prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What type of distribution do the tail distributions of price returns follow?

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Context 1: Paper Info

Title: Age and market capitalization drive large price variations of cryptocurrencies
Publish Date: 23 Feb 2023
Author List: 

Figure

Figure 3  Illustration of different effects of age and market capitalization on power-law exponents of cryptocurrencies (a) Posterior probability distributions of the linear coefficients associated with the effects of age [p(A)] and (b) the effects of market capitalization [p(C)] on power-law exponents related to large positive returns Panels (c) and (d) show the analogous distributions for the association with power-law exponents related to large negative returns In all panels, the different curves show the distributions for each of the top 20 cryptoassets by market capitalization Cryptocurrencies significantly affected by age or market capitalization are highlighted in boldface, and the numbers between brackets show their positions in the market capitalization rank 
Figure S5 There is more probability mass in the positive tail than in the negative tail of price returns (a) Probability distributions of the lower cut-offs (r min ) obtained by applying the Clauset-Shalizi-Newman method to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r min for positive and negative returns (b) Probability distributions of 90th percentiles (r 90 ) estimated from the power-law models adjusted to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r 90 for positive and negative returns (c) Probability distributions of the fraction of weeks that r 90 estimated from positive returns (r + 90 ) is larger than r 90 estimated from negative returns (r − 90 ) This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Figure S7 Robustness of the results of Fig  2(b)-(d) against considering only cryptocurrencies with fraction of rejection f r < 0 1 Panels (a) and (b) show the same distributions of Fig  S4 but after filtering out all time series of cryptocurrencies with fraction of rejections f r ≥ 0 1 As in the case related to sampling issues, we observe that these distributions barely change when considering only cryptocurrencies with f r < 0 1 Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  S4 (two-sample Kolmogorov-Smirnov test, p > 0 05) 

abstract

Cryptocurrencies are considered the latest innovation in finance with considerable impact across social, technological, and economic dimensions  This new class of financial assets has also motivated a myriad of scientific investigations focused on understanding their statistical properties, such as the distribution of price returns 
However, research so far has only considered Bitcoin or at most a few cryptocurrencies, whilst ignoring that price returns might depend on cryptocurrency age or be influenced by market capitalization  Here, we therefore present a comprehensive investigation of large price variations for more than seven thousand digital currencies and explore whether price returns change with the coming-of-age and growth of the cryptocurrency market 
We find that tail distributions of price returns follow power-law functions over the entire history of the considered cryptocurrency portfolio, with typical exponents implying the absence of characteristic scales for price variations in about half of them  Moreover, these tail distributions are asymmetric as positive returns more often display smaller exponents, indicating that large positive price variations are more likely than negative ones 
Our results further reveal that changes in the tail exponents are very often simultaneously related to cryptocurrency age and market capitalization or only to age, with only a minority of cryptoassets being affected just by market capitalization or neither of the two quantities  Lastly, we find that the trends in power-law exponents usually point to mixed directions, and that large price variations are likely to become less frequent only in about 28% of the cryptocurrencies as they age and grow in market capitalization 
Since the creation of Bitcoin in 2008 , various different cryptoassets have been developed and are now considered to be at the cutting edge of innovation in finance   These digital financial assets are vastly diverse in design characteristics and intended purposes, ranging from peer-to-peer networks with underlying cash-like digital currencies (e g 
Bitcoin) to general-purpose blockchains transacting in commodity-like digital assets (e g  Ethereum), and even to cryptoassets that intend to replicate the price of conventional assets such as the US dollar or gold (e g  Tether and Tether Gold)   With more than nine thousand cryptoassets as of 2022 , the total market value of cryptocurrencies has grown massively to a staggering $2 trillion peak in 2021  
Despite long-standing debates over the intrinsic value and legality of cryptoassets , or perhaps even precisely due to such controversies, it is undeniable that cryptocurrencies are increasingly attracting the attention of academics, investors, and central banks, around the world   Moreover, these digital assets have been at the forefront of sizable financial gains and losses in recent years , they have been recognized as the main drivers of the brand-new phenomena of cryptoart and NFTs , but also as facilitators of illegal activities, such as money laundering and dark trade  
Financial research dedicated Our results are based on daily price time series of 7111 cryptocurrencies that comprise a significant part of all currently available cryptoassets (see Methods for details)  From these price series, we have estimated their logarithmic returns 2/16 Log-return, r )  The black horizontal arrow represents a given position of the expanding time window (at t = 2004 days) used to sample the return series over the entire history of Bitcoin 
This time window expands in weekly steps (seven time series observations), and for each position, we separate the positive (blue) from the negative (red) price returns  The gray line illustrates observations that will be included in future positions of the expanding time window (t > 2004)  (b) Survival functions or the complementary cumulative distributions of positive (blue) and negative (red) price returns within the expanding time window for t = 2004 days and above the lower bound of the power-law regime estimated from the Clauset-Shalizi-Newman method  
The dashed lines show the adjusted power-law functions, p(r) ∼ r −α , with α = 4 5 for positive returns and α = 3 0 for negative returns  (c) Time series of the power-law exponents α t for the positive (blue) and negative (red) return distributions obtained by expanding the time window from the hundredth observation (t = 100) to the latest available price return of Bitcoin 
The circular markers represent the values for the window position at t = 2004 days and the dashed lines indicate the median of the power-law exponents ( α+ = 4 50 for positive returns and α− = 2 99 for negative returns)  (d) Time series of the p-values related to the power-law hypothesis of positive (blue) and negative (red) price returns for every position of the expanding time window 
The dashed line indicates the threshold (p = 0 1) above which the power-law hypothesis cannot be rejected  For Bitcoin, the power-law hypothesis is never rejected for positive returns (fraction of rejection f r = 0) and rejected in only 4% of the expanding time window positions (fraction of rejection f r = 0 04) 
where x t represents the price of a given cryptocurrency at day t  All return time series in our analysis have at least 200 observations (see Supplementary Figure for the length distribution)  Figure (a) illustrates Bitcoin's series of daily returns  To investigate whether and how returns have changed over the aging and growing processes of all cryptocurrencies, we sample all time series of log-returns using a time window that expands in weekly steps (seven time series observations), starting from the hundredth observation to the latest return observation 
In each step, we separate the positive from the negative return values and estimate their power-law behavior using the Clauset-Shalizi-Newman method   Figure (a) further illustrates this procedure, where the vertical dashed line represents a given position of the time window (t = 2004 days), the blue and red lines indicate positive and negative returns, respectively, and the gray lines show the return observations that will be included in the expanding time window in future steps 
Moreover, Fig  (b) shows the corresponding survival functions (or complementary cumulative distributions) for the positive (blue) and negative (red) returns of Bitcoin within the time window highlighted in Fig  (a)  These survival functions correspond to return values above the lower bound of the power-law regime (r min ) and dashed lines in Fig  (b) show the power-law functions adjusted to data, that is,
with α = 4 5 for the positive returns and α = 3 0 for the negative returns in this particular position of the time window (t = 2004 days)  We have further verified the goodness of the power-law fits using the approach proposed by Clauset et al  (see also Preis et al  )  As detailed in the Methods section, this approach consists in generating several synthetic samples under the power-law hypothesis, adjusting these simulated samples, and estimating the fraction of times the Kolmogorov-Smirnov distance between the adjusted power-law and the synthetic samples is larger than the value calculated from the empirical data 
This fraction defines a p-value and allows us to reject or not the power-law hypothesis of the return distributions under a given confidence level  Following Refs  we consider the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), rejecting the power-law hypothesis when p-value ≤ 0 1 
For the particular examples in Fig  (b), the p-values are respectively 1 00 and 0 17 for the positive and negative returns, and thus we cannot reject the power-law hypotheses  After sampling the entire price return series, we obtain time series for the power-law exponents (α t ) associated with positive and negative returns as well as the corresponding p-values time series for each step t of the expanding time window 
These time series allow us to reconstruct the aging process of the return distributions over the entire history of each cryptoasset and probe possible time-dependent patterns  Figures ) and 1(d) show the power-law exponents and p-values time series for the case of Bitcoin  The power-law hypothesis is never rejected for positive returns and rarely rejected for negative returns (about 4% of times) 
Moreover, the power-law exponents exhibit large fluctuations at the beginning of the time series and become more stable as Bitcoin matures as a financial asset (a similar tendency as reported by Begušić et al  )  The time evolution of these exponents further shows that the asymmetry between positive and negative returns observed in Fig  ) is not an incidental feature of a particular moment in Bitcoin's history 
Indeed, the power-law exponent for positive returns is almost always larger than the exponent for negative returns, implying that large negative price returns have been more likely to occur than their positive counterparts over nearly the entire history of Bitcoin covered by our data  However, while the difference between positive and negative exponents has approached a constant value, both exponents exhibit an increasing trend, indicating that large price variations are becoming less frequent with the coming-of-age of Bitcoin 
The previous analysis motivates us to ask whether the entire cryptocurrency market behaves similarly to Bitcoin and what other common patterns digital currencies tend to follow  To start answering this question, we have considered the p-values series of all cryptocurrencies to verify if the power-law hypothesis holds in general 
Figure (a) shows the percentage of cryptoassets rejecting the power-law hypothesis in at most a given fraction of the weekly positions of the expanding time window ( f r )  Remarkably, the hypothesis that large price movements (positive or negative) follow a power-law distribution is never rejected over the entire history of about 70% of all digital currencies in our dataset 
This analysis also shows that only ≈2% of cryptocurrencies reject the power-law hypothesis in more than half of the positions of the expanding time window ( f r ≥ 0 5)  For instance, considering a 10% threshold as a criterion ( f r ≤ 0 1), we find that about 85% of cryptocurrencies have return distributions adequately modeled by power laws 
Increasing this threshold to a more lenient 20% threshold ( f r ≤ 0 2), we find large price movements to be power-law distributed for about 91% of cryptocurrencies  These results thus provide strong evidence that cryptoassets, fairly generally, present large price movements quite well described by power-law distributions 
Moreover, this conclusion is robust when starting the expanding window with a greater   Large price movements are power-law distributed over the entire history of most cryptocurrencies with median values typically smaller than those found for traditional assets  (a) Percentage of cryptoassets rejecting the power-law hypothesis for large positive (blue) or negative (red) price returns in at most a given fraction of the weekly positions of the expanding time window ( f r ) used to sample the return series 
Remarkably, 68% of all 7111 digital currencies are compatible with the power-law hypothesis over their entire history, and about 91% of them reject the power-law hypothesis in less than 20% of the positions of the expanding time window ( f r ≤ 0 2)  (b) Probability distributions obtained via kernel density estimation of the median values of the power-law exponents along the history of each digital currency 
The blue curve shows the distribution of the median exponents related to positive returns ( α+ ) and the red curve does the same for negative returns ( α− )  The medians of α+ and α− are indicated by vertical dashed lines  Panels (c) and (d) show the distributions of these median exponents when considering the top 2000 and the top 200 cryptocurrencies by market capitalization, respectively 
We observe that the distributions of α+ and α− tend to shift toward larger values when considering the largest cryptoassets  number of return observations (between 100 and 300 days) and filtering out cryptoassets with missing observations (Supplementary Figures )  Still, it is worth noticing the existence of a few cryptoassets (9 of them) with relatively small market capitalization (ranking below the top 1000) for which the power-law hypothesis is always rejected (Supplementary Table ) 
Having verified that large price movements in the cryptocurrency market are generally well-described by powerlaw distributions, we now focus on the power-law exponents that typically characterize each cryptoasset  To do so, we select all exponent estimates over the entire history of each digital asset for which the power-law hypothesis is not rejected and calculate their median values for both the positive ( α+ ) and negative ( α− ) returns 
The dashed lines in Fig  ) show these median values for Bitcoin where α+ = 4 50 and α− = 2 99  It is worth noticing that the variance of large price movements σ 2 is finite only for α > 3, as the integral σ 2 ∼ ∞ r min r 2 p(r)dr diverges outside this interval  Thus, while the typical variance of large positive returns is finite for Bitcoin, negative returns are at the limit of not having a typical scale and are thus susceptible to much larger variations 
Figure shows the probability distribution for the median power-law exponents of all cryptoassets grouped by large positive and negative returns  We note that the distribution of typical power-law exponents associated with large positive returns is shifted to smaller values when compared with the distribution of exponents related to large negative returns 
The medians of these typical exponents are respectively 2 78 and 3 11 for positive and negative returns  This result suggests that the asymmetry in large price movements we have observed for Bitcoin is an overall feature of the cryptocurrency market  By calculating the difference between the typical exponents related to positive and negative large returns (∆α = α+ − α− ) for each digital currency, we find that about 2/3 of cryptocurrencies have α+ < α− (see Supplementary Figure for the probability distribution of ∆α) 
Thus, unlike Bitcoin, most cryptocurrencies have been more susceptible to large positive price variations than negative ones  While this asymmetry in the return distributions indicates that extremely large price variations tend to be positive, it does not necessarily imply positive price variations are more common for any threshold in the return values 
This happens because the fraction of events in each tail is also related to the lower bound of the power-law regime (r min )  However, we have found the distribution of r min to be similar among the positive and negative returns [Supplementary Figure ]  The distribution of high percentile scores (such as the 90th percentile) is also shifted to larger values for positive returns [Supplementary Figure ] 
Moreover, this asymmetry in high percentile scores related to positive and negative returns is systematic along the evolution of the power-law exponents [Supplementary Figure ]  These results thus indicate that there is indeed more probability mass in the positive tails than in the negative ones, a feature that likely reflects the current expansion of the cryptocurrency market as a whole 
The distributions in Fig  ) also show that large price variations do not have a finite variance for a significant part of cryptoassets, that is, α+ ≤ 3 for 62% of cryptocurrencies and α− ≤ 3 for 44% of cryptocurrencies  A significant part of the cryptocurrency market is thus prone to price variations with no typical scale 
Intriguingly, we further note the existence of a minority group of cryptoassets with α+ ≤ 2 (7%) or α− ≤ 2 (3%)  These cryptocurrencies, whose representative members are Counos X (CCXX, rank 216) with α − = 1 96 and α + = 1 84 and Chainbing (CBG, rank 236) with α + = 1 87, are even more susceptible to extreme price variations as one cannot even define the average value µ for large price returns, as the integral µ ∼ ∞ r min rp(r)dr diverges for α ≤ 2  We have also replicated the previous analysis when considering cryptocurrencies in the top 2000 and top 200 rankings of market capitalization (as of July 2022) 
Figures ) and 2(d) show the probability distribution for the median power-law exponents of these two groups  We observe that these distributions are more localized (particularly for the top 200) than the equivalent distributions for all cryptocurrencies  The fraction of cryptocurrencies with no typical scale for large price returns ( α+ ≤ 3 and α− ≤ 3) is significantly lower in these two groups compared to all cryptocurrencies 
In the top 2000 cryptocurrencies, 51% have α+ ≤ 3 and 26% have α− ≤ 3  These fractions are even smaller among the top 200 cryptocurrencies, with only 44% and 15% not presenting a typical scale for large positive and negative price returns, respectively  We further observe a decrease in the fraction of cryptoassets for which the average value for large price returns is not even finite, as only 2% and 1% of top 2000 cryptoassets have α+ ≤ 2 and α− ≤ 2  This reduction is more impressive among the top 200 cryptocurrencies as only the cryptoasset Fei USD (FEI, rank 78) has α+ = 1 97 and none is characterized by α− ≤ 2  The medians of α+ and α− also increase from 2 78 and 3 11 for all cryptocurrencies to 2 98 and 3 35 for the top 2000 and to 3 08 and 3 58 for the top 200 cryptocurrencies 
Conversely, the asymmetry between positive and negative large price returns does not differ much among the three groups, with the condition α+ < α− holding only for a slightly larger fraction of top 2000 (69 1%) and top 200 (70 6%) cryptoassets compared to all cryptocurrencies (66 4%)  Moreover, all these patterns are robust when filtering out time series with sampling issues or when considering only cryptoassets that stay compatible with the power-law hypothesis in more than 90% of the positions of the expanding time window (Supplementary Figures ) 
We also investigate whether the patterns related to the median of the power-law exponents differ among groups of cryptocurrencies with different designs and purposes  To do so, we group digital assets using the 50 most common tags in our dataset (e g  "bnb-chain", "defi", and "collectibles-nfts") and estimate the probability distributions of the median exponents α+ and α− (Supplementary Figures ) 
These results show that design and purpose affect the dynamics of large price variations in the cryptocurrency market as the medians of typical exponents range from 2 4 to 3 7 among the groups  The lowest values occur for cryptocurrencies tagged as "doggone-doggerel" (medians of α+ and α− are 2 38 and 2 83), "memes" (2 41 and 2 87), and "stablecoin" (2 65 and 2 79) 
Digital currencies belonging to the first two tags overlap a lot and have Dogecoin (DOGE, rank 9) and Shiba Inu (SHIB, rank 13) as the most important representatives  Cryptoassets with these tags usually have humorous characteristics (such as an Internet meme) and several have been considered as a form of pump-and-dump scheme , a type of financial fraud in which false statements artificially inflate asset prices so the scheme operators sell their overvalued cryptoassets 
Conversely, cryptoassets tagged as "stablecoin" represent a class of cryptocurrencies designed to have a fixed exchange rate to a reference asset (such as a national currency or precious metal)   While the price of stablecoins tends to stay around the target values, their price series are also marked by sharp variations, which in turn are responsible for their typically small power-law exponents 
This type of cryptoasset has been shown to be prone to failures , such as the recent examples of TerraUSD (UST) and Tron's USDD (USDD) that lost their pegs to the US Dollar producing large variations in their price series  The asymmetry between positive and negative large returns also emerges when grouping the cryptocurrencies using their tags 
All 50 tags have distributions of α+ shifted to smaller values when compared with the distributions of α− , with differences between their medians ranging from −0 74 ("okex-blockdream-ventures-portfolio") to −0 14 ("stablecoin")  Indeed, only four ('stablecoin", "scrypt", "fantom-ecosystem" and "alameda-research-portfolio") out of the fifty groupings have both distributions indistinguishable under a two-sample Kolmogorov-Smirnov test (p-value > 0 05) 
Focusing now on the evolution of the power-law exponents quantified by the time series α t for positive and negative returns, we ask whether these exponents present particular time trends  For Bitcoin [Fig  )], α t seems to increase with time for both positive and negative returns  At the same time, the results of Fig  also suggest that market capitalization affects these power-law exponents 
To verify these possibilities, we assume the power-law exponents (α t ) to be linearly associated with the cryptocurrency's age (y t , measured in years) and the logarithm of market capitalization (log c t )  As detailed in the Methods section, we frame this problem using a hierarchical Bayesian model 
This approach assumes that the linear coefficients associated with the effects of age (A) and market capitalization (C) of each digital currency are drawn from distributions with means µ A and µ C and standard deviations σ A and σ C , which are in turn distributed according to global distributions representing the overall impact of these quantities on the cryptocurrency market 
The Bayesian inference process consists of estimating the posterior probability distributions of the linear coefficients for each cryptocurrency as well as the posterior distributions of µ A , µ C , σ A , and σ C , allowing us to simultaneously probe asset-specific tendencies and overall market characteristics 
Moreover, we restrict this analysis to the 2140 digital currencies having more than 50 observations of market capitalization concomitantly to the time series of the power-law exponents in order to have enough data points for detecting possible trends  When considering the overall market characteristics, we find that the 94% highest density intervals for µ A ([-0 01, 0 06] for positive and [-0 02, 0 03] for negative returns) and µ C ([-0 02, 0 03] for positive and [-0 001, 0 04] for negative returns) include the zero (see Supplementary Figure for their distributions) 
Thus, there is no evidence of a unique overall pattern for the association between the power-law exponents and age or market capitalization followed by a significant part of the cryptocurrency market  Indeed, the 94% highest density intervals for σ A ([0 87, 0 93] for positive and [0 63, 0 70] for negative returns) and σ C ([0 57, 0 61] for positive and [0 49, 0 52] for negative returns) indicate that the cryptocurrency market is highly heterogeneous regarding the evolution of power-law exponents associated with large price variations (see Supplementary Figure for the distributions of σ A and σ C )  Figure illustrates these heterogeneous behaviors by plotting the posterior probability distributions for the linear coefficients associated with the effects of age (A) and market capitalization (C) for the top 20 digital assets, where cryptocurrencies which are significantly affected (that is, the 94% highest density intervals for A or C do not include the zero) by these quantities are highlighted in boldface 
Even this small selection of digital  currencies already presents a myriad of patterns  First, we observe that the power-law exponents of a few top 20 cryptocurrencies are neither correlated with age nor market capitalization  That is the case of Shiba Inu (SHIB, rank 13) and Dai (DAI, rank 11) for both positive and negative returns, UNUS SED LEO (LEO, rank 18) and Polkadot (DOT, rank 12) for the positive returns, and USDCoin (USDC, rank 4) and Solana (SOL, rank 9) for negative returns 
There are also cryptocurrencies with exponents positively or negatively correlated only with market capitalization  Examples include Tether (USDT, rank 3) and Dogecoin (DOGE, rank 10), for which the power-law exponents associated with positive returns increase with market capitalization, and Binance USD (BUSD, rank 6), for which power-law exponents associated with positive and negative returns decrease with market capitalization 
We also observe cryptocurrencies for which age and market capitalization simultaneously affect the power-law exponents  Polygon (MATIC, rank 14) is an example where the power-law exponents associated with positive returns tend to increase with age and decrease with market capitalization  Finally, there are also cryptocurrencies with power-law exponents only associated with age 
That is the case of Bitcoin (BTC, rank 1), Ethereum (ETH, rank 2), and Cardano (ADA, rank 8), for which the power-law exponents related to positive and negative returns increase with age, but also the case of Uniswap (UNI, rank 19), for which the exponents decrease with age  Figure systematically extends the observations made for the top 20 cryptoassets to all 2140 digital currencies for which we have modeled the changes in the power-law exponents as a function of age and market capitalization 
First, we note that only 10% of cryptocurrencies have power-law exponents not significantly affected by age and market capitalization  The vast majority (90%) displays some relationship with these quantities  However, these associations are as varied as the ones we have observed for the top 20 cryptoassets 
About 52% of cryptocurrencies have power-law exponents simultaneously affected by age and market capitalization  In this group, these quantities simultaneously impact the exponents related to positive and negative returns of 34% of cryptoassets, whereas the remainder is affected only in the positive tail (9%) or only in the negative tail (9%) 
Moving back in the hierarchy, we find that the power-law exponents of 32% of cryptocurrencies are affected only by age while a much minor fraction (6%) is affected only by market capitalization  Within the group only affected by age, we observe that the effects are slightly more frequent only on the exponents related to negative returns (12%), compared to cases where effects are restricted only to positive returns (10%) or simultaneously affect both tails (10%) 
Finally, within the minor group only affected by market capitalization, we note that associations more frequently involve only exponents related to negative returns (3%) compared to the other two cases (2% only positive returns and 1% for both positive and negative returns)  Beyond the previous discussion about whether positive or negative returns are simultaneously or individually affected by age and market capitalization, we have also categorized the direction of the trend imposed by these two quantities on the power-law exponents 
Blue rectangles in Fig  represent the fraction of relationships for which increasing age or market capitalization (or both) is associated with a raise in the power-law exponents  About 28% of all cryptocurrencies exhibit this pattern in which large price variations are expected to occur less frequently as they grow and age 
Conversely, the red rectangles in Fig  depict the fraction of relationships for which increasing age or market capitalization (or both) is associated with a reduction in the power-law exponents  This case comprises about 25% of all cryptocurrencies for which large price variations are likely to become more frequent as they grow in market capitalization and age 
Still, the majority of associations represented by green rectangles refer to the case where the effects of age and market capitalization point in different directions (e g  exponents increasing with age while decreasing with market capitalization)  About 36% of cryptocurrencies fit this condition which in turn contributes to consolidating the cumbersome hierarchical structure of patterns displayed by cryptocurrencies regarding the dynamics of large price variations 
This complex picture is not much different when considering only cryptocurrencies in the top 200 market capitalization rank (Supplementary Figure )  However, we do observe an increased prevalence of patterns characterized by exponents that rise with age and market capitalization (37%), suggesting that large price variations are becoming less frequent among the top 200 cryptocurrencies than in the overall market 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 36% of the associations are classified as mixed trends (green rectangles), 28% are increasing trends (blue rectangles), and 26% are decreasing trends (red rectangles)  We have studied the distributions of large price variations of a significant part of the digital assets that currently comprise the entirety of the cryptocurrency market 
Unlike previous work, we have estimated these distributions for entire historical price records of each digital currency, and we have identified the patterns under which the return distributions change as cryptoassets age and grow in market capitalization  Similarly to conventional financial assets , our findings show that the return distributions of the vast majority of cryptoassets have tails that are described well by power-law functions along their entire history 
The typical power-law exponents of cryptocurrencies (α ∼ 3) are, however, significantly smaller than those reported for conventional assets (α ∼ 4)   This feature corroborates the widespread belief that cryptoassets are indeed considerably more risky for investments than stocks or other more traditional financial assets 
Indeed, we have found that about half of the cryptocurrencies in our analysis do not have a characteristic scale for price variations, and are thus prone to much higher price variations than those typically observed in stock markets  On the upside, we have also identified an asymmetry in the power-law exponents for positive and negative returns in about 2/3 of all considered cryptocurrencies, such that these exponents are smaller for positive than they are for negative returns 
This means that sizable positive price variations have generally been more likely to occur than equally sizable negative price variations, which in turn may also reflect the recent overall expansion of the cryptocurrency market  Using a hierarchical Bayesian linear model, we have also simultaneously investigated the overall market characteristics and asset-specific tendencies regarding the effects of age and market capitalization on the power-law exponents 
We have found that the cryptocurrency market is highly heterogeneous regarding the trends exhibited by each cryptocurrency; however, only a small fraction of cryptocurrencies (10%) have power-law exponents neither correlated with age nor market capitalization  These associations have been mostly ignored by the current literature and are probably related to the still-early developmental stage of the cryptocurrency market as a whole 
Overall, 36% of cryptocurrencies present trends that do not systematically contribute to increasing or decreasing their power-law exponents as they age and grow in market capitalization  On the other hand, for 26% of cryptocurrencies, aging and growing market capitalization are both associated with a reduction in their power-law exponents, thus contributing to the rise in the frequency of large price variations in their dynamics 
Only about 28% of cryptocurrencies present trends in which the power-law exponents increase with age and market capitalization, favoring thus large price variations to become less likely  These results somehow juxtapose with findings about the increasing informational efficiency of the cryptocurrency market  
In fact, if on the one hand the cryptocurrency market is becoming more informationally efficient, then on the other our findings indicate that there is no clear trend toward decreasing the risks of sizable variations in the prices of most considered cryptoassets  In other words, risk and efficiency thus appear to be moving towards different directions in the cryptocurrency market 
To conclude, we hope that our findings will contribute significantly to the better understanding of the dynamics of large price variations in the cryptocurrency market as a whole, and not just for a small subset of selected digital assets, which is especially relevant due to the diminishing concentration of market capitalization among the top digital currencies, and also because of the considerable impact these new assets may have in our increasingly digital economy 
Our results are based on time series of the daily closing prices (in USD) for all cryptoassets listed on CoinMar-ketCap (coinmarketcap com) as of 25 July 2022 [see Supplementary Figure (a) for a visualization of the increasing number cryptoassets listed on CoinMarketCap since 2013]  These time series were automatically gathered using the cryptoCMD Python package and other information such as the tags associated with each cryptoasset were obtained via the CoinMarketCap API  
In addition, we have also obtained the daily market capitalization time series (in USD) from all cryptoassets which had this information available at the time  Earliest records available from CoinMarketCap date from 29 April 2013 and the latest records used in our analysis correspond to 25 July 2022  Out of 9943 cryptocurrencies, we have restricted our analysis to the 7111 with at least 200 price-return observations 
The median length of these time series is 446 observations [see the distribution of series length in Supplementary Figure   We have estimated the power-law behavior of the return distributions by applying the Clauset-Shalizi-Newman method to the return time series r t   In particular, we have sampled each of these time series using an expanding time window that starts at the hundredth observation and grows in weekly steps (seven data points each step) 
For each position of the expanding time window, we have separated the positive returns from the negative ones and applied the Clauset-Shalizi-Newman method to each set  This approach consists of obtaining the maximum likelihood estimate for the power-law exponent, α = 1 + n/ (∑ n t=1 ln r t /r min ) , where r min is the lower bound of the power-law regime and n is the number of (positive or negative) return observations in the power-law regime for a given position of the expanding time window 
The value r min is estimated from data by minimizing the Kolmogorov-Smirnov statistic between the empirical distribution and the power-law model  The Clauset-Shalizi-Newman method yields an unbiased and consistent estimator , in a sense that as the sample increases indefinitely, the estimated power-law exponent converges in distribution to the actual value 
Moreover, we have used the implementation available on the powerlaw Python package   In addition to obtaining the power-law exponents, we have also verified the adequacy of the power-law hypothesis using the procedure originally proposed by Clauset et al  as adapted by Preis et al    This procedure consists of generating synthetic samples under the power-law hypothesis with the same properties of the empirical data under analysis (that is, same length and parameters α and r min ), adjusting the simulated data with the power-law model via the Clauset-Shalizi-Newman method, and calculating the Kolmogorov-Smirnov statistic (κ syn ) between the distributions obtained from the simulated samples and the adjusted power-law model 
Next, the values of κ syn are compared to the Kolmogorov-Smirnov statistic calculated between empirical data and the power-law model (κ)  Finally, a p-value is defined by calculating the fraction of times for which κ syn > κ  We have used one thousand synthetic samples for each position of the expanding time window and the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), such that the power-law hypothesis is rejected whenever p-value ≤ 0 1 
We have estimated the effects of age and market capitalization on the power-law exponents associated with positive or negative returns of a given cryptocurrency using the linear model where α t represents the power-law exponent, log c t is the logarithm of the market capitalization, and y t is the age (in years) of the cryptocurrency at t-th observation 
Moreover, K is the intercept of the association, while C and A are linear coefficients quantifying the effects of market capitalization and age, respectively  Finally, N (µ, σ ) stands for the normal distribution with mean µ and standard deviation σ , such that the parameter ε accounts for the unobserved determinants in the dynamics of the power-law exponents 
We have framed this problem using the hierarchical Bayesian approach such that each power-law exponent α t is nested within a cryptocurrency with model parameters considered as random variables normally distributed with parameters that are also random variables  Mathematically, for each cryptocurrency, we have
12/16 where µ K , σ K , µ C , σ C , µ A , and σ A are hyperparameters  These hyperparameters are assumed to be distributed according to distributions that quantify the overall impact of age and market capitalization on the cryptocurrency market as a whole  We have performed this Bayesian regression for exponents related to positive and negative returns separately, and used noninformative prior and hyperprior distributions in order not to bias the posterior estimation  
Specifically, we have considered and ε ∼ U (0, 10 2 ) , where U (a, b) stands for the uniform distribution in the interval [a, b] and Inv−Γ(θ , γ) represents the inverse gamma distribution with shape and scale parameters θ and γ, respectively  For the numerical implementation, we have relied on the PyMC Python package and sampled the posterior distributions via the gradient-based Hamiltonian Monte Carlo no-U-Turn-sampler method 
We have run four parallel chains with 2500 iterations each (1000 burn-in samples) to allow good mixing and estimated the Gelman-Rubin convergence statistic (R-hat) to ensure the convergence of the sampling approach (R-hat was always close to one)  In addition, we have also verified that models describing the power-law exponents as a function of only age (C → 0 in Eq  3) or only market capitalization (A → 0 in Eq  3) yield significantly worse descriptions of our data as quantified by the Widely Applicable Information Criterion (WAIC) and the Pareto Smoothed Importance Sampling Leave-One-Out cross-validation (PSIS-LOO) (see Supplementary Table )  ) is larger than r 90 estimated from negative returns (r − 90 ) 
This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails  The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels  The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Sampling issues refer to missing data and problems caused by prices of cryptoassets decreasing to zero  We note that these distributions barely change when considering only cryptocurrencies without any sampling issue  Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  (two-sample Kolmogorov-Smirnov test, p > 0 05) 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 35% of the associations are classified as mixed trends (green rectangles), 37% are increasing trends (blue rectangles), and 18% are decreasing trends (red rectangles) 
Context 2: Paper Info

Title: Environmental variability and network structure determine the optimal plasticity mechanisms in embodied agents
Publish Date: Unkown
Author List: Sina Khajehabdollahi (from Department of Computer Science, University of Tübingen)

Figure

Figure2: An outline of the network controlling the foraging agent The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig 1 The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent
Figure4: The evolved parameters θ = (θ 1 ,      , θ 8 ) of the plasticity rule for the reward prediction (a ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,      , 1, and σ ∈ 0, 0 1,      , 1 in all 100 combinations) Despite the relatively small difference between the tasks, the evolved learning rules differ considerably For visual guidance, the lines connect θs from the same run 
Figure5: a The trajectory of an agent (blue line) in the 2D environment A well-trained agent will approach and consume food with positive values (green dots) and avoid negative food (red dots) b The learning rate of the plastic sensory network eta p grows with the distance between environments d e c  and decreases with the frequency of environmental change d The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network e The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food 

abstract

The evolutionary balance between innate and learned behaviors is highly intricate, and different organisms have found different solutions to this problem  We hypothesize that the emergence and exact form of learning behaviors is naturally connected with the statistics of environmental fluctuations and tasks an organism needs to solve 
Here, we study how different aspects of simulated environments shape an evolved synaptic plasticity rule in static and moving artificial agents  We demonstrate that environmental fluctuation and uncertainty control the reliance of artificial organisms on plasticity  Interestingly, the form of the emerging plasticity rule is additionally determined by the details of the task the artificial organisms are aiming to solve 
Moreover, we show that coevolution between static connectivity and interacting plasticity mechanisms in distinct sub-networks changes the function and form of the emerging plasticity rules in embodied agents performing a foraging task  One of the defining features of living organisms is their ability to adapt to their environment and incorporate new information to modify their behavior 
It is unclear how the ability to learn first evolved , but its utility appears evident  Natural environments are too complex for all the necessary information to be hardcoded genetically and more importantly, they keep changing during an organism's lifetime in ways that cannot be anticipated ;   The link between learning and environmental uncertainty and fluctuation has been extensively demonstrated in both natural ; , and artificial environments  
Nevertheless, the ability to learn does not come without costs  For the capacity to learn to be beneficial in evolutionary terms, a costly nurturing period is often required, a phenomenon observed in both biological , and artificial organisms   Additionally, it has been shown that in some complex environments, hardcoded behaviors may be superior to learned ones given limits in the agent's lifetime and envi-ronmental uncertainty ; ;  
The theoretical investigation of the optimal balance between learned and innate behaviors in natural and artificial systems goes back several decades  However, it has recently found also a wide range of applications in applied AI systems ;   Most AI systems are trained for specific tasks, and have no need for modification after their training has been completed 
Still, technological advances and the necessity to solve broad families of tasks make discussions about life-like AI systems relevant to a wide range of potential application areas  Thus the idea of open-ended AI agents that can continually interact with and adapt to changing environments has become particularly appealing 
Many different approaches for introducing lifelong learning in artificial agents have been proposed  Some of them draw direct inspiration from actual biological systems ;   Among them, the most biologically plausible solution is to equip artificial neural networks with some local neural plasticity , similar to the large variety of synaptic plasticity mechanisms ; ; that performs the bulk of the learning in the brains of living organisms  
The artificial plasticity mechanisms can be optimized to modify the connectivity of the artificial neural networks toward solving a particular task  The optimization can use a variety of approaches, most commonly evolutionary computation  The idea of meta-learning or optimizing synaptic plasticity rules to perform specific functions has been recently established as an engineering tool that can compete with stateof-the-art machine learning algorithms on various complex tasks ; ; Pedersen and Risi (2021);  
Additionally, it can be used to reverse engineer actual plasticity mechanisms found in biological neural networks and uncover their functions ;   Here, we study the effect that different factors (environ-arXiv:2303 06734v1 [q-bio NC] 12 Mar 2023 mental fluctuation and reliability, task complexity) have on the form of evolved functional reward-modulated plasticity rules 
We investigate the evolution of plasticity rules in static, single-layer simple networks  Then we increase the complexity by switching to moving agents performing a complex foraging task  In both cases, we study the impact of different environmental parameters on the form of the evolved plasticity mechanisms and the interaction of learned and static network connectivity 
Interestingly, we find that different environmental conditions and different combinations of static and plastic connectivity have a very large impact on the resulting plasticity rules  We imagine an agent who must forage to survive in an environment presenting various types of complex food particles  Each food particle is composed of various amounts and combinations of N ingredients that can have positive (food) or negative (poison) values 
The value of a food particle is a weighted sum of its ingredients  To predict the reward value of a given resource, the agent must learn the values of these ingredients by interacting with the environment  The priors could be generated by genetic memory, but the exact values are subject to change  To introduce environmental variability, we stochastically change the values of the ingredients 
More precisely, we define two ingredient-value distributions E 1 and E 2 and switch between them, with probability p tr for every time step  We control how (dis)similar the environments are by parametrically setting E 2 = (1 − 2d e )E 1 , with d e ∈ [0, 1] serving as a distance proxy for the environments; when d e = 0, the environment remains unchanged, and when d e = 1 the value of each ingredient fully reverses when the environmental transition happens 
For simplicity, we take values of the ingredients in E 1 equally spaced between -1 and 1 (for the visualization, see Fig  )  The static agent receives passively presented food as a vector of ingredients and can assess its compound value using the linear summation of its sensors with the (learned or evolved) weights, see Fig   
The network consists of N sensory neurons that are projecting to a single post-synaptic neuron  At each time step, an input X t = (x 1 ,       , x N ) is presented, were the value x i , i ∈ {1,       , N } represents the quantity of the ingredient i  We draw x i independently form a uniform distribution on the [0, 1] interval (x i ∼ U (0, 1)) 
The value of each ingredient w c i is determined by the environment (E 1 or E 2 )  The postsynaptic neuron outputs a prediction of the food X t value as y t = g(W X T t )  Throughout the paper, g will be either the identity function, in which case the prediction neuron is linear, or a step-function; however, it could be any other nonlinearity, such as a sigmoid or ReLU 
After outputting the prediction, the neuron receives feedback in the form of the real value of the input R t   The real value is computed as R t = W c X T t + ξ, where W c = (w c 1 ,       , w c N ) is the actual value of the ingredients, and ξ is a term summarizing the noise of reward and sensing system ξ ∼ N (0, σ) 
Figure : An outline of the static agent's network  The sensor layer receives inputs representing the quantity of each ingredient of a given food at each time step  The agent computes the prediction of the food's value y t and is then given the true value R t ; it finally uses this information in the plasticity rule to update the weight matrix 
For the evolutionary adjustment of the agent's parameters, the loss of the static agent is the sum of the mean squared errors (MSE) between its prediction y t and the reward R t over the lifetime of the agent  The agent's initial weights are set to the average of the two ingredient value distributions, which is the optimal initial value for the case of symmetric switching of environments that we consider here 
As a next step, we incorporate the sensory network of static agents into embodied agents that can move around in an environment scattered with food  To this end, we merge the static agent's network with a second, non-plastic motor network that is responsible for controlling the motion of the agent in the environment 
Specifically, the original plastic network now provides the agent with information about the value of the nearest food  The embodied agent has additional sensors for the distance from the nearest food, the angle between the current velocity and the nearest food direction, its own velocity, and its own energy level (sum of consumed food values) 
These inputs are processed by two hidden layers (of 30 and 15 neurons) with tanh activation  The network's outputs are angular and linear acceleration, Fig    The embodied agents spawn in a 2D space with periodic boundary conditions along with a number of food particles that are selected such that the mean of the food value distribution is ∼ 0  An agent can eat food by approaching it sufficiently closely, and each time a food particle is eaten, it is The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig   
The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent  These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent re-spawned with the same value somewhere randomly on the grid (following the setup of ) 
After 5000 time steps, the cumulative reward of the agent (the sum of the values of all the food it consumed) is taken as its fitness  During the evolutionary optimization, the parameters for both the motor network (connections) and plastic network (learning rule parameters) are co-evolved, and so agents must simultaneously learn to move and discriminate good/bad food 
Reward-modulated plasticity is one of the most promising explanations for biological credit assignment   In our network, the plasticity rule that updates the weights of the linear sensor network is a rewardmodulated rule which is parameterized as a linear combination of the input, the output, and the reward at each time step:
Additionally, after each plasticity step, the weights are normalized by mean subtraction, an important step for the stabilization of Hebbian-like plasticity rules   We use a genetic algorithm to optimize the learning rate η p and amplitudes of different terms θ = (θ 1 ,       , θ 8 )  The successful plasticity rule after many food presentations must converge to a weight vector that predicts the correct food values (or allows the agent to correctly decide whether to eat a food or avoid it) 
To have comparable results, we divide θ = (θ 1 ,       , θ 8 ) by We then multiply the learning rate η p with θ max to maintain the rule's evolved form unchanged, η norm p = η p • θ max   In the following, we always use normalized η p and θ, omitting norm   To evolve the plasticity rule and the moving agents' motor networks, we use a simple genetic algorithm with elitism  
The agents' parameters are initialized at random (drawn from a Gaussian distribution), then the sensory network is trained by the plasticity rule and finally, the agents are evaluated  After each generation, the bestperforming agents (top 10 % of the population size) are selected and copied into the next generation 
The remaining 90 % of the generation is repopulated with mutated copies of the best-performing agents  We mutate agents by adding independent Gaussian noise (σ = 0 1) to its parameters  To start with, we consider a static agent whose goal is to identify the value of presented food correctly  The static reward-prediction network quickly evolves the parameters of the learning rule, successfully solving the prediction task 
We first look at the evolved learning rate η p , which determines how fast (if at all) the network's weight vector is updated during the lifetime of the agents  We identify three factors that control the learning rate parameter the EA converges to: the distance between the environments, the noisiness of the reward, and the rate of environmental transition 
The first natural factor is the distance d e between the two environments, with a larger distance requiring a higher learning rate, Fig    This is an expected result since the convergence time to the "correct" weights is highly dependent on the initial conditions  If an agent is born at a point very close to optimality, which naturally happens if the environments are similar, the distance it needs to traverse on the fitness landscape is small 
Therefore it can afford to have a small learning rate, which leads to a more stable convergence and is not affected by noise  A second parameter that impacts the learning rate is the variance of the rewards  The reward an agent receives for the plasticity step contains a noise term ξ that is drawn from a zero mean Gaussian distribution with standard deviation σ 
This parameter controls the unreliability of the agent's sensory system, i e , higher σ means that the information the agent gets about the value of the foods it consumes cannot be fully trusted to reflect the actual value of the foods  As σ increases, the learning rate η p decreases, which means that the more unreliable an environment becomes, the less an agent relies on plasticity to update its weights, Fig   
Indeed for some combinations of relatively small distance d e and high reward variance σ, the EA converges to a learning rate of η p ≈ 0  This means that the agent opts to have no adaptation during its lifetime and remain at the mean of the two environments  It is an optimal solution when the expected loss due to ignoring the environmental transitions is, on average, lower than the loss the plastic network will incur by learning via the (often misleading because of the high σ) environmental cues 
A final factor that affects the learning rate the EA will converge to is the frequency of environmental change during an agent's lifetime  Since the environmental change is modeled as a simple, two-state Markov process (Fig  ), the control parameter is the transition probability p tr   When keeping everything else the same, the learning rate rapidly rises as we increase the transition probability from 0, and after reaching a peak, it begins to decline slowly, eventually reaching zero (Fig  ) 
This means that when environmental transition is very rare, agents opt for a very low learning rate, allowing a slow and stable convergence to an environment-appropriate weight vector that leads to very low losses while the agent remains in that environment  As the rate of environmental transition increases, faster learning is required to speed up convergence in order to exploit the (comparatively shorter) stays in each environment 
Finally, as the environmental transition becomes too fast, the agents opt for slower or even no learning, which keeps them ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,       , 1, and σ ∈ 0, 0 1,       , 1 in all 100 combinations)  Despite the relatively small difference between the tasks, the evolved learning rules differ considerably 
For visual guidance, the lines connect θs from the same run  near the middle of the two environments, ensuring that the average loss of the two environments is minimal (Fig  )  The form of the evolved learning rule depends on the task: Decision vs  Prediction The plasticity parameters θ = (θ 1 ,       , θ 8 ) for the rewardprediction task converge on approximately the same point, regardless of the environmental parameters (Fig  ) 
In particular, θ 3 → 1, θ 5 → −1, θ i → 0 for all other i, and thus the learning rule converges to: Since by definition y t = g(W t X T t ) = W t X T t (g(x) = x in this experiment) and R t = W c X T t + ξ we get: Thus the distribution of ∆W t converges to a distribution with mean 0 and variance depending on η p and σ and W converges to W c  
So this learning rule will match the agent's weight vector with the vector of ingredient values in the environment  We examine the robustness of the learning rule the EA discovers by considering a slight modification of our task  Instead of predicting the expected food value, the agent now needs to decide whether to eat the presented food or not 
This is done by introducing a step-function nonlinearity (g(x) = 1 if x ≥ 1 and 0 otherwise)  Then the output y(t) is computed as: Instead of the MSE loss between prediction and actual value, the fitness of the agent is now defined as the sum of the food values it chose to consume (by giving y t = 1)  Besides these two changes, the setup of the experiments remains exactly the same 
The qualitative relation between η p and parameters of environment d e , σ and p tr is preserved in the changed experiment  However, the resulting learning rule is significantly different (Fig  )  The evolution converges to the following learning rule: In both cases, the rule has the form ∆W t = η p X t [α y R t + β y ] 
Thus, the ∆W t is positive or negative depending on whether the reward R t is above or below a threshold (γ = −β y /α y ) that depends on the output decision of the network (y t = 0 or 1)  Both learning rules (for the reward-prediction and decision tasks) have a clear Hebbian form (coordination of preand post-synaptic activity) and use the incoming reward signal as a threshold 
These similarities indicate some common organizing principles of reward-modulated learning rules, but their significant differences highlight the sensitivity of the optimization process to task details  We now turn to the moving embodied agents in the 2D environment  To optimize these agents, both the motor network's connections and the sensory network's plasticity parameters evolve simultaneously 
Since the motor network is initially random and the agent has to move to find food, the number of interactions an agent experiences in its lifetime can be small, slowing down the learning  However, having the larger motor network also has benefits for evolution because it allows the output of the plastic network to be read out and transformed in different ways, resulting in a broad set of solutions 
The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network  e  The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) 
In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food  The agents can solve the task effectively by evolving a functional motor network and a plasticity rule that converges to interpretable weights (Fig  ) 
After ∼ 100 evolutionary steps (Fig  ), the agents can learn the ingredient value distribution using the plastic network and reliably move towards foods with positive values while avoiding the ones with negative values  We compare the dependence of the moving and the static agents on the parameters of the environment: d e and the state transition probability p tr  
At first, in order to simplify the experiment, we set the transition probability to 0, but fixed the initial weights to be the average of E 1 and E 2 , while the real state is E 2   In this experiment, the distance between states d e indicates twice the distance between the agent's initial weights and the optimal weights (the environment's ingredient values) since the agent is initialized at the mean of the two environment distributions 
Same as for the static agent, the learning rate increases with the distance d e (Fig  )  Then, we examine the effect of the environmental transition probability p tr on the evolved learning rate η p   In order for an agent to get sufficient exposure to each environment, we scale down the probability p tr from the equivalent experiment for the static agents 
We find that as the probability of transition increases, the evolved learning rate η p decreases (Fig  )  This fits with the larger trend for the static agent, although there is a clear difference when it comes to the increase for very small transition probabil-ities that were clearly identifiable in the static but not the moving agents 
This could be due to much sparser data and possibly the insufficiently long lifetime of the moving agent (the necessity of scaling makes direct comparisons difficult)  Nevertheless, overall we see that the associations observed in the static agents between environmental distance d e and transition probability p tr and the evolved learning rate η p are largely maintained in the moving agents 
Still, more data would be needed to make any conclusive assertions about the exact effect of these environmental parameters on the emerging plasticity mechanisms  A crucial difference between the static and the moving agents is the function the plasticity has to perform  While in the static agents, the plasticity has to effectively identify the exact value distribution of the environment in order to produce accurate predictions, in the embodied agents, the plasticity has to merely produce a representation of the environment that the motor network can evolve to interpret adequately enough to make decisions about which food to consume 
To illustrate the difference, we plot the Pearson correlation coefficient between an agent's weights and the ingredient values of the environment it is moving in (Fig  )  We use the correlation instead of the MSE loss (which we used for the static agents in Fig  ) because the amplitude of the vector varies a lot for different agents and meaningful The evolved parameters of moving agents' plasticity rule for the g(s) = x, identity (a ) and the step function (Eq 
4) (b ) sensory networks (the environmental parameters here are d e ∈ [0, 1], σ = 0 and p tr = 0 001)  The step function (binary output) network evolved a more structured plasticity rule (e g , θ 3 > 0 for all realizations) than the linear network  Moreover, the learned weights for the identity network (c ) have higher variance and correlate significantly less with the environment's ingredient distribution compared to the learned weights for the thresholded network (d )
conclusions cannot be drawn from the MSE loss  For many agents, the learned weights are consistently anti-correlated with the actual ingredient values (an example of such an agent is shown in Fig  )  This means that the output of the sensory network will have the opposite sign from the actual food value 
While in the static network, this would lead to very bad predictions and high loss, in the foraging task, these agents perform exactly as well as the ones where the weights and ingredients values are positively correlated, since the motor network can simply learn to move towards food for which it gets a negative instead of a positive sensory input 
This additional step of the output of the plastic network going through the motor network before producing any behavior has a strong effect on the plasticity rules that the embodied agents evolve  Specifically, if we look at the emerging rules the top performing agents have evolved (Fig  ), it becomes clear that, unlike the very well-structured rules of the static agents (Fig  ), there is now virtually no discernible pattern or structure 
The difference becomes even clearer if we look at the learned weights (at the end of a simulation) of the best-performing agents (Fig  )  While there is some correlation with the environment's ingredient value distribution, the variance is very large, and they do not seem to converge on the "correct" values in any way 
This is to some extent expected since, unlike the static agents where the network's output has to be exactly correct, driving the evolution of rules that converge to the precise environmental distribution, in the embodied networks, the bulk of the processing is done by the motor network which can evolve to interpret the scalar value of the sensory network's output in a variety of ways 
Thus, as long as the sensory network's plasticity rule co-evolves with the motor network, any plasticity rule that learns to produce consistent information about the value of encountered food can potentially be selected  To further test this assumption, we introduce a bottleneck of information propagation between the sensory and motor networks by using a step-function nonlinearity on the output of the sensory network (Eq 
4)  Similarly to the decision task of the static network, the output of the sensory network now becomes binary  This effectively reduces the flow of information from the sensory to the motor network, forcing the sensory network to consistently decide whether food should be consumed (with the caveat that the motor network can still interpret the binary sign in either of two ways, either consuming food marked with 1 or the ones marked with 0 by the sensory network) 
The agents perform equally well in this variation of the task as before (Fig  ), but now, the evolved plasticity rules seem to be more structured (Fig  )  Moreover, the variance of the learned weights in the bestperforming agents is significantly reduced (Fig  ), which indicates that the bottleneck in the sensory network is in-creasing selection pressure for rules that learn the environment's food distribution accurately 
We find that different sources of variability have a strong impact on the extent to which evolving agents will develop neuronal plasticity mechanisms for adapting to their environment  A diverse environment, a reliable sensory system, and a rate of environmental change that is neither too large nor too small are necessary conditions for an agent to be able to effectively adapt via synaptic plasticity 
Additionally, we find that minor variations of the task an agent has to solve or the parametrization of the network can give rise to significantly different plasticity rules  Our results partially extend to embodied artificial agents performing a foraging task  We show that environmental variability also pushes the development of plasticity in such agents 
Still, in contrast to the static agents, we find that the interaction of a static motor network with a plastic sensory network gives rise to a much greater variety of wellfunctioning learning rules  We propose a potential cause of this degeneracy; as the relatively complex motor network is allowed to read out and process the outputs from the plastic network, any consistent information coming out of these outputs can be potentially interpreted in a behaviorally useful way 
Reducing the information the motor network can extract from the sensory system significantly limits learning rule variability  Our findings on the effect of environmental variability concur with the findings of previous studies that have identified the constraints that environmental variability places on the evolutionary viability of learning behaviors 
We extend these findings in a mechanistic model which uses a biologically plausible learning mechanism (synaptic plasticity)  We show how a simple evolutionary algorithm can optimize the different parameters of a simple reward-modulated plasticity rule for solving simple prediction and decision tasks 
Reward-modulated plasticity has been extensively studied as a plausible mechanism for credit assignment in the brain ; ; and has found several applications in artificial intelligence and robotics tasks ;   Here, we demonstrate how such rules can be very well-tuned to take into account different environmental parameters and produce optimal behavior in simple systems 
Additionally, we demonstrate how the co-evolution of plasticity and static functional connectivity in different subnetworks fundamentally changes the evolutionary pressures on the resulting plasticity rules, allowing for greater diversity in the form of the learning rule and the resulting learned connectivity 
Several studies have demonstrated how, in biological networks, synaptic plasticity heavily interacts with and is driven by network topology   Moreover, it has been recently demonstrated that biological plasticity mechanisms are highly redundant in the sense that any observed neural connectivity or recorded activity can be achieved with a variety of distinct, unrelated learning rules  
This observed redundancy of learning rules in biological settings complements our results and suggests that the function of plasticity rules cannot be studied independently of the connectivity and topology of the networks they are acting on  The optimization of functional plasticity in neural networks is a promising research direction both as a means to understand biological learning processes and as a tool for building more autonomous artificial systems 
Our results suggest that reward-modulated plasticity is highly adaptable to different environments and can be incorporated into larger systems that solve complex tasks  This work studies a simplified toy model of neural network learning in stochastic environments  Future work could be built on this basic framework to examine more complex reward distributions and sources of environmental variability 
Moreover, a greater degree of biological realism could be added by studying more plausible network architectures (multiple plastic layers, recurrent and feedback connections) and more sophisticated plasticity rule parametrizations  Additionally, our foraging simulations were constrained by limited computational resources and were far from exhaustive 
Further experiments can investigate environments with different constraints, food distributions, multiple seasons, more complex motor control systems and interactions of those systems with different sensory networks as well as the inclusion of plasticity on the motor parts of the artificial organisms 
Context 3: A Brief History of Benjamin Franklin's Residences on Craven Street, London: 1757 - 1775 - Journal of the American Revolution
Benjamin Franklin House, 36 Craven St, London  (Photo by Elliott Brown | Wikimedia Commons)
If one looked into Benjamin Franklin’s time on Craven Street, they might initially believe he lived at 36 Craven Street the entirety of his two stays in London based on the plethora of articles on the internet that say so  If they dug a little deeper they might read that he lived at No  27 Craven Street, previously numbered 7, but now numbered 36; or that he lived exclusively at No  7 Craven Street; or that he lived in multiple residences on Craven Street; or that he moved out of No  36 to another house on Craven Street and then moved back into No  36 the last year of his residence  What is one to believe with all of the conflicting accounts? What does the historical record have to say about Franklin’s time on Craven Street?
Figure 1  Spur Alley 1685  “A map of the parish of St Martins in the Fields, taken from ye last survey, with additions (1685)”  (© The British Library Board, Shelfmark: Maps Crace Port  13 2, Item number: 2)
Before Craven Street existed there was Spur Alley, a narrow passageway sandwiched between the Hungerford Market to the north (now Charing Cross Station) and Scotland Yard and the Northumberland House and Garden to the south  It was flanked on both ends by major thoroughfares, the Strand on the west, connecting Westminster to London by road, and the River Thames on the east, not only connecting the two cities to each other and to Southwark on the south side of the Thames, but connecting the entire metropolis to the rest of the world  Being located in the City of Westminster, Spur Alley had escaped the devastation of the Great Fire of London in 1666 leaving its wooden structures, built in the early part of seventeenth century, intact, but also in dire need of restoration or demolition  “The ratebooks show that during the last thirty years or so of their existence the houses in Spur Alley were in a very bad condition  Few of them were rated at more than a few shillings and many of them were unoccupied ”[1] The landowner, William, 5th Baron Craven, desiring to increase the profitability of his assets, tore down the derelict structures on Spur Alley around 1730 and leased the newly established lots to builders  By 1735, twenty brick houses in the Georgian style had been built on the west side and sixteen on the east side of the way now called Craven Street [2]
Figure 2  Craven Street 1746  (John Rocque London, Westminster and Southwark, First Edition 1746, Motco Enterprises Limited, motco com)
Letters to Franklin during his residence with Mrs  Margaret Stevenson, his landlady on Craven Street, were addressed rather vaguely; “Craven Street/Strand”, “Mrs  Stevensons in Craven Street”, or “Benjamin Franklin Esqr ” are but a few examples  Letters from Franklin referenced “London,” or sometimes “Cravenstreet,” but never included a number  Despite the absence of numbered addresses in Franklin’s correspondence, there was a sense of one’s place in the neighborhood based on entries in the Westminster Rate Books (tax assessments)  The Rate Books did not list house numbers during Franklin’s time there, but they did list the residents of Craven Street in a particular order that became the default numbering system for the street  Number one was associated with the first resident listed under “Craven Street” in the Rate Books and was the northernmost house on the west side of the street  The numbers increased counter-clockwise down the west side and up the east side in accordance with the list of residents  In 1748, the first year of Margaret Stevenson’s (Stevens in the Rate Books for that year) residence on Craven Street, she is listed as the twenty-seventh resident, the second house north of Court Street (later Craven Court, now Craven Passage) on the east side of the street [3]
In 1766, Parliament passed the London Paving and Lighting Act (6 Geo  3 c  26), “An act for the better paving, cleansing, and enlightening, the city of London, and the liberties thereof; and for preventing obstructions and annoyances within the same; and for other purposes therein mentioned ”[4] One of the other purposes therein mentioned was the numbering of houses  With an aim to bring order to the chaotic numbering systems or lack thereof on London streets the Act provided that “… the said commissioners … may also cause every house, shop, or warehouse, in each of the said streets, lanes, squares, yards, courts, alleys, passages, and places, to be marked or numbered, in such manner as they shall judge most proper for distinguishing the same ”[5] This was quite an undertaking that took years to accomplish  It was a decade later before numbered addresses on Craven Street in the City of Westminster appeared in The London Directory (1776)  The London Directory and its competitors were published primarily by booksellers or printers to supplement their income and were highly profitable  To say they were competitive is an understatement  “Some of the most hotly disputed struggles over copyright in the century concerned guidebooks  Many were optimistically emblazoned with a royal license and a notice that the work had been entered at Stationers’ Hall  Various struggles between rival guides intensified as the potential for profits became clear ”[6] The London Directory boldly proclaimed to contain “An ALPHABETICAL LIST OF THE NAMES and PLACES of ABODE of the MERCHANTS and PRINCIPAL TRADERS of the Cities of LONDON and WESTMINSTER, the Borough of SOUTHWARK, and their Environs, with the Number affixed to each House ”[7] Kent’s Directory made a similar proclamation: “An Alphabetical LIST OF THE Names and Places of Abode OF THE DIRECTORS of COMPANIES, Persons in Public Business, MERCHANTS, and other eminent TRADERS in the Cities of London and Westminster, and Borough of Southwark WITH THE NUMBERS as they are affixed to their Houses agreeable to the late Acts of Parliament ”[8] Mrs  Stevenson wasn’t included in the directories because she didn’t meet the criteria of being a merchant or trader, not because she was a woman  Although it is rare to see women listed in the directories, some examples do exist [9] If Mrs  Stevenson had appeared in the directories in 1776 it would not have been on Craven Street as she had moved to Northumberland Court, a stone’s throw away, the previous year [10] A comparison of Craven Street residents whose names and addresses do appear in the directories with the same residents as they appear in the Westminster Rate Books determines if the numbering systems were congruent  For the most part they were  For example, Joseph Bond at No  30, William Rowles at No  31, Samuel Sneyd at No  32, and Jonathan Michie at No  35 in The London Directory coincide with their places of residence in the Westminster Rate Books; however, errors did occur  The 1776 edition of The London Directory lists Brown & Whiteford, wine merchants, at No  9 Craven Street while the Westminster Rate Books list them as the twenty-ninth residents  Obviously, it makes no sense to have Brown & Whiteford at No  9 in The London Directory and their next-door neighbor, Joseph Bond, at No  30  The same error appears in Baldwin’s The New Complete Guide for 1783  The New Complete Guide may have “borrowed” the error from The London Directory  It was not uncommon for the owner of one directory to copy entries from another to save both time and money  Beginning in 1778 and contrary to The London Directory, Kent’s Directory faithfully followed the numbering system of the Westminster Rate Books in all of its editions and listed Brown & Whiteford at No  29 as did Bailey’s Northern Directory in 1781  Perhaps realizing their error, The London Directory changed their listing of Brown & Whiteford from No  9 to No  29 in their 1783 edition and maintained that listing thereafter 
Sometime prior to 1792, the embankment on the Thames at the south end of Craven Street had been sufficiently extended allowing for the construction of ten new houses below the original houses: “ … four houses, Nos  21–24, were built on the west side, and six houses, Nos  25–30, on the east side of the way ”[11] In a note in the same report, the new numbering system is explained  “The houses in the street, which had previously been numbered consecutively down the west side and up the east side, were then renumbered on the same system to include the additional houses ”[12] Because the new houses (21-24) on the west side were built below the existing houses (1-20), houses 1-20 retained their original numbering 
Figure 4  Craven Street 1799  (Richard Horwood’s Map of London, Westminster and the Borough of Southwark 1799, Motco Enterprises Limited, motco com)
One would think that the numbers of the sixteen original houses on the east side, Nos  21 – 36, would simply increase by ten with the addition of the ten new houses, but such was not the case; they increased by nine  How could that be? The only possible explanation is that No  21 of the original houses was demolished to make way for the construction of the northernmost of the six new houses on the east side (No  30)  Evidence of No  21’s demolition appears in the lease granted to Charles Owen by William, 7th Baron Craven, in 1792, which describes No  22 as: “All that messuage in Craven Street late in the occupation of Francis Deschamps undertaker … being the Southernmost house in the Old Buildings on the East Side of the said Street numbered with the No  22 ”[13] The lease describes No  22 as being the southernmost house in the old buildings on the east side of Craven Street  Clearly the house previously at No  21 did not exist when the lease granted to Charles Owen was written in 1792 as it used to be the southernmost house  It is also worth noting that in 1790, The London Directory listed Jacob Life at No  21 (original numbering)  In 1791-2, it listed him at No  6  With No  21 vacated, it would allow for its demolition and the construction of the tenth new house  By utilizing lot No  21 for the new construction, only nine additional lots were needed to build the ten houses, hence, Margaret Stevenson’s former residence at 27 became 36 (27 + 9) in the renumbering and not 37 
For nearly a century and a half after Franklin departed London for America in March of 1775 the scales were tipped heavily in favor of his residence having been No  7 Craven Street  As early as 1807 in London; Being An Accurate History And Description Of The British Metropolis And Its Neighborhood, Volume 4, one would have read: “In Craven Street is a house, No  7, remarkable for having been the residence of Dr  Benjamin Franklin [14] In 1815, the identical phrase appeared in The Beauties of England and Wales [15] After 23 editions of not mentioning Franklin, his name finally appeared in the 24th edition of The Picture of London in 1826: “The house, No  7, Craven Street, in the Strand, was once the residence of Dr  Benjamin Franklin ”[16] In 1840, Jared Sparks referred to Franklin’s Craven Street residence appearing in London guide books in his voluminous The Works of Benjamin Franklin: “In the London Guide Books, ‘No  7, Craven Street,’ is still indicated as the house in which Dr  Franklin resided ”[17] In 1846, George Gulliver F R S , in his book, The Works of William Hewson, wrote: “She [Polly] had been upon terms of the warmest friendship with Dr  Franklin
Figure 5  No  7 Craven Street with Memorial Tablet  (Photo courtesy of British History Online, and the Survey of London)
since she was eighteen years of age  That eminent philosopher resided with her mother, Mrs  Margaret Stevenson, at No  7, Craven Street, Strand, during the fifteen years of his abode in London ”[18] Guide books mentioning Franklin at No  7 continued to proliferate throughout the century: Handbook for London; Past and Present, Volume I (1849);”[19] Handbook for Modern London (1851);”[20] The Town; Its Memorable Characters and Events (1859);”[21] London and Its Environs (1879) [22] There was an anomaly when London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition (1880) placed Franklin at 27 Craven street [23] The anomaly lasted for six years until his place of residence was changed to No  7 in the revised edition, London  Illustrated by Eighteen Bird’s-Eye Views of the Principal Streets (1886) [24] London Past and Present; Its History, Associations, and Traditions, Volume 1 (1891), copied the 1849 Handbook for London almost word-for-word and included, “The house is on the right from the Strand ”[25] In October of 1867, The Society of Arts in London declared that: “In order to show how rich the metropolis is in the memory of important personages and events, which it would be desirable to mark by means of tablets on houses, the Council have caused an alphabetical list to be prepared, … ”[26] Franklin had been elected a corresponding member to the Society in 1756 and was a popular choice among Council members deciding who they were to memorialize [27] By January of 1870, a tablet honoring him was affixed to the house they believed to have been his residence while in London, No  7 Craven Street in the Strand on the west side of the street [28] A majority of historians writing about Franklin in the nineteenth and early twentieth century placed him at No  7: O  L  Holley, The Life of Benjamin Franklin (1848); E  M  Tomkinson, Benjamin Franklin (1885); John Torrey Morse, Benjamin Franklin (1891); Paul Elmer More, Benjamin Franklin (1900); John S  C  Abbot, Benjamin Franklin (1903); Sydney George Fisher, The True Benjamin Franklin (1903)  A notable exception is D  H  Montgomery’s His Life Written by Himself published in 1896  He has Franklin at No  27 Craven Street  It seems then that depending upon the source, Franklin was thought to have lived at either No  7 or No  27, but not both, the overwhelming majority favoring No  7  As late as 2011, Franklin is still mentioned as living at No  7 [29]
In 1913, No  7 was scheduled to be torn down  An article in the March 1914 edition of The Book News Monthly, describes the situation:
As is well known to informed American pilgrims, it has been possible for all admirers of the famous philosopher and statesman to pay their respects to his memory before that house, No  7 Craven Street, just off the Strand, which was his chief home during his two sojourns in the British capital, but even as these lines are being written the London newspapers are recording that that interesting shrine is soon to be pulled down to make room for a restaurant  It is some mitigation of this misfortune to remember that at the most the Craven Street house was nothing more than a reproduction of the one in which Franklin had his suite of four rooms, for the structure has been rebuilt since Franklin’s time  When, then, some one makes a piteous plea that at least the philosopher’s bedroom shall be preserved, the soothing answer is that the apartment in question is only a replica of that in which the illustrious American enjoyed his well-earned slumbers in 1757-62 and 1764-75  The restaurant-builder, however, with an eye doubtless to possible American patronage, has assured the world that every effort will be made to preserve as much as possible of the entire structure [30]
Concerned with the possible demolition of Franklin’s residence, the Royal Society of Arts (formerly the Society of Arts[31]) initiated an inquiry into the matter [32] The London County Council, having taken over the responsibility of placing memorial tablets on notable houses from the Royal Society, was charged with the investigation  It ultimately fell to Sir George Laurence Gomme, a clerk to the Council, to come up with a response  A few years earlier Sir George had discovered Margaret Stevenson residing at No  27 Craven Street in the Westminster Rate Books  He must have wondered why No  7 on the west side of Craven Street was being celebrated as Franklin’s residence when the evidence clearly showed otherwise 
Sir George and his staff examined the various London directories discussed earlier and came up with a novel explanation for the discrepancy  They concluded that there had been two numbering systems on Craven Street  An anonymous author echoes Sir George’s conclusion about the two numbering systems in an article in The Journal of the Royal Society of Arts:
…an inspection of the directories of that time proves that there were at least two systems of numbering in Craven Street before the erection of the additional houses  According to one of these the numbers started from the top (Strand end) on the west side of the street, and ran down to the bottom to No  20, then crossed over and went back to the Strand along the east side – 21 to 36  According to the other system, the east side of the street was numbered from the bottom upwards, starting at No 1  This was not apparently in general use, but there is evidence that this numbering was at all events occasionally used 
The evidence of these two systems of numbering, and for believing that Mrs  Stevenson’s house was first No  7 under the oldest system, next No  27 under the second system, and finally No  36 under the latest and existing system, is to be found in the various directories and the Westminster rate-books [33]
The “evidence” mentioned above consisted of The London Directory’s listing of Brown & Whiteford at No



ground_truth: Power-law functions.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the transition probability of the environment affect the learning rate in the static agent?

Read the following text and answer briefly.

Context 3:  9: “The rate-books for 1781 and 1786 show the house next but one to the north of Mrs  Stevenson’s house as in the occupation of Brown and ‘Whiteford,’ while the old directories mention the business of the firm as wine merchants, and give their address as 9, Craven Street – then a little later, down to 1791, as 29, Craven Street  Curiously enough, in the years 1778 to 1780, or 1781, Lowndes gives it as No  9, and Kent as 29 ”[34] Ignoring Kent’s Directory having Brown and Whiteford as 29 and The London Directory (Lowndes) having Brown and Whiteford “a little later” as 29, and knowing that Mrs  Stevenson lived two doors south of them, Sir George concluded that her house must have been numbered 7, even though there is no listing in any of the directories of her residence ever being No  7  He surmised that the No  7 on the west side of Craven Street with the memorial tablet thought to have been Franklin’s residence had simply been confused with number 7 (27) on the east side  Again from The Journal of the Royal Society of Arts:
Taking all the evidence together, there cannot be any doubt whatever that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court, first numbered 7, afterwards 27, and finally 36, and consequently that the house in which Franklin lived was that now numbered 36, not the one now numbered 7, on which the tablet is placed [35]
A response to The Royal Society of Arts was issued: “… the London County Council … informed the Society that it had made a mistake and that No  36 Craven street was the building that deserved commemoration ”[36] The Society accepted the Council’s conclusion, and despite assurances of preservation by the restaurant builder, No  7 was torn down the following year 
Sir George’s assertion “that Mrs  Stevenson’s house, in which Franklin lodged, was the house two doors north from Craven Court” was correct, however, his assertion that it was “first numbered 7, afterwards 27”, was not  It was only by association with the errant entry of Brown & Whiteford at No  9 from 1776-1782 in The London Directory that Mrs  Stevenson’s address was conjured to be No  7  The problem with associating her address exclusively with that of Brown & Whiteford at No  9 during those years is that, as previously demonstrated, The London Directory also listed four other Craven Street residents, Bond, Rowles, Sneyd, and Michie, who’s addresses did conform to the numbering system in The Westminster Rate Books  If Brown & Whiteford at No  9 was indicative of a numbering system different from The Westminster Rate Books, Bond, Rowles, Sneyd, and Michie would have been listed as Nos  10, 11, 12, and 15, respectively  So on one hand Sir George was relying on the Westminster Rate Books to establish Mrs  Stevenson at No  27 and on the other hand he was dismissing the Westminster Rate Books to establish her at No  7  Instead of using the anomalous listing of Brown & Whiteford at No  9, he could have just as easily, and more logically, used the Bond et al  listings, or the post-1782 Brown & Whiteford listing in the London Directory at No  29 to establish Mrs  Stevenson at No  27  Even if there had been two numbering systems, his assertion that No  27 was first numbered 7 would still be false  The earliest numbering system was the Westminster Rate Books dating from the early 1730s when the houses were constructed  Brown & Whiteford at No  9 didn’t appear until 46 years later and then only for a brief period 
There is ample evidence in Franklin’s correspondence and in a memoir by Polly Hewson (Mrs  Stevenson’s daughter) that Benjamin and Mrs  Stevenson lived in not one, but two houses on Craven Street  On July 6, 1772, Polly wrote to Benjamin from her house at Broad Street North in London: “My Mother I must tell you went off last friday week, took our little Boy with her and left Mr  Hewson [Polly’s husband, William] the care of her House [27 Craven Street]  The first thing he did was pulling down a part of it in order to turn it to his own purpose, and advantage we hope  This Demolition cannot affect you, who at present are not even a Lodger [Benjamin was traveling at the time], your litterary apartment remains untouch’d, the Door is lock’d …”[37] In a memoir about her husband written after his death Polly writes: “He [William Hewson] began his Lectures Sept  30, 1772, in Craven-street, where he had built a Theatre adjoining a house which he intended for the future residence of his family ”[38] On October 7, 1772, Benjamin wrote to his son William: “I am very well  But we [Mrs  Stevenson and I] are moving to another House in the same street; and I go down tomorrow to Lord LeDespencer’s to [stay a] Week till things are settled ”[39] To his son-in-law, Richard Bache, on the same day he wrote: “We are moving to another House in the [street] leaving this to Mr  Hewson ”[40] Writing to a friend on October 30, 1772 he explained: “I should sooner have answered your Questions but that in the Confusion of my Papers, occasioned by removing to another House, I could not readily find the Memorandums …”[41] On November 4, 1772 Benjamin informed his wife Deborah of the move  “We are removed to a more convenient House in the same street, Mrs  Stevenson having accommodated her Son-in-Law with that we lived in  The Removing has been a troublesome Affair, but is now over ”[42]
An agreement had been struck between the parties  Margaret and Benjamin would move to another house on Craven Street and allow Polly and William to move into No  27, the large yard behind the house being spacious enough to accommodate the anatomy school William wished to build [43] Perhaps the idea was inspired by Margaret’s next-door neighbor at No  26, Dr  John Leake, a man-midwife and founder of the Westminster Lying-in Hospital, who had built a theater adjoining his residence in which he practiced anatomy and taught midwifery [44]
After Margaret and Benjamin vacated No  27, Polly, William, their son William Jr , and William’s younger sister, Dorothy Hewson, took up residence there [45] In the 1773 Westminster Rate Books for Craven Street, Mrs  Stevenson’s (Stephenson in the Rate Books) name has been crossed out and replaced with “William Hewson ”[46] Further proof that the Hewsons had indeed moved into 27 Craven Street has been confirmed by the discovery of human and animal remains buried in the basement of No  36 (formerly No  27 and now the Benjamin Franklin House), a by-product of the dissections that took place at William’s anatomy school [47]
So what house on Craven Street did Mrs  Stevenson and Benjamin move into after vacating No  27? An examination of the Westminster Rate Books for the years 1774 and 1775 reveal them living not at No  7 on the west side of Craven Street as one might expect from the overwhelming consensus of nineteenth century guidebooks and biographies, but surprisingly at No  1 [48] The controversy of No  7 being torn down was all for naught as it had never been Franklin’s residence  Sir George was correct on that point  Unfortunately, No  1 was torn down as well in the early part of the twentieth century  The first time No  1 is mentioned as Franklin’s second residence is in the Survey of London: Volume 18, St Martin-in-The-Fields II: the Strand published by the London County Council in 1937, ironically the same County Council that had declared No  36 as Franklin’s only residence twenty-four years earlier 
From 1748 until 1772 Margaret ‘Stephenson’ occupied this house [No  27 (36)], and it was there that Benjamin Franklin settled after his arrival in London in 1757 as Agent to the General Assembly of Pennsylvania … In October, 1772, Mrs  Stevenson and Franklin removed to No  1, Craven Street (now demolished), and No  36 was for the next two years occupied by William Hewson, surgeon, who had married Mary Stevenson [49]
In the spring of 1774, William Hewson died unexpectedly of septicemia two weeks after cutting himself while dissecting a cadaver  Polly was left to care for their two young sons and was pregnant with a daughter she would give birth to in August of the same year  Is it possible that Margaret and Benjamin moved back into No  27 to assist Polly after the death of her husband as suggested in The Americanization of Benjamin Franklin?[50]
If the Westminster Rate Books are to be believed, the answer is no  For the year 1774, the Rate Books list Margaret Stevenson at No  1 and William Hewson at No  27  For the year 1775, they list Margaret Stevenson at No  1 and Magnus Falkner (Falconer/Falconar) at No  27  Magnus was William’s assistant at the anatomy school and fiancé to William’s sister, Dorothy  On his death bed, William instructed Polly, “let Mr  Falconar be my successor ”[51] Magnus would immediately take over the running of the anatomy school and continue William’s unfinished research  Four months later, he and Dorothy would marry [52] Essentially only two things changed at 27 Craven Street after William’s death: Polly gave birth to her daughter, and Magnus replaced William as the lease holder, so even if Margaret and Benjamin had wished to move back into No  27, there would have been no room for them  It is also interesting to note that considering the multiple times Benjamin wrote of his move out of No  27 (and complained of it), he never once mentioned moving back into No  27 in any of his correspondence after Mr  Hewson’s death 
Figure 6  No  36 Craven Street  (Photo courtesy of David Ross, britainexpress com)
In sum, based on the Westminster Rate Books[53] and Franklin’s correspondence, Mrs  Stevenson is known to have resided at No  27 (36) Craven Street from 1748 to 1772  It follows that, aside from the two years Franklin spent in Philadelphia from 1762 to 1764, he resided there from 1757 to 1772  Franklin’s correspondence also reveals that in the autumn of 1772, he and Mrs  Stevenson moved to another house on Craven Street  The 1773 Westminster Rate Books show her name crossed off at No  27 and William Hewson’s inserted  The following year the Rate Books list her at No  1 Craven Street  Evidence for Mrs  Stevenson and Benjamin remaining at No  1 after William’s death appears in the Westminster Rate Books for 1775 which have Mrs  Stevenson still residing at No  1 and Magnus Falkner residing at No  27  Further evidence can be construed from the lack of any mention of a move back into No  27 in Franklin’s correspondence  Despite the many theories one could devise as to why Franklin was thought to have lived at No  7 Craven Street by so many guide books and Franklin biographers of the nineteenth century, one thing is certain; at some point after Franklin’s departure to America in March of 1775, and no later than 1807, someone mistakenly associated him with No  7 on the west side of Craven Street, and it soon became his de facto residence  Credit must go to D  H  Montgomery in 1896 and Sir George in 1913 for setting the record partially straight by placing Franklin at No  27(36)  In 1937, the London County Council gave us the first accurate account of Franklin’s residences on Craven Street in the Survey of London at No  27(36) and No  1  It has been shown conclusively that No  27 was never previously numbered 7  It was, however, renumbered 36 in 1792 after ten additional houses were built at the southern end of the street and remains No  36 to this day 
[1] “Craven Street and Hungerford Lane”, in Survey of London: Volume 18, St Martin-in-the-Fields II: the Strand, ed  G H Gater and E P Wheeler (London, 1937), 27-39, Early History of the Site 
http://www british-history ac uk/survey-london/vol18/pt2/pp27-39
[2] “England, Westminster Rate Books, 1634-1900,” from database with images, Craven Street – 1735, FamilySearch from database by FindMyPast and images digitized by FamilySearch; citing Westminster City Archives, London 
[3] Ibid , Craven Street – 1748 
[4] The Statutes at Large, From Magna Charta to the End of the Eleventh Parliament of Great Britain  Anno 1761 Continued, Vol  XXVII, ed  Danby Pickering, (Cambridge, John Archdeacon, 1767), 96 
[6] James Raven, Publishing Business in Eighteenth-Century England, (Woodbridge: The Boydell Press, 2014), 201 
[7] The London Directory For the Year 1776, Ninth Edition, (London: T  Lowndes, 1776), title page 
[8] Kent’s Directory For the Year 1778, Forty-Sixth Edition, (London: Richard and Henry Causton, 1778), title page 
[9] A listing in Kent’s Directory for the Year 1882 on p  28 reveals, “Brown Sarah, Leather-seller, 1, Westmoreland-buildings, Aldersgate-street”, and in Kent’s Directory for the Year 1883 on p  175, “Whiteland Mary, Wine & Brandy Mercht  Jermyn-str  St  James ”
[10] “The Papers of Benjamin Franklin,” Sponsored by The American Philosophical Society and Yale University, Digital Edition by The Packard Humanities Institute, 22:263a 
http://franklinpapers org/franklin
Mrs  Stevenson wrote to Benjamin Franklin a letter from her new home at 75 Northumberland Court on November 16, 1775: “In this Court I have a kind friend, Mr  Lechmoen he comes and seats with me and talks of you with a hiy regard and friendship ”
[11] Survey of London, Early History of the Site 
[12] Survey of London, Footnotes/n 10 
[13] Survey of London, Historical Notes/No  31 
[14] David Hughson, LL D



ground_truth: As the transition probability increases, the learning rate initially rises and then declines.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When did the London Paving and Lighting Act pass, which mandated the numbering of houses?

Read the following text and answer briefly.

Context 3: , London; Being An Accurate History And Description Of The British Metropolis And Its Neighbourhood, To Thirty Miles Extent, From An Actual Perambulation, Vol  IV, (London: W  Stratford, 1807), 227 
[15] The Reverend Joseph Nightingale, The Beauties of England and Wales: Or, Original Delineations, Topographical, Historical, and Descriptive, of Each County, Vol  X, Part III, Vol  II (London: J  Harris; Longman and Co ; J  Walker; R  Baldwin; Sherwood and Co ; J  and J  Cundee; B  and R  Crosby and Co ; J Cuthell; J  and J  Richardson; Cadell and Davies; C  and J  Rivington; and G  Cowie and Co , 1815), 245 
[16] John Britton, F S A  & Co , ed , The Original Picture of London, Enlarged and Improved: Being A Correct Guide For The Stranger, As Well As For the Inhabitant, To The Metropolis Of The British Empire Together With A Description Of The Environs, The Twenty-Fourth Edition (London: Longman, Rees, Orme, Brown, and Green, 1826), 479 
[17] Jared Sparks, The Works of Benjamin Franklin, Vol  VII, (Philadelphia: Childs & Peterson, 1840), 151 
[18] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xx 
[19] Peter Cunningham, Handbook for London; Past and Present, Vol  I, (London: John Murray, 1849), 245 
[20] F  Saunders, Memories of the Great Metropolis: or, London, from the Tower to the Crystal Palace, (New York: G P  Putnam, MDCCCLII), 138 
[21] Leigh Hunt, The Town; Its Memorable Characters and Events, (London: Smith, Elder and Co , 1859), 185 
[22] K  Baedeker, London and Its Environs, Including Excursions To Brighton, The Isle of Wight, Etc : Handbook For Travelers, Second Edition, (London: Dulau and Co , 1879), 133 
[23] Herbert Fry, London In 1880 Illustrated With Bird’s-Eye Views of the Principal Streets, Sixth Edition, (New York: Scribner, Welford, & Co , 1880), 50 
[24] Herbert Fry, London  Illustrated By Eighteen Bird’s-Eye Views of the Principal Streets, (London: W  H  Allen and Co , 1886), 40 
[25] Henry B  Wheatley, F S A , London Past and Present; Its History, Associations, and Traditions, Vol  1, (London: John Murray, New York: Scribner & Welford, 1891), 473 
[26] The Journal of the Society of Arts, Vol  XV, No  778, (October 18, 1867): 717 
[27] D  G  C  Allen, “Dear and Serviceable to Each Other: Benjamin Franklin and the Royal Society of Arts,” American Philosophical Society, Vol  144, No  3, (September 2000): 248-249 
Franklin was a corresponding member in 1756 because he was still residing in Philadelphia  He became an active member the following year when he moved to London 
[28] The Journal of the Society of Arts, Vol  XVIII, No  894, (Jan  7, 1870): 137 
“Since the last announcement, the following tablets have been affixed on houses formerly occupied by – Benjamin Franklin, 7 Craven-street, Strand, W C ”
[29] Franklin in His Own Time, eds  Kevin J  Haytes and Isabelle Bour, (Iowa City, University of Iowa Press, 2011), xxxvii 
“Takes lodgings with Margaret Stevenson at No  7 Craven Street ” It is unknown if the editors are referring to No  7 on the west side of Craven Street or No  36 on the east side using Sir George’s explanation of No  36 being previously numbered 7 
[30] Henry C  Shelly, “American Shrines on English Soil, III  In the Footprints of Benjamin Franklin,” in The Book News Monthly, September, 1913 to August, 1914, (Philadelphia: John Wanamaker, 1914), 325 
[31] The Journal of the Royal Society of Arts, Vol  LVI, No  2,880, (Jan  31, 1908): 245 
http://babel hathitrust org/cgi/pt?id=mdp 39015058423073;view=1up;seq=251
“His Majesty the King, who is Patron of the Society, has granted permission to the Society to prefix to its title the term ‘Royal,’ and the Society will consequently be known in future as the ‘Royal Society of Arts ’”
[32] Nineteenth Annual Report, 1914, of the American Scenic and Historic Preservation Society, (Albany: J  B  Lyon Company, 1914), 293 
http://babel hathitrust org/cgi/pt?id=wu 89072985302;view=1up;seq=4;size=150
[33] The Journal of the Society of Arts, Vol  LXII, No  3,183, (Nov  21, 1913): 18 
http://babel hathitrust org/cgi/pt?id=mdp 39015058422968;view=1up;seq=26
[36] Allen, “Dear and Serviceable,” 263-264 
[37] Papers of Benjamin Franklin, 19:20 
[38] Thomas Joseph Pettigrew, F  L  S , Memoirs of the Life and Writings of the Late John Coakley Lettsom With a Selection From His Correspondence, Vol  I, (London: Nichols, Son, and Bentley, 1817), 144 of Correspondence 
[39] Papers of Benjamin Franklin, 19:321b 
[40] Ibid , 19:314 
[41] Ibid , 19:353a 
[43] Simon David John Chaplin, John Hunter and the ‘museum oeconomy’, 1750-1800, Department of History, King’s College London  Thesis submitted for the degree of Doctor of Philosophy of the University of London , 202 
“Following Falconar’s death [1778] the lease [27 Craven Street] was advertised, and the buildings were described as:
A genteel and commodious house, in good Repair, with Coach-house and Stabling for two Horses…consisting of two rooms and light closets on each floor, with outbuildings in the Yard, a Museum, a Compleat Theatre, and other conveniences  (Daily Advertiser, 27 August 1778)”
[44] Simon Chaplin, “Dissection and Display in Eighteenth-Century London,” in Anatomical Dissection in Enlightenment England and Beyond: Autopsy, Pathology and Display, ed  Dr  Piers Mitchell, (Burlington: Ashgate Publishing Company, 2012), 108 
“Given that a nearby building at 35 [ No  26 in Franklin’s time] was occupied by the man-midwife John Leake, who advertised lectures – including lessons in the art of making preparations – at his ‘theatre’ between 1764 and 1788, it is possible that some facilities were shared  In both cases, however, the buildings [Leake’s residence at No  26 and Hewson’s residence next door at 27] served a dual function as domestic accommodation and as sites for lecturing and dissection ”
[45] George Gulliver, F R S , The Works of William Hewson, F  R  S , (London: Printed for the Sydenham Society, MDCCCXLVI), xviii 
[46] Westminster Rate Books, Craven Street – 1773, courtesy of the City of Westminster Archives 
[47] S W  Hillson et al , “Benjamin Franklin, William Hewson, and the Craven Street Bones,” Archaeology International, Vol  2, (Nov  22, 1998): 14-16 
http://dx doi org/10 5334/ai 0206
[48] Westminster Rate Books, Craven Street – 1774, 1775, courtesy of the City of Westminster Archives 
[49] Survey of London, Historical Notes/No  36, Craven Street (not sourced) 
[50] Gordon S  Wood, The Americanization of Benjamin Franklin, (New York: The Penguin Press, 2004), 261 
[51] Pettigrew, Memoirs, 146 of Correspondence 
[52] http://founders archives gov/documents/Franklin/01-22-02-0178, note 7  “Falconar married Hewson’s sister five months after the Doctor’s death; most of the Craven Street circle attended the wedding, and BF gave away the bride: Polly to Barbara Hewson, Oct  4, 1774, APS” (American Philosophical Society); “England Marriages, 1538–1973 ,” database, FamilySearch (https://familysearch org/ark:/61903/1:1:V52W-TGS : accessed September 15, 2015), Magnus Falconar and Dorothy Hewson, September 12, 1774; citing Saint Martin In The Fields, Westminster, London, England, reference ; FHL microfilm 561156, 561157, 561158, 942 B4HA V  25, 942 B4HA V  66 
[53] I chose to rely on the Westminster Rate Books for the numbering system on Craven Street  The books were consistent throughout the eighteenth century in the ordering of residents on the street and were used as the basis for the 1792 re-numbering  For the most part, commercial directories aligned with them as well  If by chance a directory didn’t initially align, it would inevitably produce future editions that did 
Benjamin Franklin, Benjamin Franklin House, London
More from David Turnquist
If one looked into Benjamin Franklin’s time on Craven Street, they might   
I think it’s very ironic that on the street maps included in your excellent article, Craven Street is so close to Scotland Yard  Because following the back and forth juxtapositions of numbers 7, 27 and 36 Craven Street (throw in 75 Northumberland Court and 1 Craven Street, too) was a case that could confound Sherlock Holmes 
Excellent job of deciphering street renumbering material spanning sixty years, including that of a wrong house number (# 7) being erroneously identified and then perpetuated in subsequent street map printings  It’s gratifying at least to know that the present day #36 Craven Street is the correct house for Ben Franklin tourists to visit  Except for #1 Craven Street for the last three years Franklin was in London, but we won’t get into that 
Again, excellent article, David!



ground_truth: 1766.