in recent years , the advent of various kinds of social networks , high - definition video streaming , and ubiquitous cloud storage entails large - scale storage in communication networks .cloud storage systems are able to meet the demand on large - scale storage capacity only with limited storage capacity of each storage node . moreover , the cloud storage systems improve reliability of data storage and recovery since they are robust to failures of individual storage nodes to a certain degree. the robustness also makes repair and maintenance easy when an appropriate network coding technique is adopted .  theoretically , successful recovery in a cloud storage system is possible if and only if the corresponding max - flow or min - cut from the storage nodes is greater than or equal to the size of the original data object . to implement this feature , given two positive integers and , a maximum distance separable ( mds ) code can be used to encode and store the original data into storage nodes such that recovery of the original data is possible with out of nodes . for practical implementations , erasure coding is known to be more reliable than duplication of the file .reed - solomon code is the most popular one for practical implementation of mds codes and reed - solomon code with information dispersal algorithm ( ida ) for distributed storage was investigated in .fountain codes and low - density parity - check ( ldpc ) are also known to have approximate mds properties .especially , raptor code , the first known class of fountain codes as well as online codes , is another example of rateless erasure codes and provides linear time complexity of encoding and decoding .when a storage node fails , the code repairing problem in a distributed storage system is addressed in where the code repairing techniques are categorized into exact repair , functional repair , and exact repair of systematic parts .the blocks newly reconstructed by the functional repair preserve the mds property and enable data recovery , but they are not the same as the original blocks . on the contrary , in the exact repair , the failed blocks are exactly reconstructed .the exact repair of systematic parts is a hybrid repair model standing between the functional repair and the exact repair .tthe functional repair problem in distributed storage systems was studied in , interpreting the problem as a multicasting problem over an information flow graph . for the exact repair ,it was shown in that the optimal minimum bandwidth regenerating ( mbr ) code can be found for , where and are the number of surviving nodes and the number of storage nodes , respectively .for the exact repair , the exact minimum storage regenerating ( msr ) code based on interference alignment was proposed in , when and where is the minimum number of nodes required for data recovery .the capacity of multicast networks with network coding was given in the pioneering work of ahlswede et al .it was also shown in that the random linear network coding over a sufficiently large finite field asymptotically achieved the multicast capacity .for distributed storage , network coding was introduced in for wireless sensor network . in ,decentralized erasure codes inspired by network coding on random bipartite graphs were proposed for distributed multiple sources and their applications to sensor networks were presented .another linear technique to increases data persistence in wireless sensor networks was proposed and compared to other codes when the positions and topology of nodes were unknown in .pyramid codes for flexibly exploiting the tradeoffs between total storage space and access efficiency in a distributed storage system was investigated in . in , partial network coding ( pnc )generalizing network coding was investigated for data collection in distributed sensor networks . for a joint storage and transmission problem, showed that a linear coding strategy with file splitting ( instead of coding ) achieved optimality in total cost including the individual cost of updating , storing and retrieving .other key issues on network codes for distributed storage can be referred to .  for an efficient utilization of limited storage capacity , resource allocation in distributed storage systems has been actively explored .a storage allocation problem under a constraint of total storage capacity was studied in , where the recovery probability at the data collector was analyzed when the link connections from each node to the data collector are modeled as independent and identically distributed ( i.i.d . )bernoulli random variables with parameter .it was found in that the maximal symmetric allocation that equally distributes the total storage capacity to the storage nodes is optimal if the total storage budget is large enough .the gap between the maximal symmetric allocation and the optimal solution vanishes as the total number of storage nodes grows , when , where is the normalized total storage capacity .if the total storage budget is small , the minimal allocation was shown to be optimal , where the total storage budget is distributed to the minimized number of storage nodes only . in ,these results were extended to a distributed storage system with heterogeneous links where the connection probability from node to the data collector is .most of the previous papers on network coding in distributed storage systems rely mainly on simple graph networks with reliable links .system design and performance analysis of distributed storage systems in fading channels is crucial .for example , diversity gains by multiple antennas or multiple nodes in distributed storage systems are required to be properly analyzed and evaluated in fading channels .however , there have been very few studies on distributed storage allocation with non - reliable fading links so far .although some works , such as , tried to take account of unreliable links with bernoulli random variables , they failed to exactly account for the key features of wireless links such as channel fading .for example , diversity gains by multiple antennas or cooperative nodes in distributed storage systems are required to be properly analyzed and evaluated in fading channels . despite the importance of system design and performance analysis of distributed storage systems in fading channels , there have been few studies on distributed storage allocation with non - reliable fading links so far .in this context , we consider a wireless cloud storage system where data storage and recovery are carried out through wireless fading channels .a personal cloud storage system might be a good application of our system model .contrary to public cloud storage on internet like dropbox or google drive , users directly access nearby wireless storage to store or retrieve their data in a personal cloud storage system . apparently , distributed storage techniques and resource allocation under limited storage space are key techniques enabling personal wireless cloud storage systems . in this regard ,the problem of storing and retrieving a file over multiple storage devices under limited sum storage capacity is also a fundamental issue to be addressed , from which we can identify the optimal file portions to store at each storage device . in system design or deployment ,the problem reveals the optimal storage size at each storage device under a limited budget of total available storage space .a cloud edge computing system can also be an extended application .cloud edge computing includes the processes of distributing input data generated from a user to nearby computing entities ( or devices ) and delivering output data from the computing entities to the user .the processes are basically identical with the processes of storing and retrieving data over distributed storage devices .therefore , our system model can be used as a framework for wireless cloud edge computing system .  in our system model, the recovery failure event occurs either when a data object is not correctly stored at the distributed storage nodes or when the data recovery at the collector fails .we analyze and characterize recovery failure probability in an asymptotic sense for high snr regime and low snr regime , and quantify the effect of limited storage capacity on the asymptotic recovery failure probability in high snr regime .the contributions of this paper are summarized as follows .* we find the _ asymptotically _ optimal storage allocation under a constraint of total storage capacity for high snr regime and low snr regime . * using exponential equality analysis, we show that the optimal allocation strategy for high snr regime is maximal symmetric spreading of the total storage budget across the distributed storage nodes . *we show that for low snr regime , the minimal allocation with complete and 1 incomplete storage nodes is optimal . in the minimal allocation , storage budget , which is the remaining storage budget after allocating budget to storage nodes ,is not required to be allocated . * in low snr regime ,the storing phase becomes the performance bottleneck and thus the recovery failure probability mainly depends on the cardinality of the decoding set of which elements are the storage nodes that have stored the object successfully .* based on numerical investigation , we show that in intermediate snr , a balance allocation between the minimal allocation and the maximal symmetric allocation would not be necessary if we properly switch them according to snr .the rest of the paper is organized as follows . in sectionii , we present our system model and notations about the exponential equality .section iii describes how the wireless cloud storage system operate .the optimal allocation for asymptotic snr region is investigated and analyzed in section iv . in sectionv , we present numerical results .finally , we conclude this paper in section vi .]  as shown in fig .[ fig : system ] , the number of storage nodes is set to .data source , data collector , and storage nodes are denoted by , and , respectively .the collector node recovers the stored data from the storage nodes .if we consider the scenario when a mobile user stores its data on cloud storage , the collector node will be the same as the data source , so the direct link from the source node to the data collector does not exist . the channel gain from node to node is denoted by and follows the complex gaussian distribution with zero mean and unit variance , i.e. , . for simplicity , we assume all channels independent and identically distributed ( i.i.d . ) and the path loss are ignored , but the result of this paper can be extended to a general case with ease . to facilitate tractable analysis without losing key insights on system design , we assume i.i.d. channels . moreover ,if storage nodes are not close one another in a rich scattering environment , the independent channel model can be effective.the analytic framework developed in i.i.d .channels will be also useful in non i.i.d . channels . for filestoring and recovery , a file is divided into many data blocks and the storing and recovery processes are carried out for each data block . because mobility of both storage devices and user is limited in personal cloud storage systems , the doppler spread is not likely to be large .for example , in low mobility environments ( i.e , moving speed is less than 5 m / sec ) , the coherent time becomes approximately 20 ms with 2.4ghz carrier frequency . on the other hand ,symbol duration is on the order of tens of microseconds and the supported data rate reaches more than 1 gbps in current wireless communication protocols such as wi - fi direct .consequently , each data block size can be as large as up to 2.5 mbytes under the coherent time requirement .this block size is large enough to implement ideal mds coding and given the fact that storage devices like hdd or ssd use block size of 4 kb 2,048 kb , the requirement of coherence time can be readily met for each data block .therefore , we assume that the coherence time is longer than each storage period and recovery period for each data block and thus the channel gains do not change during each period but the channel gains independently change between the periods .  an additive white gaussian noise ( awgn ) is denoted by and follows the complex gaussian distribution with zero mean and variance , i.e. , .the average signal - to - noise ratio ( snr ) of a link is denoted as and the amount of coded data stored in storage node is denoted as .  in our model, each storage node does not have an individual restriction on the storage size but the maximum amount of data stored at each storage node is because the total storage budget is limited to such that .we assume that the size of data object is normalized to be unit compared to the total storage capacity as in other literature , which simplifies design and analysis of distributed storage systems .a data object corresponds to a divided data block and the recovery probability to be analyzed is for each data block , not for the whole file .total storage constraint and allocated budget is also identically used for a data block consisting the file because they are already normalized values compared to the original data . in other words , can also denotes the amount of mds coded fraction stored in storage node .   as shown in fig .[ fig : system ] , the storage and recovery operation is constituted by two basic phases .the first one is for storing data object in the distributed storage nodes and the second one is for recovering the stored data .these two phases are decoupled in time since the stored data are recovered later although the first phase affects the data recovery in the second phase .that is , the storage phase and the recovery phase are not concurrently entered into .  1 .storage phase : the source node broadcasts a data object to the cloud storage nodes to store the data object during a given time period .if a storage node has successfully decoded the broadcast data object from the source , the storage node converts the decoded data object into suitable mds coded blocks as much as its allocated storage size , and stores it .storage node successfully decodes the data object from the source node only when the instantaneous mutual information between the source node and storage node is greater than or equal to the required rate of the data object , i.e. , the value of threshold depends on the adopted modulation and coding scheme ( mcs ) and bandwidth for data transmission .for example , a typical range of required snr in 802.11n wi - fi is around from 10 to 30 db .taking account of specific system parameters of the adopted wireless communication protocol , the threshold corresponding to a required snr value can be computed .the amount of mds coded data stored at node is equal to its allocated storage size , where is the decoding set whose elements are the storage nodes which have successfully decoded the broadcast data object . if , the data object is not properly stored because the data object can not be properly recovered from the stored mds coded data . on the other hand , if , the collector has a chance to properly recover the data object from the stored mds coded data , depending on channel conditions from the storage nodes to the collector .note that instead of mds coding , random linear coding over a sufficiently large field can be used for data object recovery .recovery phase : when the data collector wants to recover the data object , it requests the storage nodes in the decoding set to send the mds coded blocks of the stored data . the storage phase and the recovery phase are assumed to have the same time length for simplicity .let be the index of the storage node in the decoding set .they are indexed in descending order of the channel power gain from the storage node to the collector such that .then , storage node transmits during fraction of the time period allocated for the recovery phase , where varies with the cardinality of the decoding set and has to satisfy and which is a constraint from the property of mds code .the data collector is assumed to know the index and the stored data size of each storage node by proper signaling with the storage nodes . with the help of pilot symbols from each storage node in the decoding set , the data collector estimates the channel gains from the storage node . to focus on developing the optimal distributed caching strategy , we assume perfect channel state information at the data collector and discard the signaling overhead .based on the channel gains , the data collector determines the time fractions , , by solving the following optimization problem : the optimized value of is given by \nonumber\\ & + \left(1-\sigma_{\hat{j}=1}^{i-1}a_{\hat{j}}\right)\cdot\textbf{1}\left[\sigma_{\hat{j}=1}^{i}a_{\hat{j}}>1 \text { and } \sigma_{\hat{j}=1}^{i-1}a_{\hat{j}}\leq1 \right]\end{aligned}\ ] ] where is the indicator function which returns 1 if the argument is true or 0 otherwise , which suggests that longer transmission time is allocated to the storage node with a stronger channel gain under the storage constraint and the transmit time constraint . + according to the determined , the data collector receives the stored mds coded block from storage node during the time portion. then , it moves on to storage node and receives the stored mds coded block during . in this way, the data collector receives the stored mds coded date from the storage nodes in the decoding set .the data object is successfully recovered if otherwise , recovery of the stored data object fails .the exponential equality is denoted as the symbol , i.e. , , when where is called the _ exponential order _ of .the exponential inequalities denoted by and are similarly defined . is used to denote . is the set of real -tuples , while denotes the set of nonnegative real -tuples .for any set , the intersection of the set and is denoted by , i.e. , .assume that is a gaussian random variable with zero mean and unit variance .then , the asymptotic probability density function ( pdf ) of the exponential order of denoted by is obtained as by limiting to infinity , the pdf in is given by thus , for independent random variables distributed identically to , the probability that belongs to set can be characterized by provided that should be non - empty .in other words , the exponential order of depends on only .the list of symbols used in the paper is given in table [ table : syms ] ..list of symbols and its descriptions [ cols="^,<",options="header " , ]   [ table : syms ]for the system model described in the previous section , the recovery failure probability ( i.e. , the complimentary recovery probability ) is hard to obtain in closed form , as noted in , even if interesting wireless ingredients are not incorporated . in this section , we instead explore the optimal storage allocation in an asymptotic sense .exponential order determines the decreasing speed of recovery errors in high snr regime .consequently , large exponential order offers low recovery failure probability if snr is sufficiently high . on the other hand ,the recovery failure probability is not characterized well by exponential order in low snr regime , so the dominant order of recovery probability helps understand recovery performance in low snr regime .we analyze the recovery failure probability for high snr regime in this subsection to understand its asymptotic behavior .that is , we derive the exponential order of the recovery failure probability and find the optimal storage allocation to maximize the exponential order .the exponential order characterizes the decreasing tendency of the recovery failure probability versus snr , and is interpreted as diversity order if bit error probability or outage probability is considered in conventional wireless communication systems .contrary to the conventional diversity order , the exponential order of the recovery failure probability is determined by not only the number of independent fading paths for data storage and recovery but also the limited total storage capacity .  forgiven storage allocation , the exponential order of the recovery failure probability is lower and upper bounded , respectively , by \right)\label{thm : d_order1}\\  \text { and } & d(\mathbf{a } ) \leq \min_{\mathcal{d}\subseteq\{1,\ldots , k\}}\left ( k-|\mathcal{d}|+|\mathcal{d}|\cdot\mathbf{1}[\sum_{i\in\mathcal{d}}a_i\geq1]\right).\label{thm : d_order2 }  \end{aligned}\ ] ] [ lem : d_order ]  refer to appendix [ app:1 ] .  under a total storage capacity constraint ,the optimal storage allocation in terms of exponential order is to maximally and symmetrically allocate the sum storage capacity across all storage nodes .[ thm : allo ]  our optimization problem is formulated as for an arbitrary storage allocation , the upper bound of in is determined by the maximum value of the cardinality of a decoding set satisfying .let this decoding set be .obviously , is a set of the nodes whose allocated storage sizes are the smallest ones , i.e. , where denotes the -th smallest storage size in , and is determined such that .  to maximize the upper bound of , we have to find an allocation which yields the smallest .consider the symmetric storage allocation in which the allocated storage sizes are the same as .then , for the symmetric allocation , the following inequalities hold . where is due to the fact that an average with the smallest ones is less than an arithmetic average ; is because is determined to satisfy for the symmetric allocation . combining and , we have . since , there exists a set such that , which implies that .therefore , the allocation corresponding to the minimum is the symmetric storage allocation , .  with the symmetric allocation , the lower bound of coincides with the maximized upper bound .that is , the term in lower bound becomes with the symmetric allocation . since the maximum values of the upper and lower bounds coincide ,we conclude that the symmetric allocation maximizes the exponential order .[ col : opt_d ] for given and , the optimal exponential order of the recovery failure probability is with the optimal storage allocation policy .the proof of theorem [ thm : allo ] showed that for symmetric storage allocation , .therefore , the maximum possible cardinality of the decoding set is . plugging this in lemma [ lem : d_order ] ,we obtain .theorem [ thm : allo ] is on the same line with the result of .theorem [ thm : allo ] exhibits that the maximal symmetric spreading of the sum storage capacity yields the optimal recovery probability in terms of exponential order even for wireless distributed storage systems suffering from channel fading .[ remark:1 ]  the optimal exponential order of the recovery failure probability is bounded above and below by thus , the approximated slope of the exponential order is , which is strictly less than 1 , for the sum storage capacity .[ remark:2 ]  although the exponential order well characterizes asymptotic behavior of the recovery failure probability , we also derive a high snr approximation of the recovery failure probability for concrete understanding of recovery success and failure in high snr regime , when the sum storage capacity is maximally and symmetrically spread to the storage nodes .when snr is sufficiently high , the recovery failure probability is approximated as \approx \binom{k}{\lceil \frac{k}{t}\rceil-1}(2^q-1)^{k-\lceil \frac{k}{t}\rceil+1}\rho^{-(k-\lceil \frac{k}{t}\rceil+1 ) }  \end{aligned}\ ] ]  lemma [ lem : d_order ] and theorem [ thm : allo ] indicate that when snr is sufficiently high , the exponential order of the recovery failure probability is dominated by . for and ,the recovery failure probability when snr is sufficiently high is obtained as  &=\sum_{\mathcal{d}\subseteq\{1,\ldots , k\}}\text{pr}_{f}[q\:|\:\mathcal{d}]\text{pr}[\mathcal{d}]\\  & \overset{(a)}{\approx}\binom{k}{\lceil \frac{k}{t}\rceil-1}\text{pr}\left[|\mathcal{d}|=\left\lceil k / t\right\rceil-1\right]\nonumber\\  & = \binom{k}{\lceil \frac{k}{t}\rceil-1}\text{pr}[\log_2(1+\rho|h|^2)>q]^{\lceil \frac{k}{t}\rceil-1}\nonumber\\  & \times\text{pr}[\log_2(1+\rho|h|^2)<q]^{k-\lceil \frac{k}{t}\rceil+1}\\  & = \binom{k}{\lceil \frac{k}{t}\rceil-1}\left(\exp\left(\frac{2^q-1}{\rho}\right)\right)^{\lceil\frac{k}{t}\rceil-1}\nonumber\\  & \times\left(1-\exp\left(\frac{2^q-1}{\rho}\right)\right)^{k- \lceil \frac{k}{t}\rceil+1}\\  & \overset{(b)}{\approx}\binom{k}{\lceil \frac{k}{t}\rceil-1}\left(\frac{2^q-1}{\rho}\right)^{k-\lceil \frac{k}{t}\rceil+1}\label{eq : taylor}\end{aligned}\ ] ]  where is from the result of lemma [ lem : d_order ] with high snr assumption and is due to taylor s expansion as .although we have derived the optimal storage capacity allocation for high snr , it is not clear whether the derived solution is always optimal for all other snr regimes . as stated earlier , the exact closed form expression of the recovery failure probability is hard to obtain due to its mathematical intractability .thereby , in this subsection , we explore the optimal storage allocation strategy in low snr regime . to this end, we have to first understand the relationship between the decoding set cardinality and the recovery failure probability .we start with the following definition of a complete storage node .storage node is defined as a complete storage node if it can store a complete data object and the data object can be perfectly recovered from it without help of any other storage nodes .that is , if , storage node is a complete storage node .[ lemma:1 ] any storage allocation strategy without complete storage nodes has higher recovery failure probability in low snr than a storage allocation strategy with only one complete storage .let us consider the following storage allocation strategy : to recover the data object , at least two storage nodes are required for the allocation and the corresponding recovery probability becomes }\nonumber\\  & \times\text{pr[recovery from incomplete storage nodes]}\\  & = \sum_{k=2}^{k}\binom{k}{k}\text{pr[]}\cdot\text{pr}\nonumber\\  & \overset{(a)}{\leq}\sum_{k=2}^{k}\binom{k}{k}\text{pr[]}\cdot\text{pr}\label{eq:(a)}\\  & = \sum_{k=2}^{k}\binom{k}{k}e^{-k\cdot\frac{2^q-1}{\rho}}\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k - k}\nonumber\\  & \times\left(1-\left(1-e^{-\frac{2^q-1}{\rho}}\right)^k\right)\\  & = e^{-2\cdot\frac{2^q-1}{\rho}}\sum_{k=2}^{k}\binom{k}{k}e^{-(k-2)\cdot\frac{2^q-1}{\rho}}\nonumber\\  & \times\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k - k}\left(1-\left(1-e^{-\frac{2^q-1}{\rho}}\right)^k\right)\\  & \overset{(b)}{\leq}e^{-2\cdot\frac{2^q-1}{\rho}}\end{aligned}\ ] ] where is from the selection upper bound ( i.e. , selecting the node with the strongest channel to the data collector among complete nodes ) , and is because as increases , the rest term except goes to 0 and strictly less than 1 .on the other hand , the recovery probability for the storage allocation is given by }\cdot\text{pr[recovery from node 1]}\\  & = \text{pr[ ] pr[]}\\  & = \bar{p}_o(\mathbf{a}_1 ) \left(=\exp\left(-\frac{2^q-1}{\rho}\right)\exp\left(-\frac{2^q-1}{\rho}\right)\right)\end{aligned}\ ] ] consequently , lemma 1 is proved because in low snr regime means always shows lower recovery failure probability than any other allocation without complete storage nodes .  for any allocation strategy with complete storage nodes and incomplete storage nodes , the recovery failure probability in low snr regime is higher than an allocation with complete storage nodes .[ lemma:2 ]  let an allocation strategy with complete storage nodes and incomplete storage nodes be and an allocation strategy with complete storage nodes . similar to the proof of lemma 1 , we can easily show the recovery probability of can be upper bounded as follows : }\nonumber\\   & \times\text{pr[recovery from storage nodes]}\\   & \overset{(a)}{\leq}\sum_{k_1=1}^{k_1}\sum_{k_2=0}^{k_2}\binom{k_1}{k_1}\binom{k_2}{k_2}e^{-(k_1+k_2)\cdot\frac{2^q-1}{\rho}}\nonumber\\   & \times\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k_1+k_2-k_1-k_2 }   \left(1-\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k_1+k_2}\right)\nonumber\\   & + \sum_{k_2=2}^{k_2}\binom{k_2}{k_2}e^{-k_2\cdot\frac{2^q-1}{\rho}}\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k_2-k_2}\nonumber\\  & \times\left(1-\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k_2}\right)\nonumber\\   & = \binom{k_1}{1}\exp\left(-2\cdot\frac{2^q-1}{\rho}\right)\left(1-\exp\left(-\frac{2^q-1}{\rho}\right)\right)\nonumber\\   & + o\left(\exp\left(-2\cdot\frac{2^q-1}{\rho}\right)\right)\\   & \lesssim\:k_1\exp\left(-2\cdot\frac{2^q-1}{\rho}\right )  \end{aligned}\ ] ] where is obtained from the selection upper bound as in and is the asymptotic inequality which denotes that the inequality is valid for high snr regime . because  holds for .when the sum storage buget is allocated to complete storage nodes , the remaining storage budget should be allocated to only one storage node , to achieve the optimal performance in low snr regime.[lemma:3 ]  refer to appendix [ app:2 ] .the optimal storage budget allocation in low snr regime consists of complete storage nodes and one incomplete storage whose storage size is .the proof is straightforward from lemmas [ lem : d_order][lemma:3 ] .  in low snr regime , if the sum storage budget for storage nodes is given by , two nodes should have storage size and one node should have storage size .note that this allocation is completely different .  when the total storage budget is given by an integer value i.e. , , with the optimal storage allocation in low snr , the number of storage nodes with non - zero allocated memory is and they are all complete storage nodes .then , contrary to the maximal symmetric allocation in high snr , node selection for selection diversity in the recovery phase is possible . in this case, the recovery failure probability can be obtained exactly in closed form as in the following corollary .when the total storage budget is an integer value , the recovery failure probability with the minimal allocation is obtained as =\sum_{k=0}^{t}\binom{t}{k}\left(e^{-\frac{2^q-1}{\rho}}\right)^k   \left(1-e^{-\frac{2^q-1}{\rho}}\right)^{t}.  \end{aligned}\ ] ]  =\sum_{k=0}^{t}\text{pr}[|\mathcal{d}|=k]\cdot\text{pr}_f[q\:\big|\:|\mathcal{d}|=k]]\nonumber\\   & = \sum_{k=0}^{t}\prod_{i\in\mathcal{d}}\text{pr}\left[\log(1+\rho|h_i|^2)>q\right]\nonumber\\   & \qquad\times\prod_{j\notin\mathcal{d}}\text{pr}\left[\log(1+\rho|h_j|^2)<q\right]\nonumber\\   & \qquad\times\text{pr}\left[\log(1+\rho\max_{i\in\mathcal{d}}|h_i|^2)<q\right]\\   & = \sum_{k=0}^{t}\binom{t}{k}\exp\left(-\frac{2^q-1}{\rho}\right)^k   \left(1-\exp\left(-\frac{2^q-1}{\rho}\right)\right)^{t - k}\nonumber\\   & \qquad\times\left(1-\exp\left(-\frac{2^q-1}{\rho}\right)\right)^k\end{aligned}\ ] ]to focus on channel fading effects only , we assume simple topology where the distances from data source / collect to storage nodes are the same .the channel gains of each link are i.i.d .exponential random variables with unit mean .the channel gains do not change during each period but independently change along the periods .an recovery failure occurs if either data object is not properly stored or accumulated information at data collector is less than threshold .  fig .[ fig : sim1 ] shows the exponential order of the recovery failure probability versus the number of storage nodes for various sum storage constraints .it is shown that the exponential order increases with the number of storage nodes and the increasing slope depends on the sum storage constraint .for example , when , the increasing slope of the exponential order is as noted in remark [ remark:2 ] .contrary to the conventional multiple relay or antenna diversity order , the exponential order dose not increase with the number of storage nodes since the gain provided by multiple storage nodes is hampered by limited storage capacity . in other words , adding storage node can improve reliability of storage operation but its effect is marginal without sufficient storage capacity .[ fig : sim2 ] shows the effect of the sum storage constraint on the exponential order of the recovery failure probability when the number of storage nodes is fixed as or . as predicted in corollary [col : opt_d ] , the exponential order increases and converges as the sum storage budget grows . according to theorem [ thm : allo ] , with the optimal storage allocation in high snr , the storage size allocated to each node becomes smaller as the number of storage nodes increases .for example , to obtain 80% of the maximum exponential order which is marked as the black circles , the storage sizes required at each node ( i.e. , ) are and for and , respectively . fig .4 reveals that if sum storage capacity is relatively small compared to the number of storage nodes , only a small amount of increment in storage capacity can considerably increase the exponential order .this phenomenon is more noticeable when the number of storage node is large , which implies that full exponential order is achievable with only a very small storage budget per storage node if the number of storage nodes is large enough .[ fig : sim3 ] figure verifies that the asymptotic analysis of recovery failure probability well approximate the recovery failure probability in high snr regime .as shown in the analysis of exponential order , the recovery failure probability is degraded as the sum storage capacity is smaller .it is also verified that the high snr approximation matches well with the simulation result if snr is greater than 10 db .given that a typical range of required snr in 802.11n wi - fi is around from 10 to 30 db , the high snr approximation would be useful in performance evaluations .[ fig : sim4 ] exhibits that the minimal allocation is strictly better than the symmetric allocation in terms of the recovery failure probability in low snr regime . for , although both of the allocation strategies have the same exponential order , the minimal allocation outperforms the symmetric allocation in terms of recovery failure probability . in the symmetric allocation ,all the storage nodes have to decode the data object from the data source for the optimal performance , but this is unlikely in low snr .consequently , the performance of the symmetric allocation is restricted at the storing phase and worse than that of the minimal allocation in low snr . for ,the exponential orders of the symmetric allocation and the minimal allocation are and , respectively . in high snr ,the symmetric allocation is definitely better but in low snr , the recovery failure probability of the minimal allocation is much lower than that of the symmetric allocation .the crossing point between the two allocation schemes is around db .although the optimal storage allocation is unknown for intermediate snr regime , the optimal storage allocation strategies for high and low snr regimes may suggest that a balanced allocation between the maximal symmetric allocation and the minimal allocation be effective in intermediate snr .for example , when there are 6 storage nodes and the sum storage budget is 2 , i.e. , , the symmetric allocation is and the minimal allocation is .possible balanced allocation strategies between them could be and .both of the allocation strategies require at least 2 storage nodes to recover the data object but can choose 2 among 4 storage nodes while can choose 2 among 3 storage nodes . as a result , is expected to outperform , which is verified in fig .[ fig : sim5 ] . comparing with the maximal symmetric allocation and the minimal allocation ,the minimal allocation is the best until snr is around 4 db at which the minimal allocation and the maximal symmetric allocation cross , whereas the maximal symmetric allocation is the best after the crossing .this result strongly implies that selection between the minimal allocation and the maximal symmetric allocation suffices .in this paper , we introduced a new wireless distributed storage model with a sum storage capacity and investigated its performance in terms of the recovery failure probability . using exponential order analysis , we proved that the maximal symmetric allocation is the optimal allocation strategy for high snr regime . for the maximal symmetric allocation, we also presented an approximated representation of the recovery failure probability based on a high snr approximation . on the other hand , using asymptotic analysis for low snr , the minimal allocation with complete storage nodes and one incomplete storage node was shown to be optimal in low snr regime . if the sum storage capacity is given as an integer value , we derived the exact recovery failure probability of the minimal allocation in low snr regime .based on the numerical investigation , we also showed that a proper selection between the minimal allocation and the maximal symmetric allocation would make any balance allocation unnecessary .by the law of total probability , the recovery failure probability is given by =\sum_{\mathcal{d}\subseteq\{1,\ldots , k\}}\text{pr}_{f}[q\:|\:\mathcal{d}]\text{pr}[\mathcal{d } ]  \end{aligned}\ ] ] where is the rate of the data object , which is equivalent to the data object size .the probability for the decoding set is obtained as \nonumber   & = \prod_{i\in\mathcal{d}}\text{pr}\left[\log_2(1+\rho|h_{s , i}|^2)>q\right]\nonumber\\   & \times\prod_{i\in\{1,\ldots , k\}\setminus\mathcal{d}}\text{pr}\left[\log_2(1+\rho|h_{s , i}|^2)<q\right]\\   & = \prod_{i\in\mathcal{d}}\text{pr}\left[\rho|h_{s , i}|^2>2^q-1\right]\nonumber\\   & \times\prod_{i\in\{1,\ldots , k\}\setminus\mathcal{d}}\text{pr}\left[\rho|h_{s , i}|^2<2^q-1\right]\\   & \overset{(a)}{\doteq}\prod_{i\in\mathcal{d}}\text{pr}\left[\rho^{1-v_{s , i}}>\rho^0\right]\prod_{i\in\{1,\ldots , k\}\setminus\mathcal{d}}\text{pr}\left[\rho^{1-v_{s , i}}<\rho^0\right]\\    & \doteq\prod_{i\in\mathcal{d}}\rho^0\cdot   \prod_{i\in\{1,\ldots , k\}\setminus\mathcal{d}}\rho^{-1}\\   & = \rho^{-(k-|\mathcal{d}|)}\label{eq : second }  \end{aligned}\ ] ] where is the cardinality of the decoding set and is the exponential order of ; comes from the definition of the exponential order .assuming that are i.i.d .exponential random variables , for a given decoding set , the conditional recovery failure probability is upper bounded by =\ : & \text{pr}\left[\sum_{i\in\mathcal{d}}t_i\log_2(1+\rho|h_{i , c}|^2)<q\right ]   \\&\mathbf{1}[\sum_{i\in\mathcal{d}}a_i\geq1]+\mathbf{1}[\sum_{i\in\mathcal{d}}a_i<1]\\   & \overset{(a)}{\leq}\ : \text{pr}\left[\sum_{i\in\mathcal{d}}t_i\log_2(1+\rho|g_i|^2)<q\right]\nonumber\\   & \times\mathbf{1}[\sum_{i\in\mathcal{d}}a_i\geq1]+\mathbf{1}[\sum_{i\in\mathcal{d}}a_i<1]\\   & \overset{(b)}{\doteq}\left\ {     \begin{array}{ll }     \rho^{-\min_{i\in\mathcal{d}}{t_i}^{-1 } } , & \textrm{for } \sum_{i\in\mathcal{d}}a_i\geq1,\\     1 , & \textrm{for } \sum_{i\in\mathcal{d}}a_i<1     \end{array }     \right.\label{eq : cond1 }   \end{aligned}\ ] ] where is because is replaced by ; the optimal transmit time allocation depends on the ordered channel gains of , but not on , which yields higher recovery failure probability due to unoptimized transmit time allocation for . follows from the definition of the exponential order and ( see for more details ) such that &\doteq\rho^{-\inf_{\mathbf{v}\in o^+}\sum_{i\in\mathcal{d}}v_{i } } \nonumber\\ & \doteq \rho^{-\min_{i\in\mathcal{d}}{t_i}^{-1 } } \label{eq : example}\end{aligned}\ ] ] where and is the exponential order of .[ fig : out_region ] illustrate a simple example of when there are only 2 nodes .   on the other hands ,if the channel gains for each node are replaced by the highest channel gain among them during the recovery phase , a lower bound on the conditional recovery failure probability is obtained as \nonumber\\   & \geq\:\text{pr}[\log_2(1+\rho\max_{i\in\mathcal{d}}|h_{i , c}|^2)<q ]   \mathbf{1}[\sum_{i\in\mathcal{d}}a_i\geq1]+\mathbf{1}[\sum_{i\in\mathcal{d}}a_i<1]\\   & \doteq \rho^{-|\mathcal{d}|}\mathbf{1}\left[\sum_{i\in\mathcal{d}}a_i\geq1\right ]   + \mathbf{1}\left[\sum_{i\in\mathcal{d}}a_i<1\right]\label{eq : cond2 }    \end{aligned}\ ] ]    \\   & \overset{(a)}{\approx}\text{pr}\left[(1-a_j^{\downarrow})|h_{i , c}|^2\rho+a_j^{\downarrow}\max\{|h_{i , c}|^2,|h_{j , c}|^2\}\rho > q\right]\\   & = 1-\text{pr}\left[(1-a_j^{\downarrow})|h_{i , c}|^2+a_j^{\downarrow}\max\{|h_{i , c}|^2,|h_{j , c}|^2\}\leq q/\rho\right]\\   & = 1- \text{pr}\left[(1-a_j^{\downarrow})|h_{i , c}|^2+a_j^{\downarrow } |h_{i , c}|^2 \leq q/\rho ~\mid  |h_{i , c}|^2>|h_{j , c}|^2\right]\text{pr}\left[|h_{i , c}|^2>|h_{j , c}|^2\right]\nonumber\\   & \quad - \text{pr}\left[(1-a_j^{\downarrow})|h_{i , c}|^2+a_j^{\downarrow } |h_{j , c}|^2 \leq q/\rho ~\mid  |h_{i , c}|^2 \leq |h_{j , c}|^2\right]\text{pr}\left[|h_{i , c}|^2 \leq |h_{j , c}|^2\right]\\   & = 1-\frac{1}{2 } \left(1- \exp\left(-\frac{q}{\rho}\right)\right ) \nonumber\\   & \qquad- \frac{1}{2}\int_0^{q/\rho}\text{pr}\left[(1-a_j^{\downarrow})t+a_j^{\downarrow}\max(t,|h_{j , c}|^2)<q/\rho\right]\text{pr}\left[|h_{i , c}|^2=t\right ] \:dt\\   & = 1-\frac{1}{2 } \left(1- \exp\left(-\frac{q}{\rho}\right)\right)-\frac{1}{2}\int_0^{q/\rho}\text{pr}\left[|h_{j , c}|^2<\frac{q/\rho-(1-a_j^{\downarrow})t}{a_j^{\downarrow}}\right]\exp(-t ) \:dt\\   & = 1-\frac{1}{2 }\left(1- \exp\left(-\frac{q}{\rho}\right)\right)-\frac{1}{2}\int_0^{q/\rho}\left(1-\exp\left(-\frac{q/\rho-(1-a_j^{\downarrow})t}{a_j^{\downarrow}}\right)\right)\exp(-t ) \:dt\\   & = \exp\left(-\frac{q}{\rho}\right)+\frac{a_j^{\downarrow}}{2(1 - 2a_j^{\downarrow})}\left(\exp\left(-2\cdot\frac{q}{\rho}\right)-\exp\left(-\frac{q}{\rho } \cdot \frac{1}{a_j^{\downarrow}}\right)\right)\label{eq : cvx}\\   & = f_{\text{prob}}(a_j^{\downarrow})\label{eq : out_form}\end{aligned}\ ] ]    for the allocation defined in the proof of lemma [ lemma:2 ] , let be and then , the sum of the storage sizes allocated to incomplete storage nodes is equal or less than .there can be up to incomplete storage nodes in this storage allocation .let this allocation be where .then the recovery probability for is given by }\label{eq : lemma3 - 12}\\   & + \sum_{k_1=1}^{k_1}\sum_{\mathcal{i}\neq \emptyset}\binom{k_1}{k_1}\frac{e^{-(k_1+|\mathcal{i}|)\cdot\frac{2^q-1}{\rho}}}{\big(1-e^{-\frac{2^q-1}{\rho}}\big)^{-(k_1+k_2-k_1-|\mathcal{i}|)}}\label{eq : lemma3 - 21}\\   & \times\text{pr [ recovery from com . andnodes in ] } \label{eq : lemma3 - 22}\end{aligned}\ ] ] where is a subset consists of the incomplete storage nodes in .the first term consisting of and has a dominant scale in the recovery probability and is common regardless of in .therefore , we have to focus on the second term consisting of and to analyze the effect of incomplete storage allocations .now we prove that in the second term , the case when leads to a dominant scale as ; }\nonumber\\   \overset{(a)}{\geq}&\sum_{|\mathcal{i}|=1}k_1e^{-2\cdot\frac{2^q-1}{\rho}}\left(1-e^{-\frac{2^q-1}{\rho}}\right)^{k_1+k_2 - 2}\nonumber\\   & \times\text{pr [ recovery from 1 com .node ] } \nonumber\\   \overset{(b)}{\gtrsim}&\ : ( k_1 k_2-\delta_1)e^{-2\cdot\frac{2^q-1}{\rho}}e^{-\frac{2^q-1}{\rho}}\quad\text{for arbitrary small } \delta_1\nonumber\\   = & \theta\left(\exp\left(-3\cdot\frac{2^q-1}{\rho}\right)\right)\label{eq : l31}\end{aligned}\ ] ] where is from that recovery from 1 complete storage only without help of incomplete storage nodes yields worse recovery probability ; is satisfied because goes to 1 as increases . for ,the followings hold : }\nonumber\\   \overset{(a)}{\leq}&\sum_{k_1=1}^{k_1 }   \frac{\sum_{|\mathcal{i}|\geq 2}\binom{k_1}{k_1}e^{-(k_1+|\mathcal{i}|)\cdot\frac{2^q-1}{\rho } } }   { \underbrace{\big(1-e^{-\frac{2^q-1}{\rho}}\big)^{-(k_1+k_2-k_1-|\mathcal{i}|)}}_{\geq1}}\nonumber\\   & \underbrace{\times\text{pr [ recovery from com . nodes ] } } _ { = 1-(1-\exp(-(2^q-1)/\rho))^{k_1+|\mathcal{i}|}\:\:\leq\:\ : ( k_1+|\mathcal{i}|)\exp(-(2^q-1)/\rho)}\nonumber\\\overset{(b)}{\lesssim}&\sum_{k_1=1}^{k_1}\quad\sum_{|\mathcal{i}|\geq 2}\binom{k_1}{k_1 }   e^{-(k_1+|\mathcal{i}|)\cdot\frac{2^q-1}{\rho } }   \left(k_1+|\mathcal{i}|\right)e^{-\frac{2^q-1}{\rho}}\nonumber\\   = & \theta\left(\exp\left(-4\cdot\frac{2^q-1}{\rho}\right)\right)\label{eq : l32}\end{aligned}\ ] ] where is from that a recovery from complete nodes is always better than that from complete storage nodes and incomplete storage nodes ; is because } \\ & = 1-\left(1-\exp\left(-\frac{2^q-1}{\rho}\right)\right)^{k_1+|\mathcal{i}|}\nonumber\\ & \leq \left(k_1+|\mathcal{i}|\right)\exp\left(-\frac{2^q-1}{\rho}\right).\nonumber\end{aligned}\ ] ] comparing with , we verify that the case when and is dominant in the second term consisting of and . consequently , in low snr regime , the optimal allocation strategy for the remaining budget can be identified by solving following optimization problem : }\nonumber\\ & \text{subject to } \sum_{j=1}^{k_2}a_j^{\downarrow}=t-\lfloor t \rfloor \nonumber\end{aligned}\ ] ] where is the largest value among the storage sizes allocated to incomplete storage nodes , that is , the allocation to storage nodes is rewritten in descending order such that . removing common terms and making the problem concise, we can reduce the optimization problem to }.\nonumber\\  & = \max_{a_j^{\downarrow}}\sum_{i=1}^{k_1}\sum_{j=1}^{k_2}\text{pr } \big[(1-a_j^{\downarrow})\log\left(1+|h_{i , c}|^2\rho\right)\nonumber\\  & \qquad\qquad + a_j^{\downarrow}\log\left(1+\max\{|h_{i , c}|^2,|h_{j , c}|^2\}\rho\right)>q\big]\\  & \text{subject to } \sum_{j=1}^{k_2}a_j^{\downarrow}=t-\lfloor t \rfloor.\end{aligned}\]]where and are i.i.d . exponential random variables .note that the probabilities for the summation are based on independent events and have the form in .note that is a convex function with respect to  ] ; 2 ) for , the limit of exists and is continuous on  ] is convex on a real interval $ ] because a sum of convex functions is also convex .moreover , it is symmetric for and hence it is a schur - convex function .therefore , with the constraint , the strong majorization holds for the allocation vector with only one non - zero element .that is ,   10 [ 1]#1 url [ 2]#2 b. hong and w. choi , `` asymptotic analysis of failed recovery probability in a distributed wireless storage system with limited sum storage capacity , '' _ in proc .ieee international conference on acoustics , speech , and signal processing ( icassp ) _ , florence , italy , may 2014 .j. kubiatowicz , c. wells , b. zhao , d. bindel , y. chen , s. czerwinski , p. eaton , d. geels , r. gummadi , s. rhea and h. weatherspoon , `` oceanstore , '' _ acm sigarch computer architecture news _ , vol .190 - 201 , 2000 .r. bhagwan , k. tati , y .- c .cheng , s. savage and g. m. voelker `` total recall : system support for automated availability management , '' _ symp .networked systems design and implementation ( nsdi ) _ , 2004 .k. v. rashmi , n. b. shah , and p. v. kumar , `` optimal exact - regenerating codes for distributed storage at the msr and mbr points via a productmatrix construction , '' _ ieee trans .inf . theory _ ,57 , no . 8 , 2011 .a. dimakis , v. prabhakaran and k. ramchandran , `` ubiquitous access to distributed data in large - scale sensor networks through decentralized erasure codes , '' _ international symposium on information processing in sensor networks _ ,2005 .   c. huang ,m. chen and j. li , `` pyramid codes : flexible schemes to trade space for access efficiency in reliable data storage systems , '' _ ieee international symposium on network computing and applications ( nca ) _ , 2007 .j. n. laneman , d. n. c. tse , and g. w. wornell , cooperative diversity in wireless networks : efficient protocols and outage behavior , _ ieee trans .theory , _ vol .3062 - 3080 , dec . 2004 .k. azarian , h. el gamal , and p. schniter , `` on the achievable diversity - multiplexing tradeoff in half - duplex cooperative channels , '' _ ieee trans .inf . theory _4152 - 4172 , dec .

Abstract: in wireless cloud storage systems , the recovery failure probability depends on not only wireless channel conditions but also storage size of each distributed storage node . for an efficient utilization of limited storage capacity and the performance characterization of allocation strategies , we asymptotically analyze the recovery failure probability of a wireless cloud storage system with a sum storage capacity constraint for both high snr regime and low snr regime . then , we find the optimal storage allocation strategy across distributed storage nodes in terms of the asymptotic recovery failure probability . our analysis reveals that the maximal symmetric allocation is optimal for high snr regime and the minimal allocation ( with complete storage nodes and an incomplete storage node ) is optimal for low snr regime , where is the sum storage capacity . based on the numerical investigation , we also show that in intermediate snr regime , a balance allocation between the minimal allocation and the maximal symmetric allocation would not be required if we select one between them according to snr .   cloud storage system , wireless storage , maximum distance separable coding , recovery failure , storage allocation .


how genotypes map into phenotypes counts amongst the most essential questions to understand how evolutionary innovations might come about and how evolutionarily stable strategies are fixed in populations . with some of its featuresseemingly dependent on the system studied and on the description level considered , the genotype - phenotype ( gp ) map appears far from trivial .many studies have addressed the effect of mutations on phenotype : point mutations  , genome fragment deletion  , duplication or inversions , or the knockout of specific genes  among others may or may not have an effect at the molecular , metabolic , regulatory , or organismal level  . also , the ability of genotypes to yield more than one phenotype is a main resource of molecular adaptation  .the probability of expressing different phenotypes or of experiencing mutations that modify the current phenotype depends on the structure of the gp map , which eventually determines how the space of function is explored , and what are the chances that a population survives or innovates in the face of endogenous or exogenous changes  .most models are restricted to the many - to - one realisation of the gp map , and thus assume that adaptation is dominated by mutations .there is a plethora of different model systems studied under this assumption . despite seemingly relevant underlying molecular differences , those models present a remarkable number of common properties .exhaustive research on the gp map was pioneered by studies of rna sequence - to - secondary - structure mappings .most topological properties identified in rna spaces are shared by other simple systems , such as the existence of huge genotype networks , the increase in phenotype robustness with the size of the latter , and a very skewed distribution of network sizes .the set of genotypes that yield the same phenotype typically forms a network , since those genotypes are pairwise connected through mutations .sufficiently large genotype networks so defined were postulated as a condition for the navigability of sequence space long ago  .subsequent studies have shown that such large networks do exist , and that the difference in sequence between genotypes in those networks can be as large as the difference between two random sequences  .phenotype robustness refers to the average effect of point mutations in the genotypes of a specific genotype network .it has been shown to grow logarithmically with the size of the phenotype in rna  , in a self - assembly model of protein quaternary structure  and in simple models for protein folding  .the existence of qualitative and quantitative statistical properties of the gp maps shared by apparently dissimilar systems suggests that they might arise from basic universal features  .though genotype networks are not always fully connected , they do traverse the whole space of genotypes for sufficiently abundant phenotypes , thus ensuring high navigability  . even in cases where genotype networks are fragmented ,those fragments could be mutually reached if the gp map is many - to - many .the existence of `` promiscuous '' sequences that map into more than one phenotype enhances navigability and promotes fast adaptation  .the statistical property of gp maps that has attracted the most attention is very likely the distribution of genotype network sizes , or phenotype sizes for short . due to the astronomically large sizes of genotype spaces , initial estimations of the size of phenotypeswere performed through random samplings of genotype space .the results were often represented as frequency - rank plots , with phenotypes ordered according to their sizes .random samplings of genotype spaces in many - to - one gp maps invariably yielded some very abundant phenotypes and a large number of phenotypes represented by a few or just one genotype  .often , a frequency - rank plot was fitted to a generalized zipf s law  , implying a power - law - like distribution of phenotype sizes .however , subsequent studies demonstrated that the frequency - rank plot of phenotype sizes actually had a more complex functional shape  , and specific functional fits were avoided .subsequent studies have exhaustively mapped the complete sequence space to its corresponding phenotypes , among which rna sequence - to - minimum energy secondary structure map  , the hydrophobic - polar ( hp ) model for protein folding  , or toylife , which includes a sequence - to - structure - to - function description  . as a result , complete phenotype size distributions ( for short sequences ) are now available .fitted shapes range from power - law - like curves  to lognormal distributions  .it has been argued that , among other generic properties , a skewed distribution of phenotype sizes results from the organization of biological sequences into constrained and unconstrained parts . in , the authors introduce the fibonacci gp map , a many - to - one artificial model , where sites in a sequence can be coding or non - coding , and either lead to new phenotypes under mutations ( coding sites ) or yield the same phenotype ( neutral , non - coding sites ) .the model can be analytically solved and yields a power - law phenotype size distribution , in qualitative agreement with some observations .  in this contribution, we attempt an identification of the elements in the organization of sequences that characterize the _ quantitative _ properties of the distribution of phenotype sizes .we show in a constructive fashion that the model in  is an example of a broad spectrum of sequence - to - structure gp models . starting with the simplest case , where sequences are separated into constrained and neutral parts , and adding subsequent elements in the organization of the sequences and versatility levels of the sites ,we show how the distribution of phenotype sizes changes from pure power - law ( with an exponent dependent on how genotypes are distributed among phenotypes ) to lognormal .this functional form is independent of whether the gp map is many - to - many ( sequences are promiscuous ) or many - to - one ( the phenotype can be uniquely predicted from the sequence ) .our final example corresponds to the rna sequence - to - secondary structure map , where we demonstrate that the combinatorial properties of the distribution of sites of variable neutrality along sequences causes the distribution of phenotypes to follow a lognormal distribution , with parameters that can be traced to properties of the genotype set .our main result is that a lognormal distribution of phenotype sizes is the expected result in any gp map where sufficient variation in the number of phenotypes of similar size is present .we will study four models that interpolate between the simplest case of sequences divided into neutral and non - neutral sites separated into two groups and a general case ( represented by rna ) , and calculate for each of them the size of a phenotype given the sequence organization of its corresponding genotypes , the number of phenotypes with the same size , the frequency rank ordering of phenotypes , and eventually the distribution of phenotype sizes .table [ table : definitions ] summarizes the nomenclature and definitions used in this work , and figure [ fig : quantities ] illustrates some relevant quantities ..summary of symbols used in this work and their short definitions .[ cols="^,<",options="header " , ]   of a phenotype as a function of its rank .here represents the number of constrained sites and works as an intermediate variable to simplify calculations ; the first three values of are indicated in the figure . , corresponding to the length of the horizontal segments ( arbitrary in this representation ) , is the number of phenotypes with constrained sites and is their size . ]the genotype space is made of sequences of length letters from an alphabet of size .two examples of alphabet sizes are for a binary alphabet and for dna or rna , or .the _ versatility _ of site is defined as the average number of different letters of the alphabet that can occupy a given sequence position . in general for all sitesthis is a quantity closely related to neutrality .we will study the simplified case where sites can take one out of two different values , , with .sites are called constrained if , and neutral if .we will use to count the number of sites with low versatility .the size of a phenotype is the number of different genotypes compatible with that phenotype . from the definition of follows that is a nonincreasing function of . in the literature , phenotype frequency  , number of sequences for a phenotype  or neutral set size  been used with a meaning identical to phenotype size here .the set of -genotypes is defined as the number of genotypes compatible with -phenotypes , .the rank of the first phenotype in size class is . note that the total number of phenotypes coincides with the maximum rank .if is the probability density that a phenotype has size , then we can count phenotypes as to first order we can approximate the integral as ( the approximation gets better the smaller ) .thus , up to a normalisation constant , .the probability density yields the probability of finding a phenotype with size when uniformly sampling over phenotypes .this corresponds to the distribution , as defined in other studies  .finally , we will also introduce a factor to represent the fraction of -genotypes that actually go to a given -phenotype .this factor arises from additional restrictions in the assignment of genotypes to phenotypes which are not made explicit in the models . in general ,if the models we are going to introduce assign the same genotypes to several -phenotypes .this would correspond to a many - to - many gp map a sort of maps suitable to describe molecular promiscuity .incidentally , molecular promiscuity strongly enhances navigability in genotype space  .other choices may account for specific restrictions in the models ; in particular , a suitable choice of may render the gp map many - to - one .we will return to this point when we provide details of the models .a succint definition of the hierarchy of models introduced in this work is as follows :  * * model 1 : * _ constrained and neutral sites occupy fixed positions ._ sequences are separated in two parts , the first one of length occupied by constrained sites , , and the second part of length occupied by neutral sites , .two minor variants considered are ( i ) phenotypes are all viable and ( ii ) lethal mutations occur independently of the site class . ** model 2 : * _ constrained and neutral sites occupy variable positions ._ this is illustrated by means of two examples : ( i ) constrained sites are split into two fragments at the beginning and at the end of the sequence and ( ii ) constrained sites can occupy arbitrary positions in the sequence . ** model 3 : * _ versatile sites occupy fixed positions ._ two different types of sites with fixed versatilities and are considered . ** model 4 : * _ versatile sites occupy variable positions : rna . _ in a first approximation , rna sequences contain two types of sites that occupy different positions in the sequence subject to secondary structure constraints : those forming pairs ( stacks ) in the secondary structure have average versatility , and those unpaired ( loops ) have average versatility .the model can be generalized to an arbitrary number of site classes .figure [ fig : models ] schematically represents the different models analysed here and some properties that will be of relevance to understand the distributions of phenotype sizes they yield .  . from left to right , we depict the organization of sites in phenotypes , examples of how sequences are assigned to phenotypes with example mutations highlighted in bold red , and a brief explanation on possible constraints in that assignment .( a ) in model 1 a given sequence can be assigned with equal probability to any phenotype with constrained sites .a mutation in one of those sites changes the phenotype , and there is only one possible structural arrangement given . the bold line in the phenotype associated to the mutated sequence is meant to represent a nwe phenotype with the same arrangement as above .( b ) two examples analysed in this work in the class of model 2 are when constrained sites are split into two parts at the beginning and at the end of the sequence ( above ) and when they can occupy arbitrary positions ( below ) .different ways in which the constrained sites can be arranged define different phenotypes to which the same sequence can be assigned . as in model 1 , a mutation in a constrained site changes to a new phenotype with the same structural arrangement .models 1 and 2 are unconstrained , many - to - many maps .( c ) as it occurs in model 1 , defines the structure of the phenotype in model 3 , though there might be constraints in the assigment of sequences . in the example , assuming that less versatile sites admit only two letters , for instance a and c , implies that the sequence shown can not be assigned to any phenotype with letters g or u in less versatile sites ( positions 4 to 6 in the example ) .( d ) model 4 includes elements of models 2 and 3 : the order of sites with different versatilities matters in the definition of phenotype and there are restrictions in the assignment of sequences to phenotypes . in the example shown , corresponding to rna , mutations may or may not change the phenotype , depending on a non - trivial relationship between structure and sequence composition . still , in general , also in models 3 and 4 a genotype can be assigned to multiple phenotypes.,scaledwidth=85.0% ]this is probably the simplest non - trivial model in the class of gp maps , very similar in spirit to that presented in  .phenotypes are characterized by constrained sites in the first part of the sequence . for a fixed ,mutations in a constrained site change the phenotype , and mutations in neutral sites yield genotypes compatible with the phenotype .therefore ,   note that , if , the complete genotype space is partitioned among -phenotypes for every value of .this implies that , if we consider all possible phenotypes ( i.e. all values ) , a particular genotype is simultaneously compatible with many different phenotypes representing a highly promiscuous sequence .specifically , if the total number of genotypes compatible with -phenotypes is , so the total amount of genotypes .this result clearly shows the many - to - many nature of the gp map of this model with this choice of genotypes are assigned to all phenotypes they are compatible with and , therefore , are repeatedly counted .a minimal rule to avoid multiple assignments is to think of as the probability that a genotype is actually assigned to an . when this probability is uniform , , and we choose , the total number of genotypes becomes , the size of the genotype space , so the resulting map is effectively many - to - one .other examples in which depends on will appear later .now , to obtain size as a function of rank we must eliminate in and substite it into to get . in this case , from eq .  and assuming , \approx \log_k\left[(k-1)r\right],\ ] ] and substituting in    to obtain the probability density we first notice that eq .implies , hence . on the other hand , thus the probability distribution is a power - law with exponent .   in the same scenario as above ,let us assume that a fraction of mutations is lethal , thus leading to a non - viable genotype .in this case , eqs .to  are identical , with substituted by .therefore , and are as above with the latter change .this result shows that the existence of a non - viable class to which viable genotypes can mutate does not necessarily imply relevant functional changes in the distribution of phenotypes , which is in either case of the form , with .the effect of uniformly distributed lethal mutations could be therefore absorbed as a constant into .the situation changes if mutations are not distributed uniformly , but their likelihood depends on .this would be a particular realisation of model 3 introduced below .   in any realistic model (e.g. the case of rna ) the position of constrained and neutral sites should matter in the definition of a phenotype . while does not change its functional form as a result , does ( and as a consequence ) , causing potentially relevant modifications in and . in general , the number of different phenotypes would take the form , where accounts for changes in the letter of the constrained site ( yielding a different phenotype , as assumed ) and is a model - dependent combinatorial number that counts the different ways in which the sites can be arranged to yield meaningful ( and different ) phenotypes . in general ,the factor in stems from mutations in neutral sites , while the arrangement of constrained and neutral sites along the sequence is weighted by , with effects on the functional form of that , in general , depend on the permitted arrangements .as will be shown , might enormously increase the number of phenotypes and , especially , the relative abundances of -phenotypes .   as a way of example ,let us consider one of the simplest situations where the position of the constrained sites matters .suppose that those sites can be split into two groups with lengths and and placed at the beginning and at the end of the sequence ( such that and ) .this gives different phenotypes with constrained sites , and from these expressions we can obtain ( see appendix a ) the asymptotic ( for large ) rank distribution and the size probability density with and some constants .therefore , even in this simple case with quite a limited number of possible organization of constrained sites , and are no longer pure power - laws , though the dominant term of the phenotype size distribution ( size still dominated by mutations in neutral sites ) is characterized by an exponent .the total number of genotypes compatible with -phenotypes is also modified , , and is seen to increase linearly with .we now assume that the constrained and unconstrained sites can occupy any site of the chain .in that case with no simple expression for .let us focus , however , on the size distribution , and consider the case where .asymptotically for  changing to through  and writing , we finally obtain ^ 2\right\}\ ] ] a log - normal distribution with mean and variance , very different from the distribution of the previous cases .this section presents an example of a main result of this study .it shows that , when the definition of the phenotype depends on the specific position of constrained and neutral sites in sequences , the functional form of ( and , in consequence , of ) qualitatively changes .in particular , the exponential growth of with dominates , which takes the form of a lognormal distribution .other quantities defining the gp map , such as or , change now the parameters of the distribution , but do not modify its shape .   the models analysed above demonstrate that when sites are either constrained or neutral , the exponent associated to the power - law part of is . as we show next , this exponent is modified when the sites in the sequence show intermediate degrees of versatility , which causes the number of -genotypes to depend on .let us consider the case where the sites are just less constrained than the sites , such that the former admit an average of different letters of the alphabet and the latter admit , with .relevant functions read with .as it can be readily seen by substitution , these expressions reduce to model 1 for and .now , yielding for large this scales as , where depends on and as yielding in the limit of model 1 . substituting this expression into eq . , hence , up to a constant factor , again maintains its power - law shape but its exponent depends on and .the number of -genotypes now becomes this number can either increase or decrease with depending on whether is larger or smaller than 1 .both situations are possible under the constraint .the values of and change in response to possible enrichements or depletions in the total number of assigned genotypes with .this is a first example of similar cases encountered later in this work and in the literature , as we discuss later .   in a first approximation ( which has been shown to yield acceptable fits to data  ), rna sequences can be divided into two classes of sites : those in stacks ( bound ) and those in loops ( unbound ) , characterized by different degrees of neutrality ( see e.g.  and fig . 4 in  ) .changes in the position of loops and stacks means a different phenotype .additionally , the composition of each site in the sequence bears a significant correlation with the structural element it will preferentially represent in the phenotype ( see fig . 7 in  ) .therefore , a first approximation to a gp representation of rna involves elements in our previous models 2 and 3 . in the following ,the abundances or phenotypes will be ruled by ( averaged ) values and of the number of letters that can be changed in stacks or loops , respectively ( see fig . [fig : models ] ) , without affecting the phenotype .studies of rna neutral networks and their related properties are usually restricted to the many - to - one mapping between sequence and structure . despite the fact that any rna sequence is compatible with multiple structureswhose relative weight in an ensemble of identical sequences is defined by their folding energy  , it is common practice to select only the minimum energy fold as the associated phenotype .this decision transforms an intrinsic many - to - many gp map where alternative phenotypes can be reached through mutations or promiscuity , into a many - to - one map where navigability is limited to the effects of neutral drift .analytical approaches can not include , in general , energetic considerations , so they implicitly work in the many - to - many unrestricted case .this situation is comparable to the assignation of sequences to structures we have performed in our models , where every sequence is assigned to all phenotypes it is compatible with , while possible restrictions in the assignments are encompassed in .the distribution of secondary structure sizes for the unrestricted map ( i.e. all sequences compatible with a given secondary structure ) fixing the number of stacks or loops has been derived in  for the general case of structures with pseudoknots , in  and  , and in  in a form that will be used here .   in this case will denote the number of pairs of nucleotides in stacks ( , with if is odd and if is even ) , hence will be the number of nucleotides in loops ( , which is the size of the minimal hairpin loop ) ; is the probability distribution for secondary structures with paired nucleotides , for sequences of length ( in the limit ) .it has been shown  that this distribution behaves as a normal distribution in with mean and standard deviation . in the case that structures with stems with less than two base pairs or loops with less than threeunpaired bases are forbidden accounting for minimal energetic constraints we obtain , , , and note that different constraints will lead to different values of these quantities , but otherwise will not change the fact that is a normal distribution .finally , the number of different phenotypes of a sequence of length with paired bases is given , in the limit , by with ( see ) .   in the case that the unpaired sites admit different letters and the paired sites letters ( ) , the size of a phenotype is given by . here, we will consider that a phenotype is formed by all sequences compatible with that phenotype , thus setting .we have denoting and noting that , substitution of into yields the log - normal distribution    in the same two - sites approximation the functional form of the rank is derived in appendix b. after some algebra we arrive at with constants and depending on parameters of the combinatorial factor , see appendix b.the functional shape of the distribution of phenotype sizes is strongly dependent on the sequence organization within phenotypes . in a first approximation that discards the heterogeneity among genotypes in the same phenotype, one may describe that ensemble of sequences through a prototypic sequence whose sites admit a phenotype - dependent , variable number of letters of the alphabet , a quantity that we have dubbed versatility .the substitution of each sequence in a phenotype by the average over the phenotype seems a strong approximation .however , there is evidence that deviations from the average within a phenotype are small : the number of neutral neighbours of genotypes within a phenotype are tightly clustered around an average value characteristic of that phenotype size  . with this proviso, two main elements determine the corresponding distribution of phenotype sizes .the first one , generic for all systems , is the relationship between the size of a phenotype and the versatility of each site . in the framework used in this work, the size of a phenotype can be written in general as this product yields an intrinsic allometric relation between the size of a phenotype and the length of the sequence .the second element , specific of each sequence - to - structure map , is the number of phenotypes with similar size .this quantity takes the overall form with the combinatorial factor accounting for the number of ways in which an ensemble of sites with values can be arranged into meaningful phenotypes , and the product accounting for the number of neutral sequences within the phenotype .if the values of the combinatorial factor are constrained enough such that the asymptotic behavior of with is subdominant with respect to that of the product as in models 1 and 3 the distribution of phenotype sizes is a power - law .if , on the contrary , the dominant term is the combinatorial factor in particular when the distribution of structural motifs converges to a gaussian the distribution of phenotype sizes becomes a lognormal .our calculations make it explicit that variations in the precise values of versatility , in the number of different classes of sites , or in particular constraints on structures ( as , e.g. the minimum number of base pairs required to form a stack ) have a quantitative effect on the parameters of the lognormal , but do not affect the shape of the distribution .  in the case we should expect a power - law - like distribution of phenotype sizes characterized by an exponent .the actual value of stems from a combination of the number of genotypes compatible with a given phenotype and the total number of phenotypes with the same ( or similar ) size .variations in the functional form of with could be responsible for changes in . in a general scenario ,let us assume that phenotype sizes can be ordered according to a certain variable ( in our case the number of low versatility positions ) , and let us define the total number of genotypes compatible with -phenotypes as , formally generalizing the quantity calculated in the specific models tackled in this work .the behaviour of with determines the value of the exponent : if is constant , then .however , if is exponentially enriched ( depleted ) in genotypes as grows , the value of becomes larger ( smaller ) than 2 . in the case of model 3 , for example , with and .two examples of enrichment or depletion in the number of genotypes compatible with -phenotypes are , with and , and , with and . in a very explicit way now , changes in the actual assignment of genotypes to phenotypes through ( embedded in ) will affect the probability density distribution .another example in the class of model 3 , yielding power - law - like with non - trivial is the model in  . besides the division of sequences into neutral and constrained sites ,the authors introduce a stop codon which causes an transition rate to alternative phenotypes , that being the eventual reason for a non - trivial value of . in that case , , which corresponds to a value of and , consistently , , with .the stop codon represents a particular instance of a decreased tolerance to mutations in less versatile sites .another formal example could be a rate to lethal mutations increasing with .this class of mechanisms skew the assignation of genotypes to phenotypes or , equivalently , deplete the amount of genotypes associated to phenotypes as grows : larger values of imply that there are more positions where non - neutral mutations can occur , and this leads to and .figure [ fig : powerlaw ] summarizes the sequence organization of different models with a power - law distribution of phenotype sizes , the origin and functional form of the function , and the corresponding value .dependent rate of lethal mutations ; ( e ) model 1 with a uniform distribution of lethal mutations .colour codes for sites as in fig .[ fig : models ] . ]  in fig .[ fig : distributions ] we represent schematically the functional form of and for the class of our model 3 and a possibly general class of models analogous to rna ( class 4 ) . at present , it is difficult to clearly match all models in the literature to classes 3 or 4 .for example , the hydrophobic - polar ( hp ) non - compact model seems to be characterized by a distribution of phenotype sizes similar to a power - law  , while other models for heteropolymers that have been compared to hp yield broad distributions with a maximum  .even rna with a two - letter alphabet apparently yields power - laws  , so it might belong to a non - trivial combination of models 3 and 4 as well .this is a very intriguing and complex question that we have to leave for future studies .these considerations notwithstanding , the situation where the combinatorial factor converges to a gaussian distribution is expected to be very general for sequence - to - structure gp maps  , implying that a lognormal distribution of phenotype sizes might be a generic property of such maps .up to now , there are few quantitative results supporting this statement , very likely due to the impossibility to exhaustively fold genome spaces for large .a remarkable exception is  , where the lognormal distribution has been suggested as the best fit to computational distributions of rna secondary structure sizes for lengths up to .it is interesting to highlight that our results have been obtained under a uniform assignment of genotypes ( represented through our variable ) to phenotypes .however , the many - to - one gp map in rna assigns the minimum energy structure to each sequence . in the language of our function , the correlation between energy and in rna will preferentially assign genotypes to phenotypes with a large number of pairs ( large ) since , on average , the larger the number of pairs the lower the folding energy  .it can not be discarded that genotype - to - phenotype assignment rules based on quantities not considered here might skew the distribution or eventually yield different functional forms .though this is a possibility that has to be kept in mind , results in  reveal that , at least in the case of four - letters rna , deviations from lognormality can not be numerically detected .we suspect that this is likely due to a dominant effect of over both in the many - to - many and in the many - to - one representations of the rna sequence - to - structure map .   and expected for different model classes .( a ) rank ordering of sizes in model 1 yields . in the class represented by our model 3 ( with additional examples in figure [ fig : powerlaw ] ) an enrichment or depletion of with causes deviations towards smaller or larger values of , respectively , in absolute value .( b ) probability density : recall that .colours code for equivalent curves in ( a ) .( c ) possible shape of for rna ( see , e.g.  ) and expected ( d ) .only sufficiently large phenotypes can be found under random sampling of genotypes ( compare with results in  ) ; they would occupy the shaded blue regions in ( c ) and ( d).,scaledwidth=80.0% ]  simple models as those presented here can be used as well to estimate other relevant quantities of gp maps , and to determine if they are almost universal or model - dependent .one such quantity is the relationship between phenotypic robustness and the size of a phenotype . in our scenario , and similarly to other examples  , phenotypic robustness coincides with genotypic robustness , which is calculated straight forward as the ratio between the number of neutral neighbours , and the total number of neighbours of a sequence , .this yields a function of .next , is obtained easily from its relationship with , and it takes the general form , where is a model - dependent quantity .therefore , the relationship between phenotype robustness and the logarithm of phenotype size consistently appears in very generic sequence - to - structure models .the relationship between phenotype robustness and evolvability can not be derived unless a explicit rule linking possible mutations to phenotypes with different is introduced . in our models 1 , 2 , and 3 ,such a rule , which could take a form analogous to the stop codon of the fibonacci map  , is not defined .the case of rna is particularly interesting and has received significant computational attention since long ago . only partial explorations of the accessibility of alternative phenotypes have been performed due to the huge sizes of phenotypes  .hopefully , further extensions of our model 4 could help in the analytical treatment of this highly complex problem .advances in empirical techniques , such as the intensive use of microarrays , should allow in the near future an exhaustive characterization of actual genotype spaces , as has been done for short transcription factor binding sites  .we believe that analyses of empirical gp maps will reveal strengths and weaknesses of the approach here presented , and likely suggest ways of improvement , regarding in particular a formal description of phenotype networks ( networks of genotype networks ) and evolvability in natural systems .in order to derive and for model 2 with constrained sites split into two groups at the extremes of the sequence ( section [ sec : model2a ] ) it will prove convenient to use the affine transformation of =\ell\log k - c_k , \qquad c_k\equiv\frac{\log k}{k-1}.\ ] ] then eq .  can be rewritten ^{\ell}=\frac{x}{c_k}e^{\ell\log k } = \frac{x}{c_k}e^{x+c_k},\ ] ] from which ^{-c_k}. \label{eq : productexp}\ ] ] inversion of this equation yields ^{-c_k}\big),\ ] ] with lambert s product - logarithm function ( * ? ? ?4.13.1 ) .now , and using eq ., finally , since when ( * ? ? ?4.13.10 ) , when the rank is large +\log c_k - c_k\sim \log r+a,\ ] ] with a constant. then , for large we obtain eq . .as for , from eqs .  and , differentiating with respect to yields .therefore , eliminating from this same equation we end up with with another constant .this is eq . .the rank function for the case of rna sequences whose sites may take two values of neutrality and , a number of secondary structures of length with sites with neutrality and a total number of different secondary structures of lenght is where now , since will be negative for all , we can use the asymptotic expansion of the complementary error function to write   in order to find how the size of a phenotype depends on its rank value it is convenient to introduce new parameters .let us denote and , and with .the size of a phenotype is given by , therefore now , taking logarithms in and neglecting subdominant terms in , hence and therefore which implies and jac designed the study , carried out the calculations , interpreted the results and wrote the manuscript .both authors read and approved the final text .the authors acknowledge the thorough revision of three anonymous reviewers , which has helped improving this paper .this work has been supported by the spanish ministerio de economa y competitividad and feder funds of the eu through grants viraless ( fis2014 - 57686-p ) and variance ( fis2015 - 64349-p ) .t. baba , t. ara , m. hasegawa , y. takai , y. okumura , m. baba , k.a .datsenko , tomita m. , b.l .wanner , and h. mori .construction of _ escherichia coli _ k-12 in - frame , single - gene knockout mutants : the keio collection ._ , 20:0 2006.0008 , 2006 .sam f. greenbury , steffen schaper , sebastian e. ahnert , and ard a. louis .genetic correlations greatly increase mutational robustness and can both reduce and enhance evolvability ._ plos comp ._ , 12:0 e1004773 , 2016 .w. grner , r. giegerich , d. strothmann , c. reidys , j. weber , i. l. hofacker , p. f. stadler , and p. schuster .analysis of rna sequence structure maps by exhaustive enumeration .i. neutral networks ._ monatshefte f. chemie _ , 127:0 355374 , 1996 .matthew c. cowperthwaite , evan p. economo , william r. harcombe , eric l. miller , and lauren ancel meyers .the ascent of the abundant : how mutational networks constrain evolution ._ plos comp ._ , 4:0 e1000110 , 2008 .s. f. greenbury and s. e. ahnert .the organization of biological sequences into constrained and unconstrained parts determines fundamental properties of genotype - phenotype maps . _j. royal soc .interface _ , 12:0 20150724 , 2015 .erik schultes , peter t. hraber , and thomas h. labean .global similarities in nucleotide base composition among disparate functional classes of single - stranded rna imply adaptive evolutionary convergence . _rna _ , 3:0 792806 , 1997 .

Abstract: an essential quantity to ensure evolvability of populations is the navigability of the genotype space . navigability , understood as the ease with which alternative phenotypes are reached , relies on the existence of sufficiently large and mutually attainable genotype networks . the size of genotype networks ( e.g. the number of rna sequences folding into a particular secondary structure , or the number of dna sequences coding for the same protein structure ) is astronomically large in all functional molecules investigated : an exhaustive experimental or computational study of all rna folds or all protein structures becomes impossible even for moderately long sequences . here , we analytically derive the distribution of genotype network sizes for a hierarchy of models which successively incorporate features of increasingly realistic sequence - to - structure genotype - phenotype maps . the main feature of these models relies on the characterization of each phenotype through a prototypical sequence whose sites admit a variable fraction of letters of the alphabet . our models interpolate between two limit distributions : a power - law distribution , when the ordering of sites in the prototypical sequence is strongly constrained , and a lognormal distribution , as suggested for rna , when different orderings of the same set of sites yield different phenotypes . our main result is the qualitative and quantitative identification of those features of sequence - to - structure maps that lead to different distributions of genotype network sizes .   * keywords : genotype - phenotype map , neutrality , rna , phenotype size , evolution *