low - density parity - check ( ldpc ) codes constructed on high - order finite fields show remarkable coding gains with respect to ( w.r.t . )binary ldpc / turbo codes .the gain is especially visible in the moderate - to - short block length ( bits ) regime , where binary iteratively - decodable codes tend to exhibit either high error floors or non - negligible coding gain losses w.r.t .available benchmarks ( e.g. , db with respect to the random coding bound ( rcb ) ) .code , for moderate - long block it provides a tight estimation of the performance of the best codes . in the short - length regime, properly - designed codes can surpass the rcb even remarkably .however , it will be kept as a reference through the paper together with the sphere packing bound ( spb ) of .] ultra - sparse non - binary ldpc codes based on fields of order for short block lengths allow approaching the average performance of random codes by db down to medium - low codeword error rates ( cers ) .  in this paper , we provide a novel turbo code construction , which leads to _ non - binary _ turbo codes based on convolutional codes on a finite field of order .the proposed construction bridges rate- non - binary turbo codes and regular non - binary ldpc codes , where and are the check node ( cn ) and variable node ( vn ) degrees , respectively .more specifically , the turbo code construction follows from a specific expansion of a protograph .the so - obtained turbo codes have performance close to that of their non - binary ldpc counterparts , with the advantage of an efficient encoding structure .even more important , the proposed construction allows building families of rate - compatible turbo codes with flexible block size , by adopting combinatorial ( on - the - fly ) interleaver constructions .convolutional turbo(-like ) codes over non - binary finite fields / rings have been investigated previously , e.g. in . with respect to the past works ,the main novelties of our construction are listed next .most of the previous contributions devote attention to fields of low order , whereas we focus on high order fields , e.g. of order .moreover , our construction is based on codes with memory ( in symbols ) , whereas in codes with larger memories are considered .the convolutional codes adopted by the proposed construction are time - variant , while in the feed - forward / feedback polynomials are fixed .to our knowledge , the only non - binary iterative schemes adopting time - variant convolutional codes are the irregular repeat accumulate ( ira ) codes of .however , in a turbo code structure is not considered explicitly .we further propose an alternative serial turbo code interpretation of the regular protograph ensemble which allows to efficiently encode rate- regular ldpc codes as a serial concatenation of a non - binary time - variant differentiator , and interleaver , and a non - binary time variant accumulator .the proposed construction features a nice graph interpretation which provides useful insights for the interleaver design .high rates can be obtained by suitably puncturing the code symbols , whereas low rates can be easily obtained by using the multiplicative repetition approach of .a discussion on how decoding can be performed either in a turbo - like fashion ( i.e. , by iterating a bcjr decoder on the -states trellises of the component codes ) or via the classical belief propagation ( bp ) algorithm for ldpc codes is provided . in both cases , the decoder complexity growth w.r.t .the field order can be limited to by adopting fast fourier transforms .the contribution is structured as follows .section [ sec : structure ] describes the code structure . in section [ sec : decoding ]the iterative decoding algorithm is discussed .numerical results and conclusions follow in sections [ sec : results ] and [ sec : conclusions ] respectively .regular non - binary ldpc codes are characterized by excellent iterative decoding thresholds . in this paper , we shall focus on the rate- case .the iterative decoding threshold over the binary - input additive white gaussian noise ( awgn ) channel for the unstructured ensembles with random choice of the non - zero coefficients in the parity - check matrix is db , will denote the energy per information bit and the one - sided noise power spectral density . ]only db away from the shannon limit .the turbo codes described in this paper are constructed on a specific protograph sub - ensemble of the regular ensemble , depicted in fig .[ fig : proto](a ) .a protograph is a tanner graph with a relatively small number of nodes .a protograph consists of a set of variable nodes , a set of check nodes , and a set of edges . each edge connects a vn to a cn .a larger _ derived _ graph can be obtained by a copy - and - permute procedure : the protograph is copied times , and the edges of the individual replicas are permuted among the replicas .a protograph can be described by a _ base matrix _ of size .the element of represents the number of edges connecting the vn to the cn . the base matrix associated with the protograph in fig .[ fig : proto](a ) is given by \ , .\ ] ] we proceed by expanding the protograph into the derived graph as follows .we first replace the upper - left ` ' in with a identity matrix , .the lower - left ` ' is replaced by a permutation matrix .each of the ` ' entries of is replaced by a cyclic matrix .\ ] ] the derived graph adjacency matrix has the form .\ ] ]   protograph as parallel ( a ) and serial ( b ) concatenation . ]we then construct a parity - check matrix on by replacing each ` ' in with a suitable element of .the full - rank non - binary parity - check matrix is given by \label{eq : h}\ ] ] where  * is a matrix with non - zero entries only on the main diagonal ( _ pseudo - identity matrix _ ) .more specifically , for , otherwise .* is a matrix with non- entries only for .in particular , .thus , can be described as the product between a permutation matrix and a pseudo - identity matrix with diagonal coefficients , with being the permutation function . *both are double diagonal matrices with form ,\qquad z=1,2.\ ] ]   the parity - check matrix may be interpreted as the parity - check matrix of the parallel concatenated convolutional code ( pccc ) .the information word of symbols in , , is input to a rate- , memory- time - variant recursive systematic convolutional ( rsc ) tail - biting encoder ( _ non - binary accumulator _ ) .the first set of parity symbols is obtained as \label{eq : rsc1}\ ] ] and with properly initialized .( and consequently of ) can be obtained as indicated in ( * ? ? ?* lemma 4 ) . ] here , , and all operations are in .the second set of parity symbols are obtained by first permuting the symbols of according to the interleaving rule . the permuted information word ( with )is then fed in a second rate- , memory- time - variant rsc tail - biting encoder .the second set of parity symbols is obtained as \label{eq : rsc2}\ ] ] and with .the systematic codeword is given by  ] , and the code has length symbols .the proposed construction brings gives a turbo code with code rate . we will refer to this construction as differentiate accumulate ( da ) code .a lower rate can be obtained by providing at the output of the encoder the intermediate vector as well , i.e. by setting  ] with =\pr\{u_i=\omega|\mathbf{y}\} ] ( ) .the computation of the a posteriori probability for the symbol can be accomplished by evaluating the operator returns the label associated with for the trellis edge connecting the state at time to the state at time , denotes the forward metric for the state at time , is the backward metric for the state at time , and is the transition metric between states at time .we normalize the metrics such that }\ } , \qquad \varphi_{0}(0)=1,\nonumber \\\beta_{i}(s)&=\pr\{s_i = s|\mathbf y_{[i+1:k]}\ } , \qquad \beta_{k}(0)=1,\nonumber \\\gamma_i(s , s')&=\pr\{s_{i-1}=s , s_i = s'|\mathbf y_i\}\nonumber \\ & = \pr\{u_i=\omega , p_i=\nu|y^u_i , y^p_i\},\label{eq : gamma_1}\end{aligned}\ ] ] with , . assuming independent outputs , can be factored into where depends on only since .the forward / backward metrics can be computed recursively as \gamma^u_{i+1}(s , s').\label{eq : bwd_recursion}\end{aligned}\ ] ] note that involves a convolution since are related by .similarly , for are related by .we introduce the p.m.f .vectors ,\\ \mathbf{\beta}_{i}&=\left[\beta_{i}(0),\beta_{i}(1),\beta_{i}(\alpha),\ldots,\beta_{i}\left(\alpha^{q-2}\right)\right],\\ \mathbf{\gamma}^p_i&=\left[\gamma^p_i(0),\gamma^p_i(1),\gamma^p_i(\alpha),\ldots,\gamma^p_i\left(\alpha^{q-2}\right)\right],\\ \mathbf{\gamma}^u_i&=\left[\gamma^u_i(0),\gamma^u_i(1),\gamma^u_i(\alpha),\ldots,\gamma^u_i\left(\alpha^{q-2}\right)\right],\\\end{aligned}\ ] ] where in the last expression ( with a slight abuse of the notation ) we re - defined . in vector form , we can re - arrange , into \nonumber \\ \mathbf{\beta}_{i}&=\pi^{-1}_{f_{i+1}}\left\{\left[\mathbf{\beta}_{i+1}\cdot\mathbf{\gamma}^p_{i+1}\right]\circledast\pi_{g_{i+1}}\left(\mathbf{\gamma}^u_{i+1}\right)\right\}. \label{eq : vect_fwdbwd}\end{aligned}\ ] ] in , denotes the permutation , induced by the multiplication by a scalar of a random variable with p.m.f .vector , on , while denotes the inverse permutation ( or equivalently the permutation induced by the multiplication by ) .furthermore , denotes the ( point - wise ) multiplication of two vectors , and denotes the convolution of the two vectors .the a posteriori p.m.f .vector of given the channel output is finally given ( up to a normalization factor ) by \right\}}_{\mathbf{\mu}_i^u}\cdot \mathbf{\gamma}^u_i\ ] ] where represents the extrinsic information , .the message update can be easily followed on the normal factor graph of a section of the trellis provided in fig .[ fig : fg ] . the complexity is here dominated by the convolution operations , and thus scales as .is the field order , and hence the length of the vectors involved in the convolutions of . ]the algorithm can be simplified by applying the ( fast ) fourier transform ( ft ) for finite abelian groups on the vectors involved in the convolutions . assuming extension fields with characteristic , the ft reduces to the walsh - hadamard transform , i.e. given a function , , its fourier ( walsh - hadamard ) transform , , is obtained as  being the inner product over between the length- binary vector representations of . by employing ffts , the decoding complexity is reduced to .simulation results on the awgn channel for codes on are presented next . in all the simulations , we adopted the bp decoding over the tanner graph of the codes with a maximum number of iterations set to .binary antipodal modulation has been considered .[ fig : perf128_all ] shows the performance for a rate - compatible code family with input block size bits . the mother code is a code , whose parity - check matrix coefficientshave been selected according to the method of .a lower code rate has been obtained by repeating each code symbol twice , and by multiplying the replicas by random elements in as for the multiplicative repeat ( mr ) approach of .higher code rates have been obtained in two different ways , i.e. ( i ) according to the parallel concatenation scheme , by periodically - puncturing parity symbols at the output of the two accumulators and ( ii ) by puncturing the vns of type ( thus , a rate da code is obtained , and further higher rates can be achieved by puncturing symbols periodically at the output of the accumulator in the da encoder ) . in both cases , symbol - wise puncturing pattern ( spp ) has been applied .the interleaver has been designed according to a circulant version of the progressive edge growth ( peg ) algorithm .the rate mother code does not show floors down to , performing within db from the rcb .similar results are obtained by the lowest - rate code . for the two schemes with rate ,the performance is still within from the rcb down to , with a slight advantage for the da construction .the advantage is more visible for the rate case . here, the pccc performance suffers for a lack of steepness , which is not due to a low minimum distance ( low - weight error patterns have not been detected ) , but to a slow decoding convergence associable with the large fraction of punctured symbols . for the da case ,the rate code parity - check matrix of has been used for the tanner graph , and hence the higher rates have been obtained with a reduced fraction of punctured symbols .the same plot provides the performance of the double - binary turbo code of the dvb - rcs standard .the pccc outperforms the double - binary one by more that db at .[ fig : spb ] depicts the minimum required to achieve for several rate parallel concatenated convolutional codes and rate da codes , with block sizes spanning from bits to bits .the performances of rate binary irregular protograph - based ldpc and accumulate repeat accumulate ( ara ) codes from are provided too . the chart is completed by the spb for the continuous - input awgn channel .the da codes have been again obtained by puncturing the -type nodes of the pccc graph .the interleavers have been generated on the fly according to .additionally , the rate and  bits codes associated with the cycle graphs of fig . [ fig : petersen_graph ] have been simulated .the rate pcccs perform within db from the spb all over the block sizes ( with the exception of ) . for the largest ( ) block length ,the gap is reduced to db . for the rate , the gap w.r.t .the spb is slightly larger ( db more ) .the gain of the proposed non - binary turbo codes over the binary ldpc codes is remarkable ( db or more)for the shortest block sizes . for the largest ( ) block length ,the gain is reduced to db .the performance of two short codes from are provided as well .the first is a extended bch code under maximum likelihood ( ml ) decoding , which achieves at db , only db away from the spb with a coding gain of db over the da code .we shall consider in the comparison that the da code does not perform a complete ml decoding , and hence provides an error detection mechanism that may be required by critical application , e.g. telecommand in the up - link of space communication systems .the second code is a terminated binary convolutional code with constraint length .this code performs close to the da code , which however has a slightly higher code rate ( vs. ) and a lower block size ( vs information bits ) .  , bits . ]   for achieving for various codes with rates , compared with the corresponding spbs . ]two novel classes of turbo codes constructed over high - order finite fields have been presented .the codes are derived from a protograph sub - ensemble of the regular ldpc ensemble .one of the proposed construction is based on the serial concatenation of a non - binary , time - variant differentiator and a of non - binary , time - variant accumulator , and provides a highly - structured flexible encoding scheme for ldpc ensembles .symbol map decoding of the component codes has been illustrated , together with its fft - based simplification .the proposed codes allow efficient decoding either as ldpc or as turbo codes .remarkable gains ( db ) w.r.t .binary ldpc / turbo codes have been demonstrated in the moderate - short block regimes .this research was supported in part by the esa project antares and in part by the ec under seventh fp grant agreement ict optimix n. infso - ict-214625 .g. white and j. costello , d.j ., `` construction and performance of -ary turbo codes for use with -ary modulation techniques , '' in _ proc . of ieee int .symp . on information theory _ , 2000 ,p. 220 .w. lin , b. bai , y. li , and x. ma , `` design of q - ary irregular repeat - accumulate codes , '' in _ proc .int . conf . on advanced information networking and applications _ ,los alamitos , usa , 2009 , pp .201206 .k. kasai , d. declercq , c. poulliat , and k. sakaniwa , `` rate - compatible non - binary ldpc codes concatenated with multiplicative repetition codes , '' in _ proc .symp . on information theory _, austin , texas , usa , jun . 2010 .s. crozier , j. lodge , p. guinand , and a. hunt , `` performance of turbo codes with relative prime and golden interleaving strategies , '' in _ proc .6th international mobile satellite conference _ , ottawa , ontario , canada , jun .1999 , pp . 268275 .

Abstract: two classes of turbo codes over high - order finite fields are introduced . the codes are derived from a particular protograph sub - ensemble of the low - density parity - check code ensemble . a first construction is derived as a parallel concatenation of two non - binary , time - variant accumulators . the second construction is based on the serial concatenation of a non - binary , time - variant differentiator and of a non - binary , time - variant accumulator , and provides a highly - structured flexible encoding scheme for ensemble codes . a cycle graph representation is provided . the proposed codes can be decoded efficiently either as low - density parity - check codes ( via belief propagation decoding over the codes bipartite graph ) or as turbo codes ( via the forward - backward algorithm applied to the component codes trellis ) . the forward - backward algorithm for symbol maximum a posteriori decoding of the component codes is illustrated and simplified by means of the fast fourier transform . the proposed codes provide remarkable gains ( db ) over binary low - density parity - check and turbo codes in the moderate - short block regimes .


pattern formation is one of the most fascinating and challenging problems in natural science .complexity of nature has been making various kinds of beautiful patterns .we are surrounded by such beautiful patterns , e.g. , honeycomb structure in a convection system , diffusion limited aggregation pattern , self - affine mountain profiles , and so on .pattern formation phenomena can be observed not only in physical systems , but also in chemical , and biological systems .for instance , our human body structure is a result of highly controlled pattern formation . even in our everyday life, fluid system usually makes beautiful patterns such as a milk - crown structure . in order to understand the complex nature of such pattern formations ,more comprehensive studies have to be carried out .especially , the pattern formation of soft matter is a growing research field since it relates to various scientific and industrial issues . here, we focus on the pattern formation of polymer gel .polymer gel is a typical example of soft materials which shows complex rheological property .it is well known that the pattern formation is caused by a volume phase transition of polymer gel .since the pattern formation was reported by tanaka group , it has been studied both by experimental and theoretical ways . in this phenomenon , the origin of pattern formation is thought a mechanical instability owing to the abrupt volume change .in contrast , we have found a novel pattern formation appearing on a gelation surface .the pattern itself looks similar to the previous ( volume phase transition based ) one .however , physical mechanisms behind them are completely different .the pattern formation we found occurs without volume phase transition .furthermore , no other sudden change can be observed during pattern formation .poly - acrylamide ( aa ) gel slabs have been mostly used to make the pattern .poly - aa gel is very popular for biologists because they often use it for electrophoresis. radical chain reaction ( polymerization ) governs the gelation of poly - aa .we found that ambient oxygen is necessary to make the pattern .the pattern formation occurs when the top of the sample is open to ambient air .then oxygen is absorbed from a surface of slab , and diffuses in pre - gel solution during polymerization process .when we decrease the concentration of ambient oxygen , the pattern becomes thin .the oxygen is known as an inhibitor in the radical polymerization ( gelation ) .the necessity of inhibitor in a diffusion system reminds us of the turing instability .it is caused by diffusion and nonlinear competing reactions between an activator and an inhibitor .such system is called reaction diffusion ( rd ) system .the importance of inhibitor ( oxygen ) suggests the system might be governed by the rd system .in addition , the typical length scale ( wavelength of the pattern ) were measured and analyzed based on the framework of rd system .then , the length scale was consistent with the rd system .  in our previous study , we demonstrated the universality of this kind of pattern formation .we found that dimethylacrylamide ( dmaa ) and sodium acrylate ( sa ) polymers are also able to create surface pattern formation under similar experimental conditions .thus we think this pattern formation is universal at least for quasi 2-dimensional ( 2d ) free surface radical polymerization ( gelation ) process .we have measured the effective surface roughness ( esr ) using 2d photo data of the pattern .the esr characterizes the degree of pattern formation .it indicates a large value for vivid pattern formation , and a small value for slight pattern formation .we varied three essential parameters ( oxygen concentration , initiator concentration , and temperature ) , and measured the esr of resultant patterns in order to examine the parameters dependence of the pattern formation .finally , we obtained an empirical scaling which combines three parameters into a single scaling parameter .  although the actual pattern formation is 3-dimensional ( 3d ) surface deformation , all analyses mentioned above are based on 2d photo data .typical photos of the surface patterns are shown in fig .[ fig : rand_stripe ] . as shown in fig .[ fig : rand_stripe ] , 3d deformation can be visualized by 2d gray scale pattern by an inclined illumination .however , 3d direct measurement should be applied for the real quantitative analysis of 3d deformation .one main focus of this paper is to introduce a 3d measurement system to analyze the pattern formation directly .a line laser displacement sensor and an automatic x - stage are used for this purpose .there is another research focus concerning the ordering of pattern formation . in fig .[ fig : rand_stripe ] , two kinds of patterns are shown , ( a ) random and ( b ) stripe .as reported in the previous paper , random pattern appears in relatively low temperature regime .and stripe pattern appears in relatively high temperature regime .usually , random structure should be observed in high temperature regime , since temperature is a source of thermal noise . in this sense , it is a counterintuitive result .the reason of this trend has been left unsolved .very recently , we realized the substrate rack of the apparatus ( constant temperature chamber ) , which is underneath the sample , has stripe shape .besides , the pattern aligns to that direction .therefore we guess that the stripe pattern might come from the substrate based external temperature field . in order to check this assumption , we use a heating wire array , and make a poly - aa gel slab on it .direction , wavelength , and amplitude of the resultant patterns are measured by the 3d measurement system , and controllability of them are discussed in this paper .finally , we investigate more general possibility of pattern control , using various shapes of external temperature field .the materials and methods to make gel slabs are basically same as previous studies . aa monomer ( = ) constitutes sub - chains , and methylenbisacrylamide ( bis , = ) constitutes crosslink .ammonium persulfate ( aps ) is used as an initiator of radical reaction , and tetramethylethlyenediamine ( temd ) is used as an accelerator of the radical polymerization .most of produced gels are composed by g aa , 6 mg bis , - mg aps ,  temd , and ml deionized water .the experimental procedure is very simple .pre - gel solution is made in a beaker and degassed under ultrasonic vibration .right after this procedure , the initiator is added and the solution is poured onto a petri - dish ( cm in diameter ). then surface deformation ( pattern formation ) occurs spontaneously within about 2 hours .this pattern remains permanently after the gelation .typical thickness of gel slabs is approximately mm .the gel slabs are thin enough to be regarded as quasi-2d .we develop a 3d measurement system to directly measure the surface deformation of gel slabs .schematic image of the system is shown in fig .[ fig : surface3d ] . a line laser sensor ( keyence ljg-030 )is mounted above a sample .this sensor is composed by a line laser unit and a 2d cmos detector so that the sensor is able to acquire an intersectional profile of a 3d deformed surface . the sample is held on an automatic x - stage ( coms pm80b-100x ) .the axis of this stage is placed perpendicular to the line of laser .sample is shifted by the x - stage after taking every intersection data .all instruments are controlled by a pc ( with labview 8.6 ) .collected data are reorganized to the 3d surface profile .this system has  m horizontal resolution ( both of x- and y - direction ) , and  m vertical resolution .recently , the system is also applied to measure the drop - granular impact cratering .first , we make a gel slab on a heating wire array as shown in fig . [fig : wires_array ] .temperature of the array is set as or 27-.3em and environmental ( room ) temperature is 27-.3em .when we make a gel slab on it , clear stripe structure can be observed .direction of the stripe completely agrees with that of the wire array .this means that the direction of stripe pattern can be controlled by an external temperature field substrate .this is a qualitative check of the pattern controlability .next , we are going to try to control the characteristic length scale ( wavelength ) of the stripe pattern . in this experiment ,interval distance between two wires is a main control parameter .we varies as , = , , , and mm . temperature of the wire is fixed or 27-.3em .typical patterns ( photos ) and 3d measured results are shown in fig . [fig : hw_3d ] .3d measurement data corresponds to central ( mm ) region of samples . as seen in fig .[ fig : hw_3d ] , the surface becomes rather flat with slight stripes when the interval distance is small .by contrast , independent stripe patterns appear if the interval distance is large enough .however , it is hard to confirm the correspondence between the interval distance and the pitch of stripe patterns . in other words, we are not able to see clear evidence of wavelength controllability in these data .using 3d height map data , we now analyze the characteristic length scale of the patterns more accurately .we apply the fast fourier transform ( fft ) analysis to the 3d data ( fig .[ fig : hw_3d](e - h ) ) . the peak wave number obtained by fft analysisis translated to the wavelength ( characteristic length scale ) .measured is shown in fig .[ fig : lambda ] . horizontal axis in fig .[ fig : lambda ] indicates experimental conditions ( wire temperature and initiator amount ) .symbols represent the interval distance of heating wires as denoted in the legend . in fig .[ fig : lambda ] , we can not confirm clear trend of the wavelength data .it seems to fluctuate around an average value of mm .this value certainly agrees with the previous result .the result suggests that it is hard to control the wavelength of the patterns .the stripe patterned external temperature field makes a stripe patterned gel slab .however , it is impossible to control the wavelength ( pitch ) of the pattern . in the previous paper , we considered that the origin of the pattern formation is a sort of rd based instability .if it is true , the characteristic length scale ( wavelength ) should be determined by the diffusion length scale of the inhibitor . in rd system, the wavelength is intrinsically determined rather by internal conditions than by external conditions .the result obtained here is consistent with that consideration .only the direction can be controlled by the external temperature field .next , we measure the amplitude of surface deformation .the surface roughness ( standard deviation of height map data ) is computed on 3d data that are same as those used in fig .[ fig : lambda ] computation .the result is shown in fig .[ fig : w ] .although the data scatter a lot , there is not any clear trend , again .the characteristic amplitude order ( surface roughness ) is about mm .  according to the data in figs .[ fig : lambda ] and [ fig : w ] , controllability of the pattern formation is very limited .we are able to control global structure ( direction ) of the pattern , whereas both of the wavelength and amplitude controls are impossible .the length scales seem to be determined by other physical mechanisms like diffusion of inhibitor , and/or polymerization dynamics , etc .   here, we try a little bit more challenge of the pattern control .since global structure of the pattern formation is controllable , we make gel slabs on several shapes of external temperature fields .the fields are generated by heated metal molds that have square , circle , and heart shapes .the metal molds are heated by a heating wire binding the molds .  in fig .[ fig : molds ] , photos of the obtained gel slabs and corresponding temperature field shapes are displayed .as expected , we can control the global structure of resultant patterns .they are clearly affected by external temperature fields .furthermore , the wavelength seems to be independent of the temperature field .these characteristics are consistent with the result obtained by the wire array experiment .the obtained results so far conclude the difficulty of the length scale control .this pattern formation phenomenon itself is very robust .the fine parameter tuning is not necessary to make the pattern . in this sense ,the pattern formation is a kind of self - organized phenomenon .it is hard to control the length scale due to the robustness of self - organization . on the other hand, we can easily control the global structure of the pattern formation .surface pattern deformation simply obeys external temperature field shape .  in ref. , we considered the transition from random to stripe patterns at a higher temperature region . in the previous experiments, temperature is controlled by a constant temperature chamber with a stripe shaped substrate rack . thus the environmetal temperature was controlled .in low temperature reigme , heat condution and gelation process is rather uniform .contrastively , heat conduction from the substrate rack can not be negligible in high temperauture regime .that is a reason of transition like behavior at a certain temperature . in the current experiment, we use a heating wire array to actively control the temperature field , and the environmental temperature is room temperature ( 27-.3em ) . as a result, the created pattern obeys the temperature field .therefore , we think that the temperature gradient plays an essential role in this phenomenon . however , the precise measurement of the temperature field is not easy .the current temperature control system is too rough to measure and discuss such a delicate temperature field .the precise temperature field measurement is a future problem .the typical length scale of the pattern formation is in an order of - mm .such macroscopic self - organized pattern is not so common in polymer science . in this report, we do not discuss the microscopic dynamics of polymerization , or microscopic polymer network structure .the relation between them ( macroscopic pattern and microscopic dynamics / structure ) is an open problem .numerical model may be helpful to understand the physics of this pattern formation .however , it will not be so easy . obviously , diffusion coefficients of monomer , growing polymer blobs , and oxygen , decrease drastically during polymerization ( gelation ) . andfinally they will be almost arrested by the elasticity of gel .it is difficult to model such time dependent diffusion problem .in addition , large amount of memory must be needed to reproduce macroscopic pattern formation from microscopic polymerization dynamics model .macroscopic phenomenological model might be suitable for this type of problem .moreover , macroscopic pattern formation of soft matter might relate to the patterning problem of the bio - materials , such as brains , reptiles skin , and so on .wrinkling on old human s skin is also a kind of pattern formation .the pattern formation reported here is more or less similar to such bio - related pattern formation .but it just looks similar .it never means underlying physics conformity .we can not conclude anything about this similarity at this moment .we have examined the controllability of the spontaneous pattern formation occurring on a gelation surface .a heating wire array was utilized to make external temperature field .pre - gel solution was put on it during gelation . then the ordered structure that aligns to the array direction appeared .this implies that we can control the global structure of the pattern formation .we next varied interval of heating wires and put pre - gel solution on it to investigate the controllability of the length scale ( wavelength and amplitude ) of the pattern formation .these surface patterns were measured by a 3d surface deformation measurement system .that system is composed by a line laser sensor and an automatic x - stage and has  m resolution .  from the data analyses, we found that both of wavelength and amplitude can not be controlled by external temperature field .these length scales seem to be determined by internal parameter such as diffusion length scale .[ 1 ] mm diameter , but the contact is almost linear since the wire surface is rigid . in other words ,the contact is 1-dimensional .the is interval distance between two successive wires .temperature of the wire can be controlled by applying voltage to the wire.,title="fig : " ]  [ 0.7 ] ( distance between wires ) cases respectively , ( a , e ) mm , ( b , f ) mm , ( c , g ) mm , and ( d , h ) mm . in all cases ,wire temperature is set as 30 27-.3em , and initiator amount is 9 mg .scale bars correspond to mm.,title="fig : " ]

Abstract: we report the controllability of a gelation surface pattern formation . recently , we have found and studied a novel kind of pattern formation that occurs during a radical polymerization ( gelation ) process . the pattern formation is observed in an open top boundary of quasi two dimensional gelation . in previous studies , we have used two dimensional photo based image processing to analyze the patterns . however , the actual pattern is a three dimensional surface deformation . thus we develop a three dimensional measurement system using a line laser displacement sensor and an automatic x - stage . patterns measured by the system are analyzed and discussed by means of pattern controllability . in particular , we focus on the possibility of the pattern control using an external temperature field . as a result , we reveal that the global structure can be controlled , whereas the characteristic length scales ( wavelength and amplitude ) are not controllable .   * keywords : * pattern formation , directional control , three dimensional measurement


the globalization of human interactions have created a fertile ground for the fast and broad spread of infectious diseases , potentially leading to worldwide epidemics .we are thus force to understand the spreading of infectious diseases within this global scenario . yet , the study of worldwide epidemics is challenging given the heterogeneity of the populations involved .the first sign of heterogeneity is given by the variability of the reproductive number within or across populations .the reproductive number is defined as the number of secondary cases generated by a primary infected case within a population of susceptible individuals . in the case of sexually transmitted diseasesthe reproductive number is proportional to the rate of sexual partner acquisition and it exhibits wide fluctuations . in network based approachesthe reproductive number is proportional to the node s degree and it exhibits wide fluctuations as well . in the absence of biases among the connections between agents this heterogeneity is completely taken into account by the reproductive number distribution .there are other properties beyond the reproductive number requiring the subdivision of a population in different classes or types .this includes but is not limited to age , geographical location , social status and sexual behavior . in generalthese heterogeneities can not be characterized by a single probability distribution .they require a multi - type approach with probability distributions characterizing each type and a mixing matrix describing the patterns of transmission among them .multi - type models are difficult to deal with and are generally tackled using multi - agent simulations .the advantage of multi - agent simulations is that we can consider several details and study their impact on the spreading dynamics . on the other hand ,given the large number of variables and model parameters it is difficult to understand which are the key variables driving the system s dynamics .therefore , analytical calculations are required to funnel the multi - agent simulations into specific regions of the parameters space .  in this worki study the spreading of infectious diseases on multi - type networks .i take as starting point the static problem formulation developed by newman and the theory of age - dependent multi - type branching process .i develop these mathematical approaches to accommodate some distinctive properties of real networks that have not previously considered . in section [ sec : multi ] i introduce the basic framework . focusing on the population structurei consider the contact - graph characterizing the detailed interactions among agents and , at a metapopulation level , the type - network characterizing the interactions among agent s types . through some simple examplesi illustrate the properties of the mixing matrix and its type - network representation .this section ends defining a branching process modeling a spanning tree from an index agent to all other agents in the contact - graph . in section [ sec : spreading ] i characterize the local spreading dynamics from an agent to its contacts , taking the susceptible , infected , and removed ( sir ) model as a case study . bringing together the underlying network structure and the local transmission dynamics in section [ sec : spreading ] i define a branching process that models the disease spreading dynamics . in section [ sec : iterative ] i extend the iterative approach for a single type to accommodate the particularities of the multi - type case .focusing on the expected behavior , in section [ sec : average ] i obtain general equations determining the progression of the expected number of cumulative and new infections .starting from these equations i analyze some limited cases .first , i derive the final expected outbreak size and , second , i analyze the time progression of the expected outbreak size for the case of a time homogeneous local transmission . in section [ sec : intervention ] i discuss the impact of the population heterogeneity on intervention strategies .i emphasize the role of the characteristic distance between agents to quantify the impact of intervention strategies on small - world populations .i also illustrate interventions targeting specific agent s types using a bipartite population as a case study .finally , in section [ sec : discussion ] i provide an overview of the main results and discuss future directions .consider a population of agents that are susceptible to an infectious disease . by _i mean any entity that could host and transmit the disease .since we are interested on the transmission of infectious diseases among humans an agent is a human in the first place . for vector - borne diseases we could have in addition agents representing the intermediary host while for airborne diseasesan agent could also represent a public place .the agents are assumed to be heterogeneous meaning that there are different agent classes or _types _ according to their pattern of connectivity to other agents and/or to the speed at which they could potentially transmit an infectious disease .for instance , human can be divided according to their age , social status and geographical location .furthermore , in the case of vector- and air - born diseases there is an additional type given by the non - human intermediary .more precisely , let us assume that the agent population is divided in types and there are agents of type , satisfying the normalization condition    note that within this work i use the indexes for the agent s type . in the followingi introduce two representations of the population structure at the agent and type levels , respectively .the contact - graph takes precisely into account who could potentially transmit the disease to whom .more precisely ,  the contact - graph is a labeled graph where vertices represent agents , edges represent the potential disease transmission channels among them , and the vertices are labeled according to the agent s type .[ cg ]  the contact - graph represents the population mixing at the agent s level . since there is a one - to - one relation between vertices and the corresponding agents i use these two terms interchangeable .all the information necessary to characterize a given graph is provided by its adjacency matrix .yet , we should take into account the large size of real populations and their change in time .in general , the only way to achieve such a detailed description relies on agent - based simulations .my scope is to bypass this detailed description and focus on statistical properties that does not depend on the population structure details or their change in time . yet , to achieve that i need to specify the time scale where these statistical properties are measured .excluding the effect of patient isolation or any other intervention , the time scale that matters is the time interval from the infection of an agent to its death or recovery , i.e. the disease life time within an agent .at this point i intentionally exclude the effect of interventions , such as patient isolation , in order to achieve a more general approach .their influence is taken into account when defining the disease spreading dynamics ( see section [ sec : spreading ] ) .it is also worth mentioning that the disease life time is a random variable .therefore , the statistical properties introduced below are the expectation after averaging over the disease life time distribution .the degree of a node is the total number of edges emanating from it regardless the type of the node at the other end .let be probability distribution that a type node has degree and denote by    its mean .note that by allowing to take values larger than one we are already taking into account the existence of concurrency .  to characterize the spreading process it is also relevant to determine the same distribution but for a vertex found and the end of an edge selected at random .this sampling introduces a bias towards nodes with higher degree resulting in the probability distribution    with average excess degree    where the minus one subtracts the edge from where the node was reached .associated with these two probability distributions we introduce the generating functions      from the derivatives of and we obtain the moments of and , respectively . for instance    since the agent population is finite there is a typical distance between every two agents on the contact - graph . social experiments such as the kevin bacon and erds numbers or the milgram experiment reveal that social actors are separated by a small number of acquaintances ( `` small - world '' property ) .this observation is supported by theoretical approaches demonstrating that grows at most as in random graphs .more recently it has been shown that for several real networks actually decreases or remains constant as the network evolve and increases its size .thus , i explicitly take into account that is finite .let us assume that type agents establish connections with other agents at a constant rate and that the disease life time is constant and equal to .in this case we obtain a poisson distribution for the agent s degree    furthermore , , , and .[ poisson1 ]     at the metapopulation level the population structure of the is determined by the mixing patterns among the different agent s types . given a type agent and one of its edges let be the probability that the agent at the other end is of type ( _ mixing matrix _ ) . from the mixing matrix we can construct the type - network characterizing the metapopulation structure .  _type - network : _ in the type - network a node represents a type , an arc is drawn from type to if , and the arc s weights are given by .[ tn ]  note that since may be nonzero the type - network may contain loops .[ typegraph ] shows some simple type - networks .the single - type case is represented by a node with a loop ( fig .[ typegraph]a ) .a bipartite population is represented by two nodes with an incoming and an outgoing arc ( fig .[ typegraph]b ) .this example could model a heterosexual population with no other distinction than gender or a metapopulation given by people and public places .a fully mixed population is represented by a complete network ( fig .[ typegraph]c ) .a less intuitive example is the type - network shown in fig .[ typegraph]d , representing a population divided in two cities and the commuters between them .   given a contact - graph ,let us consider an epidemic outbreak starting from a single agent ( index case ) . in the worth case scenariothe disease propagates to all the agents that could be reached from the index case using the network connections .thus , the outbreak is represented by a spanning or causal tree from the index case to all reachable agents . on this tree, the generation of an agent corresponds with the topological or hopping distance from the index case .this picture motivates the introduction of the following branching process :  multi - type annealed spanning tree ( ast )  consider a labeled contact - graph characterized by and the type - network .the multi - type annealed spanning tree ( ast ) is the branching process satisfying the following properties :  1 .the process start from an index case of type at generation .the index case generates sons with probability distribution .each son is of type with probability .each son at generation generates sons with probability distribution .each son is of type with probability .3 .  a son at generation does not generate new sons .[ ast ]  the term annealed means that we are not analyzing the true ( quenched ) spanning tree on the graph but a branching process with similar statistical properties .this approximation is particularly good if the contact - graph is continuously changing in time albeit the constancy of its statistical properties .a similar mathematical construction has been previously introduced by newman .the main difference here is the explicit consideration of the truncation distance .finally , it is worth noticing that all results derived below are exact for the multi - type ast but an approximation for the original population structure .to proceed further we should specify how the disease is transmitted from an agent to its neighbors in the contact - graph .let be the probability that an infected agent of type infects a susceptible neighbor of type . within this worki assume that if then .indeed , the absence of transmission between two types is taken into account by the corresponding matrix element of . upon infectionwe also need to specify when it takes place . given a type agent ( primary case ) and one of its neighbors of type ( secondary case ) , we define the generation time as the time elapse from the infection of the primary case to the infection of the secondary case provided it happens .i assume that the generation times are independent random variables with the distribution function    parameterized by the type of the primary and secondary cases .[ sir ]  in the sir model agents can be in the three exclusive states susceptible , infected and removed . a _susceptible _ agent is one that have not become infected but it is susceptible to acquire the infectioninfected _ agent is one that have already acquired the disease and can potentially transmit the disease .a _ removed _ agent is one that has been previously infected but it is already excluded from the spreading process . within this work the removal of an agent takes into account intervention strategies resulting in the isolation of infected individuals from the disease transmission chain .the death or `` natural '' recovery of infected agents was already taken into account during the definition of the contact - graph in subsection [ sec : cg ] .consider an agent of type and one of its neighbors of type .let be the infection time of agent by in the absence of agent s removal and let be its distribution function .furthermore , let be the removal time of agent in the presence of agent s removal and let be its distribution function .the probability that agent is infected by agent by time is given by  \ . \label{bab}\ ] ]  from this magnitude we obtain the probability that agent gets infected by agent no matter when    and the distribution of generation times    the sir model could be further generalized taking immunization into account . in this casenon - infected agents are divided into susceptible and immune .if is the probability that a type agent is immune then the probability that agent is infected by agent by time reads  \ . \label{babi}\ ] ]  furthermore , the transmission probability and the generation time distribution are obtained substituting this equation into ( [ rabsir ] ) and ( [ gabsir ] ) , respectively .these examples illustrates how to calculate the transmission probability and the generation time distribution from the standards models characterizing the spreading of infectious diseases .more important , by encapsulating the model details into and we can obtain general results that are independent of these details .later on , we can analyze the particularities of each model .   at this pointthe local spreading dynamics has been completely specified and we can super - impose it on the multi - type ast .multi - type age - dependent ast  the multi - type age - dependent ast is composed of two elements , a multi - type ast [ ast ] and a local spreading dynamics defined by .the global dynamics is then specified by the following rules  1 .the process starts with an infected agent of type while all other agents are susceptible .an infected agent of type infects each of its neighbors of type with probability and generation time distribution .[ adast ]  the age - dependent ast is a generalization of the bellman - harris and crum - mode - jagers multi - type age - dependent branching processes .the key new element is the truncation at a maximum generation , allowing us to consider the small - world property of real networks . in spite of the similaritiesthe mathematical framework i implement deviates substantially from these previous approaches .indeed , i exploit this truncation making a backward iteration from the final generation to the index case .consider a branch of the ast rooted on a type agent , at generation , that was infected at time zero .let be the probability distribution to find infected type agents at time on that branch .in particular is the probability distribution of the total number of infected type agents at time on the whole ast , given the index case was of type .based on the tree structure we can develop an iterative approach to compute recursively .consider a type infected agent at generation of the multi - type age - dependent ast .this agent has degree with probability for and excess degree with probability for .let us index by its neighbors on the next generation , where for , for , and for .then   \nonumber\\ & + & \sum_{k=1}^\infty p^{(a)}_k \sum_{n_1=0}^\infty\dots\sum_{n_k=0}^\infty \delta_{\sum_{\alpha=1}^k n_\alpha + \delta_{ab } , n } \prod_{\alpha=1}^k \sum_{c=1}^m e_{ac } \nonumber\\ & \times & \left [ r_{ac}\int_0^t dg_{ac}(\tau )  p^{(1,c , b)}_{n_\alpha}(t-\tau ) + \delta_{n_\alpha,0 } [ 1- r_{ac}g_{ac}(t ) ] \right ] \label{pn0}\end{aligned}\ ] ]   \nonumber\\ & + & \sum_{k=2}^\infty q^{(a)}_k \sum_{n_1=0}^\infty\dots\sum_{n_{k-1}=0}^\infty \delta_{\sum_{\alpha=1}^{k-1}n_\alpha + \delta_{ab } , n } \prod_{\alpha=1}^{k-1 } \sum_{c=1}^m e_{ac } \nonumber\\ & \times & \left [ r_{ac}\int_0^t dg_{ac}(\tau )  p^{(d+1,c , b)}_{n_\alpha}(t-\tau ) + \delta_{n_\alpha,0 } [ 1- r_{ac}g_{ac}(t)]\right ] \label{pnd}\end{aligned}\ ] ]    let be the number of infected type agents on a branch rooted at type agent , and let be the infected type agents on the branches rooted at each of its neighbors . then    where takes into account if the root agent is or it is not of type .the probability distribution of is given by the sum of all the possible combinations of the random variables that satisfy ( [ ninj ] ) .now , the root agent and its neighbors lie on a tree and therefore are independent random variables . furthermore , all agents at generation has the same statistical properties , i.e. are identically distributed random variables .therefore , the probability of each combination is decomposed into the product of the probability distribution of the number of infected agents of type on the sub - branches rooted at each neighbor .thus , taking into account that each neighbors is of type with probability we obtain   \nonumber\\ & + & \sum_{k=1}^\infty p^{(a)}_k \sum_{n_1=0}^\infty\dots\sum_{n_k=0}^\infty \delta_{\sum_{\alpha=1}^kn_\alpha + \delta_{ab } , n } \prod_{\alpha=1}^k \sum_{c=1}^m e_{ac } q_{n_\alpha}^{(d+1,a , c , b)}(t)\ , \label{pn0q}\end{aligned}\ ] ]   \nonumber\\ & + & \sum_{k=2}^\infty q^{(a)}_k \sum_{n_1=0}^\infty\dots\sum_{n_{k-1}=0}^\infty \delta_{\sum_{\alpha=1}^{k-1 } n_\alpha + \delta_{ab } , n } \prod_{\alpha=1}^{k-1 } \sum_{c=1}^m e_{ac } q_{n_\alpha}^{(d+1,a , c , b)}(t)\ , \label{pndq}\end{aligned}\ ] ]  where is the probability distribution of which we proceed to calculate .  let us focus on one neighbor and let us assume that it is of type . with probability this agent is not infected at any time and with probability  ] and . then    where is independent of .the conditions of the perron - frobenius theorem are valid beyond positive defined matrices and holds for the mixing matrix representing a strongly connected network .thus , the largest eigenvalue of is simple and all the entries of its corresponding eigenvector are different from zero and have the same sign .in particular we choose all the components of to be positive . based on this fact we can write ( [ cij ] ). there may be , however , some entries of and thus of ( [ aij ] ) and that are zero .indeed we can rewrite ( [ cij ] ) as    thus whenever .otherwise , we obtain the inequalities    where      from ( [ nijfm ] ) and ( [ noo ] ) we obtain    from this equation we obtain ( [ nijsc ] ) with    where the inequality follows from ( [ aijminsc ] ) .  .the region is indicated by the dotted pattern.,height=288 ]  figure [ fig : noo ] illustrates the predictions of theorem [ nijfm ] for complete type - networks .when the expected outbreak size is of the order of the prefactor which is not expected to be large .different behaviors are observed , however , for depending on . for is a dramatic increase in the expected outbreak size .as soon as a significant fraction of the agent population becomes affected .in contrast , when is not so large it becomes clear that the expected outbreak size changes smoothly with increasing , including the region around .this fact becomes relevant when analyzing the impact of intervention strategies ( see section [ sec : intervention ] ) .finally , it is worth mentioning that a similar picture is obtained for the more general case of strongly - connected type - networks , albeit some corrections given by the missing terms the sum in ( [ nijsc ] ) .now let us consider the particular case where the spreading dynamics is homogeneous , i.e. .in this case , from ( [ iijt ] ) we obtain the incidence    in particular when can be diagonalized we rewrite ( [ iijtht ] ) as    where is a time dependent diagonal matrix with diagonal entries    [ sec : twocities ]  consider the case with the reproductive number matrices  \ , \ \ \ \\tilde{r } = \left [ \begin{matrix } k_1 & k_2\\ k_2 & k_1\\ \end{matrix } \right ] \label{trtc}\ ] ]  since is symmetric it can be diagonalized and , where is the transpose of . in this case with  \ , \ \ \ \  p = \frac{1}{\sqrt{2}}\left [ \begin{matrix } 1 & 1\\ 1 & -1\\ \end{matrix } \right ] \label{ptc}\ ] ]  where    are the eigenvalues of . assuming an index case is of type from ( [ it d ] ) we finally obtain      this example shows that in some cases we can exactly calculate the expected progression of an epidemic outbreak .more generally we obtain the following asymptotic behaviors .[ ntsc ]  consider a strongly connected type network and a homogeneous and exponential distribution of generation times , where is the transmission rate .let be the largest eigenvalue of ( [ bij ] ) and let     : if and then     : if then  \ , \label{iijthtfc}\ ] ]  where is the same as in theorem [ fsc ] . : following the same guidelines of the theorem [ fsc ] proof we arrive to the inequality    where    the laplace transform of is given by    when this series converges only for .therefore , when . : the demonstration of this case is straightforward . from theorem [ fsc ]it follows that is of order for .therefore , for the sum in ( [ iijtht ] ) is dominated by the term .corrections are given by the ratio between the and the preceeding term satisfying , which is at most .the case provides the connection between this work and multi - type age - dependent branching processes with an infinite number of generations .indeed , mode have already demonstrated the exponential growth regime for the case ( see , chapter 3 ) .theorem [ ntsc ] shows that on the other limit the spreading dynamics is instead characterized by a gamma distribution , which is also the case for the single - type case .   for different values of .( c ) number of secondary cases generated by a primary case for a sars outbreak in singapore , as reported in ( bars ) .the solid line is the best fit to the gamma probability density function times a pre - factor , resulting in .( d ) plot of the parameter dividing the exponential and power law initial growth regimes as a function of the number of intermediary steps.,width=480 ]  theorem [ ntsc ] can be extended to consider other generation time distributions , such as a gamma distribution    where .the gamma distribution can be interpreted as the existence of intermediary steps before an agent becomes infectious ( see fig .[ fig : gtau]a , b ) . for we recover the exponential distribution which corresponds with the absence of intermediary steps .the gamma distribution can be also obtained from the fit to some empirical distribution of generation times ( see fig . [fig : gtau]c ) .  in this casethere are two important modifications to theorem [ ntsc ] .first , the parameter is now given by   ^{\frac{1}{\alpha}}}{\rho}\ , \label{thetaalpha}\ ] ]  which increases approximately linearly with increasing ( see fig . [fig : gtau]d ) .second , in the regime although the fraction of infected agents is still given by a gamma distribution the exponent of the initial power law growth is given by , i.e.    therefore , the existence of intermediary steps reduces the the small - world effect by a factor given by the number of intermediary steps .for instance , by a factor of about four for sars ( fig .[ fig : gtau]b ) .the expected outbreak size is a monotonic increasing function of ( [ nijsc ] ) , which plays the role of the basic reproductive number in homogeneous populations .therefore , the aim of intervention strategies is to reduce the characteristic reproductive number . on the other hand ,intervention strategies implies an economical cost , including but not limited to the development of new vaccines and their deployment through vaccination campaigns .our task is to design optimal intervention strategies that minimize the expected outbreak size with a feasible economical cost .  to be more preciselet us consider a scenario where the disease is transmitted at constant rate from infected to susceptible agents , infected agents are isolated at a rate and a fraction of the population is immune to the disease . in this casethe infection and removal times follows the exponential distribution functions and , respectively .thus , from ( [ rabsir ] ) we obtain where    is the blocking fraction , i.e. the fraction of potential disease transmissions that are blocked either because of immunization or patient isolation .since is independent of the primary and secondary case types we can write the reproductive number matrices ( [ aij ] ) and ( [ bij ] ) as and , respectively , where    in turn , the largest eigenvalue of is given by    where is the largest eigenvalue of .  from the analysis made in section [ sec : finalsize ]it follows that there are two different scenarios depending . for simplicitylet us focus on the complete type - network case where .when the target of intervention strategies is , which is the consensus in the literature . the blocking fraction to achieve thisis obtained from ( [ lk ] ) , resulting in    this result has been already reported , at least for the case of two types . when is small , however , the expected outbreak size is a smooth function of ( see fig . [fig : noo ] ) .therefore , does not represent a threshold value in small - world populations .so far we have considered homogenous intervention strategies .now let us assume that the rate of patient isolation and the immunized fraction are now different for each agent s type and given by and , respectively . in this casethe blocking fraction is given by    and , which depends on the type of both the primary and secondary case .from the perron - frobenius theorem it follows that is a continuous increasing function of all the entries of the corresponding matrix  . since then is a continuous decreasing function of for all .the goal is to determine which strategy leads to the largest reduction of .consider the spread of hiv on an heterosexual population with no further distinction beyond gender . in this casethe type - network is bipartite ( see fig .[ typegraph]b ) .let and be the average excess degree for the connections from women to men and biceversa .let also assume that the rate of patient isolation is zero and that we could immunize a fraction of the overall population , distributed between a fraction and of immunized women and men , respectively .the question is to determine the value of representing the best intervention strategy . in this casethe reproductive number matrix is given by  \\ ( 1-xs)k_1 & 0 \end{matrix } \right ] \label{trb}\ ] ]  and it has the largest eigenvalue   k_1 k_2}\ . \label{lb}\ ] ]  it results that is minimum for or , i.e. the best intervention strategy is to direct all the immunization resources to only one of the sub - populations .there is significant evidence that social networks are characterized by ( _ i _ ) wide connectivity fluctuations and ( _ ii _ ) the small - world property .the variability in the number of contacts ( _ i _ ) has a direct impact on the reproductive number .this fact has been taken into account since the seminal works of may and anderson considering the variability in the rate of sexual partner acquisition .more recently it has gained attention for other infectious diseases as well , following the observation of super - spreading events in the 2002 - 2003 sars epidemics . yet , the small - world property ( _ ii _ ) has been completely neglected .from my studies of the single type case i have shown that intervention strategies are modulated by the average distance between agents in the corresponding contact - graph . in this worki have demonstrated that this result is also valid for heterogeneous populations . in this last casethe characteristic reproductive number is given by the largest eigenvalue of the reproductive number matrix .the good news is that in spite of this modulation by the target of intervention strategies is still the characteristic reproductive number .that is , the expected outbreak size still decreases with decreasing the characteristic reproductive number .the bad news is that to quantify the impact of the intervention strategies we need to estimate .there are different paths to estimate .first , we can use a direct approach as the milgram s experiments .second , we can measure other network properties such as the degree distribution and then try to estimate using network models .finally , i have shown that the progression of the expected number of new infections is modulated by ( see and section [ sec : spreading ] ) .more precissely , in small world populations the incidence is expected to grow as a power law and we can estimate from the power law exponent .further work is required to test the validity of the coarse grained description of the type - network approach .this can be done by running agent based simulations where we can have a strict control of the different statistical properties characterizing the population structure .these statistical properties can be then plug in into the type network approach to obtain qualitative and quantitative predictions that can be compared with the simulations results .  in conclusion , this work opens new avenues to future research on the spreading of infectious diseases on heterogeneous populations .it allows for a qualitative understanding through the analysis of the type - network representation of the mixing matrix .more important , it leads to general results that can be tackled case by case using exact or approximate calculations and numerical computations .

Abstract: i study the spreading of infectious diseases on heterogeneous populations . i represent the population structure by a contact - graph where vertices represent agents and edges represent disease transmission channels among them . the population heterogeneity is taken into account by the agent s subdivision in types and the mixing matrix among them . i introduce a type - network representation for the mixing matrix allowing an intuitive understanding of the mixing patterns and the analytical calculations . using an iterative approach i obtain recursive equations for the probability distribution of the outbreak size as a function of time . i demonstrate that the expected outbreak size and its progression in time are determined by the largest eigenvalue of the reproductive number matrix and the characteristic distance between agents on the contact - graph . finally , i discuss the impact of intervention strategies to halt epidemic outbreaks . this work provides both a qualitative understanding and tools to obtain quantitative predictions for the spreading dynamics on heterogeneous populations .


the prediction of financial markets has long been the object of keen interest among both financial professionals and academics .the widely , - if not universally - , accepted , provides a powerful argument that markets are inherently unpredictable , in particular on the basis of prior price data : because all information about the future is incorporated into the current price ( for all practical purposes immediately ) , price changes must follow a random walk .there is considerable evidence however that prices do not perfectly follow a random walk and that some price inefficiency is present , varying over time , perhaps enough at times to be exploitable .however , recent assessments of the performance of hedge - funds and of mutual funds cast doubt on the reality of the gains resulting from the practical implementation of these inefficiencies , if they exist . as illustrated in the approaches of and , deviations from the emh are searched in the form of anomalous performance , beyond what can be explained by risk premia associated with exposures to a few dominating risk factors .the near - absence of predictability in financial markets , or more precisely of risk - adjusted arbitrage opportunities , is truly remarkable .a rich academic literature has clarified the zen - like nature of the in the sense that , the more intelligent are the investors and the harder are their efforts to gather information to make the best possible investment decisions , the fewer trading opportunities there are , and the more efficient is the market .the fact that markets are close to efficient can thus be understood as a macroscopic organization that result from the collective actions of the active investors .borrowing from the jargon of complex system theory , market efficiency is an emergent phenomenon .emergence , the existence of qualitatively new properties exhibited by collections of interacting individuals , is often taken to be the defining characteristic of complex adaptive systems .reciprocally , we ask here how the observation of the large scale behavior of a macroscopic system can ( i ) uncover the internal properties of a system and the organization among its constituents and ( ii ) be used for its prediction .following richard feynman , we argue that , in order to really understand a system , we need to be able to strip things down , then rebuild them in order to play with the reconstructed simplified system and analyze variants , from which understanding can emerge .we address this question of `` reverse engineering '' in the context of one - dimensional financial ( market ) time - series .the challenge consists in building a virtual stock market with artificial interacting software investors .the method presumes that real - world discrete market price changes may be in principle modeled as the aggregated output of a large number of interacting boundedly rational agents .these agents have limited knowledge of the detailed properties of the markets they participate in and create , have access to a finite set of strategies to take only a small number of actions at each time - step and have restricted adaptation abilities . giventhe time series data , our method of reverse engineering determines what set of agents , with which parameters and strategies , optimizes ( in the sense of various robust metrics ) the similarity between the actual data and that generated by an ensemble of virtual stock markets peopled by software investors .we provide a validation step by testing the performance of the reverse engineered artificial market in predicting out - of - sample directional moves of the real - world time series . using only some of the simplest strategies and agents, the -value for the statistical significance of the prediction of the directional moves for more than 600 trading days of the nasdaq composite index is smaller than .the results are robust with changes of the styles of agents strategies and for different market regimes .our work uses the extensive literature on agent - based models that has been developing at least since the 1960s ( see and references therein ) . in ,a system is modeled as a collection of autonomous decision - making entities , called agents .repetitive competitive interactions among agents generate complex behavioral patterns . due to the evolutionary switching among strategies , are highly nonlinear . the aggregation of simple interactions at the micro level may generate sophisticated structures at the macro level which provide valuable information about the dynamics of the real - world system which the emulates .the main benefits of are that they ( i ) capture emergent phenomena ; ( ii ) provide a natural description of a system ; ( iii ) are flexible .have already been successfully applied in real - world problems , such as , flow simulation , organizational simulation , diffusion simulation and market simulation . in this articlewe focus on financial market simulation . shows that can explain the main statistical regularities observed in financial time series - their so - called `` stylized facts '' - such as excess volatility and volatility clustering , high trading volume , temporary bubbles and trend following , sudden crashes and mean reversion , and fat tails in the distribution of returns .toy models such as the , described in detail in , capture key features of one generic market mechanism ( competition for a scarce resource ) .the basic interaction between agents and public information is described in .details of the abms we employ will be introduced as we describe the implementation of our reverse engineering process . in brief , we concentrate on the so - called and its key variants and on the so - called -game introduced by .4 .  * * .this game is the analog of the , except for the minority payoff , whereby each agent is rewarded according to how the return at the next time step is compared with her decision taken at the previous time step . in other words ,the delgcmg is a delayed .5 .  * * . in the version of the mixg used here, we consider a mix of agents , with 50% of the agents obeying the rules of the and the other 50% obeying the rules of the .the which best reproduces the external time series provides the solution to our reverse engineering problem .this is determined from a search in the space of parameters of the abm using a as shown in algorithm [ alg : sga ] .first a population of is initialized , whereby the number of agents , the number of strategies an agent obtains , the size of her memory , and her threshold are constant in the current version of the .the only aspect in which the differ is the which is the crucial parameter set over which we optimize the fit to an external time series .for the first generation , the are initialized randomly .then for every , its fitness - reflecting how well the time series generated by the matches the external time series - is determined .this value is computed via a fitness function using different metrics , such as the and -norms , the hamming distances ( with binary and ternary coding ) between the two time series .are selected to produce offspring according to their fitness , with the fittest yielding more offspring .each new generation of is obtained as a mixture of the agents and of the strategies of the previous parent generation .many generations evolve until a convergence criterion is reached , which leads us to finally identify a which best represents the external time series within the in - sample period .the search is performed ten times to obtain ten .the differences between these ten solutions provide a measure of the quality of the reverse engineering method .the ten are also used to quantify the uncertainty in the next - day out - of - sample prediction .  .in the figure , the actual data ( nasdaq return illustrated as red crosses ) and that generated by the reconstructed virtual stock market ( the best 10 - in this sample consisting of agents - illustrated as gray dots and their average in blue ) are plotted , whereby the vertical line separates the in - sample period during which their similarity is optimized ( here 25 days ) from the out - of - sample period ( one - step prediction ) . ]figure [ fig:3rdpgvsextreturn ] shows the excess demand obtained from the aggregate decisions of all the agents of one selected best for a given time window of the nasdaq composite index .the data point to the right of the vertical line is the next - step , out - of - sample , prediction , whereas all points to the left of it belong to the in - sample period during which the has been trained on the external time series and has been optimized in terms of its .every run of the results in one best - according to its fitness - which then can be used to predict the next day return .  for each time window of the nasdaq composite index ,we obtain the best for each of the five types of defined in subsection [ sec : abmtypes ] , using the above .this provides us with five different `` lenses '' to examine the nasdaq composite , that reveal its different characteristics .in order to test the predictive value of the reverse engineered for each of the five , we report the success rate , that is , the fraction of days out - of - sample for which the predicted and realized returns have the same sign .  in order to assess the statistical significance of the obtained success rates , we compare them to those of 1000 _ random strategies _ , obtained by predicting with equal probability the rise or decline of each next - day market price . using random strategies has been shown to provide the most robust estimations of the statistical significance of strategies in the presence of biases and trends . in table[ tab : accrates ] , we report the -value of the for each , calculated as the fraction of random strategies that perform better .table [ tab : accrates ] reports the success rates and their corresponding p - values for each type of averaged over ( i ) all 606 days , ( ii ) for the trending periods ( 202 days ) and ( iii ) for the non - trending periods ( 404 days ) . in the second column , the success rates are averaged over all parameter sets of the .the third and fourth column report the minimum and maximum success rates over the parameter sets of the .+ gcmg & ( 0.01 ) 0.55 & 0.51 & 0.60 + gcmjg & ( 0.00 ) 0.57 & 0.54 & 0.60 + delgcmjg & ( 0.00 ) 0.57 & 0.54 & 0.59 + delgcmg & ( 0.02 ) 0.54 & 0.51 & 0.57 + mixg & ( 0.00 ) 0.56 & 0.53 & 0.58 +   + gcmg & ( 0.02 ) 0.57 & 0.54 & 0.63 + gcmjg & ( 0.00 ) 0.66 & 0.63 & 0.68 + delgcmjg & ( 0.00 ) 0.67 & 0.64 & 0.70 + delgcmg & ( 0.01 ) 0.58 & 0.55 & 0.61 + mixg & ( 0.00 ) 0.67 & 0.62 & 0.68 +   + gcmg & ( 0.07 ) 0.53 & 0.49 & 0.58 + gcmjg & ( 0.13 ) 0.53 & 0.49 & 0.55 + delgcmjg & ( 0.24 ) 0.52 & 0.49 & 0.54 + delgcmg & ( 0.15 ) 0.52 & 0.50 & 0.55 + mixg & ( 0.33 ) 0.51 & 0.48 & 0.54 +  over all days independently of the presence or absence of trends , the success rates of the reverse engineered are superior to all random strategies ( ) for the , , and . for the and , the results are still very significant with -values given respectively by and .  decomposing the 606 test days into trending and non - trending periods , we find that the success rates are very significant for the former periods and less so for the later periods .the reverse engineering procedure is thus a good trend detection method .  while it is expected that the , , and would perform in trending period due to their majority incentive, it is a priori quite surprising that the and also perform very significantly .we interpret this result as follows .first , the reverse engineering process applied with the , , and selects the trend - following strategies which , when used by a majority of agents , allow a good fit to the trend .second , the fact that and also perform well in trending periods implies that these trending periods are not just simple trends , but are decorated with cycles or alternating correction phases that the minority mechanism is able to pick up .  in contrast , all show a strong drop in performance in the non - trending periods , with the best performing game being the .this later result can be rationalized by the minority incentive of this game , which is known to lead to oscillation prices resulting from the frustration inherent to the minority payoff .in conclusion , we have shown that reverse engineering a real financial time series with simple selected by using a genetic algorithm might be possible and provide novel insight in the properties of financial time series .notwithstanding the simplicity ( some would say `` naivety '' ) of the used , the aggregation of simple interactions at the micro level is sufficient to generate sophisticated structures at the macro level , which is probably the explanation for the good performance obtained in the validation step .* acknowledgements * : we acknowledge financial support from the eth competence center coping with crises in complex socio - 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Abstract: using virtual stock markets with artificial interacting software investors , aka , we present a method to reverse engineer real - world financial time series . we model financial markets as made of a large number of interacting boundedly rational agents . by optimizing the similarity between the actual data and that generated by the reconstructed virtual stock market , we obtain parameters and strategies , which reveal some of the inner workings of the target stock market . we validate our approach by out - of - sample predictions of directional moves of the nasdaq composite index .   `` what i can not create , i can not understand '' : on physicist richard feynman s blackboard at time of death in 1988 ; as quoted in the universe in a nutshell by stephen hawking .