustering with disease spread using network models .the attempts that have been made  typically use approximations whose errors are not quantified , resulting in apparently contradictory results . a few papers  have considered clustering and heterogeneities , rigorously showing that increased heterogeneity tends to decrease and , but without quantitative predictions .recently  considered the spread of epidemics in a class of random networks for which the number of triangles could be controlled . it may be inferred from their figure 3 that clustering decreases the growth rate and that sufficient clustering can increase the epidemic threshold . however , at small and moderate levels clustering appears not to alter the final size of epidemics significantly .similar observations have been made by  . at first glance, this contradicts observations of  that clustering significantly reduces the size of epidemics , but that sufficiently strong clustering reduces the epidemic threshold ( see also  ) , allowing epidemics at lower transmissibility . the discrepancy in epidemic sizemay be resolved by noting that the networks in  have low average degree .we will see that clustering only affects the size if the typical degree is small or clustering is very high .the apparent discrepancy in epidemic threshold with strong clustering may be resolved by noting that the form of strong clustering considered by  forces preferential contacts between high degree nodes .the reduction in epidemic threshold is perhaps better understood in terms of degree - degree correlations than in terms of clustering .  in this paperwe develop techniques to incorporate general small - scale structure ( beyond triangles ) into the calculation of , , and . to calculate , we develop a systematic series expansion which allows us to interpolate between unclustered and clustered results by including more terms . to calculate and , we use a similar approach , but only give estimates on the size of correction terms .our methods give us a rigorous means to understand how unclustered results relate to more realistic populations , and our results resolve the apparent discrepancies mentioned above .our theory accurately predicts epidemic behaviour in a more realistic contact network derived from an agent - based simulation of portland , oregon by episims  .we expand this to investigate the interplay of clustering , heterogeneities in individual infectiousness or susceptibility , and variation in edge weights in their effect on , , and .the paper is organised as follows : section [ sec : formulation ] describes our model and networks and summarises earlier work on unclustered networks .these results will be the leading order terms for our expansions for clustered networks in the remainder of the paper .section [ sec : homogeneous ] considers how epidemics spread in a clustered network assuming homogeneous transmission .we derive the corrections to and show that the corrections to and are insignificant unless the typical degree is small or clustering very high . section [ sec : heterogeneous ] considers epidemics in clustered networks with heterogeneous infectiousness or susceptibility , building on section [ sec : homogeneous ] .section [ sec : weighted ] extends this further to consider epidemics spreading on clustered networks with weighted edges. edges with large weights tend to occur in family or work groups which magnifies the impact of clustering .finally section [ sec : discussion ] discusses the implications of our results , particularly for designing interventions .we conclude that in general , heterogeneity significantly impacts and , but not , while clustering impacts significantly , but not and . heterogeneity or edge weights may enhance the impact of clustering .we consider the spread of a disease using a discrete sir model on a static network .nodes of represent individuals and edges represent ( potentially infectious ) contacts .the contact structure of the network is fixed during the course of the outbreak . the _ degree _ of a node is the number of edges containing .figure [ fig : sample ] shows a sample outbreak .a single infection , the _ index case _ is chosen uniformly from the population to begin an _outbreak_. infection spreads along an edge from an infected node to a susceptible node with probability , the _transmissibility_. the time it takes for infection and recovery to occur may vary but does not affect our results .once recovers it can not be reinfected . typically for a large random network with a population of nodes ,the final size of outbreaks is either large , with cumulative infections , or small , with infections  .large outbreaks are _ epidemics _ and small outbreaks are _ non - epidemic outbreaks_.   a number of factors influence the transmissibility from to such as the viral load and duration of infection of , the vaccination history and general health of , the duration and nature of the contact between and , and characteristics of the disease .  for each node denote its ability to infect others by and its ability to be infected by .each edge has a weight .the parameter measures disease - specific quantities . in most of our calculations we assume theseare scalars and follow  , setting if all contacts are identical , may be absorbed into  note that is a number assigned to an edge , while is a function which states what the transmissibility between two nodes would be if they shared an edge .  with mild abuse of notationwe denote the probability density functions ( pdfs ) of , , and by , , and respectively .we assign and independently , but allow to be assigned either independently or based on observed contacts ( _ i.e. _ , by observing contacts in a population we may create a static network with edge weights assigned based on the observed contact ) . if is assigned independently , then it is possible to eliminate edge weights from the analysis by marginalising over the distribution of weights . however , if weights are not independent ( for example work or family contacts tend to have correlated weights ) then the details of the distribution and the correlations are important .  given the infectiousness of node , we follow  and define its _ out - transmissibility _ this is the marginalised probability that infects a randomly chosen neighbour given . from the definition of and the pdf can calculate the pdf .we symmetrically define the _ in - transmissibility _ and its pdf .we denote the average of a quantity by .the average transmissibility is    rather than studying outbreaks as dynamic processes on networks , we may consider them in the context of epidemic percolation networks ( epns )  .the epn framework allows us to study epidemics as static objects and is useful for quickly estimating , , and . in this sectionwe summarise properties of epns ; more details are provided in  and [ app : epn ] .once the properties of the nodes and edges are assigned , an epn is created as follows : we place each node of into . for each edge in we place directed edges and into independently with probability and respectively .the nodes infected in an outbreak correspond exactly to those nodes that may be reached from the index case following edges of .more specifically , the distribution of out - components of a node in different epn realisations matches the distribution of outbreaks resulting from different epidemic realisations in the original model with as the index case .it may be shown that the distributions of out- and in - component sizes give us information about the probability of nodes to start an epidemic or become infected in an epidemic .we will see that in a large population the structure of a single epn can be used to accurately estimate , , and .once we create an epn and choose the index case , we define the _ rank _ of node as the length of the shortest directed path from the index case to .if no such path exists , is never infected .  interchanging all arrow directions interchanges and .this means that if we can calculate , then may be calculated by the same technique , but with the direction of infection reversed . because of this , we focus our attention on calculating , and apply the same methodology to calculate .an important consequence is that if is constant , then  .we expect that epidemics are possible if and only if the _ basic reproductive ratio _ is greater than .that is , if an average infection causes more than one new case , an epidemic may occur , but otherwise the outbreak dies out quickly .however , this use of is not consistent with the typical definition : _ the average number of new infections caused by a single infected individual introduced into a fully susceptible population _ , which gives .a more appropriate definition is _ the average number of new infections caused by infected individuals early in outbreaks_. the distinction is subtle , but results from the fact that whether an outbreak can grow depends on whether the people of low rank infect more than one person each  .low rank individuals may be different from the average individual .most obviously , they have more contacts  ; but with clustering , they also have a disproportionately large fraction of neighbours infected or recovered .  in order to quantify more rigorously , we first define to be the number of people of rank for a given outbreak simulation .we then define the _ rank reproductive ratio _ }{{\mathbb{e}}[n_r ] } \label{eqn : rog}\ ] ] to be the expected number of new cases caused by a rank node ( averaged over all possible outbreak realisations ) . corresponds to the usual definition of . in practise , we find that reaches a plateau quickly as increases before eventually decreasing as the finite size of the population becomes important .consequently , an improved definition of is the limit of as grows , subject to the assumption that is unaffected by the finite size of .this gives ( _ cf _ , ) and generalises the definition given by  for ode models .under this definition , epidemics are possible if , but not if .we discuss this further in [ app : r0 ] . in a large population considering multiple index cases with a single epngives a good estimate of  ] for , , using an epn from the ( fixed ) episims network with a homogeneous population , compared with the unclustered prediction . at small ( right panel ) match the unclustered prediction.,title="fig : " ] /{\mathbb{e}}[n_r] ]we first note that = t{\left \langle k \right \rangle} ] is more difficult : consider all pairs of nodes and with at least one path of length between them .let be the number of paths of length between and and be an indicator function : if is an edge and if it is not ( see figure [ fig : en2 ] ) .the probability that an infection of results in infection of in exactly two steps is [1-t]^{\chi_{uv}} ] for by & =  \frac{1}{n } \sum_u\sum_{v\neq u } t^2n_{uv}(1-t)^{\chi_{uv } } -  \binom{n_{uv}}{2 } t^4 + { \mathcal{o}}(t^5 ) \, , \\  & = t^2{\left \langle k^2-k \right \rangle } - 2t^3{\left \langle n_\triangle \right \rangle } - t^4{\left \langle n_\square \right \rangle } +  { \mathcal{o}}(t^5 ) \ , , \end{aligned}\ ] ] where is the average number of triangles each node is in , and is the average number of squares each node is in ( _ cf _ , ) .higher order terms involve more complicated shapes .this gives at leading order we recover the unclustered prediction for , reflecting the fact that at small the probability the outbreak follows all edges of a cycle is negligible .as increases , the first corrections are due to triangles , then squares , then pairs of triangles sharing an edge , and sequentially larger and larger structures made up of paths of length two .a comparison of these approximations with the exact value is shown in figure [ fig : r01approx ] .   from equation  with the exact value ( solid ) for the episims network .the right panel shows the comparison at small .,title="fig : " ]  from equation  with the exact value ( solid ) for the episims network .the right panel shows the comparison at small .,title="fig : " ] +  although we have defined for an ensemble of realisations , figure [ fig : simulations ] shows that accurately predicts the observed ratio for individual simulations once the outbreaks are well - established .early in outbreaks , the behaviour is dominated by stochastic effects , and so the ratio of successive rank sizes is noisy .once the outbreak has grown large enough , random events become unimportant and the ratio settles at .   and ( right ) in the episims network .the left panels show against rank and right panels show the cumulative fraction of the population infected.,title="fig : " ]  and ( right ) in the episims network .the left panels show against rank and right panels show the cumulative fraction of the population infected.,title="fig : " ]    and attack rate of epidemics for the ( clustered ) episims network ( ) versus , compared to the prediction derived from the degree distribution assuming no clustering .each data point is from a single epn , ( the variation in resulting from different epns is negligible ) . ]  in order to assess the effect of clustering on and , we compare epidemics on the episims network with the analytic predictions derived assuming a cm network of the same degree distribution in figure [ fig : probcomp ] .the epidemic threshold is not noticeably altered , and the values of and are almost indistinguishable from the predictions made assuming no clustering , despite the large amount of clustering in the network .  although initially surprising , these results may be understood intuitively as follows : if is large enough that the disease follows all edges of a short cycle then some other edge from a node of that cycle is likely to start an epidemic and the cycle does not prevent an epidemic .on the other hand , if is smaller so that it does not follow all edges of a cycle , then the disease never sees the existence of the cycle , and the outbreak progresses as if there were no cycle .  to make this more rigorous , we first look at the epidemic threshold .we assume is well - approximated by .let be the threshold without clustering and be the threshold found by including the correction due to triangles . from equationit follows that ^ 2 \right)\ , .\label{eqn : homog_cond}\ ] ]because a given node of degree is contained in at most triangles , we conclude .so if is small the leading order term of equation  is small and triangles do not significantly alter the epidemic threshold regardless of the density of triangles . for the episims network, takes the value , and so we do not anticipate clustering to play an important role in determining the threshold .  above threshold , we assume that may be expanded much like  where is the epidemic probability in a cm network of the same degree distribution . although calculating only requires information about nodes of distance at most two from the index case , may depend on effects occurring at larger distance , and so the expansion has many additional terms . in general , we expect that if the average degree is large , then the various coefficients of the correction terms are all small .the larger a structure is , the smaller we expect its corresponding coefficient to be. the coefficient for triangles may be found by where is the probability that a given triangle prevents an epidemic if is the index case ( regardless of whether is part of the triangle ) . reversing the order of summation we get where is the number of triangles in and is the average of the given quantity taken over all triangles .thus and we can find by considering the average effect of a single triangle in an unclustered network .  to calculate the impact of a triangle with nodes , , and on for a given network, we consider that triangle and a randomly chosen edge elsewhere in the network .if we replace the edges and with and , then we have a new network without the triangle , but with the same degree distribution .we must estimate the expected change in caused by switching the edges .we begin by assuming is the index case .the triangle can affect only if the infection tries to cross all three edges , that is , if the infection process ` loses ' an edge because of clustering .this may happen in three distinct ways .in the first , node infects both and , and then and/or tries to infect the other . in the second infects but not , then infects , and finally tries to infect .the third is symmetric to the second ( with infecting ) .  to leading orderwe can ignore other short cycles , so the probability that an edge leading out of ( not to or ) will not cause an epidemic is , where ( as before ) is the probability that a randomly chosen secondary case does not cause an epidemic in an unclustered network and can be calculated using equation .   and with and breaks the triangle and allows more infections , without affecting the degree distribution.,title="fig : " ] and with and breaks the triangle and allows more infections , without affecting the degree distribution.,title="fig : " ]  we perform a sample calculation with the first case : infects both and .assume that has degree , has degree , and has degree .the probability that infects both and without some other edge leading from , , or starting an epidemic is . if the edge were broken and and were joined to and respectively ( see figure [ fig : trianglebreak ] ) , then the new probability of to infect both and without an epidemic becomes .the difference is , which is the product of three terms , all at most .if the sum is moderately large , then either or ( if is not close to then the first term is small , otherwise the second term is small ) .thus the triangle has little impact on the epidemic probability in this case . is small , thenthe _ relative _ change may be large , but the absolute change is small .] similar analysis applies to the other two cases where the to or to infections are lost .provided the typical sum of degrees of nodes in a triangle is relatively large , the probability of an epidemic when the index case is in the triangle is not impacted significantly .if the index case is not part of the triangle , then the above analysis is modified because we must also consider each node in the path from the index case to the triangle .we must first calculate the probability that infection reaches a node in the triangle while simultaneously no intermediate node sparks an epidemic , and then we calculate the probability as above that the triangle prevents an epidemic .if the index case is and the path from to the triangle goes through , , and then reaches , then the probability that the triangle prevents an epidemic is given by .this falls off very quickly , and so nodes not in the triangle are unimportant , unless typical degrees are small .  in contrast , in a network with small average degree and a significant number of trianglesthis becomes significant .this explains observations of  who use networks with average degree less than and find that clustering significantly alters .  it is tempting to generalise our conclusion and state that if the average degree is large , clustering has no impact on or . however , there are a number of counter - examples : consider a network made up of isolated cliques with nodes , then in expansion  the coefficient for cliques of nodes will not be small .consequently care must be taken when using such an expansion to ensure that neglected terms resulting from larger scale structures are in fact negligible . for social networks, we generally anticipate this highly segregated situation to be unimportant .we conclude that for most reasonable networks , clustering is only important for and if the typical degrees of nodes are low in which case is small .a consequence of these results is that if is moderately large , then and are effectively unaltered by clustering . if is small , however , clustering may or may not play a role in determining and , depending on whether is small because the degrees are small or because is small .when we drop the assumption of constant transmissibility , disease spread becomes more complicated .if is heterogeneous and infects a neighbour , then the _ a posteriori _ expectation for becomes higher : it is likely to infect more neighbours .this accentuates the effect of short cycles , enhancing the impact of clustering on , , and .a similar argument applies with heterogeneity in : if is not infected by one of its neighbours , then the _ a posteriori _ expectation for becomes lower : it is less likely to be infected by other neighbours , and so has multiple opportunities to prevent an epidemic .again this accentuates the effect of short cycles .  in this sectionwe investigate how varying the infectiousness and susceptibility of nodes in the episims network enables clustering to alter the values of and .we will make use of the _ ordering assumption _ and its consequences from  : if is `` more infectious '' than in a given instance [ or `` more susceptible '' than , then is always more infectious than [ or always more susceptible than .more specifically , the ordering assumption states that if , then for all , and the corresponding statement for .the results of  show that if the ordering assumption holds , heterogeneity tends to reduce and , and the upper bounds on and correspond to homogeneous populations ( constant ) ..for the calculations of sections [ sec : heterogeneous ] and [ sec : weighted ] we determine using equations  and  with the distributions of and given in the first four rows , or by considering a maximally heterogeneous population for which of the population infects all neighbours and infect no neighbours .the function is the dirac delta function . [cols="^,^,^",options="header " , ]   for simulations in this section , we consider five different illustrative cases , which will be denoted throughout by the symbol given in table [ table : symbols ] . in the first four , we use equation  so that with the distribution of and varying for each .we vary to change the average transmissibility . in the fifth casethe out - transmissibility is maximally heterogeneous : a fraction of the population infect all neighbours , while the remaining infect no neighbours .the fifth case gives a lower bound on for a homogeneously susceptible population  .it is hypothesised to remain a lower bound on if susceptibility is allowed to vary  .we could also consider maximal heterogeneity in susceptibility , but the results for and merely correspond to interchanging their values for maximal heterogeneity in infectiousness , and so we do not need to consider it explicitly .we use simulations to calculate the rank reproductive ratio for the cases of table [ table : symbols ] and plot the result for in figure [ fig : hetr0 g ] .note that remains a good approximation to . in the first four cases , is again again asymptotic to the unclustered approximation as .there are small kinks for [ 0pt][0pt ] and [ 0pt][0pt ] at and respectively , resulting from the nature of those distributions .the heterogeneities act to enhance the effect of clustering on , but the effect is relatively small ./{\mathbb{e}}[n_r] ] calculated from simulations for the heterogeneous examples of table [ table : symbols ] .the final panel ( lower right ) compares for all of the different cases , including both unclustered and homogeneous.,title="fig : " ] + /{\mathbb{e}}[n_r] ] calculated from simulations for the heterogeneous examples of table [ table : symbols ] .the final panel ( lower right ) compares for all of the different cases , including both unclustered and homogeneous.,title="fig : " ] +  in the final , maximally heterogeneous case [ 0pt][0pt ] , remains a good approximation to . at small values of ,the heterogeneity causes clustering to have a larger impact than in a homogeneous population as seen in the lower right panel of figure [ fig : hetr0 g ] , and so this is not asymptotic to the unclustered approximation . at larger values of heterogeneous and homogeneous growth rates are similar .as before , we can calculate analytically , which helps explain our observations .if the ordering assumption holds , we may use a simplified notation to denote the transmissibility from a node with out - transmissibility to a node with in - transmissibility . from and from .if the ordering assumption fails , similar results hold , but the notation is more cumbersome . ] we have = { \left \langle t \right \rangle } { \left \langle k \right \rangle} ] .this may be calculated analytically in the homogeneous case ( constant ) .when heterogeneities are included , the calculation becomes harder , and when edge weights are included it becomes largely intractable .however , these are easily estimated through simulation .these observations show that using to predict will generally be inadequate . in a homogeneous but clustered population , reduced but is unaffected , and so predictions of based on will be too small . in networks that are not clustered buthave heterogeneities in susceptibility , is unaffected but is substantially reduced .consequently , the value of predicted from will be too large.  perhaps our most important conclusion about clustering is that it plays an important role in altering the growth of an epidemic , but it only plays a small role in determining whether an epidemic may occur or how big it would be .if the relevant questions are , `` how likely is an epidemic and how large would it be ? ''then the modeller may proceed ignoring clustering . if however , the question is `` how fast will an epidemic grow ? ''then clustering must be considered , but only enough to calculate .our results have implications for designing intervention strategies .a number of strategies are available to control epidemic spread , including travel restrictions , quarantines , and vaccination .most of the mathematical theory predicting the effects of these strategies has been developed under the assumption of no clustering .most immediately , if we measure at the early stages of an epidemic , traditional approaches will suggest that vaccinating just over half of the population will bring the epidemic below threshold .however , if the population is clustered , then the observed was already affected by the fact that some transmission chains were redundant . following vaccination, some of these chains will no longer be redundant and the disease may still spread with .  achieving a better understanding of the effect of clustering further helps to guide our intuition when choosing between strategies . for example , let us assume that we have the choice between two strategies : in the first , we stagger work schedules in such a way that a typical person s contacts is reduced by ; in the second , we implement population - wide behavior changes so that the same reduction in number of contacts is achieved , but the work contacts are unaltered .the first reduces clustering while the second increases the relative frequency of clustering .the value of is much smaller in the second case than in the first because of the larger clustering , but and are reduced by a comparable amount in both cases .which strategy is best depends on our goals and relative costs .strategies that enhance heterogeneity in infectiousness or susceptibility can be important to help reduce or , even when there is little impact on . depending on which quantitywe want to minimize , different choices will be optimal .consider a choice between vaccinating all individuals with a vaccine that reduces by a factor of for all pairs and or a contact tracing strategy that will remove of all new infections before they have a chance to infect anyone .both strategies reduce by a half .however , the first reduces uniformly , while the second increases heterogeneity in . thus if we have the choice of the two strategies , contact tracing is more likely to eliminate the disease before an epidemic can happen .if our choice is instead between a global vaccine reducing by a factor of for all individuals , or a completely effective vaccine that is only available for of the population , the latter choice will be more effective for reducing .this work was supported by the division of mathematical modeling at the ubc cdc under cihr ( grants no .mop-81273 and ppr-79231 ) and the bc ministry of health ( pandemic preparedness modeling project ) , by doe at lanl under contract de - ac52 - 06na25396 and the doe office of ascr program in applied mathematical sciences , and by the rapidd program of the science & technology directorate , department of homeland security and the fogarty international center , national institutes of health .lus m. a. bettencourt contributed greatly to the early development of this work .i am grateful to sara y del valle for providing the episims network data .in this appendix , we describe the _ epidemic percolation network _ ( epn ) , a tool that allows us to consider an epidemic as a static object rather than a dynamically changing process .this eases the understanding of certain key features and provides an improved technique to efficiently estimate .epns have received moderate use recently  , and a precursor appeared in  .a sample epn for an erds rnyi network of average degree and is shown in figure [ fig : epn ] .   and an epn that leads to the same outbreak .nodes in the are denoted by large circles , nodes in the ( but not in the ) are denoted by pentagons , nodes in the ( but not in the ) are denoted by triangles , and nodes not in any of these components are denoted by small circles.,title="fig : " ] and an epn that leads to the same outbreak .nodes in the are denoted by large circles , nodes in the ( but not in the ) are denoted by pentagons , nodes in the ( but not in the ) are denoted by triangles , and nodes not in any of these components are denoted by small circles.,title="fig : " ]  typically to estimate in an sir model many monte carlo simulations are performed .this process requires many iterations to have confidence in the results .representative results from such simulations are found in figure [ fig : er_vb_sim ] .note that there is considerably more noise in the estimates of than in the estimates of .   and in an erds rnyi network of nodes and .theory ( curves ) compare well with results of simulations ( symbols ) .we take , with distributions of and as given in table [ table : symbols].,title="fig : " ]  and in an erds rnyi network of nodes and .theory ( curves ) compare well with results of simulations ( symbols ) .we take , with distributions of and as given in table [ table : symbols].,title="fig : " ]  instead we generate a single epn .we first assign and to each node and ( if necessary ) to each edge .. if the infectiousness of depends on the infectiousness of the node that infected , then these results fail .this is the _ time - homogeneity _ assumption of  and is also used by  .some effects that can occur when this assumption is false appear in  . ]then for each node and neighbour we calculate and place the directed edge into with probability .the distribution of out - components of a given node is the same as for the final outbreak following an introduced infection of that node in the original epidemic model .  , but calculated through a single epn for each .the noise is substantially reduced in the calculations , but slightly increased in the calculations.,title="fig : " ] , but calculated through a single epn for each .the noise is substantially reduced in the calculations , but slightly increased in the calculations.,title="fig : " ]  if the system is above the epidemic threshold , then will ( almost surely ) have a giant strongly connected component  .we follow  and define the set of nodes ( including ) from which may be reached following the directed edges to be the giant in - component .we symmetrically define to be the set of nodes reachable from .note that . if the initial infection is in , an epidemic occurs , and all nodes in infected .thus the size of corresponds to the probability of an epidemic and the size of corresponds to the size of an epidemic .this may be seen by comparing the epn in figure [ fig : epn ] with the outbreak shown in figure [ fig : sample ] .are infected , but the proportion vanishes as . ]thus in the limit of large networks , epidemic probability is well - approximated by while the fraction infected is well - approximated by .this observation allows us to estimate from a single epn ( figure [ fig : er_vb_epn ] ) , rather than from hundreds of simulations ( figure [ fig : er_vb_sim ] ) . if the structure of the network is sufficiently random , the error in and from a single epn is ( see , _e.g. _ ,  ) , and so in a large population a single simulation will provide a sufficiently good estimate .in this appendix we provide examples demonstrating the need for the more careful definition of in section [ sec : formulation ] , and we explore properties of this definition .  a pair of simple examples demonstrates the difficulties with the standard definition . in our first example , the standard definition suggests no epidemic is possible ( ) , while in fact they are . in our second example , the standard definition suggests epidemics are possible ( ) , while in fact they are not .  for the first example , consider a fully - connected population of nodes .we add isolated nodes and consider a disease for which .a node in the connected component will infect on average nodes , while an isolated node infects none . on average therefore , a random index case infects other nodes . under the standard definition and epidemics should be impossible .however , if the index case is in the connected component , the introduction is likely to lead to an epidemic .  alternately , consider a population of nodes with each node having three neighbours .for simplicity we assume no short cycles .assume that a disease spreads with probability to a given neighbour .the average number of secondary infections caused by a single introduced infection is , giving under the standard definition .however , each secondary infection has only two susceptible neighbours , and so infects on average neighbours , and the outbreak dies out .some of these issues have been dealt with by  , who considered compartmental deterministic models of several types of individuals . at early timenonlinear terms are unimportant , and the profile of the infected population aligns with the eigenvector of the `` next - generation '' matrix . in stochastic settings ,the same alignment occurs , but it may do so more quickly or slowly than predicted and for some realisations it may instead die out . to make a more rigorous definition of , we turn to statements about the average behaviour .we set }{{\mathbb{e}}[n_r]}\ ] ] to be the ratio of the expected number of infections in rank to the expected number in rank .this value is affected by local small - scale structures .if the network is small , it is also affected by the finite size of the network , but if the network is large enough relative to , we expect that the value will be unaffected by large - scale structure. in more concrete terms , the early growth of a disease in a neighbourhood is unaffected by whether that neighbourhood is part of a city of 100000 , 1 million , or 10 million .as the disease spreads further , the effect of the finite city size will be noticeable for the smaller cities first .if the population is large enough , the ratio converges before the finite size has any impact .we define mathematically as   this definition is similar to that of  , who used ^{1/(r+1 ) } \nonumber\\ { \ifx\relax\relax \ensuremath{\mathcal{r}_0 }  \else \ensuremath{\mathcal{r}_{0,\relax } } \fi}&=\limsup_{r \to \infty } \limsup_{|g|\to\infty }  \hat{r}_r\ , , \label{eqn : trapman}\end{aligned}\ ] ] which is the limit as of the geometric mean of ( assuming the limit exists ) .this definition is more general and will converge in some cases where  does not .however , if  does converge ( and typically we see that it does ) , then it reaches the same value , but does so sooner .so to clearly see from , we must have a larger network ., , and for epidemics in the episims network ( ) , an unclustered bimodal network ( with each node s degree coming either from a poisson distribution peaked at or a poisson distribution peaked at ) , and an erds rnyi network ( , average degree ) .the calculations used simulations for each network . note the difference in vertical scales.,title="fig : " ] , , and for epidemics in the episims network ( ) , an unclustered bimodal network ( with each node s degree coming either from a poisson distribution peaked at or a poisson distribution peaked at ) , and an erds rnyi network ( , average degree ) .the calculations used simulations for each network . note the difference in vertical scales.,title="fig : " ] , , and for epidemics in the episims network ( ) , an unclustered bimodal network ( with each node s degree coming either from a poisson distribution peaked at or a poisson distribution peaked at ) , and an erds rnyi network ( , average degree ) .the calculations used simulations for each network . note the difference in vertical scales.,title="fig : " ]  another suitable definition would be \nonumber \\ { \ifx\relax\relax \ensuremath{\mathcal{r}_0 }  \else \ensuremath{\mathcal{r}_{0,\relax } } \fi}&=\lim_{r \to \infty } \lim_{|g| \to \infty } \overline{r}_r \ , , \label{eqn : rhat}\end{aligned}\ ] ] where the expectation is taken over realisations with .this will tend to require more steps to converge because it counts small outbreaks equally with large outbreaks , and so outbreaks which have not yet grown and are dominated by stochastic effects would be as important to the average as well - established epidemics .  a comparison of these three definitions of is shown in figure [ fig : r0defn_comp ] .they all result in similar values for . for a clustered network ,equation  converges more quickly . for large unclustered networks , andboth converge to at while takes longer . in an erds rnyi network ,all three definitions give for all ; only noise due to insufficient simulations affects the calculation .  to be fully rigorous , the limit must be appropriately defined .it does not make sense to talk about for a given network , and we can not simply add nodes to the pre - existing network .we must take a sequence of networks in such a way that the small - scale structure is preserved , and as the network size grows , the size of the preserved structure increases .  to make this rigorous, we follow  .take a sequence of finite networks , with as .we define to be the network induced on the set of nodes within distance of a central node .the sequence of networks is taken so that the probability that the structure surrounding a randomly chosen central node is isomorphic to a given is the same for all if .this means that the small - scale structure in the different networks is the same , and the size of what is considered `` small - scale '' increases with .we note that although the limit may be well - defined , it is possible that the limit in does not converge .this may occur because , for example , growth within a neighbourhood may happen at one rate , while spread between neighbourhoods in a suburb may happen at another , and spread between suburbs in a city may happen at yet another .if the rate of spread continues to change as the grouping size changes , then the limit may not exist .an effect analogous to this may appear in  which considered disease spread in italy .two distinct growth rates are seen depending on whether the disease is spreading in the general country or in rome .  , , and for the same networks and conditions as in figure [ fig : r0defn_comp ] except using a single epn for each data point with different index cases within that epn rather than distinct simulations [ actually at submission the first two plots used only index cases , more will be added when calculations complete].,title="fig : " ] , , and for the same networks and conditions as in figure [ fig : r0defn_comp ] except using a single epn for each data point with different index cases within that epn rather than distinct simulations[ actually at submission the first two plots used only index cases , more will be added when calculations complete].,title="fig : " ] , , and for the same networks and conditions as in figure [ fig : r0defn_comp ] except using a single epn for each data point with different index cases within that epn rather than distinct simulations [ actually at submission the first two plots used only index cases , more will be added when calculations complete].,title="fig : " ]  finally , it is possible to estimate using a single epn rather than multiple simulations . this is a much faster process , and so it is possible to do many more simulations to reduce the noise .however , because it is chosen from a single epn , there may be a small systematic error . in figure[ fig : epn_r0defn_comp ] , we plot the values for the same conditions as in figure [ fig : r0defn_comp ] .we use the same number of index cases and so the noise is comparable .the value of is not noticably affected by choosing a single epn rather than multiple simulations . in the calculations in the paper , we have used a single epn rather than multiple simulations .we briefly review previous work for epidemic spread in cm networks .these are the simplest networks to investigate , and so the theory has been developed further than for other networks  .see  for some discussion of more arbitrary unclustered networks .we extend the earlier theory by allowing independently assigned edge weights .early in the spread of an infectious disease on a cm network , the probability of a node becoming infected is proportional to its degree , and so the pdf for the degree of infected nodes is .we choose an infected node with degree uniformly from nodes of rank .if the network is large enough that we can ignore short cycles , then all of s neighbours are susceptible except the node which infected .thus may infect up to neighbours .the probability that will infect a randomly chosen neighbour is chosen from , and so the probability infects exactly neighbours is ^{k-1-j}$ ] . integrating this over possible values of and summing over and , we find that for the rank reproductive ratio is and so thus we find that for cm networks .the best - known such networks are erds rnyi networks which have a poisson degree distribution in the limit of large network size . ] .we look for the probability that a single infected node causes a chain of infections leading to an epidemic .because interchanging edge direction in an epn interchanges and , we may focus on calculating .equivalent techniques replacing by below give .our analysis is performed in the infinite network limit .we set to be the probability a randomly chosen index case does not start an epidemic .we find ^k p(t_{out } ) dt_{out } \right ) \, , \ ] ] where is the probability a randomly chosen secondary case does not start an epidemic .the value of satisfies the recurrence relation ^{k-1 } p(t_{out } ) \ , dt_{out } \right)\ , .\ ] ] if , the trivial solution is the only solution .for an additional solution appears and is the physically relevant root . from thiswe can calculate .note that depends on the distribution of , but is not affected by the distribution of .similarly , depends on the distribution of but is not affected by the distribution of .this result holds for unclustered , but not for clustered , networks .we have shown that for cm networks , .in particular it depends only on the network properties and the average transmissibility .in contrast , the probability and size are affected by the details of the distribution . intuitively , this is easy to understand .for example , if we consider in populations with varying , at early times the rate of growth is governed by the average number of new infections created , which depends on the average transmissibility . however , a disproportionate number of highly susceptible nodes are infected , and so the average of remaining nodes drops . by the end of the epidemic nodesare much harder to infect than they would have been if all were equally susceptible initially , and so the epidemic infects fewer people .a consequence of this is that we can not predict based only on the early growth rate .although this is frequently done ( see for example  and references therein ) , these calculations usually assume that the population is homogeneously susceptible , which is not always the case , particularly when a vaccine or previous exposure to similar diseases exists .+   we consider a network produced by episims for portland , oregon  . this simulation uses census data , road structure , building locations , and population surveys to construct a virtual population that travels through the city . from the activity of individuals in the simulation , we may reconstruct who was in contact with whom and for how long .there are nodes in the network , of which are in the giant component .the average degree is approximately , and the average squared degree is approximately .the degree distribution has an exponential tail , and clustering is concentrated in the low - degree nodes . for our approximations of , we also need information about length paths .we calculate the number of pairs of nodes with each value of for which and .large values of are more frequent when .the distribution of edge weights is fairly broad .many contacts are very short , but the number of long contacts is not negligible .sara y del valle , phillip d. stroud , james p. smith , susan m. mniszewski , jane m. riese , stephen j. sydoriak , and deborah a. kubicek . :epidemic simulation system .technical report laur06 - 6714 , los alamos national laboratory , 2006 .o. diekmann , j. a. p. heesterbeek , and j. a. j. metz . on the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations . , 28:365382 , 1990 .stephen eubank , hasan guclu , v s anil kumar , madhav v marathe , aravind srinivasan , zoltn toroczkai , and nan wang .modelling disease outbreaks in realistic urban social networks . , 429(6988):180184 , 2004 .neil m. ferguson , derek a. t. cummings , simon cauchemez , christophe fraser , steven riley , aronrag meeyai , sopon iamsirithaworn , and donald s. burke .strategies for containing an emerging influenza pandemic in southeast asia . , 437(7056):209214 , 2005 .jol mossong , niel hens , mark jit , philippe beutels , kari auranen , rafael mikolajczyk , marco massari , stefania salmaso , gianpaolo scalia tomba , jacco wallinga , janneke heijne , malgorzata sadkowska - todys , magdalena rosinska , and w. john edmunds . social contacts and mixing patterns relevant to the spread of infectious diseases ., 5(3):381391 , 2008 .

Abstract: networks of person - person contacts form the substrate along which infectious diseases spread . most network - based studies of the spread focus on the impact of variations in degree ( the number of contacts an individual has ) . however , other effects such as clustering , variations in infectiousness or susceptibility , or variations in closeness of contacts may play a significant role . we develop analytic techniques to predict how these effects alter the growth rate , probability , and size of epidemics and validate the predictions with a realistic social network . we find that ( for given degree distribution and average transmissibility ) clustering is the dominant factor controlling the growth rate , heterogeneity in infectiousness is the dominant factor controlling the probability of an epidemic , and heterogeneity in susceptibility is the dominant factor controlling the size of an epidemic . edge weights ( measuring closeness or duration of contacts ) have impact only if correlations exist between different edges . combined , these effects can play a minor role in reinforcing one another , with the impact of clustering largest when the population is maximally heterogeneous or if the closer contacts are also strongly clustered . our most significant contribution is a systematic way to address clustering in infectious disease models , and our results have a number of implications for the design of interventions .


since its origins , the new science of complex networks has been primarily driven by the need to characterize the properties of real - world systems  .the introduction of new ideas and concepts in the field has been very often associated to the availability of new , more accurate , or larger data sets , and to the discovery of new structural properties of complex systems from the real world  .this is the reason why a lot of interest has been recently devoted to the study of multiplex networks , i.e. networks in which the same set of nodes can be connected by means of links of qualitatively different type or nature .several data sets of real - world systems that can be represented and studied as multiplex networks have appeared in the recent literature  , and we expect that many more will arrive in the next few years . the first papers on the subject have focused on the characterization of the structure of multiplex networks  , and on modeling the basic mechanisms of their growth  . in parallel to this , some effort has been also devoted on investigating various kinds of dynamical processes on multiplex topologies , including diffusion  , epidemic spreading  , cooperation  , opinion formation  , and percolation  .there is today a general agreement on the fact that multiplex networks represent the ideal framework to study a large variety of complex systems of different nature .and there are already some numerical and analytical results showing that the dynamics of processes on multiplex networks is far richer than in networks with a single layer .a comprehensive review of the main advances in this new vibrant field of research can be found in a few recent survey papers  .  in this articlewe focus on an issue that has revealed of great importance in single - layer networks , but has not yet been investigated thoroughly in multiplex networks , i.e. that of correlations  . in networks with a single layer it has been found that there are correlations in the properties of connected nodes .namely , the degree of a node can be either positively or negatively correlated with the degree of its first neighbors . in the first case , the hubs of the networksare preferentially linked to each others , while in the second case they are preferentially connected to low - degree nodes  .  in multiplex networksthe very same concept of correlations is far richer than in a network with a single layer .in fact , on one hand it is still possible to explore the standard degree - degree correlations at the level of each layer of the network , but on the other hand it is more interesting to introduce a truly multiplex definition of correlations , for instance by looking at how a certain property of a node at a given layer is correlated to the same or other properties of the same node at another layer .we present here a complete and self - consistent study of correlations of node properties in multiplex networks . in doing this , we follow the usual steps of the typical approach to complex networks : _i ) _ we first explore empirically correlations in real multiplex networks , _ ii ) _ we introduce various measures to characterize and quantify correlations in multiplex networks , _ iii ) _ we propose a series of models to reproduce the correlations found in real multiplex systems , or to assess their relevance .we find that multiplexity introduces novel levels of complexity .in particular , in real - world multiplex networks the patterns of presence and involvement of the nodes at the different layers are characterized by strong correlations . andthis has to be taken into account when it comes to model such systems .the article is organized as follows . in sect .[ why ] we focus on two small real - world networks and we use them as examples to explain why a description in terms of multiplex networks captures more information on a system than a single - layer projection . in the remaining sections we study the structure of five real - world multiplex networks ( additional information about the networks are provided in appendix ) , with the main attention to the concept of correlations . in particular , in sect .[ sec : activity ] we focus on the patterns of node activity and involvement at the various layers .we say that a node is active at a given layer if it has at least one link at that layer , and we introduce various quantities to characterize the distribution and the correlations of node activities .we also investigate the activity correlations between pairs of layers .we find that real - world multiplex networks are quite sparse , with only a few nodes active in many layers , and are characterized by strong correlations : interestingly , the activity of a node in a particular layer is very often correlated with its activity in some other layer . in sect .[ sec : activity_models ] we introduce the first null - models to assess the significance of the observed patterns of node activity . in sect .[ sec : cartography ] we investigate correlations between the activity and the degree of the nodes of a multiplex network , while in sect .[ sec : degree - degree ] we show how to quantify inter - layer degree correlations ( degree correlations between layers ) .in particular , we focus our attention on measuring correlation in the node degrees of a pair of layers , either by using the spearman s rank correlation coefficient of the two degree sequences , or by plotting , as a function of , the average degree at one layer of nodes having degree at the other layer .we find that there exist significant correlations among the degree of the same node at different layers , and such correlations can be either positive , meaning that nodes tend to have similar roles across layers , or negative , meaning that nodes with a large degree in one layer tend to have small degrees in another layer . finally , in sect.[sec : model_degree - degree] , we propose two algorithms based on simulated annealing which allow to construct multiplex networks with tunable inter - layer degree - degree correlations , and in sect . [sec : conclusions ] we report our conclusions .the details on the five multiplex networks constructed from data sets of biological , technological and social complex systems , and analyzed in the paper , are describe in the appendix .the networks and the software implementations of the algorithms described in this paper are available at  .the aim of this work is to identify , measure and model the different kinds of correlations among node properties which can be found in a multiplex network .for such a reason we constructed                  a. cardillo , m. zanin , j. gmez - gardees , m. romance , a. j. garca del amo , s. boccaletti , `` modeling the multi - layer nature of the european air transport network : resilience and passengers re - scheduling under random failures '' _ eur .phys . j. spec .top . _ * 215 * , 23 - 33 ( 2013 ) .a. halu , s. mukherjee , g. bianconi , `` emergence of overlap in ensembles of spatial multiplexes and statistical mechanics of spatial interacting networks ensembles '' , _ phys .* 89 * , 012806 ( 2014 ) .( 2013 ) .p. klimek , s. thurner , `` triadic closure dynamics explains scaling - exponents for preferential attachment- , degree- and clustering distributions in social multiplex data '' , _new j. phys . _ * 15 * 063008 ( 2013 ) .s. boccaletti , g. bianconi , r. criado , c. del genio , j. gmez - gardees , m. romance , i. sendia - nadal , z. wang , m. zanin , `` the structure and dynamics of multilayer networks '' , _ phys .rep . _ * 544*(1 ) , 1122 ( 2014 ) .             v. nicosia , p. verts , w. schafer , v. latora , e. bullmore , `` phase transition in the economically modeled growth of a cellular nervous system '' , _ proc ., usa _ * 110*(19 ) , 7880 - 7885 ( 2013 ) .

Abstract: the interactions among the elementary components of many complex systems can be qualitatively different . such systems are therefore naturally described in terms of multiplex or multi - layer networks , i.e. networks where each layer stands for a different type of interaction between the same set of nodes . there is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single - layer projection . here , we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks . correlations in node properties , especially degree - degree correlations , have been thoroughly studied in single - layer networks . here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network . such correlations are intrinsically multiplex , and we first study them empirically by constructing and analyzing several multiplex networks from the real - world . in particular , we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers , and between activities and degrees . we show that real - world networks exhibit indeed non - trivial multiplex correlations . for instance , we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees , while other two layers are negatively correlated . we then focus on constructing synthetic multiplex networks , proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance .


air temperature is a measure of how hot or cold the air is .the knowledge of the mean air temperature of the future seasons is very helpful for individuals , but even more for organizations whose workers and machines have to operate outdoors ( agriculture , armed force , railways , maintenance of roads , tourism , etc . ) and for gas and electricity companies .the seasonal temperature values , in fact , have a great economic impact for both organizations and companies , and their prediction would allow a more efficient activity planning .the accurate prediction of the air temperature is very complex and usually requires a processing of a large amount of data on the weather conditions as well as the scientific understanding of atmospheric processes and their possible spatial and temporal variations  .moreover , it is not possible to perform the prediction of the seasonal temperature with a minimum error for all the months and places in the world .generally , the quality of seasonal forecasting has reached a good reliability over the most of the tropical regions , while the extra tropical latitudes have achieved reliable results only in certain regions ( for example , in northern america ) .more specifically , in europe the predictability of temperature is quite complicated due to the variability and external influences of the local climate  .however in the recent decades a considerable effort has been made to improve the understanding of physical phenomena responsible for the seasonal variability of climate parameters  .  in order to forecast the temperature that will occur in future seasons , climatological models are frequently used .some of them follow an empirical approach based on several series of historical data , useful for building predictive models ; others develop a theoretical approach on complex calculations with parameters and functions that are able to understand the behaviour of the atmospheric system .the first approach considers a set of minimum , maximum and mean temperatures recorded for several decades ( at least thirty years ) in representative meteorological stations in the area , while the second approach uses numerical analysis techniques which involve data observed by radiosondes , weather satellites and surface weather observations of land and oceans processed by powerful supercomputers , which are necessary because of the huge amount of data . for the latter approach , the horizontal spatial domain is global , if it covers the entire earth , or regional , if it covers only a small part of the planet . in the first approach ,the forecast is in somewhat approximate , although it is indicative of the future trend of the seasonal temperature in a certain area ; in the second approach the values of minimum and maximum temperature of the foreseen seasons can be estimated very reliably , because it takes into account the correlations of anomalies in the atmospheric circulation at large scale .however , it is difficult for operators to arrange the numerical approach on a smaller scale for planning purposes .  in order to make the forecasting of foreseen seasonal temperatures less sophisticated , in this worktwo soft computing methods are applied : artificial neural networks ( anns )  and genetic programming ( gp )  .  as annwe use a feedforward multi layer perceptron ( mlp ) neural network , a computational structure made by many processing elements ( units ) , the neurons , operating in parallel , aiming to approximate an unknown function of its inputs  . to configure the mlpwe use the training procedure called `` back propagation ''  .it uses a subset of the data set feature vectors , each one labeled with the correct output , as examples of the correct input / output relationship .artificial neural networks have already been successfully applied in different contexts  , including the atmospheric researches , such as the prediction of maximum daily precipitation  , the monthly precipitation and evapotranspiration index  and air temperature  . generally , in these works , in addition to the change of the reference framework , which can be on a local or regional scale , a wide range of parameters is used .the use of parameters depends on the availability of data .it is not always possible to have them at the local level as well as the reduction in a local scale of the information available at the regional scale can not be free from errors and approximations .we also employ gp in order to improve the understanding of the neural method performance .gp is an evolutionary computational technique proposed by koza  in order to extract automatically intelligible relationships in a system without being explicitly programmed .it has been used in many applications such as symbolic regressions  , and classifications  .it works by using genetic algorithms  to generate and evolve automatically composite functions , traditionally represented as tree structures  .gp is then able to show the relation between input data and output data by means of an explicit formula .also this soft computing method has been successfully applied to many problems of practical order  .this paper is organized in the following way : in the next section we describe the application area and in section 3 the environmental data used for tuning the methods .the results are reported in the experiments section .the area in which soft computing techniques are applied is that of the surroundings of the town of benevento having the geographic coordinates ( lat .41 07 50 14 47 13 e ) .it is located equidistant from the seas and protected by the hills located in the west that is an inner area of the italian peninsula .therefore , the area of this application is not affected by maritime influences , nor by winds coming from the west .nevertheless , its geographic location falls in the context of a mediterranean climate with a tendency to continental features . more specifically , along the italian peninsula it is observed , in winter , the interaction between the area flows from the northwest and west and those who move from the east , which causes heavy rains , while in summer the presence of subtropical anticyclone determines a prolonged drought interrupted by sporadic thunder phenomena .the north - western winter air streams can bring down the minimum temperature to a few degrees above zero , and the maximum in summer exceeds 30 .this synoptic situation is made evident by the data recorded in the meteorological station of benevento  .it is located in the hilly area , which characterizes the territory surrounding this town , at 144 m a.s.l .although it has recorded rainfall and temperature data since 1882 , the official ones published on the hydrological annals are available only from 1925 until 1998 with occasional missing data .the decommissioning of the benevento station forced us to examine the data of another site , at a distance of less than twenty kilometers north - west from benevento .so the data corresponding to the last 13 years come from the castelvenere station ( lat .41 ; long .14 ) , that is located in a similar geographical position ( 140 m a.s.l . ) and it presents a minimal difference with the data of temperature and precipitation recorded in benevento .however , in order to avoid mistakes , these data have been scaled according with this difference . in both stations , for each year the available data are the daily minimum temperature , the daily maximum temperature , the daily mean temperature , the daily amount of rainfall and the number of rainy days that occurred during each month .  in order to better show the weather characteristics of the area , we provide some information obtained from the considered series .the mean maximum temperature recorded during the summer season varies from 25.6 to 32.1 .the mean minimum and maximum value has been reached respectively in 1976 and 1928 .the mean minimum temperature in the winter season ranges from a low of -0.8 in 1982 to a maximum of 7.8 in 1979 .also in 1982 , the highest annual temperature range of 31.1 was reached , although the yearly mean temperature remains below 17 .the daily mean temperature is slightly below 15 .  just to complete the data relating to the maximum temperatures recorded in the last decades, castelvenere station shows a tendency to increase with peaks that have exceeded abundantly 40 during summer months , whereas in the same station the minimum temperature has gone down to 7 below zero during the winter months .the mean maximum temperature of the summer months is within the same range defined before , but basically the values are higher .even the mean minimum temperature of the winter months falls between the extremes defined at a general level , however , it should be noticed that it is not unusual to have periods of more than one week with temperatures below zero ( 2009 and 2012 ) . as for precipitations, it can be seen clearly in the annual distribution of the data that they are highest in autumn and winter seasons and at a minimum in the summer . in particular, the range of the annual precipitation varies from a minimum of 256 mm in 2003 to a maximum of 1127 mm in 1955 , with an average value which is near to 800 mm .the largest quantities were mainly recorded in the autumn months ( nearly 600 mm in 2010 and over 500 mm in 1966 ) , while the lowest ones were recorded in the summer months ( 20 mm in 2000 and 21 mm in 1987 ) .this effect is confirmed by the frequency of rainfall , which sees a concentration of rainy days in the autumn months ( 56 rainy days in 2002 ) , in contrast to summer months in which it is reduced dramatically , as in the months of the beginning of this century with one day in 2000 .in the proposed application we want to predict the mean seasonal temperature of the subsequent years .therefore , a number of parameters ( features ) to assess the seasonal conditions of this area are calculated from the available data from 1925 to 2012 .such parameters are then used for the prediction . according to the data availability in the series of the meteorological stations in benevento area , we consider 9 features :  ll feature & range + year & from 1925 to 2012 + season & winter , spring , + & summer , autumn + mean seasonal temperature & c degrees + mean seasonal of daily maximum temperatures & c degrees + mean seasonal of daily minimum temperatures & c degrees + mean yearly temperature & c degrees +mean seasonal rainfall & mm + number of seasonal raining days & positive integer + mean seasonal temperature in the next year & c degrees +  [ ta1 ]  with the exception of the season all the features are expressed as numeric variables .they should contribute to the prediction of seasonal mean temperature of the year following the year in progress .in addition to temperature values , which express the most features used , two sets of data concerning the precipitation are considered .they have been inserted in the input data , since the temperature is to be put in relation to the general circulation and disturbances that could be incurred . in some casesthe occurrence of rainfall can be enforced by local situations , such as mountain range and vegetation , which determine the amount of moisture in the air .such amount is to be put in relation with the temperature that is observed .furthermore , rainfall is also related to the number of rainy days , useful to detect the frequency with which this phenomenon occurs .missing data are restricted to the period of the second world war ( 1942 1948 ) and the failure of the stations for certain periods ; however , this has not compromised the quality of information used for the application . to obviate these data ,it has not been made any interpolation , while for the data of the second station an appropriate correction has been introduced taking into account the local influences on the recorded data ( different exposure to the solar radiation , moisture conditions , etc . ) .the resulting dataset is composed of 248 labeled examples , where each example has the following structure to the 10-fold cross - validation methodology , the whole data set has been partitioned into 10 non overlapping subsets . of these 10 subsets , one can be chosen to evaluate the neural network performance ( validation set ) , and the remaining 9 can be used to instruct it ( training sets ) , i.e. setting the network weights .this procedure has been repeated 10 times , corresponding to all possible choices of the validation set and the corresponding average performance has been gauged in terms of an average % -misclassification error .cross - validation is important in testing hypotheses suggested by the data , especially where further samples are costly or impossible to collect .  for our experiments we adopted an excel - based system which simulates a neural network developed by angshuman saha , that is freely available all over the web .by using this tool , we can define feed - forward back - propagation networks with 1 or 2 hidden layers , enter training data , set various learning parameters , start a learning phase and see the results ( e.g. error rate ) presented in various ways , for example graphically ( see fig .categorical values are automatically converted into numerical ones .the weights are initially randomly chosen in a fixed range .the error back propagation algorithm has several parameters , and the most important ones are called , respectively , `` learning rate '' and `` momentum term '' .the first term is a measure of the influence degree , in the formula for updating weights , of the error term , whereas the latter one determines the influence of the past history of weight changes in the same formula ., width=404 ]  one of the most important characteristics of a mlp is the number of neurons in the hidden layer(s ) .if an inadequate number of neurons are used , the network will be unable to model complex data , and the resulting fit will be poor . if too many neurons are used , the network generalizes poorly to new , unseen data .so the hidden layers have been configured by a random initial choice and  * pruning , i.e. eliminating nodes , when the network learns easily the training set but shows bad performances on the validation set ; and * growing , i.e. adding nodes when the network shows bad performances on the training set .  in our casethe resulting architecture is made up of 8 inputs , a single hidden layer of 2 neurons , and one output .the number of training cycles ( epochs ) has been fixed to 500 .to value the performances , we use the error percentage resulting from the 10-fold cross - validation methodology and the coefficient of determination , denoted , that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable .the results on the whole data set have been reported in table 2 .the mean error is of 8,71% while the coefficient of determination is 0,97 , a very high value .[ f5 ] reports comparatively the predicted and expected temperature values .  , width=291 ]  @ trial & validation error & + 1 & 3,68% & 0,97 + 2 & 8,31% & 0,97 + 3 & 7,70% & 0,97 + 4 & 9,28% & 0,97 + 5 & 7,32% & 0,97 + 6 & 6,15% & 0,97 + 7 & 16,41% & 0,97 + 8 & 10,35% & 0,97 + 9 & 10,41% & 0,97 + 10 & 7,51% & 0,97 + & 8,71% & 0,97 +  [ ta2 ]  then , since the data originated from two different stations , we partitioned the dataset , according to its origin , in two subsets , i.e. from 1925 al 1998 and from 2000 to 2012 , and the ten fold cross validation procedure has been repeated for each one .the results on the two subsets have been reported respectively in table 3 and table 4 .@  trial & validation error & + 1 & 10,76% & 0,97 + 2 & 7,73% & 0,97 + 3 & 8,85% & 0,97 + 4 & 6,79% & 0,97 + 5 & 8,21% & 0,97 + 6 & 8,63% & 0,96 + 7 & 7,10% & 0,97 + 8 & 12,52% & 0,97 + 9 & 11,87% & 0,97 + 10 & 13,82% & 0,97 + & 9,63 % & 0,97 +  [ ta3 ]  @  trial & validation error & + 1 & 4,19% & 0,97 + 2 & 5,89% & 0,97 + 3 & 8,78% & 0,97 + 4 & 7,22% & 0,97 + 5 & 7,48% & 0,97 + 6 & 3,29% & 0,97 + 7 & 3,21% & 0,97 + 8 & 5,03% & 0,97 + 9 & 1,04% & 0,97 + 10 & 3,79% & 0,97 + * mean * & 4,99% & 0,97 +  [ ta4 ]  [ [ gp ] ] gp ~~  given the data set characteristics , we are looking for a formula that satisfies we limit the set of component functions to the arithmetic operators plus some trigonometric functions ( sine , cosine and tangent ) , the exponential and the natural logarithm , the logistic function , the hyperbolic tangent and the minimum function .  as fitness measure we use the mean square error .the exemplars are split in two groups for training ( 90% ) and validation ( 10% ) .the training set is created by randomly selecting samples from the whole dataset .we use a genetic programming software tool called eureqa , for detecting equations and hidden mathematical relationships in a given data set .eureqa works in order to reduce the error function given by the discrepancy between the data and the generated model  .eureqa is a proprietary a.i .- powered modeling engine originally created by cornell s artificial intelligence lab and later commercialized by nutonian , inc .the software uses evolutionary search to determine mathematical equations that describe sets of data in their simplest form .it also allows to evidence the behaviour of each solution with respect to its size .  after about 700000 generations we found a set of solution , as reported in table 5the performance of each solution are reported in table 6 .a plot of expected vs predicted temperatures on the whole data set by using solution 10 is reported in fig .[ f6 ] .@  solution num . &size & fit & solution + 1 & 1 & 1,000 & + 2 & 3 & 0,985 & + 3 & 6 & 0,806 & + 4 & 5 & 0,982 & + 5 & 10 & 0,762 & + 6 & 14 & 0,445 & + 7 & 38 & 0,309 & + & & & + 8 & 32 & 0,340 & + & & & + 9 & 28 & 0,350 & + & & & + 10 & 26 & 0,436 & + & & & +  [ ta5 ]  , width=291 ]  @  solution & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 + * *&0,92 & 0,92 & 0,92 & 0,94 & 0,94 & 0,96 & 0,97 & 0,97 & 0,97 & 0,98 + * * & 2 & 2 & 2 & 2 & 2 & 1,00 & 0,98 & 0,78 & 0,76 & 0,69 +  [ ta6 ]  as it was evidenced by the solution expressions , just 3 parameters determine the solution .we value the relevance of each variable in terms of  * sensitivity * : the relative impact within this model that a variable has on the target variable .* % positive * : the likelihood that increasing this variable will increase the target variable .* positive magnitude * : when increases in this variable lead to increases in the target variable , this is generally how big the positive impact is .* % negative * : the likelihood that increasing this variable will decrease the target variable  * negative magnitude * : when increases in this variable lead to decreases in the target variable , this is generally how big the negative impact is .the results , for the best solution found are reported in table 7 .@  variable & sensitivity & % positive & positive mag . &% negative & negative mag .+ & 3,0141 & 46% & 3,3674 & 54% & 2,7143 + & 1,3642 & 67% & 1,1636 & 33% & 1,7654 + & 0,1883 & 100% & 0,1883 & 0% & 0 +  [ ta7 ]  then the experiment has been repeated by adding the gaussian function to the set of functions . in this casethe best solution considers more and different variables , as reported in table 8 .@  variable & sensitivity & % positive & positive mag . &% negative & negative mag .+ & 1,1779 & 67% & 1,1357 & 33% & 1,2622 + & 0,55971 & 100% & 0,55971 & 0% & 0 + & 0,41929 & 0% & 0 & 100% & 0,41929 + & 0,16345 & 80% & 0,12137 & 20% & 0,33176 +  [ ta8 ]in this study , two soft computing methods , as artificial neural networks , in the multi - layer perceptron approach , and genetic programming , were applied to predict next year s seasonal temperatures . in both methods we considered the same weather variables extracted from the records of two meteorological stations of benevento and castelvenere . while the traditional statistical methods need several a priori assumptions , and are non - parametric tools and suitable for any situations and data sets .   as regard the application of , the results on the whole data set show a mean error of 8,71% with a very high value of coefficient of determination of 0,97 . if we consider only the subset 2000 - 2012 , the ten fold cross validation result is lower than 5% with the same coefficient of determination .however , from a customer manager s point of view , the classification results from the are sub - symbolic and are often difficult to comprehend .the proposed approach offers an explicit rule representation ( a formula ) while maintaining the low error rates and high correlation values .as it has been evidenced in the last solution reported in table 5 , a formula which has a relatively low size and a rather good fitness considers just the mean yearly temperature of the previous year , the actual year and the relative season .this solution reaches a very good performance in terms of and ( the former is 0,98% and the latter is 0,69 , as reported in table 6 ) .the comprehension of the gp results has markedly improved in the next serial of tests and it has pointed out that the foreseen of seasonal temperature is related to the tendency of previous mean season temperature and subordinately to the mean yearly temperature ( see table 8) .this result could be linked to the atmosphere circulation acting in the application area , otherwise these variables would have a higher negative sensitivity  . a major impact of the mean seasonal of daily maximum temperatures and the mean seasonal of daily minimum temperatures , as also the mean seasonal rainfall and number of seasonal raining days in the solution test, would be evidence of anomalies in the atmosphere circulation ( i.e. higher content in greenhouse gas , frequent cloud burst ) . in our application ,developed in the area of benevento , the evolutionary trend appears almost linear , so evidencing the dominance of the natural forces .probably this was to be expected for the natural and mainly agricultural land use , affecting to a lesser degree on the temperature behaviour with respect to industrial areas .these good results encourage the improving of these methods , by exploring and fine - tuning the various parameters , whose exact determination may be quite complex ( e.g. temperature , pressure , moist and so on ) . in the last decadethese estimations have been further complicated by anthropogenic forces that make less comprehensible the behaviour of the temperature  .anderson j , van den dool h , barnston a , chen w , stern w , ploshay j ( 1999 ) present - day capabilities of numerical and statistical models for atmospheric extratropical seasonal simulation and predication .meteorol . soc .80:13491361     cramer nl ( 1985 ) a representation for the adaptive generation of simple sequential programs in : grefenstette jj ( ed ) proc .international conference on genetic algorithms and their applications , carnegie mellon university , pittsburgh , pp 183187   deo rc , ahin m ( 2015 ) application of the artificial neural network model for prediction of monthly standardized precipitation and evapotranspiration index using hydrometeorological parameters and climate indices in eastern australia .atmospheric research 161 , 162:6581         makkeasorn a , chang nb , zhou x ( 2008 ) short - term streamflow forecasting with global climate change implications a comparative study between genetic programming and neural network model .journal of hydrology 352:336354  nastos pt , paliatsos ag , koukouletsos kv , larissi ik , moustris kp ( 2014 ) artificial neural networks modelling for forecasting the maximum daily total precipitation at athens , greece .atmospheric research 144:141150       saha s , nadiga s , thiaw c , wang j , wang w , zhang q , van den dool hm , pan h - l , moorthi s , behringer d , stokes d , pena m , lord s , white g , ebisuzaki w , peng p , xie p ( 2006 ) the ncep climate forecast system. j. climate 19:34833517    shiri j , kii o , landeras g , lopez jj , nazemi ah , stuyt lcpm ( 2012 ) daily reference evapotranspiration modeling by using genetic programming approach in the basque country ( northern spain ) ._ journal of hydrology_ 414 415:302316      venkadesh s , hoogenboom g , potter w , mcclendon r ( 2013 ) a genetic algorithm to refine input data selection for air temperature prediction using artificial neural networks . applied soft computing 13:22532260

Abstract: in this work two soft computing methods , artificial neural networks and genetic programming , are proposed in order to forecast the mean air temperature that will occur in future seasons . the area in which the soft computing techniques were applied is that of the surroundings of the town of benevento , in the south of italy , having the geographic coordinates ( lat . 41 ; long.14 ) . this area is not affected by maritime influences as well as by winds coming from the west . the methods are fed by data recorded in the meteorological stations of benevento and castelvenere , located in the hilly area , which characterizes the territory surrounding this city , at 144 m a.s.l . both the applied methods show low error rates , while the genetic programming offers an explicit rule representation ( a formula ) explaining the prevision .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore