Variational approach in modeling the rural historical settlement: a model of a two-level hierarchy of centers

 Introduction

Historical settlement systems have long been the subject of study by historians.

In particular, scientists are trying to understand the nature of the medieval settlement structure, to give an approximate estimate of its population. As a rule, these tasks are solved within the framework of traditional research, through the involvement and analysis of historical sources. Only recently, methods from the exact sciences have been used to solve such issues. Thus, for the analysis of the spatial type of rural Russian settlement of the XV–XVII centuries, A.Ya. Degtyarev uses the method of the nearest neighborhood from geographical science ^[1]. The probabilistic approach is used by O.N. Trapeznikova and A.A. Frolov to model the rural population of the Valdai Upland at the turn of the Middle Ages and Modern times ^[2]. It is based on the idea of the spontaneity and random nature of the settlement process in a territory with homogeneous natural and climatic conditions. Since such conditions cannot be guaranteed for large territories, this approach is limited to analyzing only microregions ^[3].To remove such a restriction, the author of this article develops a variational approach proposed by S.M. Huseyn-Zadeh ^[4].

It is based on the idea of settlement, placement of the population on the territory as a process aimed at achieving certain goals. Participants in the process are economic entities. For example, in the conditions of modeling rural settlement, these are yards, the number of which can be hundreds or thousands of units. And the goals are dictated by natural considerations of households, participants in the settlement process, for example: maximum accessibility to the cultivated area (arable land, fire), maximum proximity to the central settlement.

Note that these goals can be mutually opposite, as in this example. So, if the first goal is realized, the population will be evenly distributed throughout the territory (dispersion). If only the second goal is realized, on the contrary, it will concentrate at one point – the central settlement. If we combine both goals, then it is natural to assume that a compromise between them should result. In terms of historical geography, it takes the form of a "nest" the nature of settlement – the condensation of settlements around its center, followed by their dispersion along the periphery:

Fig.1 Types of spatial placement of settlements: nesting, random, ordered (source: ^{[1, p. 33]})

Within the framework of the variational approach, goals are formalized in the form of functionals F, that is, functions that depend on other functions as their arguments. The argument functions are, for example, population density p(x) at point x of the studied territory, the density of the centers q(x), the distance r(x) from point x to the nearest center. These functions can be set (varied) in an infinite number of ways, but the optimal relations between them are defined as the solution of the variational problem. This means that no matter how we vary the functions next to the fixed (desired) one, the linear part of the increment of the corresponding functional (its variation) does not change and remains equal to zero: ?F=0 ^[5].  To the reader familiar with the basics of mathematical analysis, it is clear that this is a complete analogy with the necessary condition of the extremum – the equality of the first derivative to zero.

Earlier, the author proposed a model of a one–level hierarchy of centers: a central settlement is a populated district ^[6,7]. That is, the entire territory under study is a system of disjoint zones, each of which contains a central settlement and "pulling" localities belong to it. Variational modeling reveals the power nature of the optimal relationships between the parameters of the settlement system. Thus, the following relationship is observed between the population N and the area S of the zone of influence: N ? S ^?, where ? = 0.75. That is, the larger the area, the greater the number of people living in this territory.A less trivial conclusion can be drawn regarding the relationship between the population density p and the area of the territorial district:.

That is, the relationship between these parameters is reversed: territories with a high population density occupy smaller areas.If we translate the above model parameters to a logarithmic scale for convenience, then the relationship between them goes from power to linear:

 and , where C is some constant.The theoretical results of the simulation are very well supported by real data.

As such, the material of scribal books on the Village Pyatina of the Novgorod land of the turn of the XV–XVI centuries, reconstructed by A.A. Frolov and N.V. Piotukh, was used ^[8]. The correlation coefficient calculated using the Statistica program (the strength of the connection) between both values is quite high (the significance level is over 0.95). And the linear regression coefficient (0.7308) differs from the theoretical one (0.75) by less than 0.02 units.

I. Two-level model of center placement

 In connection with the above-mentioned one of the tendencies of the system towards concentration, the following question arises: is there a concentration of settlements exclusively to their central settlements, or is it part of a more general movement towards other goals?

In other words, is it possible to have some hierarchy of goals in the system?  Further study of the materials of scribal books and historiography confirms the need to test this hypothesis, and, consequently, to modify the original model. Namely, we are talking about introducing intermediate centers into the model and thus considering the following hierarchy: the central settlement of the first level – the central settlement of the second level – a locality. 

Fig.2 Two-level system of placement of centers on the territoryThis system is characterized by three parameters: density p(x)

placement of courtyards (settlements) at point x of the studied territory (area) D, density q 1_(x) placement of the centers of the first level and density q 2_(x) placement of second-level centers. Naturally, all these values vary from point to point, which is due to the heterogeneity of the territory.

Participants in the settlement process are guided in their actions by three natural considerations. Firstly, they try to place courtyards (villages) near their cultivated areas (zones of influence). Secondly, these settlements are located as close as possible to the central settlements of the 2nd level. And those, in turn, are as close as possible to the centers of the 1st level. The aspirations of each of the participants, taken into account together, are composed of three goals that characterize the system as a whole.

Let us now proceed to the formalization of all these concepts in terms of the calculus of variations. Denote by ?(x) the area of the zone of influence of the yard (settlement) near the point x in the studied territory D. Without limiting the generality of reasoning, we assume that the zones of influence of all settlements, including the centers of the first and second level, has the shape of a circle. It can be shown that all further conclusions can be extended to all other cases, for example, when the zones of influence have a rectangular shape. Denote by r(x) the radius of the zone of influence near the point x. Then ?(x) and r(x) are related by the formula of the area of the circle: , or . In turn, the area ?(x) can be expressed in terms of density p(x) as:. From here we get .The indicator of the first goal will be the total radius of the zones of influence of settlements.

Denote by dS a small element of the area near the point x. The radius of the zone of influence of the settlement located on the element dS near the point x is proportional to r(x). Then the total radius on this element is equal to r(x) dS, and on the entire studied territory – , where the integral is taken over the entire area D. Taking into account the formula connecting the radius and density, for the first functional we obtain the formula:. The indicator of the second goal is the total distance from settlements to their central settlements of the 2nd level.

Repeating the same reasoning as in the first case, we get the distance r ₂(x) from x to the nearest center of the 2nd level proportional. Then the total distance from settlements near point x to its center of the 2nd level will be proportional. If we sum up this value over the entire domain D, we come to the formula for the second functional: .

Finally, as an indicator of the third goal, let's take the total distance from the centers of the 2nd level to their centers of the 1st level. Denote by r ₁(x) the distance from the center of the 2nd level, located near the point x, to its center of the 1st level. By analogy with the previous case, this distance will be proportional. The role of density p(x) now q ₂(x) is playing.Then the total distance from the centers of the 2nd level near the point x to its center of the 1st level will be proportional. As a result, the third functional is represented as: .

So, the functionals F ₁, F ₂, F ₃ depend on the arguments of the functions p(x), q ₁(x), q ₂(x). Depending on the specific type of functions, these functionals take on certain values. In theory, it is necessary to select these functions so that when they are substituted as arguments, the value of the functional F ₁ remains unchanged, and F ₂ and F ₃ take minimum values. It is clear that such a solution is not feasible, due to the multidirectional goals of the system. Therefore, within the framework of the proposed variational approach, they proceed to the problem of finding the optimal solution in the sense of Pareto ^[4]. Its essence lies in the fact that there is no other solution that would simultaneously _{give the best result for all functionals F} 2, F ₃, subject to the equality constraint for the functional F ₁. In other words, a compromise is being sought as a solution that would take into account, to one degree or another, all the goals of the system. Formally, the resulting task is to find such functions p(x), q ₁(x), q ₂(x) so that the linear combination of functionals takes a minimum value, and the value of the functional F ₁ is constant, where ?, ? are some positive constants.The Pareto optimum is sought from the equation , or .

Here ?F 2, ?F ₃, ?F ₁ are variations of the functionals corresponding to the variations _?q 2(x), ?p(x) _functions q 2(x), _p(x), and n is a constant. Applying the apparatus of calculus of variations, from here we come to a system of two equations:

Denote by ? ₁(x) – the area of the zone of influence of the center of the 1st level near the point x, and after n ₁(x) – its population size. Similarly, via ? ₂(x) and n ₂(x) let's denote the area and population of the zone of influence of the center of the 2nd level near the point x. We obtain formulas that relate the values of each pair to each other.

From equation (2) we have , where C is some constant.        (3).

Since ? ₂(x)=1/_q2(x), then from the last relation we obtain a power dependence, and the inverse, between the population density and the area of the zone of influence of the 2nd level: . That is, territorial districts with a high density of residence have a smaller area. The area and population of the zone of influence of the 1st level have a power dependence. Indeed, from the last relation and the formula n ₂(x)=p(x)? ₂(x) we have . When translated to a logarithmic scale, this dependence is transformed into a linear one with the same coefficient: , where A is some constant.

Similar conclusions, up to the exponent, can be made with respect to the values of the 1st level of the hierarchy. Namely, from equation (1) we have , where D is some constant. Hence, taking into account equation (3), we get: . Using similar reasoning as in the previous case, we get the dependence between the population density and the area of the zone of influence of the 1st level: Note, here is also a power dependence, but with a different exponent. Similar reasoning leads us to the conclusion that there is a power dependence between the population of the 1st level zone and its area: . Unlike the previous ratio with density, here is a direct relationship: the larger the area, the larger the population of the corresponding territorial district. Logarithm leads to a linear relationship between the parameters with a coefficient of 6/7: , where B is some constant.

The obtained theoretical results are tested on the example of a rural medieval settlement structure.

II.                Settlement structure of the Novgorod lands at the turn of the XV-XVI centuries. based on the materials of the scribe census of the Shelonskaya Pyatina After joining Moscow at the end of the XV century .

the entire Novgorod land for the convenience of the census and collection of taxes was divided into five pyatins. Almost simultaneously, there was a change of the ruling elite in this region. The possessions belonging to the Novgorod boyars and large monasteries were taken away by the Moscow Grand Duke. Some of them were distributed to Moscow serving people on estates, some were transferred to the disposal of the Novgorod Palace.

The first complete description of the Shelonskaya Pyatina was compiled in 7006 (1498/1499) by Matvey Ivanovich Valuev. The census covered all categories of land, including palace lands, and established a new, monthly salary of land.

Fig.3 Novgorod pyatins at the end of the XV–beginning of the XVI century. (source: ^{[9, p. 33]})

The scribal books of the Shelonskaya Pyatina were introduced into scientific circulation by K.A. Nevolin in 1853 ^[10]. The publication of their text began in the NPK series (Novgorod Scribal Books), the fourth volume of which was published in 1886 under the editorship of A.I. Timofeev ^[11]. The volume contains three manuscripts from the repository of the Moscow State Archive of the Ministry of Foreign Affairs (MGAMID), including the scribe book by M.I. Valuev. As modern researchers managed to find out, they were brought to Moscow from Novgorod by royal decree back in 1688 ^{[12, p.76]}. Now these manuscripts are stored in the fund 137 of the Russian State Archive of Ancient Acts (RGADA) "Boyar and city books".

The scribe book of M.I. Valuev (the first in order in the publication) begins with the preface: "The books of Novogrotsky Sholonsky pyatiny by Matthew Ivanovich Valuev of summer 7000 sixth. And in them are written suburbs and volosts and rows and churchyards and villages and villages of the Grand Duke, and for boyars and for boyar children, for landlords and svoezemtsov, and Kupetsky villages, and vladychni and manastyr villages and villages, in ploughs, according to Novogorodsky. And there are three obzhi in sokha."

The description of pyatina goes by counties and census districts, which almost always coincide with the churchyards-districts. The division of the lands of each census district into categories (rent, "for estates", "svoezemtsovye", monastic, as well as "vopchago" possessions) is carried out very strictly and consistently.

At the same time, the compilers of the scribe's book use even more "old books", the creation of which researchers attribute back to the times of Novgorod independence. "Old letter" it consists only of data on the number of villages, courtyards, people and household in individual volosts (possessions) and from a detailed enumeration of monetary and natural duties that peasants carried in favor of the former landowners-Novgorodians.

In the new letter, the number of "people" is significantly greater than in the "old" one. Most researchers explain this fact not by natural population growth, but by a new principle of counting "people". Namely, if in the "old" if the bill went to the households, then in the "new" – by the heads of families (several families could live in one yard).

 This book, unlike similar descriptions of the Village and Votskaya pyatinas, has come down to us in a very sad state: the description of only 24 census districts (out of 71) has been completely preserved. However, quite soon several more manuscripts of the scribal books of the Shelonskaya Pyatina were discovered in the archives. In 1905, the fifth volume of the NPC prepared by S.K. Bogoyavlensky was published, which was called by the publisher himself a "continuation and supplement" of the fourth volume ^[13]. The sources of the publication were the same MGAMID (at the expense of newly discovered manuscripts of scribal books). In addition, after the discovery of the materials of the Novgorod Palace Order in the Moscow Archive of the Ministry of Justice, in the manuscripts of which late copies of scribal books of the XVI century were found, which were also included in the fifth volume. Now these manuscripts are part of the fund 1209 "Local order, Patrimonial Board and Patrimonial Department" RGADA.

First in order (columns 1-70) as part of the fifth volume, a monument is published, which the publisher himself calls an "abridged" book. This is a 7006 payment book, as indicated by the mention of the payment term. She, reporting very little geographical information, gives a complete list of all land holdings in pyatina with an indication of their size in inches (units of area for land tax in Novgorod land in the XV–XVI centuries.) and the names of landowners and former owners-Novgorodians and allows you to fill in, at least in part, those gaps that occur in the description complete.

The monument next in order (columns 71-286) as part of the fifth volume is a scribe's book of vowels, church, local and foreign lands 7006 letters of M.I. Valuev. It is a previously unpublished part of the same code of M.I. Valuev, published in the fourth volume of the NPC.

The scribal books of the palace volosts were preserved only in a copy of the middle of the XVIII century . From the composition of this manuscript, the publisher identifies five monuments composed at different times. The first monument (columns 287-314) is the scribe book of the palace lands of M.I. Valuev's letter of 7006 (1498/1499). The date in the table of contents of this book is different – 7009, that is, 1500/1501. But, according to researchers, it rather means the date of sending to Novgorod the list that became the protograph for subsequent copies ^{[14, p. 18]}.

The third published monument (columns 332-406) includes two scribal books of the palace lands: the book of the letter of M.I. Valuev 7006, which is a continuation of the first monument (columns 332-387), and the book of G. N. Volosaty 7032 (1523/1524).

The fourth monument in a row contains the title: "The books of the palace of Shelonsky Pyatina were written by Matvey Ivanovich Valuev in the year 7007 of those volosts that he attributed to the Novo Palace." Apparently, having completed in 7006 (1497/1498) the description of all categories of lands in Pyatin, M.I. Valuev in the following year, 7007 (1498/1499), was instructed to re-describe the lands given to the Novgorod Palace and establish other norms of payments from the land. As A.M. Andriyashev writes, this census was made unsuccessfully, excessive taxes were imposed on the population. This caused the need to re-pay taxes, which was reflected in subsequent scribal books ^[15].

The process of finding new excerpts from M.I. Valuev's scribe book continues even after the publications of the Archeographic Commission. Separate sheets with a description of the settlement of the city of Rus and the Surrounding area were discovered by A.A. Zimin and published in 1959 ^[16]. More recently, K.V. Baranov discovered and published 6 more sheets as part of the collection of manuscripts of the fund 1209 RGADA: one of them is a fragment of the description of the Rusa posad, and the rest belong to separate estates of the Shelonskaya Pyatina.

From written sources of the XV–XVI centuries. (especially from the scribal descriptions of Pyatin) it follows that the churchyards-districts were the primary cells of the system of state taxation and administrative division of the Novgorod land at that time. The oldest evidence of churchyards contains the "Charter of Prince Vladimir Svyatoslavich on tithes, courts and church people", which "I have given to all cities, and to churchyards, and to freedoms where peasants are" ^{[17, pp.18-20]}. Then this term referred to settlements where the local population brought tribute for their prince. So, in the news of the "Tale of Bygone Years" about Princess Olga, it is reported: "in the summer of 6455, Ida Olga Novugorodu. And lay down the duties and tributes according to Myst, and the taxes and tribute according to the Pocket; and her traps are all over the earth, and signs and places and posts. And her sleigh has been standing in Pleskovo up to this day, and along the Dnieper there is a preponderance and along the Desna, and there is a village of her Olzhichi up to this day. Having dressed up, return to your son in Kiev and stay with him in love" ^{[18, p.25]}.As the country becomes Christianized, churchyards also become centers of spiritual life.

A temple is being built in them, where the surrounding population regularly converges on church services and holidays.

We read about churchyards as centers of local legal proceedings in the Novgorod birch bark certificates. Thus, letter No. 154 (XIV–mid-XV century.) informs about the interrogation of a witness (named Omant) by the ruler and the summons (announcement of the summons of the accused to court in Novgorod) on the spot: "The question of the ruler of Omant. Rostyagalesfipe (Filipe) with Ivan Stoick…Dale Filipe Stoic 3 rubles in silver and 7 hryvnia kun and a horse. And the Stoic is led away in viri and with the posadnik and with the sochki (sotskiye). And then it was a long day. And that was lost in the churchyard at the auction" ^{[19, p.33]}.

The use of this word in other meanings, for example, a certain community of people, should be distinguished from a churchyard-settlement. Thus, S.B. Veselovsky also wrote that "initially, not the territory was assigned to the churchyard, but the subject and the subject population, and only later the territory occupied by this population acquired the certainty of a subject and judicial district" ^{[20, p.15]}.

In this regard, we will give an example that allows us to attribute one of the stages of the evolution of the term "churchyard" by the period, at least, no later than the middle of the XIII century. We are talking about the text of the Novgorod birch bark charter No. 211 (layer 1268-1281), which informs about the purchase by some feudal lord from the community-churchyard of the right to use mowing in the village of Voiegieh and reconstructed by A.A. Zaliznyak in the form: "For the village of Voiegieho, there were about 5 hryvnia in total (total). From otskos and from between daho sarati (to plow) the churchyard. 2 hryvnia was carried ..." ^{[21, C.235–236]}.

The beginning of work on mapping and observing the nature of the boundaries of churchyards and pyatinas according to the Novgorod scribal books was laid in the works of K.A. Nevolin. At the beginning of the XX century. this direction of research was continued in the works of A.M. Andriyashev. The localization of the settlements of the Shelonskaya Pyatina carried out by him with a high degree of accuracy made it possible to draw a number of important qualitative conclusions about the nature of the settlement system of the Novgorod land of the era of scribal books ^[15]. According to A.M. Andriyashev, the borders of churchyards (districts) in rare cases coincided with the flow of large rivers, but willingly passed through vast swamps and forests, being a natural obstacle to the communication of residents among themselves.  These territorial districts were formed gradually and organically in the course of economic development of the area, not allowing through the stripes of the lands that make up them. At the same time, the size of churchyards depended on their proximity to cities and trade routes. A more dense and centralized population required the creation of more administrative units. In Shelonskaya Pyatina, the closer to Novgorod, Lake Ilmen and the lower reaches of the Shelona River, the smaller the churchyards-districts are in size. On the contrary, as they move away from the centers of economic life, their sizes increase, exceeding one thousand square kilometers in area for Sumer, Belsky, Dremyatsky and other remote districts.

The fifth division of the country, according to the general opinion of researchers, was an innovation and came simultaneously with the establishment of the Moscow government and its orders in the Novgorod land. Unlike churchyards, the borders of the pyatins passed along large rivers and reservoirs. For the Shelonskaya Pyatina, these are the rivers Lovat and Meadows flowing into the Ilmen and the Gulf of Finland, respectively. Sometimes these boundaries were drawn regardless of the churchyard division (see the following figure):

Fig.4 Map of churchyards-districts of the Shelonskaya Pyatina of the turn of the XV–XVI centuries. (built by means of QGIS on the map from ^[15]). Districts are highlighted in a separate color, some of which belonged to other five regions (green – to the Votskaya, purple – to the Village).

Scribal books of the XV–XVI centuries, along with other written sources, allow us to roughly recreate the economic system of the Novgorod lands of that transitional era. In his brief description, necessary for understanding further modeling, the author of this work follows the conclusions of A.I. Nikitin, a well-known historian of that period ^[22].

In the era preceding the Moscow conquest, the vast majority of the lands of Veliky Novgorod were in the hands of private owners, primarily boyars and monasteries. Their possessions consisted of a mass of plots scattered throughout the Novgorod land. Neighboring villages were reduced into one whole – volosts and volosts. In some of them, a special (large) owner's yard was arranged, which was a place for both storage and management.

The owners themselves, living permanently in the main city, entrusted local management to their trusted persons – keepers, (or poselsky, if we are talking about the possessions of monasteries). These latter were usually placed in estates of the largest size, from where they were in charge of the district subordinate to them, going around individual villages. The functions of the keepers included the distribution of free land plots to the peasants for use and the collection of duties imposed by the owners from them.

At the same time, the land was surrendered to the peasants not under the conditions of payment of a certain rent, but a certain share of the product – polovya, polovnichestvo (that is, half of the total income). In addition to the half, a small income was also collected – a constant levy in favor of the owner with other products of the peasant economy (bread, meat, cheese, eggs), which meant the expense of receiving owners in the villages. Finally, the third type of extortion – the housekeeper's income – repeated the owner's income, only to a much lesser extent.

 The Moscow conquest was marked not only by the mass seizure of boyar and monastery lands and the eviction of their owners, but also by the radical breakdown of the local economic structure. All the seized lands were transferred to the Grand ducal fund (the so-called palace and toll lands were made up), from which the Grand Duke endowed his servants with estates). At the same time, unlike the previous era, these lands did not go to the landlords as private, but as temporary property for the duration of service. As for the peasant population, in addition to the previous dues and duties, the landlords and dues volosts paid state (obezhnaya) tribute.  The following are excerpts from the description of a landowner's estate from the scribe book of Shelonskaya Pyatina in 1495 ^{[11, columns 7-9]}:

"In Paozerye zho, for Ondrej for Fedorovich, the villages of Lukinsky Isakovo Fedotina, the village of Yurovichi: dv. in the big man his Petrushka Lyatchko; and Christian: dv. Fomka Zinovov, dv. Ontonko Ivanov, dv. Ileyka Ivanov, Ondreev man's arable land 4 boxes of hay 60 kopen, obzha; and khrestyanskaya arable land 18 boxes; hay one and a half hundred kopen, 4 obzhi. D. Terebushino: dv. Ivashko Trofimov, dv. Smeshko Mitrofanov, arable land 8 boxes, hay 40 kopen, one and a half obzh. D. Pelesha: dv. Mikulka Mikheev and Ovsevko Kuzmin, 6 boxes of arable land, 50 kopecks of hay, obzha. D. Ilovo: dv. Vasko Orefin, 2 boxes of arable land, 50 kopecks of hay, half an obzha..."

The above fragment describes the villages inherited by the landowner, Andrei Fedorovich, from the former owner, the Novgorodian Luka Isakovich Fedotin. As we can see, peasant plowing prevailed, although there was also a small boyar: in the village of Yurovichi there is a large owner's yard, a yard man Petrushka Lyatchko lives in it, plows the land and mows hay exclusively for his landowner.  Then there is a description of two more such boyars: one went to Andrey Fedorovich from Ivan Vazgunov, the other from Fyodor Yuryev. 

The description of the estate concludes with a summary indicating the old (under the former boyars) and the new landowner's income:

"And all according to the old letter for Ondrej villages 6 in all three boyars, and yards 10, and people 11 people, and the run 11. And the old income went from those villages to Luke and Ivan and Fedor with three rounds of money and for bread 5 hryvnia, and with seven rounds of bread, and with burn a third of the bread. And 3 yards arrived, and 5 people, and 2 passengers, and the village was gone. And according to the new letter, there are 5 villages behind Ondrey in all three boyars, and 13 courtyards, and 16 people, and 23 run-around, and 4 ploughs with a third. And from those obezh Ondreev, a person plows obzhu, and from two hundred obzhi income emlet money 2 hryvnia Novgorodsky and 4 dengi, and from bread a third. And the housekeeper's income is 5 boxes of rye and oats without a quarter, 5 shoulder blades of boran, 9 cheeses, 50 eggs… Yes, hay is mowed for Ondrey on the Lukinsky fires for a Veryazhoy 500 kopecks."

As follows from the scribal books, the main type of peasant settlement of that time was a village with from one to four courtyards. Of the parts that made up the peasant estate, first of all, we will call the courtyard (with a yard and a cage enclosing the property) and the vegetable garden. The root layer of arable land followed immediately behind the yard, which is why it was called "backyard". Also, as part of the estate, there were also arable lands away from the yards (called "leshiye", that is, in the forests) and lands newly cleared for arable land (the so-called "priterebs").

Having received the estates, the new owners completely moved into them, started a boyar smell and tried to give their possessions some organization. According to A.I. Nikitsky, this organization consisted in rounding up its lands in order to achieve predominance in a well-known district and eliminate harm from the mixing of possessions. This goal could be achieved in several ways, for example, by bartering (exchange) with the owners of neighboring plots. In the case of the desolation of peasant yards and villages, it could also be the settlement of their yard people in them. Finally, in order to increase their own incomes, landlords started peasant plots where agriculture did not play a primary role. Subsequent scribal books are full of the presence of all kinds of repairs – the beginnings of villages that either had no arable land at all, or clearly in insufficient quantity. The following are excerpts from the description of one such landowner's possession, taken from the census book of the Shelonskaya Pyatina in 1539 ^{[11, columns 292-297]}."The churchyard of Mikhailovskaya on the Polona River, which happened from the time of St. George's Monastery, and in it the church of the Most holy Theotokos.

For Fedor for Ivanov Lazarev's son in the churchyard of the yard. big, yard. Fedor himself. And there are no weak people: dv. pop Fyodor Ortemyev, deacon of the church Ivanko Mikulin, dv. proskurnitsa Marfa. And his people: dv. Prokop Fomin, dv. Tarasko melnik, dv. Nekrasko Rubin, dv. Kubas Litvin, dv. Makar Surovtsov. Yes, on the church land, the Bobyl courtyards are: Grishka Babin, Luchka Ortyushin, Petrushka, Mitka without arable land, so they give two altyns to the priest to the church, boyar and servant and peasant arable land in one field 40 peddlers, and in two fields, hay 500 kopen, 3 obzhy. And the lands on the yard on the river on the Strip are melnitsa and the latrine field is raked, 3 boxes of arable land, 30 kopen of hay."

In this passage we see a description of the central settlement of the Mikhailovsky churchyard-district –the churchyard of the same name-the place of Mikhailovsky on the Polonoi River. More than 40 years have passed since the compilation of the scribal books by M.I. Valuev, but even for the second census (1539), the volosts and volosts of the former Novgorod owners remain the economic unit of description. In this case, it is the former land of the Yuriev Monastery, one of the largest and oldest in Veliky Novgorod, located on the estate of Fyodor Ivanovich Lazarev. In the churchyard there is a church of the Intercession of the Holy Virgin and houses of church servants are located: a priest, a deacon and a prosvirnitsa.

In addition to the parish center, the churchyard is also the focus of the landlord administration. There is a large owner's yard on it, in which the landowner himself lives, and the yards of his people are located. Among them is the miller's yard, which is in charge of the landowner's mill on the river. The landowner's lands are also used by the priest who is attached to the estate. These "servants" of the land differed from the land cultivated by landlords on the landowner, in that the income from them did not go to the benefit of the landowner. In most cases, the servants cultivated the land given to them by their own efforts, but sometimes, like landowners, they surrounded themselves with their own people. So, on church land we see bobylsky yards (family-free), who paid the land tax to the priest.

Then there is a part with a description of landowner villages, similar to the above example of the estate of Andrei Fedorovich in Paozerye. Of interest is the following part with a description of the landlord 's repairs:

"... Yes, Fyodor Lazarev put repairs on the forest after pisma novo, and those repairs were given as an appendage to Nechai and Tretyak by Kostyantinov's children Valov and his brother; and Nechai and Tretyak the Oxen and their brothers left those repairs because they were thin and at odds. Poch. Shevelov: dv. Shevel Ivanov, there are 2 boxes of arable land in one field, and 10 kopecks of hay in two fields. Perished; dv. Kuzemka Ondreev, the arable land in one field is half-2 boxes, and in two fields, hay is 20 kopecks ...".

The number of boxes given in this passage indicates the amount of grain being sown, which is insufficient for conducting a full-fledged economy without additional assistance. The use of the expression "and in two-man" allows us to conclude that the three-field system of agriculture was reproduced even in such "embryonic" areas.

The final results of the description allow us to assess the extent of the expansion of the cultural area within the estate over 40 years: "And there are only 20 repairs, and there are 24 yards in them, and there are 25 people in them, there are 62 boxes of arable land in one field, and there are two fields, there are only 251 piles of hay, and 11 fields are marked in them, and 4 ploughs without a third; and the income from them is not shol, they are sitting on a lot".III. Numerical experiments

Based on the above, as a model of the settlement structure of the Shelonskaya Pyatina of the turn of the XV–XVI centuries, this article proposes to use a system of 2-level hierarchy: churchyard-district – volostka – yard.

The original data are the preserved data on the number of households and yards in 24 churchyards-districts of the Shelonskaya Pyatina, collected by A.M. Andriyashev (see Appendix 1). 

Recall that one of the theoretical results of the proposed model is a linear relationship between the logarithm of the number of yards and the logarithm of the area of the churchyard-district (zone of influence of the 1st level) with a coefficient of 6/7. Verification of the correspondence of this theoretical result to empirical data was carried out using a package of statistical programs Statistica:

Fig.5 Scattering diagram for the logarithm of the number of yards (ln dvor) and the logarithm of the area of the churchyard (ln obj)

Fig.6 Correlation coefficient for the logarithm of the number of yards (ln dvor) and the logarithm of the area of the churchyard (ln obj)

 As follows from Fig.5 and Fig.6, the hypothesis of a linear relationship between the above-mentioned features (ln dvor and ln obj) is confirmed with a significance level of over 0.95. The coefficient of linear regression between them is 0.8476, differing from the theoretical one (6/7) by 0.0095, which even slightly improves the previous result with the model of a single-level hierarchy ^[6,7]. The correlation coefficient between the features is 0.93. That is, changes in the dependent attribute are explained by changes in the independent by about 87% (coefficient of determination).

Based on the proposed model, the next step can be taken – to start the task of data replenishment. As already mentioned, only 24 of the 71 churchyards-districts of the Shelonskaya Pyatina have data on the total number of yards. Substituting into the linear regression equation as a variable the total number of visits to the remaining churchyards-districts, we get the corresponding number of their yards. The results of this reconstruction are given in Appendix 2 (column "number of yards").

From estimating the total number of yards, you can proceed to the next step – reconstruction of the total population of Pyatina. It would seem that you can get this figure by multiplying the total number of yards by the average number of families (father, mother and children). However, not one family could live in the yard, but two, and even three (for example, two brothers with families lead a common, undivided household under one roof). This fact is eloquently indicated by the number of "people" (that is, family men), which does not coincide with the number of households (see Appendix 2). So, for the Shelonskaya pyatina, according to the new letter, there were 1.5 "people" per yard.

 Later and more complete documents (the census book of the Old Russian churchyards of 1669 and the household lists of the patrimony in the Poshekhonsky patrimony of Bestuzhev-Ryumin for 1731) allowed researchers to establish how many families there were in each yard and what the sizes of the yards and families were. The analysis of these figures led A.L. Shapiro to find some regularity and build a corresponding table that correlates the number of "people" marked with scribal books in the yard and the number of people in the yard (including women). So, for the Shelonskaya Pyatina, this conversion factor is equal to 6.25 (see Appendix 3). Multiplying it by the reconstructed total number of yards (14249 units), we come to an approximate estimate of the rural population of the churchyard, equal to 90 thousand people (the reconstructed number of the Vshelgsky churchyard is added to this figure based on the area occupied by it in square kilometers, and not the number of obezh, for which there are no data from scribal books). Interestingly, this figure is very close to A.L. Shapiro's own reconstruction (93 thousand 773 people) of the rural population of the Shelonskaya Pyatina at the turn of the XV–XVI centuries, differing from the latter by no more than 4% ^{[23, p. 124]}.

ConclusionThe high degree of conformity of the obtained theoretical and empirical data allows us to consider mathematical modeling as an adequate and convenient auxiliary tool for studying the nature and dynamics of historical rural settlement systems.

So, for the North-West of Russia at the turn of the XV–XVI centuries, we can state the predominance of the "nest" type, which consisted of a combination of two multidirectional trends.

 The first of them was the desire of the population for the most convenient access to the controlled area, which led to its uniform distribution throughout the territory.

The second, on the contrary, led to the concentration of the population around its centers. Moreover, such concentration was hierarchical in nature. Firstly, the surrounding population settled near their central settlements (centers of the 2nd level). As such, there were owner villages in which the landowner's manager lived and management and control over the dependent population was concentrated.

The centers of the 2nd level, in turn, concentrated around the central settlements of a higher hierarchy (centers of the 1st level). As such, there were churchyards-settlements that were the centers of parish life and state administration.

Further, in the situation of the absence of some documentary sources, the proposed approach can also be used in the task of filling in the missing data, which was demonstrated by the example of the reconstruction of the rural population of Shelonskaya Pyatina.

Appendix 1Territorial districts of the Shelonskaya Pyatina.

General characteristics according to the new letter

 (source: ^[15])nn

Appendix 2Territorial districts of the Shelonskaya Pyatina.

Reconstruction of the population at the turn of the XV–XVI centuries. (source: ^[15])nn