Some Fundamental Principles of Living Systems’ Functioning and Their Impact on Human Psychological Systems

: This theoretical article provides a brief description of the model of living systems’ functioning by deﬁning them as self-reproducing information or as self-reproduction of resource ﬂows patterns. It reviews the living systems growth limitation between their development cycles by the Fibonacci sequence. Besides, there are presented systems resource base criteria, necessary for accumulating the resources and their investment. The article also considers the conditions for the formation of various systems strategies. Then we reviewed the principles of elemental analysis of information by a person as a living system according to the considered model. The study also shows the possibility of forming priorities in analyzing information for 16 combinations as maximum. At that, it remains crucial to divide a human’s information analysis between the two hemispheres of the brain. The described combinations of priorities in a person’s information analysis are compared with the existing differential personality models, such as the big ﬁve personality traits, the Myers–Briggs type indicator, temperaments model and Honey and Mumford Learning styles.

Introduction
Before studying a human, it is essential to determine what a human is and his aim according to the model reviewed in this article. At that, we should research not so much the complex of current expressions of human existence as the characteristics of the environment a person originated. To do that we have to introduce several concepts that will form the base of our further study presentation. Let us start by describing the nature of human as a living system and the conditions from which the environment was developed according to our model.
Let us define the concept of living systems. We will not analyze in detail what a system is since its definition varies from author to author [1,2] and a definition of living systems. However, since in this article we consider the issue of living systems in particular, so that any definition of a system will be appropriate if it does not contradict the definition of a living system used hereinafter. For instance: A system is an arrangement of parts or elements that together exhibit behavior or meaning that the individual constituents do not [3,4]. Living systems show obvious differences from non-living systems, most notably in that they preserve themselves by dissipating energy and decreasing entropy [5,6]. Since living systems use low-entropy resources, they avoid increasing their own entropy [7]. Most living systems use other living systems as resources, however, there are such living systems that receive low entropy through other sources, which is possible due to the existence of resources flows from which they could obtain negative entropy, for instance, from solar radiation [8]. Thus, flows of matter and energy constantly run through living systems.
At the same time, entropy and information are inversely related, which also inevitably associate living systems with information, as these systems have to retain some information to secure their existence [9,10]. Living systems are often considered autopoietic systems [11], however, since the term "autopoietic" has no precise definition at present [12], we will avoid using it to define living systems. Along with that, it is worth pointing out an important part of the definition of "autopoietic", that is "regeneration of structure", or it would be better to use the word "reproduction" since it sounds more neutral and it fits the range of living systems better.
It is interesting to mention that NASA uses the following definition of life: "Life is a self-sustaining chemical system capable of Darwinian evolution" [13]. The capability to evolve could also be understood as the ability to copy a structure since without it a subsequent selection among various copies of a living system is impossible. However, in addition to the ability to reproduce in new copies living organisms can replicate a structure within one copy, replacing one of its matter with another. Snakes that shed their skin or the process of cell regeneration in the human body [14] are good examples of such replication, and the change of human generations in any community, including the closed ones. Such manifestations of living systems represent a transfer of the structure that is its reproduction albeit with the removal of the structure from the original matter. The regeneration of damaged or lost parts in living systems could be considered functionally similar. Such phenomena cannot be attributed to the mechanisms of evolution, although they are associated with the dynamic structure of living systems. Apparently, such phenomena can be called self-sustaining. Thus, the reproduction of a structure includes both the reproduction of the structure of a system in new copies and the sequential transfer of one unit of the system from one matter to another. In general, most definitions of the term "autopoietic" include these statements.
It is important to mention that the NASA definition of the term "life" probably refers to living organisms. To expand the concept of "chemical system" so we could apply it to all living systems, which also comprise groups of organisms and all social systems including states [15][16][17], it is convenient to replace it with a definition like "structure" or "availability of information".
Based on the above, in this article, we will use the definition of living systems as selfreproducing information, which are structures that reproduce themselves. Along with that, in this article, we will not consider the necessary requirements for living systems in terms of elements or functions since these issues have already been properly examined [9,10,18]. All information reproduced by a system is considered to be its component. If some losses or distortions of information occur, and a system does not reproduce all previously available information, but the stored information is capable of self-reproducing, then the system exists. At that, self-replication should not be confused with autonomous existence. As mentioned earlier, living systems often use matter or energy from other living systems for their reproduction. They can also use other living systems as information carriers. According to the proposed definition, a living system will be considered as such if it is capable to give to an external matter the structure relevant to a particular system.
If a system requires for its reproduction an external infrastructure to change matter to the structure it needs, then such a system forms part of another living system. Thus, if some part of the living system structure was excluded from reproduction, then it ceases to be a part of that living system, even if that structure itself is still preserved within some matter. For instance, after the fall of the Western Roman Empire, numerous technological advances have remained to be in use but ceased their existence as an industry that is as a technological system since at that time they could not be reproduced. Another example could be some modern breeding technologies that enable producing sterile seeds or insects [19,20]. Such organisms are not living systems but they are information carriers for the social system they were created in. The same remark applies to seeds that in subsequent generations lose many of their properties [21]. The part of the living system for such plants will be only those properties that persist after several generations, and the rest of the properties forms part of a social living system in which these features are reproduced. In the same way, you can draw something on a turtle shell. Regardless of this drawing, the turtle itself represents a separate living system, but a turtle with a drawing is not since the picture forms part of another living system within which it was created and can be identified. Along with that, original agricultural species that transmit all information with losses that are of a normal random nature, and that depend on natural selection and resource endowment, can be considered living systems that use people as sources of resources and carriers [22]. The same is true of zoo animals, including those that are on the verge of extinction and are preserved only through people's efforts. According to such a definition, viruses are not-living systems but only part of a living system of organisms on which they parasitize albeit they represent a destructive part. Information transmitted by people through non-living or living objects is a part of their social living systems. However, in theory, if somebody could create a mechanism capable of completely recreating itself or restoring any of its parts when it is lost, it will be a new life, in the way living systems are defined in the present article.
It is also worth noting that self-reproduction of information is equivalent to selfreproducing patterns in resource flows. Along with that, to maintain such patterns, that is to transfer information across the matter, several flows of different resources are required since it is impossible to create a reproducible structure from one resource. The possible units of information for one resource are only the presence or absence of a given resource, which is impossible to identify and, therefore, to transmit it using only this resource itself. Thus, there are needed several flows of resources exchanged in a system. Then that system will be able to give the proper structure sequentially to each of its constituent resources. Therefore, only self-reproducing patterns in several flows of resources can be living systems. At that, the pattern of resource flows is associated with the exchange of these resources. The concept of exchange is necessary since a modification in one resource occurs due to the use of the expenses of another one and vice versa. A unidirectional change of just one type of resources is impossible since then the second type of resources can be changed only by itself, which as we explained above is impossible.

Fundamental Principles of Living Systems Functionality
A system is considered as regular interaction between resource flows, including complex resources. Such interactions in a system are not totally symmetric and depend on the mutual distribution of resources in the environment where that particular system exists. Thus, let us introduce two main parameters on which such interaction depends. The first one is H: it indicates each resource source's intensity or the amount of resource received in the result of each interaction. The second is F: it represents the density of resources or the probability of interaction with a resource's source within one cycle.
Each interaction exchanges two types of resources; however, such resources are often complex and depend on several more plain resources. Both parameters mentioned above are relative, so it is convenient to assess them regarding the counter resource parameters.
In practice, interaction in a system of two flows of resources with equal parameters is impossible. That means that one of the resources is always excessive to support one interaction, due to either frequency or intensity. Such a resource is invested in the interaction of resource flows that exceed the volume required to recover the counter resource volume obtained within one interaction.
Then, in a case of two interacting resource flows, defined as n and k, it is convenient to use two indicators to characterize their correlation-the ratio of their quantities, that is S n , defined by: and A n , defined by the formula: If S n > 1, then the resource n is excessive, and vice versa. The essentiality A n shows in which of two resource flows the intensity of resource's source makes a more significant contribution to the exchange process than their frequency. When A n > 1, the intensity is more important for a resource flow n, and vice versa. It is convenient to define S n as a relative excess of resource n, while A n is its relative essentiality.
When S n > 1, then excessive resource n is invested to increase the resource flow k, passing through a system. If A n > 1, then a more essential resource n is accumulated in a system to maintain further interactions of resource flows. It must be noted that parameters S n and A n are quite independent. Thus, an excessive resource could be essential and therefore requires accumulation.
When interacting with resource k that is scarce but frequently encountered in small amounts, more essential but intensive resources n and accumulations are invested in changing the infrastructure of interaction with resource k to increase its intensity. Additionally, vice versa, when interacting with scarce and essential resource k, excessive and less important resources n right after being obtained are sent to expand the infrastructure of search of more useful resources k to increase the frequency of interaction with them. In this case, resources k are subject to accumulation. Relative distribution of resource flows determines the direction in which system will develop, which is extensive outward or intensive into the system itself.
Thus, the permanent core of a system (resource exchange node) is formed by accumulated essential resources that are scarce or excessive that are constantly passing through themselves less essential and simpler resources, regardless of their redundancy or insufficiency for interaction. That is the way a living systems' structure works as patterns in resource flows. That is the way the structure of living systems works as patterns in resource flows. Roughly, it could be said that accumulated essential resources represent an "inner" part of a system. In contrast, more common resources that are constantly flowing through their structure constitute an "external" part. Such division is not entirely correct, since all interacting flows compose that system's structure regardless of their accumulation. Nevertheless, such a division is convenient for the brevity of description.
Lets display these relative parameters on a graph on coordinates of relative intensity and frequency ( Figure 1). In that case, both interacting flows' parameters will correspond to the opposite quadrants formed by S n and A n lines.
The ratios H n /H k = 1 and F n /F k = 1 are also important threshold values. They show how much priority one of the resources flows has in forming a system's structure. Depending on S n and A n parameters, either H n /H k > 1 or F n /F k > 1 show that resource flows n are of higher priority for a system in terms of the evolution of its structure. With that, resource flows k are formed in a system more depending on the structure of resources sources n. Thus, according to their relative characteristics, it is convenient to refer the resources of living systems not only to the opposite quadrants formed by the lines A n = 1 and S n = 1, but to the opposite octants, which were formed by additional lines H n /H k = 1 and F n /F k = 1.
We shall notice that it is impossible to invest resources only in expansion by creating a new structure. The reason is that with a system growth its border with the external environment grows slower than the relative inner amount of accumulated resources. That means that the flow of new resources will usually increase slower than the costs of maintaining a relative internal volume. Thus, a simple expansion of a system has certain limits. The easiest way to increase the flow of resources through the system while approaching the simple expansion's limits is to split the system into multiple identical copies. The emergence of such algorithms also increases competitive advantages, hence the probability to distribute exactly such systems. In addition to the simple development of mechanisms for the direct acquisition of resources, as the latter's efficiency decreases, systems invest resources in self-replication that is one of the expansion methods. 1a. those that are used immediately (A n > 1), with a larger contribution of intensity, than frequency, and 1b. accumulated resources (A n < 1), with a larger contribution of frequency, and 2a. resources with higher negentropy (S n > 1), which are spent on the redistribution of the counter 2b. resources with less negentropy, which are redistributed, adjusting to the structure of the system (S n < 1).
At present, the list of models of various living systems strategies that could be an alternative to the ones considered in the article is not large. For example, there is a Grime model that describes three types of strategies in plants [23,24]. More often, this model is used to describe the plants' competition on relatively unproductive habitats that appear after some accidences [25]. However, some researchers are expanding its application to include humans [26]. There is also a similar scheme where the fourth variant of the strategy is added [27]. Besides, there are several business models, two of which are major marketing models. The first one is the BCG Product Portfolio Matrix [28], which describes strategies for different types of company products. The second one is Porter's generic strategies model [29] of competitive strategies, where the main factor is the relative competitive characteristics of the company regarding the rest of the market. These models generally have a similar principle of resource provision in their basis. The model considered in this article as a whole only refines the existing empirical models, bringing them to a more schematic view, so they probably could be combined in the future. Now let us introduce the concept of a system development cycle. In the context of our model, it means the time between two interactions with a source of accumulated essential resources. This period is essential since, after each interaction with sources of an essential resource, a system not only accumulates it but also creates a structure on its basis, which it uses then while interacting with the counter flow of resources. However, the growth of living systems is limited to its specific maximum rate. Let us define it.
When two resource flows interact in a system, the amounts of these resources should be approximately equal. The equation of the number of resources n and k we will define as follows: the amount of resource n equal to E n = H n • F n corresponds to the amount required to maintain the counter flow of resource k in the amount of E k = H k • F k needed to keep the flow of resources n in an amount E n . In this case, the condition E n = E k must be satisfied. However, such an equation cannot be maintained at each particular moment since the exchange of resources is uneven due to the inequality of resource flows' parameters.
As mentioned earlier, excess resources flowing through a system during a specific cycle are invested in developing the system and acquiring more scarce resources. Thus, the system seeks to equalize resource flows. It should be noted that the system interacts only with that amount of resources for which a corresponding structure exists in that system. The system does not use the rest of the resources. That means that they do not flow through it. At each period, resources flow through the system in the amount for which there is a structure in it. Therefore, for the "excessive" amount of one of the resources that are still participating in the exchange, for example, n, the structure also exists at each period.
The amount of excess resource n available at the beginning of the cycle i equal to E n(i) is invested in the growth of the counter resources flow, which normally should result in a corresponding increase in resource k. However, it will be impossible to proceed increment by the beginning of the next cycle, since the amount of resource flows at the beginning of cycle i + 1 corresponds to the structure at the beginning of cycle i. The new structure obtained from investments in cycle i will appear only by the beginning of cycle i + 1. Therefore, the resource flows growth at its expense is possible only by cycle i + 2 beginning. At that increase in the amount of resource k corresponding to resource n investment in cycle i will appear only at the beginning of cycle i + 2. Then we have or The inverse expression is also true since the new amount of resource k is directed to create a new structure to obtain resource n, which will also give the result only after one cycle. Then the total amount of resources flowing in the system at the beginning of cycle i is defined as That results in that the structure of living systems in norm grows at a rate corresponding to the Fibonacci sequence that is confirmed by multiple examples from nature [30,31].
We have one more issue that should be considered when speaking about the fundamental principles of living systems functionality before analyzing humans' example. That is the resources themselves that are exchanged in the complex living systems. As mentioned above, with a sufficiently high evolution level, the two systems develop ways to coordinate actions between several individuals. Over time, complicating the system, they create a large one with various individual participants' specializations. A good example is a living organism composed of individual cells and a group of social insects or animals. In such groups, the main part of resources exchange taking place in living systems that are part of them is an exchange between the complex of resources that the living system itself can obtain and the complex of resources that the living system can receive with the help of the group. They could be roughly divided into physical and social. Additionally, further, considering humans' case, we will define such resources as "physical" and "social". It is a rough division, and we use it only for the brevity of description.
According to the proposed model, these are the fundamental principles of living systems' functionality that could also be applied to humans. Now let us consider some of the particular features of human functioning as a living system.

Basic Principles of Human Functioning as a Living System
Simple living systems use relatively primitive reflex-based instruments to analyze the environment. However, with evolution, more sophisticated analytical tools based on experience and modeling are increasingly used to analyze the environment instead of simple reflexes. The most advanced organisms have two main analytical centers that significantly increase interaction efficiency with the environment. Humans having two cerebral hemispheres also belong to such living systems.
The two hemispheres of the brain functionally represent two quite independent computing centers. Let us make assumptions about what they are needed for.
They are often called "logical" and "imaginative", or according to Kahneman, they represent a slow and a fast system [32]. Like most similar sufficiently advanced living systems, including social systems of multiple people, a human requires long-term planning and prompt response to current events. It is impossible to perform complex calculations through logical associations to ensure a quick reaction. It is more effective to do a simple search in memory for analogues to determine whether similar images are associated with positive or negative events and what actions were usually performed in such situations. This is a rough description of how the right "imaginative" hemisphere works to ensure quick reaction "here and now". In other cases, there is time for structuring information and analyzing the logical connections between its components. Among them are questions of low importance for the current situation, which is basically the system's medium and long-term strategy. Since it is more efficient to separate these two types of information processing at the hardware level (or at the analytical centers level), one common organ is not sufficient. For this reason, among others, two computing centers are required, that is two hemispheres.
Herewith, if we consider the specifics of work from the point of the analyzed information, then for the right "imaginative" hemisphere is vital to have a variety of information to expand the range of situations in which a prompt reaction is possible. This is possible in the case of interaction with resources with relatively numerous sources. Then it is required to choose the most suitable ones. At the same time, logical connections are essential for the left hemisphere to plan the achievement of goals, which is required only for resources whose sources are relatively more essential, and therefore need an increase of the probability of interacting with them. Thus, the left hemisphere primarily analyses the environment's components related to relatively more essential resources at A n > 1. Accordingly, for the cases related to the right hemisphere's specialization, the indicator is A n < 1.
It is also more efficient to use different processing centers that are the brain's hemispheres for separate control of two directions. The first one is (a) investing relatively excess resources in obtaining relatively less available resources, and the second is (b) developing interaction with relatively excess resources themselves by structuring relatively less available ones. It will be convenient to define these processes as (a) "extraverted" and (b) "introverted". The first ones are aimed at changing any parameters of "external" resources by investing "internal" ones, and the second ones are aimed to adapt to the parameters of "external" for better compliance. "Extroverted" processes correspond to the condition S n > 1, and vice versa, "introverted" ones correspond to S n < 1.
Finally, the third and last parameter of our model that should be mentioned is the difference between "physical" and "social" resources among the objects with which a human interacts. The "physical" in general retain their qualities over time, while the "social" objects are associated with other people, that is with subjects that have qualitatively less permanent functions. This means that it is possible to build static models for "physical" objects during analysis, but for "social" objects is required modeling based on some complex functions. However, the variety of "physical" objects considered constant by their characteristics is available in greater numbers, although their analysis is more superficial. That is, such division also requires a significantly different way of information structuring. That is also much more efficient with two considerably autonomous processing centers, or cerebral hemispheres.
Other ways of separating information are possible, not only into two types but also into more; however, the separation options described above are crucial for systems' effective functioning [33].
At the same time, in this article, we considered a human not as an organism, but as a person. That is, we do not consider the human body, which is a node for the physiological resources exchange that proceeds at a more primitive level than the one considered in this article that requires an analysis of the environment. In contrast, we investigate the exchange process of more complex resources, which are required not so much physiologically as psychologically.
Thus, we considered three pairs of parameters the control over which is favorable to divide between two hemispheres of the brain. It would be appropriate to consider how these parameters are divided between the hemispheres, besides operational and strategic information principle. The mechanism of the latter ones works under clear criteria and has already been described above.
"Extraverted" and "introverted" processes of a human as of a system correlate during a person's analyzed information. Therefore, they correspond to human actions with "physical" and "social" resources, depending on their relative redundancy. However, the ratio of these resources differs for each person. For this reason, the analysis of the processes associated with them cannot be so strictly divided between "extraverted" and "introverted". Likewise, due to the inevitable differences in a person's information environment concerning "physical" and "social" volumes, the analysis of relevant information cannot be divided in advance between the cerebral hemispheres.
Depending on which resource relatively more excess, "physical" and "social" resources sources, it will be determined, which resources will participate in "extraverted" processes, and which ones in "introverted" ones. Along with that, how much more essential are the sources of "physical" and "social" resources, that is, information about which kind of resources requires a more structured analysis will affect the distribution of the cerebral hemispheres in the processing of relevant information. Thus, due to the fact in which two quadrants are formed by the lines S n and A n , how the human brain will work is determined by the rate of essentiality the sources of relatively excess resources. Additionally, the way the analysis of "physical" and "social" resources is distributed between these quadrants is determined by how the flows of these resources relate to each other in terms of essentiality and redundancy.
As we can see, there are several combinations in dividing information analyzed by a person between the hemispheres of a brain, depending on the parameters of the environment at the beginning of ontogenesis. For instance, let us assume the situation when relatively scarce resources coincide with relatively more widespread resources, for which the right hemisphere, which is more useful in direct activity is responsible. Then, one can expect that a person will show signs that are characteristic of "introverts". That is, in short-term situations, he or she will be more adaptive and vice versa. Moreover, if these resources are "physical" resources, then, at first, a person will probably adapt to the current physical conditions. In the opposite case, a human will adapt to the needs related to the social environment's requirements at the moment.
As in the case of other living systems, important threshold values are the ratios H n /H k = 1 and F n /F k = 1, showing in the case of a person how much the structure of one of the resources has priority in the formation of the structure of the analysis of all information.
After describing the basic principles of combinations of analytical accents of a person as a living system, it should be briefly clarified which combinations could be formed according to this model.

Development of Human Information Analysis Mechanisms
According to our model, we considered above that the combinations of analytical accents between the brain's hemispheres depend on the relative distribution of resources used by a person. Although the characteristics of resources are not constant, the analytical mechanisms cannot be reformed depending on a particular situation. In such a case, the entire structure of accumulated information will lose its effectiveness, and it will have to be collected almost from scratch. There is a greater possibility that it is less effective than analyzing data based on outdated accents but on essential experience. For this reason, it is possible to consider the distribution of analysis functions between the hemispheres of the human brain as quite fixed from a particular moment. In addition, it can be assumed that in all living systems associations established early in the life of a system are more permanent than those established later [15].
Besides, it is essential to note that a person is a living system with a minimum ratio of the number of innate mechanisms of activity to the number of acquired ones based on analysis and modeling. However, analysis requires a minimum of initially specified information and a maximum of new data. Yet, many complex phenomena cannot be understood without prior acquaintance with simpler ones. Thus, it is impossible to understand social objects (or subjects) without a preliminary examination of generally static physical objects. In this regard, the distribution of the analysis of "social" and "physical" resources between the cerebral hemispheres is impossible initially but becomes possible after their sufficient examination. Until that moment, both hemispheres are probably analyzing flows of "physical" resources that differ in abundance and essentiality.
Besides, to manipulate something, you need to get some prior knowledge since manipulation involves different properties of one object. To operate them, you need to learn to perceive a single object as one object in various conditions, not as several different objects. Moreover, it is necessary to find out which properties various objects could have.
The last temporary limitation worth mentioning is that it requires a broad experience to determine any patterns in the environment. This could be manifested in that all rather complex aspects of information analysis initially could be a specialization of only the right "imaginative" hemisphere. After having collected information base sufficient for structuring in the relevant field, it becomes possible to implement the "logical" approach of the human brain's left hemisphere.
Along with that, in our model, we will assume consciousness only as a tool for the environment modeling and its system itself that depends on the general accents of information analysis. Therefore, it does not play a significant role in the distribution, it only complies to it.
Then, assuming that in addition to a small number of innate zoo-social programs other methods of analysis and modeling of the environment are acquired and developed in the process of ontogenesis, then the general sequence of stages of the environment exploration can be as follows.

The Model Sequence of Stages of a Human Environment's Examination
On the first stage, a human acquires knowledge about the surrounding environment's basic features, such as the signs of various "physical" objects, their types, usefulness or harmfulness, location and dislocation. It is quite easy to assess resource sources' relative characteristics for a given stage of human evolution as a living system. Since "social" resources are almost unknown to a person, all interaction with them can occur only due to innate zoo-social programs that manifest themselves with regard to relatively more essential "social" objects. Therefore, the sources of "physical" objects as resources are relatively excessive and not so essential, which could be defined as S n > 1 and A n < 1. At that, typically the "social" objects with which a person interacts at this stage are much more intense, and therefore H n /H k < 1. Thus, it is possible to determine the octants, which include the "physical" and "social" resources of a person as a living system, at this first (I), stage of development ( Figure 2). For convenience, the right hemisphere's development stages will be designated by Roman numerals, and the ones of the left hemisphere by Arabic numerals. Figure 2. Relative characteristics of resources of a person as a living system. The parameter A n > 1 corresponds to the resources analyzed by the left hemisphere of the brain, A n < 1 corresponds to the ones analyzed by the right. The parameter S n > 1 characterizes the resources with which the system interacts in an extraverted way, by changing them, S n < 1 described an introverted interaction adapted to the counter resources.
The relative parameters of the resources studied at this stage should relate to the "specialization" of the right "imaginative" hemisphere. This is quite expected since at first there will not be enough information to detect any patterns in such knowledge. Therefore, until acquiring the experience necessary to systematize information about the surrounding environment's basic properties, the left "logical" hemisphere will not thoroughly analyze this information. Thus, the right hemisphere should do the primary accumulation and analysis of information in this sphere. It can be expected that a similar order should be observed in other spheres.
With the accumulation of basic knowledge about "physical" objects, it becomes possible to examine their variable properties and the possible ways to adapt to such changes. Now let us evaluate their relative characteristics as sources of human resources. To do that, let us make an assumption about what kind of resources they should be assessed against.
With the expansion of empirically acquired knowledge and the environmental model's development, zoo-social programs' influence should decrease, replaced by activities specified by the acquired knowledge. Consequently, the initial "social" sources of primary resources are replaced by new ones, and the interaction with them is defined by acquired mechanisms. It can be assumed that a change of the source of these resources that is relatively more essential, as shown above, means the beginning of a new cycle of human evolution as a living system. Thereby, we can assume that after a person has accumulated sufficient basic knowledge about "physical" objects, the left "logical" hemisphere can structure this knowledge by determining patterns in them. However, in the first place, these patterns should define the "social" objects that previously were objects of innate zoo-social programs, since their importance cannot decrease sharply. Nevertheless, the interaction with them is specified by basic knowledge of "physical" objects. In other words, the left hemisphere enters the first (1) acquired stage of development.
There are significantly less variable "physical" objects, hence related to them sources of resources, than the total number of objects. However, at the same time, there are much fewer objects for which significant patterns could be found. Therefore, new to the analysis objects with variable properties, as before are determined as A n < 1 and F n /F k > 1. However, the probability that their intensity as sources of resources will exceed the intensity of "social" objects is not high. Thus, they can be determined as S n < 1. So, the octants of the relative characteristics of "physical" and "social" resources of this second (II) stage of development could be assessed for most cases.
After understanding the fundamental properties of "physical" objects and their change mechanisms, a person can naturally review objects' functions. By functions, we mean changes in objects' properties when interacting with other objects with some other properties. There are usually significantly fewer objects with any functions than just variable objects, but more than "social" objects, even if the latter's range expands. Therefore, A n < 1. However, with the development of interaction mechanisms with the environment, a person's independence as a living system increases. Thus, a person receives fewer resources from the "patronizing" him or her "social" objects and more resources from direct interaction with "physical" sources of resources. In this case, only the specified shift in focus of analysis will occur. Then it is possible to cancel the redundancy of such objects S n > 1 and H n /H k > 1, which allows determining the octants for the third (III) stage of development. As one may assume, the change in the significance of the former "social" objects, which are relatively more essential sources of resources means their replacement with functionally new ones (not necessarily practically new)-consequently, a new change in the development cycle. Then the left hemisphere will probably move to the second (2) acquired stage of evolution and will be able to analyze "social" objects based on their variable characteristics. However, the analysis will continue to consider them as "physical" objects.
Having understood the concept of variable properties of "physical" objects and their possible functions, we can move on to reviewing the issue dealing with the variability of "physical" objects in time. As before, for such objects relative to the former "social" objects will be true A n < 1. However, such objects are already quite rare. They mainly include living objects and some artificial ones. For this reason, for "physical" objects at this fourth (IV) stage of development is true S n < 1 and F n /F k < 1. Thus, we found the stage of evolution related to the two remaining and previously not mentioned octants.
The properties emphasized during the analysis of "social" objects were changed. Now they are functional properties, which depend on other objects. In fact, a person again deals with functionally new "social" objects, which means another shift in the development cycle and the transition of the left hemisphere to a new the third (3) level of analysis.
Having reviewed everything that we suppose a person should learn when sequentially exploring the environment on the first four model stages, we came to an important point. Now there is everything necessary to separate "social" objects from "physical" objects with variable properties, including functional ones, that sometimes change in time, the main property of which is a subjective relevance with the rest of the set of objects. At this stage, in addition to the historically distinguishing of several "social" objects in the environment, their properties start to be percept as a class of objects.
At the same time, now "social" objects belong to the broader set of "physical" objects as rarer and either less abundant or just as abundant. That could be defined as S n < 1 and F n /F k < 1. Since now in the described development sequence, the focus has shifted from "physical" to "social" objects then "physical" ones should become the priority of the left "logical" hemisphere. That is quite expected since the earlier its attention was directed primarily to previously significant "social" objects. Obviously, they were analyzed as "physical" objects. Still, it is now possible to expand the scope of attention to all physical objects of such a kind, regarding their objective essentiality, not their retrospective importance. As a result, the relative essentiality of the narrowed set of "social" objects decreased, which could be expressed by A n < 1. That keeps the system at this new the fifth (V) cycle in the same octants, but with an inverted arrangement of "social" and "physical" objects. Despite the change in relatively more essential resources and the beginning of a new cycle, it cannot be claimed that the left hemisphere has shifted to a qualitatively new level of analysis. It would be more correct to assert that the previous third (3) stage's analysis mechanisms were improved.
Further exploration of the environment by a human is associated with a better understanding of "social" objects. The next stage that naturally follows from the accumulated knowledge is the shift of attention to the analysis of specific human functional properties, often associated with emotional impact. That is, knowing about the variable functional properties of "physical" objects, their variability over time and the main features of "social" objects that distinguish them among "physical" ones, we can expect that the system will need to influence the "social" objects with greater efficiency. Such mechanisms are much more favorable than interaction with "social" objects on a par with other "physical" objects, as it happens in the initial four stages due to the work of the left "logical" hemisphere.
Typically, the benefit from emotional impact was higher than from adaptation to the variable properties of "physical" objects. Along with that, the number of "physical" objects to which it is favorable to direct long-term focus of the left "logical" hemisphere was also lower than the number of people around who can be influenced through emotions. Therefore, the influence of intensity in the "social" sources of resources in this cycle was lower than the impact of frequency, and therefore their relative essentiality was lower. Then for the sixth (VI) stage of development of the right hemisphere was correct S n > 1, A n < 1, and H n /H k > 1. Then the left hemisphere during the transition to the fourth (4) stage of evolution corresponded to the opposite octant.
The next stage in the development of mechanisms for analyzing the environment, as can be assumed, will be the study of large social systems. Such a shift of focus in the analysis of information is possible only in quite large societies that have been able to create relatively stable social subsystems within themselves. First, the system studies the fundamental properties of groups of "social" objects, which are rather homogeneous in some aspects, and uses them in its activities. Such mechanisms of interaction with "social" objects allow enhancing standardization of "social" objects for members of the corresponding groups. However, with the expansion of the number of resources sources to such groups' size, their advantages and intensity still decrease. At the same time, we considered sources of resources not entire large social systems, but only their members individually, because a person communicates with a social system only through its participants, not through it as a whole. Thus, in the case of reaching the seventh (VII) stage, the characteristics of the resource sources analyzed by the right hemisphere will be A n < 1, S n < 1 and F n /F k > 1.
Along with that, the focus in the analysis of the environment on the properties of objects corresponds to "introverted" mechanisms defined as S n < 1. In other words, it corresponds to the sources of resources for the properties of which the system adapts, investing counter resources to develop appropriate interaction mechanisms. In the path of development of information analysis mechanisms reviewed above, the left and right hemispheres alternated in "extroversion" and "introversion", successively going through evolution stages. First, it was the right, and after having acquired a sufficient amount of information, the left. However, in the cycle, when the right hemisphere proceeds to the seventh (VII) "introverted" stage, the left hemisphere will not be able to move to the next fifth (5) stage, since, firstly, it is also "introverted" and secondly is also "social". Nevertheless, it should not stay at the "introverted" fourth (4) stage either. Therefore, in this case, the left "logical" hemisphere can only shift the focus of its work to those corresponding to the third (3) stage. Obviously, during the analysis, particular "social" objects will not be prioritized anymore; a single approach will be applied to all objects. Thus, the left hemisphere's mechanisms for information analyzing, which are similar to the previous ones in general principles, in this cycle are shifted to a different octant than before, that is S k > 1, A k > 1 and F k /F n < 1.
At the next stage of studying the environment, a person should probably shift his or her focus from the basic properties of social groups to their functional properties, so that to be able to change such properties. That, while maintaining the frequency of resource sources, will slightly increase their intensity and give relative characteristics at the eighth (VIII) stage, corresponding to the octant S n > 1, A n < 1 and H n /H k < 1. As expected, the left hemisphere will return to the fourth stage (4) but changing the octant again to the opposite of the right hemisphere's current octant.
We had to assume that further stages of the development of methods for analyzing the environment are also possible, for instance, by switching to the study of the time variables of the functional properties of social groups. Still, such options will not be considered in this article.
The described model sequence of development of the environmental analysis mechanism fits well with known periodizations of human development [34].
Since the main external manifestations are associated precisely with the mechanisms of the environment analysis by the right hemisphere [35], it can be assumed that the changes related to its transition between various stages will be more noticeable to researchers. In general, this is true. In the works of famous researchers such as Piaget, Elkonin, Erikson Vygotsky and others [36][37][38][39][40][41][42][43] the stages of child development corresponding to the stages of studying the environment by the right hemisphere from the first to the fifth stages are described in sufficient detail. However, since at later stages of the study the sample of children will have a significant proportion of those who have avoided the model sequence of study, the results of the studies will be less accurate. For this reason, the steps from the sixth onwards are significantly less described in the literature. In fact, the sixth and seventh stages are described with sufficient accuracy only in the work of Lichko [44], where they are identified as one of the accentuations of character at teenagers, manifested at a certain age.
The works describing age-related changes are based on extensive research. Thus, the stages from the first to the seventh can be considered relatively well confirmed, which indicates a high probability of at least partial accuracy of the model considered in the article. The authors did not find any literature that examines similar experimental descriptions for the eighth stage considered in the article, but it is an inevitable part of the general model.

Other Sequences of Stages of a Human Environment's Examination
As we reported above, we considered the model sequence of development of mechanisms for analyzing the environment by a human. This sequence is the most comfortable to understand the possible ways of attention focusing during environmental analysis. The considered model sequence of development should ultimately lead to a combination of analysis mechanisms VIII-4, which are developed further not intensively, but extensively through the expansion of information without changing its objects of study.
However, VIII-4 is only one of several possible combinations. Suppose there are not many sources of "social" resources. In that case, it is unlikely that the information will be sufficient to proceed according to the presented model sequence to the sixth (VI) stage or any of the subsequent ones. In such cases, it is impossible to increase the flows of "social" resources at the expense of such transitions. Therefore, depending on the human environment's complexity during the corresponding evolutional cycles according to the presented model sequence, the final combination may be one of V-3, VI-4 and VII-3, which corresponds to the reduced model sequence.
At that, the inability of transition from the "physical" stages to the "social" ones is low. This is related to the fact that a person functioning as a living system through which "physical" and "social" resources are exchanged is impossible without thoroughly analyzing information about one of the resource flows. It is also unlikely that both hemispheres will evolve further than the fourth stage.
There may be enough "physical" resources at the initial stages of the model sequence of development to obtain the information required for the right hemisphere to transit to the next stages. Still, the priority "social" resources that the left hemisphere initially focuses on when analyzing information are not increasing in quantity when interacting with them as with "physical" resources that possess functional properties. In this case, the left hemisphere will remain in the first two stages of development, alternating them depending on the stage of development of the right hemisphere. Such a developmental sequence differs from the model one, and in its case, the final combination may be one of V-1, VI-2, VII-1 and VIII-2. In this case, the model sequence will not be symmetrical regarding developing both brain's hemispheres.
Other cases that differ from the model sequence will be cases of development in which, for whatever reason, the transition of the right hemisphere between any stages of development, that is, a change in the mechanisms of interaction with the human environment, does not result in a qualitatively greater increase in the flow of resources. For instance, this is possible if the previous mechanisms give an extremely large flow of any resources or if it is impossible to develop new mechanisms due to the lack of corresponding sources of resources. Then the right hemisphere can stop at one of the first four stages I-IV. Nevertheless, as mentioned earlier, it is unlikely that both hemispheres will remain in the "physical" stages.
In such cases, humans' study of the environment should proceed by developing the left hemisphere's analyzing mechanisms. Then, depending on the right hemisphere's permanent stage, the left one will change priorities following stage's sequence 1-3-6-8 or 2-4-5-7. Such a sequence of development will be inverse to the model sequence. However, abbreviated versions are also possible when there is not enough information to change the initial "social" stage for the next one. Thus, the final combination may remain one of the following: I-5, I-7, II-6, II-8, III-5, III-7, IV-6 and IV-8.
Thus, we assumed how all possible combinations of the analytical focus of a person as a living system can develop. We got 16 such combinations. These combinations of "social"/"physical" and "extraverted"/"introverted" priorities between the "logical" and "imaginative" cerebral hemispheres are formed in the process of ontogenesis by rather different algorithms with different duration.
Besides, each change in the development stages means a change in the cycle of human development as a living system. Therefore, the number of resources flowing through the system should increase at the rate mentioned above, hence the amount of analyzed information. Suppose any hemisphere does not change the mechanisms of analysis. In that case, the amount of data should continue to increase, which means the extensive development of analysis mechanisms due to their coverage growth. This also allows the right hemisphere to provide sufficient experience for generalization by the left hemisphere, even if it is fixed at one of the initial stages.
Let's briefly list the main types of analysis priorities reviewed above and assume the final goals of actions based on the analysis of these emphasizes.
The right hemisphere: I-an emphasis on the features of various objects as "physical", their types, usefulness or harmfulness, and location and displacement. Striving to surround oneself with safe and useful objects (with positive associations) in a secure order, preserving and propagating such order, and eliminating objects with negative associations.
II-an emphasis on the variable properties of objects as "physical", and how to adapt to such changes. Striving for maximum preparation for possible manifestations of the environment, to adjust to the environment's current properties or to level their effect.
III-an emphasis on the functional properties of objects as "physical". Striving to maximize the instrumentation for influencing objects and changing the properties of objects with unwanted manifestations.
IV-an emphasis on the temporal variability of objects as "physical" through time. Striving to be prepared as much as possible for potential environmental changes and smooth out the consequences of current undesirable events.
V-an emphasis on "social" objects and their variability over time. Striving to be prepared as much as possible for potential changes in manifestations of people around and to smooth out the consequences of current unwanted emotions.
VI-an emphasis on the analysis of specific human functional properties, often associated with emotional effects. Striving for maximum emotional impact on the surrounding "social" objects and keeping them within the framework of beneficial emotional relationships.
VII-an emphasis on the fundamental properties of "social" groups, and how to adapt to them. Striving for maximum compliance with the group's requirements regarding behavior and proof of compliance with the group when communicating with others, to appeal to such a group in case of conflicts with others.
VIII-an emphasis on the functional properties of "social" groups, and how he or she can influence them. Striving to attract others to his or her most beneficial group.
The left hemisphere: 1-an emphasis on various objects' features as "physical", their relationship and relative location. Striving to create an environment for yourself that is as safe as possible regarding priority "physical" aspects.
2-an emphasis on objects' variable properties as "physical", identification of the most important trends. Striving for maximum preparation for the most likely positive future changes or deceleration of negative changes.
3-an emphasis on the functional properties of objects as "physical", a search for common functional properties. Striving to maximize the instrumentation for influencing surrounding objects and avoid being affected by others in the future.
4-an emphasis on variability of objects as "physical" over time, a search for long-term trends. Striving to achieve the most stable positive state and increase the probability of eliminating negative factors in the future. 5-an emphasis on "social" objects, their variability through time. Striving to achieve positive communication with the most promising people around him or her.
6-an emphasis on the analysis of human emotional characteristics, the patterns of their changes. Striving to establish and maintain a positive attitude towards oneself with the maximum number of people.
7-an emphasis on the fundamental properties of "social" groups and their motives. Striving for maximum association with the group in the perception of others.
8-an emphasis on the functional properties of "social" groups and their patterns. Striving to form the most comfortable group.
It should be noted that the relative characteristics of S n , A n , Q n /Q k and F n /F k can be of any essentiality in the specified ranges (within the octants), which will affect the degree of intensity of the corresponding priorities in information analysis.

How the Considered Model Relates to Other Models
The sixteen combinations of human analytical priorities described in this study, if they occur, should manifest themselves in human actions since most of the human activity is the result of information analysis. In the following, we provide some well-known models to consider how they can coincide with the model presented in the article.
Let us consider the most basic temperament model that characterizes the response to extrinsic stimuli. We can conditionally determine four types of temperament regarding the speed of emotional excitement and inhibition [45], which affect the reaction time of a person as a system to the current resource sources manifestation or the risk of their disappearance. Since the right hemisphere is responsible for direct interaction with resource sources, it is logical to assume that it is its characteristics of information analysis that will affect a person's temperament. The more intensive resource sources prioritized by the right hemisphere's attention, the more beneficial it is to react quickly to obtain them. Respectively, when S n > 1, the processes of emotional arousal will proceed faster. In this case, the closer to the line A n = 1, the higher the risks of missing a source of resources, therefore, the more prolonged activity is required, and, therefore, the slower emotional inhibition after excitation, which corresponds with the octants A n < 1 and Q n /Q k > 1 or F n /F k < 1 due to their proximity to the An line. Nevertheless, in some cases, depending on the specific relative characteristics, the information analysis mechanisms corresponding to the mentioned above octants may be placed further from the An line than the mechanisms related to octants as Q n /Q k < 1 and F n /F k > 1. Thus, the octant corresponding to the III and VI stages' priorities has the highest rate of excitement and, more often, a low rate of inhibition that corresponds to a choleric temperament. The octant that includes the I and VIII stages' priorities also has a high excitement rate and usually a higher inhibition rate, which characterizes a sanguine temperament. The octant of the II and VII stages' priorities in most cases has a rate of excitement lower than that of a sanguine temperament. Still, it more often characterizes a high rate of inhibition that is designated as phlegmatic temperament. Finally, the octant, which corresponds to the IV and V stages' priorities, usually has the lowest excitation and inhibition rates, which are manifestations of a melancholic temperament.
As it was suggested, temperament reflects the mechanisms of the right hemisphere's activity. Accordingly, it is logical to assume that some kind of model exists representing the left hemisphere's mechanisms. The Honey and Mumford Learning styles model [46] might be considered such a model since the learning process is associated with the long-term accumulation of knowledge as resources, which corresponds to the criterion A n > 1, which is to the characteristics of the resource sources analyzed by the left hemisphere. According to this model, the four types of learning are conventionally characterized by their propensity for active experimentation or observation and the preference for confirmed specific results over hypothetical estimated data. Investing resources in experiments is possible only when there is an excess of resources, that is when S n > 1. Whereas the accumulation of information for developing hypotheses is necessary only at distancing from the state when A n = 1. Then the octant corresponding to the parameters S n > 1 and F n /F k > 1 should be characterized by a greater tendency to experiment with attention to specific results without complex hypotheses generating, which in the Honey-Mumford model is conceptualized as an "activist". The neighboring octant when F n /F k < 1 is also probably more inclined to experiment, but more often with greater involvement in framing hypotheses about how to obtain resources, which defines "pragmatists". Further let us consider an octant with the parameters when S n < 1 and Q n /Q k > 1, for which experimentation is unlikely, but a tendency to generate hypotheses should also appear, which corresponds to a "theorist". The last octant with Q n /Q k < 1 corresponds to the cases with the lowest probabilities of both experimentation and hypothesis, the Honey-Mumford model marks as a "reflector".
It is also worth mentioning the Myers-Briggs model [47] that is the most appropriate for the described model regarding the number of types. It describes sixteen personality types and is widely used in non-professional practice. At the same time, the validity of this model is relatively low. Still, it should be noted that this might be due to the low quality of its theory, based on a more subjective description that is overly discrete and claims that differences are innate. According to the Myers-Briggs model, which declares the development of C. Jung's theory, the personality types are divided into four pairs of parameters: extroversion-introversion (E-I), sensation-intuition (S-N), thinking-feeling (T-F) and judgment -perception (J-P). Each type is designated by the corresponding letters. Besides, each type has several "functions". The main ones are dominant, while the characteristics of auxiliary "functions" are opposite in extroversion/introversion and thinking-feeling/sensation-intuition parameters. At that, the judgment-perception parameter indicates which parameter from which pair of thinking-feeling or sensation-intuition belongs to the dominant "function". Thus, for the type ESTJ (extroversion, sensation, thinking and judgment) the dominant "function" is Te (extroversion and thinking), as indicated by J, then the auxiliary "function" has the Si (introversion and sensation) characteristics.
We will not conduct a full review of the Myers-Briggs model given parameters and "functions" since you could see the references for all details. However, we asserted that although they were considerably more subjective, in general, they correspond to the model proposed by us. Thus, the dominant and auxiliary "functions" were close to the brain's right and left hemispheres' operation principles, respectively. The Myers-Briggs model extraversion-introversion personality type was fundamentally similar to those proposed in this article regarding the general direction of activity. The description of the pair of sensation-intuition parameters was identical to the difference between the priorities I, II, 1, 2 and III, IV, 3, 4, proposed by our model where the former were aimed at direct interaction with "physical" objects, and the latter are at more indirect. Finally, according to their descriptions, the pair of parameters thinking-feeling corresponded well to the specifications of which the priorities proposed in this article-V, VI, 5, 6 or VII, VIII, 7, 8-are primary in a person's analysis of information. Thus, each Myers-Briggs model type can be compared with a combination of priorities of the model reviewed in this article. That allows us to apply to the Myers-Briggs model less subjective justifications and descriptions we offer.
In a professional environment, more complex differential psychology methods are used. One of the most common is the big five personality model [48,49] representing complex factors of a larger number of personality traits. These five main traits are conscientiousness, emotional stability, openness to experience, agreeableness and extraversion. This model for describing the human psyche's characteristics has a relatively long practical verification history with high statistical accuracy. Let us assume how this model and the one proposed in the article can be related if the latter is correct.
The most intuitively obvious coincidence of the big five model with others is, obviously, the extraversion factor. It is reasonable to assume that this parameter coincides with the case of "extraverted" orientation of the right hemisphere proposed in our model. The characteristic of neuroticism in the big five model is also well defined. The main features that it includes (according to the 45 AB5 facets model) are associated with poor emotional control, which correlates quite well with the octants we assigned to choleric and melancholic temperaments, corresponding to priorities III, VI and IV, V. The next big five model factor to consider is agreeableness. The traits associated with it conform to the manifestations expected in the "social", not in "physical" orientation of the right hemisphere. The conscientiousness factor, in turn, according to its features, can characterize the octants with priorities 1, 8 and 2, 7 associated with the accumulation of a relatively more complex analytical base in the goals area of the left hemisphere. Despite some possible complex intersection of the big five factors with the proposed model, the four considered factors were sufficient to describe any possible pair of accents regarding the proposed model. However, one of the big five factors we did not consider yet is openness. It is impossible to assume that this factor depends on the others' combination since the close dependence would hardly remain unnoticed by numerous model researchers, and such an element would be reduced. Along with that, the factor is associated with many features (like intellect, competence, etc.) that characterize the general cognitive development more than the characteristics of information analysis features. Thus, according to our assumption, the presented model is also consistent with the big five model.

Conclusions
In this article, we considered fundamental principles of the living systems' functioning, which are identified when they are defined as self-reproduction information or as self-reproduction of patterns in resource flows. These principles refer to the relative characteristics of resource flows passing through systems. In the article, significant restrictions were given for these characteristics, both at a specific time and during systems growth. The Fibonacci sequence was applied for the latter since it appropriately describes systems' growth rate between particular cycles. The cycle duration was also specified, and its dependence on the system's most essential resource was revealed.
After giving a general definition and description of living systems, a person's differential characteristics were considered on their basis in the information analysis. Mechanisms for forming possible priorities in a person's analysis of information were described, and quite objective definitions were given to them based on living systems' operation. The resulting model is reasonably consistent; however, it is purely theoretical. Herewith, a small comparison of the derived 16 types of human information analysis priorities with several existing models is given in the article's final part. The mentioned comparison indirectly confirms the given theoretical model's correctness regarding its consistency with existing empirical schemes.
The given sequence of shifts of the system's attention between the types of resource sources is subject to further verification for its applicability to other living systems. This may be true for the reason that the shifts described in the model are carried out between non-human-specific objects. The change of the analyzed objects proceeds from sources that differ as much as possible from the system itself to similar ones and then to their groups. Thus, the sequence has quite common characteristics to expand its applications.


extfieldsofstudy:
Computer Science
Medicine
Physics

text:
Hybrid localized graph kernel for machine learning energy-related properties of molecules and solids

Nowadays, the coupling of electronic structure and machine learning techniques serves as a powerful tool to predict chemical and physical properties of a broad range of systems. With the aim of improving the accuracy of predictions, a large number of representations for molecules and solids for machine learning applications has been developed. In this work we propose a novel descriptor based on the notion of molecular graph. While graphs are largely employed in classification problems in cheminformatics or bioinformatics, they are not often used in regression problem, especially of energy-related properties. Our method is based on a local decomposition of atomic environments and on the hybridization of two kernel functions: a graph kernel contribution that describes the chemical pattern and a Coulomb label contribution that 1encodes finer details of the local geometry. The accuracy of this new kernel method in energy predictions of molecular and condensed phase systems is demonstrated by considering the popular QM7 and BA10 datasets. These examples show that the hybrid localized graph kernel outperforms traditional approaches such as, for example, the smooth overlap of atomic positions (SOAP) and the Coulomb matrices.

Introduction
The past decade has seen an impressive growth in the development and application of machine learning techniques 1 in quantum chemistry and computational condensed matter physics [2][3][4] . These methods are of a great interest for theoreticians because they allow for the analysis, classification and prediction of various properties conventionally requiring a large amount of data generated via computationally demanding quantum mechanical calculations 5 . Indeed, machine learning techniques can be applied to a broad range of problems, including, among the others, potential energy surface fitting [6][7][8][9] , ab initio molecular dynamics [10][11][12] , prediction of various scalar properties [13][14][15] (e.g atomization energies, polarizability coefficients, highest occupied molecular orbital energies, electronic structure correlation energies, etc), and vectorial and tensorial quantities (e.g forces, polarizability tensors, etc) 16,17 .
The chemical compound space is characteristic by a huge dimensionality and complexity.
Datasets for molecules and solids proposed in the literature, which span only a small part of the chemical space, already contain impressively large numbers of compounds. The GDB-17 molecular dataset, for instance, contains 166 billions of molecules 18 . An example of a rich dataset of crystal structures is the set of elpasolites of Faber et al. 19 containing 2 millions crystals. This shows clearly that it is impossible to analyze or screen all these compositions and structures by demanding electronic structure calculations 2 . A solution to efficiently explore the chemical compound space (or at least significant parts of it) can be found in the development of new machine learning approaches to complement ab initio calculations. This field of research is known as quantum machine learning 20,21 (QML). The main idea of the QML approaches is to train a machine learning model on a subset of chemical structures (the smallest possible) for which simulations at a quantum mechanical level were done. The trained machine is then used to predict the target properties of the rest of the systems. This technique can be applied not only in the chemical composition space but also in the conformation space of a given system. For example, recently, a ∆-machine learning approach 22 based on thermodynamic perturbation theory has shown impressive results in the prediction of the adsorption enthalpy of small molecules adsorbed in chabazite at the random phase approximation (RPA) level of theory using only 10 training conformations 23 .
In general, the ultimate goal of QML would be to find a universal model which unifies both chemical and conformational spaces 21 .
One of the main challenges in the use of QML methods is the choice of proper descriptors -i.e. features -to represent the molecular and/or condensed matter systems. In this work we will focus on structural descriptors, namely approaches that assume the knowledge of the precise geometry of the systems under consideration beyond their chemical composition.
Currently, a large number of those descriptors have been proposed in the literature, with the symmetry functions 24 , the Coulomb matrices [25][26][27] , the smooth overlap of atomic positions (SOAP) 28,29 and the many-body tensor representation (MBTR) 30 being some of the most prominent examples. From a general point of view an efficient structural descriptor should satisfy certain prerequisites 3 . For instance, the descriptor should be unique (one-to-one correspondence between features and systems) and invariant with respect to overall rotations and rigid translations, as well as permutations of atoms of the same type.
In the present work we introduce a new descriptor based on the notion of graph that satisfies all the above-mentioned conditions. Graphs are already largely used in the domain of cheminformatics 31-34 and bioinformatics 35,36 to predict the activity or the toxicity of particular drugs or to recognize particular patterns of binding site of biomolecules (with the quantitative structure-activity relationships (QSAR) model 37 ). Nevertheless, in these cases, the use of graphs is restricted to classification problems and their applications in regres-sion problems are still rare, especially as far as energy-related properties are concerned [38][39][40][41] .
These quantities are particularly sensitive to inner changes of the molecular structure which cannot be described by simple graphs whose edges are represented by 0's and 1's for disconnected and connected nodes (namely atoms) 42 . Indeed, this simplified representation leads to isomorphic graphs even for rather different structures and this represents an issue for learning properties (e.g. the energy) which are very sensitive on the specific geometry of a system. To overcome this difficulty in the present work we used weighted graphs, whose values on the edges were obtained as superposition of atom-centered Gaussian functions.
In order to develop a machine approach applicable to both molecules and solids we considered graphs defined in localized environments around each atom of a given system. These localized graphs were used as input features of a machine learning approach based on the kernel ridge regression 43 (KRR). This method scales cubically with the size of the training set but typically requires fewer data point to be trained than other traditional approaches (e.g. neural networks). The role of the kernel in KRR is to provide a measure of similarity between different localized graphs belonging to different systems. A global "comparison" of two systems (molecules or solids) can then be obtained by summing localized contributions from each atomic environment 29 . The specific kernel used in this work combines two different parts: The first, based on a variant of the shortest-path kernel, is suited to describe structural motifs and chemical bonding between atoms; the second includes finer geometric details through a label enrichment of the nodes with Coulomb vectors. This approach can be viewed as an hybrid kernel function 44 and we will show that the introduction of both contributions provides a high level of accuracy. As a proof of principle this new methodology was applied to predict molecular atomization energies (QM7 dataset) and enthalpies of formations of solids (BA10 dataset).
The paper is organized as follows. In Sec. 2 the concept of labeled localized graph is introduced, which is represented trough maximum probability paths and enriched with a set of vectors that describe the Coulomb potential of the atomic environment. In the same section, the new kernel approach is also presented. In Sec. 3 the datasets QM7 and BA10, used to evaluate our method are reviewed. Finally, our results for the prediction of the atomization energies of molecular systems (QM7 dataset) and the formation enthalpies for the solid state systems (BA10 dataset) are presented in Sec. 4.

The Labeled Localized Graph Descriptor.
In this section we introduce the concept of labeled localized graph (LLG) for molecules and solids. By considering a local atomic environment X defined by a radius r cut , a LLG, de- -where N and M are associated with the number of atoms and the number of chemical bonds in the atomic environment, respectively 45,46 . The LLG is said "labeled " if a label l corresponding to the chemical symbol is attributed to each node Fig. 1).
A simple unlabeled graph is usually represented through an adjacency matrix A(G X ) ∈ R N,N , whose elements A ij are set to one if two different nodes i and j are linked by an edge or to zero otherwise 45,47 . However, the same graph would correspond to an infinite number of conformations through this definition: by changing the interatomic distances, as long as the edges are preserved (i.e. bonds are not broken), the corresponding graph will not change.
This issue is particularly problematic for the prediction of total energy related properties (e.g. atomization energy), which are strongly influenced by the precise atomic conformation. For this reason in this work we use weighted adjacency matrices 38 which include significantly more information on the structure of the atomic environment. Specifically, a Gaussian function is attributed to each atomic position r i in the atomic environment. Instead of considering only binary values for the off-diagonal elements of the adjacency matrix, weights w ij between each pair of vertices v i and v j are set to correspond the overlap of these Gaussian functions, such that: Hence the weights depend on covalent radii r cov i (coming from crystallographic data 48 ) of all elements and on the hyperparameter γ controlling the width of the Gaussian atomic distributions. Finally, a threshold is applied as follows: i.e., an edge between the two nodes v i and v j is defined when w ij is larger than epsilon (see Fig. 1). These weights can be interpreted intuitively as probabilities to "jump" from one atom to another atom.
Weigthed adjacency matrix : Labels of vertices : N, H 1 , H 2 , C Figure 1: Illustration of a labeled localized graph (LLG) for a nitrogen atomic environment defined by a radius r cut . The overlaps of the Gaussian functions associated to each atom in this environment determine the weights in the adjacency matrix of the graph.

Maximum Probability Paths and Label Enrichment.
From the definition of the weighted adjacency matrix A(G X ) in Eq. 2, we can determine a path π (v i , v j ) between each pair of vertices (v i , v j ) (we suppose here that each node is connected to at least one another node, as it is the case for all the systems studied in this work). This path is defined by a finite-length sequence of vertices 31,32,49 . In this paper we will consider a machine learning approach based on a kernel that measures the similarity between paths.
Specifically, we will focus on a variant of the shortest path (SP) kernel. While determining all paths of a graph is a NP-hard problem, computing the shortest path between pairs of vertices is a problem solvable in polynomial time (O(n 3 )) 49 .
Typically, the determination of the shortest path is based on the distance and on the triangular inequality (Floyd-Warshall algorithm) 50 . Rather then directly using an approach based on shortest paths, in the context of the present work we found more natural to introduce the analogous concept of maximum probability path (MPP). In particular, interpreting the elements of the weighted adjacency matrix as transition probabilities between two nodes, a path is identified with the greatest overall probability expressed as the product of the weights associated with all the edges that are involved in the path. Mathematically, the probability of a certain path can be written as: where the intermediate indexes in the product correspond to all the nodes visited along the path. In order to find the MPP between two vertices, we modified the original Floyd-Warshall algorithm 49 (see Alg. 1).
Through our variant of the Floyd-Warshall algorithm, the matrix A(G X ) is transformed into a new matrix P(G X ) that can be interpreted as a fully connected graph in which the

Algorithm 1 The Floyd-Warshall algorithm for MPPs
Require: Weighted adjacency matrix A(G X ) Ensure: MPPs matrix P(G X ) for k = 1 to N do end for end for end for matrix elements P ij correspond to maximized overall probabilities of transition from the node i to j. In our method, the MPPs are also labeled by a sequence of labels l(π(v i , v j )) = (l(v i ), . . . , l(v k ), . . . , l(v j )). As discussed below (see Sec. 2.3 A.III), this is necessary in order to include information on atomic species so that only the MPPs involving the same sequences of atoms are compared (see Fig. 2).
The concepts described up to this point (weighted graphs and MPPs) allow for a rather accurate description of geometric motifs and bonding between atoms. However, these approaches are not sufficient to capture all the geometric details in atomic environments. Specifically, modifications (e.g. rotations) which preserve interatomic bond distances in the local environment leave the matrices A(G X ) and P(G X ) unchanged. These finer structural details, which are not captured by the MPP approach, might have different relevance depending on the specific dataset but, nevertheless, contribute to the non-uniqueness of the connection between localized graph and property to predict. To overcome this difficulty, we introduced a label enrichment of the vertices 31,32 . Inspired by the Coulomb matrix descriptor introduced by Rupp et al. 25 , we defined for each node a Coulomb vector c (i) sorted according to the distances in increasing order whose components are defined as Weigthed adjacency matrix : Path matrix : The weighted adjacency matrix A(G X ) is transformed into a path matrix P(G X ). Each path between pairs of atoms is characterized by a sequence of labels l(π(v i , v j )) and a maximum probability path π(v i , v j ).
where Z i , Z j are the nuclear charges. The superscript i refers to the node which is labeled, while the subscript j is an index that indicates all the other atoms in the environment of i (see Fig. 3).
Because the atomic composition of environments can be different, to prevent any issue with dimensionality mismatch, the Coulomb vectors are padded with zeros to set their dimension to be equal and transferable. Finally, we obtain a new set of labels that takes into account the missing geometric information in the description of atomic environments. Figure 3: Illustration of the labels enrichment of nodes. In the atomic environment X to each node it is associated a sorted Coulomb vector c (i) . Example with the carbon atom :

The Hybrid Maximum Probability Path (HMPP) Kernel.
The main idea of kernel approaches consists in finding an accurate measure of the similarity between pairs of data points (atomic environments in our case). For example, in the SOAP kernel, the similarity is determined through an overlap calculation of the atomic density of two atomic environments 28,29 . When normalized, the kernel is close to one when two atomic environments are nearly identical and tends to zero if they are dissimilar.
In our approach, an atomic environment X is characterized by a set of MPPs be- . Being inspired by the iterative similarity for molecular graphs introduced by Rupp et al. 51 , where the use of a linear combination of two kernels applied to vertices and edges was proposed, the following hybrid kernel is introduced to compare the atomic environments (see also Fig. 4): This kernel is normalized via Tanimoto's normalization 45 : which is commonly used also in other graphs application [45][46][47]52 . This normalization can be seen as a Jaccard's distance -i.e. measuring how two sets intersect -between two discrete collections of paths and vertices. The kernel k(X , Y) is positive semidefinite 52 , as required for use in kernel-based machine learning algorithms.
To compare graph paths, we have chosen an approach similar to the labeled shortest path graph kernel 49 . In its original definition this kernel could take values different from 0 only when applied to pairs of (shortest) paths with the same initial and final labels 49 . In this work we found that this formulation does not reach a satisfactory level of accuracy. Indeed, even if the initial and final nodes of a pair of MPPs have the same "atomic" labels (namely they correspond to the same atomic species), the intermediate paths could be significantly different from a chemical point of view (i.e. involve different elements) and still be considered as similar according to this original definition of the kernel. For this reason in the present work we used an MPP kernel which is strictly 0 unless the two paths to be compared have exactly the same sequence of atomic labels l(π). This kernel can be expressed as follows: where the δ Label (l (π X ) , l (π Y )) is a delta kernel applied on the path labels such that: This kernel is equal to one if, and only if, the sequences of labels along π X and π Y are identical. The term k Path (π X , π Y ) is the Laplacian kernel 53 used to compare pairs of MPPs. Similarly, the Coulomb labels are compared through a sum of Laplacian kernel functions: where N and N denote the number of atoms in the atomic environments X and Y, respectively, and β 1 and β 2 are the hyperparameters that control the decay of the Laplacian kernel functions. They play a central role in the prediction quality and are estimated through a grid search using a validation dataset. The relative importance of the two kernels k MPP and k Coulomb in the description of a given system is controlled by the hyperparameter α in Eq. 5.

Benchmarks
To demonstrate the accuracy of our hybrid graph kernel approach we applied it to two regression problems of energy-related properties and compared its performance to that of the popular SOAP descriptor 28,29 and graph approximated energy (GRAPE) 38 . The machine learning models were based on in-house programs written in Python3 employing the   Using the five-fold partitioning originally proposed by Rupp et al. 25 and following the work of Ferré et al. 38 , we considered the first N molecules (with N = 100, 300, 500 and 1000 points) of the first fold as a training set. The second and third folds were combined to create a validation set for the grid search of the optimal hyperparameters, which, in the case of the HMPP kernel include the parameters r cut , γ, β 1 , β 2 and α. Finally, the two last folds were used as a test set to evaluate the accuracy of the model through the mean absolute error (MAE) and the root mean square error (RMSE) of predicted energies.

The BA10 dataset.
The BA10 dataset contains standard enthalpies of formation for a set of ten binary al-

Optimization of the hyperparameters
The determination of the hyperparameters was conducted on a hold-out validation set through a five-dimensional grid search for the parameters r cut , γ, β 1 , β 2 , and α. The cutoff radius r cut , that defines the size of each atomic environment, was evaluated in the interval between 1.6 and 2.4Å with an increment of 0.2Å for the QM7 dataset, and between 2.0 to 5.0Å with an increment of 1.0Å for the BA10 dataset. The values of 1.8Å and 3.0Å were identified as optimal for the QM7 and BA10 datasets, respectively. The scaling factor γ, that governs the width of the atomic Gaussian functions, was tested in the range from 0.1 to 0.9Å with a step of 0.2Å; the optimal values of 0.5Å and 0.5Å have been identified for the QM7 and BA10 datasets, respectively.
The two hyperparameters β 1 and β 2 that define the decay rate of the two exponential functions in the k MPP and k Coulomb kernels were evaluated on a decimal logarithmic grid defined on the interval between 10 −1 to 10 −4 . For both the datasets considered here, the optimal hyperparameters values were found to be 0.1 and 0.001 for β 1 and β 2 , respectively.
Compared to the other three parameters in the model, β 1 and β 2 seem to have a weaker system dependence and could be possibly transferred across different datasets. Although this observation should be confirmed by future investigations on several different datasets, the possibility of fixing β 1 and β 2 a priori could help to simplify the hyperparameter optimization procedure.
The α parameter, that describes the relative contribution of the kernels k MPP and k Coulomb , was tuned by considering values between 0 to 1 with a step of 0.1. For the QM7 dataset, a relatively small α (0.2) exhibited the best performance, while a significantly higher value (0.9) was identified for the dataset BA10. It is important to notice that the hybridization brings important improvements with respect to the sole use of the best performing between the k MPP and k Coulomb kernels (for the smallest training sets considered below the hybridization lowers the mean absolute error by 35 % and 17 % for QM7 and BA10, respectively).

Prediction of the atomization energies.
In this section we test the performance of the HMPP kernel in predicting the atomization energies of the molecules in the QM7 dataset. Table 1 and Figure 5 show the variation of the MAEs and RMSEs as a function of the training set size. For the sake of comparison we also show results for some other well established approaches, namely the SOAP kernel and GRAPE, the latter being a localized graph kernel method previously reported in the literature 38 and applied to regression problems. The results for the different methods have been generated using the same training sets and analogous procedures for the optimization of the hyperparameters. We observe that the HMPP kernel outperforms the other two methods regardless of the training set size. It is important to notice that in Ref. 37 the results reported for the GRAPE kernel slightly improved over the SOAP method (which is in contrast to our results from Table 1 and Figure 5). However, the authors of this work clearly stated that the tuning of the hyperparameters was limited and that, accordingly, it was not possible to conclude "that one method outperforms the other". The MAEs and RMSEs reported in Table 1 for GRAPE and SOAP are also sizeably smaller than those in Ref. 37, demonstrating the importance of a fine tuning of the model hyperparameters to fully establish the accuracy of a certain approach.  Table 1).
The GRAPE kernel uses a random walk kernel directly applied on the weighted adjacency matrix. Within this approach, each component of the adjacency matrix is defined through an overlap of atomic Gaussian functions of the same width. However, we can observe that this use of graphs does not reach a satisfactory level of accuracy, as shown by the high MAE and RMSE values in Table 1 and Figure 5. This is likely related to the lack of labels in the GRAPE approach. Tang and de Jong have proposed in 2019 a marginalized graph kernel to predict the atomization energies and have obtained impressive results with the introduction of labels 41 . However, the reported model was based on a global description of the molecular graphs and, differently from our present approach, the corresponding graph kernel cannot Table 1: Variation of mean absolute error (MAE) and the root mean square error (RMSE) (in kcal/mol) of atomization energies (QM7 dataset) obtained using the SOAP, GRAPE and HMPP kernels with the training set size (cf. Figure 5).    4.3 Prediction of the standard enthalpies of formation. Table 3: Variation of MAEs and RMSEs (in kcal/mol) with the training set size for enthalpies of formation of the BA10 dataset predicted using the SOAP and HMPP kernels (cf. Figure 6).  Table 3 and Figure 6 show the values of MAE and RMSE averaged over all the structures of the 10 binary alloys. The training sets, which are always excluded in the error evaluation, are obtained by randomly selecting 100, 300, 500, and 1000 configurations for each binary compound among the 1595 structures -i.e. without stratification of the data.
As for the dataset QM7 discussed in the previous section (see Sec Table 3).
is sizeably larger than that of SOAP (in this comparison it should also be kept into account that hyperparameters for the HMPP model are not reoptimized for each alloy). In Figure   7 we also report results from the previous work of Nyshadham et al. 60    1000 SOAP · · · GPR 0.10 · · · 60

Conclusions
In conclusion, we introduced in the framework of a local decomposition kernel a new similarity measurement based on molecular graphs. This kernel is composed by two parts: one that describes the local molecular pattern through a labeled graph, and a second one, that keeps into account some finer geometric information using Coulomb labels. These two kernels are hybridized through a hyperparameter α which controls their relative contributions.
The accuracy of this new kernel was tested on two datasets: The molecular QM7 dataset and the BA10 dataset, containing ten binary alloys. The first of these applications involves the prediction of the atomization energies of molecules. In this case our method outperforms previous descriptors proposed in the literature such as, for example, the smooth overlap of atomic positions and a previous approach based on unlabeled graph kernel introduced by Ferré et al. 38 . In the case of enthalpies of formations of solids (BA10 dataset), for small training set size -up to 300 datapoints -the accuracy of our new HMPP kernel is lower with respect to the SOAP approach. However, when the number of training configurations increases the learning ability of the SOAP descriptor saturates. The HMPP kernel does not suffer of this limitation and outperforms SOAP for 1000 and more training configurations. From these case studies the labeled graph kernel seems a particularly promising tool to improve the accuracy in machine learning regression problems in chemistry and materials science.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.