# 1.3 Mathematical models of distribution of innovations

Modelling and forecasting of distribution of innovations became actual a direction of researches since 1960th years, after occurrence of innovative works of L.Forta and J. Vudloka, E.Mensfilda, A.Flojda, E.Rogers, T.Hegerstranda and F.Bass [4, 5, 15, 18, 19, 27].

The carried out comparative analysis of existing models of distribution of innovations has allowed to allocate six classes of models which were formed in time how is shown in drawing 1.7.

Drawing 1.7 - Classes of models of distribution of innovations

Formation of classes and their filling by concrete realisations goes after growth of requests to adequacy of models to research problems and workings out. The description of classes is given more low, and the characteristic is resulted in a tabular kind in appendix A.

As criteria of an estimation have been chosen:

• a used mathematical apparatus and the borrowed analogy for construction of mathematical model;

•! Used key parametres and their characteristic: availability to measurement, direct (by quantity of sales) or indirect (by quantity of followers), measurement cost (high, average, low);

•! The account of mutual influence of innovations in the course of distribution;

•! Possibility of construction of algorithm of management of process

Distributions of innovations on the basis of model;

•! Readiness for embedding in a management contour.

Spatial models. Geographer T.Hegerstrand in 1950th years has suggested to consider spatial aspect of distribution of innovations [19]. Hegerstrand has presented the concept of wavy distribution of innovations as predicted existential process, using thus for modelling methods of Monte-Carlo. He has assumed, that for innovation dispersion in time and in space there should be a certain mechanism of contact and belief to transmit the given phenomenon. In its opinion, social contact is localised also distribution defined by the sizes of "an average information field» potential followers of an innovation [19]. The offered imitating model describes structure of such contact: with certain probability during each period of time the follower comes into contact to other people, depending on various restrictions, for example, geographical barriers, distance [96].

On Hegerstrandu speed and a direction of distribution of innovations depend on distance from the centre of origin of an innovation and internal characteristics of region, and also from "throughput" of channels of transfer. In that case, other innovations in the market are not considered. For different territories the constructed model of distribution will differ with the distribution beginning, the form of a curve and a potential maximum [97].

Model Hegerstranda had significant influence on sotsialnoyoekonomicheskuju geography in 1960 and 1970th years, was improved by such researchers as L.Baudenom, B.Johansenom, S.P.Zemtsov, A.Kliffom, R.Morrillom, D.Strengom, E.Sheppardom [96-102].

In spite of the fact that model Hegerstranda has certain successes in forecasting [103], however has essential restrictions [104-107] in the form of a constancy and availability of the cost parametres, dominating personal contact at decision-making and others, that does not allow to construct algorithm of management of distribution process.

For the description of process of distribution of innovations the models which have received the name of models of dispersion, or diffuznye models are widely enough used. More often in Russian-speaking sources such class of models is called diffuznym, however substitution of terms not only can narrow considered area, but also can yield incorrect results in connection with restrictions in data application of models. We will notice, that this class of models for more exact definition and for terminological difference from the phenomenon of physical diffusion to name so - dispersion models more correctly.

By 1962 most full to generalise and systematise experience of studying of distribution of innovations it was possible to sociologist E.Rogers who has offered model of the description of process of penetration of innovations on the market [18]. It has defined diffusion of innovations as «process thanks to which innovations are transmitted on certain channels among elements of social system during time» [18, with. 5]. Then F.Bass has offered a mathematical substantiation of model of Rogers in [4] and, having adapted equation Ferhjulsta, has applied it to forecasting of dynamics of consumption of the new goods. Bass's model is described by the equation in which dependence of growth of quantity of followers of an innovation on an advertising weight and on effect of interpersonal communications is reflected, and does not allow to construct algorithm of management of distribution process. Such models aspire to show relations between level of distribution and number of potential followers according to the nature of innovations, communication channels and

Characteristics of social system, also do not consider other innovations which are present in the market.

Further the given models developed actively and improved by such researchers, as R.Vejberom, the Yak. Goldenbergom, S.Kalishem, P.Kotler, G.Lilienom, V.Mahadzhanom, N.Midom, J. Moore, E.Muller, R.Peres, R.Peterson [6, 8-10, 12, 14, 108-110], etc. the Majority of researches has been aimed at expansion of structure of mathematical model of Bass to include various marketing variables (change of an advertising campaign and the price policy in time) [111] to consider such phenomena connected with a buying behaviour as repeated purchases, comprehension stages in acceptance of a new product [109, 112], etc. After 1980th years in researches the accent was displaced from the aggregated level (makro) on individual (mikro): in focus there was a behaviour of the potential follower of an innovation, its personal motives at the decision on innovation acceptance [113, 114]. Such models have ceased to answer desirable dimension, availability of data has seriously limited practical application [115]. After 2000th years, not looking at unsolved problems at individual level, researchers began to offer variants of association of models of two levels, mikro and makro [110].

In arrow networks innovation distribution occurs on a network, since one or several initial sets of tops (named early followers). Innovations extends «on edges» with certain probability, and such distribution proceeds until all tops in a network will not receive the information or any more there will be no tops-candidates for distribution.

It is possible to carry to arrow networks:

- Models with thresholds (Linear Threshold Model) [116-118], threshold significances based on use for tops of the count, transition is possible only from an inactive condition of top in the active;

- Models of independent cascades (Independent cascade model) [6, 119] when the next tops can be with certain probability are activated;

- The infiltration and defeat model (SIR, Suspectible - Infected - Recovered) [120, 121] in which the probability as top activation is set, and probability of its deactivation, and can be presented by means of chains Markova;, etc. models.

In work [122] distribution on casual columns is considered, affirms, that the probability of successful distribution does not depend on number of early followers. In work [119] association of model with a linear threshold and models of independent cascades is offered and their equivalence is shown. Used parametres are indirect, do not allow to construct algorithm of management of distribution process.

The given model develops in T.Valente's works, a yak. Goldenberg, M.Gomez-Rodrigez, M.Grannoveter, D.Kempe, J. Klejnberg, A.Krauze, J.Leskoves, N.Meganatan, S.Melnik, S.Morris, P.Fennel [118-124].

Teoretiko-game models of distribution of innovations are based on a hypothesis, consisting that at acceptance of new behaviour each person makes a rational choice to maximise the prize. In these models players accept new behaviour when also their neighbours in a social network have accepted it, thus, innovations extend, because there is a stimulus to conformity. The social network is represented the count, in which each knot - the agent in system. Each agent should make a choice between two alternative variants, and payment of each of two variants for the agent is increased with increase in number of the neighbours who have accepted the same variant.

One of the first works, begun the given approach, is a work in J. Reinganum [125] in which the behaviour in the competitive market of innovations is modelled. It is possible to allocate and other works, for example, in [126] the models based on base of coordination games are considered, speed of convergence is characterised as function of structure of a network of interaction. Thus received forecasts hardly differ from predicted epidemiological models. In work [127] estimations which do not depend on structure are introduced and

The size of a network so that distribution of innovations proceeds fast when the prize from innovation acceptance is high enough.

In the same place the method of calculation of the top border of the expected time necessary that an innovation is resulted became accepted in any final network. In work [128] the competitiveness question between two companies, and also the compromise between investment bolshego quantities of money for improvement of quality of a product and increase in quantity of people in a social network is studied. However used parametres and analogies do not allow to construct algorithm of management of the given process.

Various aspects of teoretiko-game models are shined in works L.Blum, M.Magnani, D.Montesi, R.Ramezaniani [125-130], etc.

Other tendency of researches in the field of distribution of innovations - use of imitating modelling on base agentnogo the approach, in particular cellular automatic machines. Cellular automatic machines can be defined as model which simulate global consequences on the basis of local rules of behaviour [115]. Models of the cellular automatic machine represent discrete dynamic system, for the first time have been developed S.Ulamom and J. A background Neumann in 1940th years for research of behaviour of difficult systems [131]. The cellular automatic machine consists of final set of the cages definitely connected among themselves and forming a uniform lattice. The condition of each cage at the moment of time is set by some parametre, and set of conditions of all cages of a lattice defines a lattice condition. Thus the lattice condition can be changed discretely according to the established rules.

The given models allow to expand structure of model of Bass, having set separate rules of behaviour of potential followers, that is as a matter of fact obviously to consider effect of influence of specific features of the person on process of distribution of innovations.

Cellular automatic machines have been applied to overcoming of such restrictions of models of dispersion of macro-level, as individual probability of acceptance

Innovations in [132], resistance to innovations [133], early forecasting of success of an innovation in [134], effects of network external factors [124] etc. the Interrelation between Bass's model and models of cellular automatic machines was studied by various researchers, for example, [135-137], and possibility of aggregation of data about micro-level of acceptance of an innovation for creation of more adequate model of Bass has been shown. The given models remove such defect of model of dispersion as postojannost a potential market capacity, however do not allow to construct algorithm of management of distribution process.

The S-shaped curves received as a result of modelling by means of cellular automatic machines, correspond with curves of model of Bass [115]. Nevertheless, the question on association mikro and makro level of acceptance of innovations is not decided till now.

The idea of application of cellular automatic machines is new enough to the theory of distribution of innovations, in domestic scientific publications the given questions meet seldom, however it is possible to allocate following researchers: O.N.Lobodina, Of this year Lomakin, R.M.Nizhegorodtsev, A.M.Fedotov, JU.D.Schmidt [20, 138, 139].

The following class of models of distribution of innovations - ekonofizichesky, studying economic systems is natural-scientific methods. The formation intensification ekonofizicheskogo the approach is observed last 10-15 years. The given models describe distribution of innovations characteristic for physical environments similar dependences.

In work [11] the analogy between the phenomenon of molecular diffusion and diffusion of innovations with the purposes of definition of factor of distribution of innovations in the market is conducted. In works [140, 141] for modelling of distribution of innovations model Izinga intended for the description of magnetisation of a material is used. In work [103] processes of interaction of waves of the innovations which are starting with different sources are revealed, such interaction is named by an interference. In work [142] the theory of Brown movement is applied to modelling of movement of the agents accepting an innovation.

In the given models it is offered to use the direct parametres which are responsible for distribution of innovations, namely innovation sales.

The given approach has that mathematical apparatus which will allow to consider not only a condition of a considered innovation, but also parametres of environment that will allow to construct algorithm of management of distribution process.

Generalisation by results of the analysis of models

First five classes of models consider process of distribution on the basis of some behavioural models of buyers, have for an object to give only less or more exact description of natural movement of an innovation, to explain the reasons influencing decision-making. Such models are not formalized enough for management of distribution process, not based on accessible metrics, do not consider possible mutual influence of innovations and parametres of environment. The resulted restrictions do not allow to put widely into practice the given models for purposeful management of processes of distribution of innovations.

The is natural-scientific approach is free from defects first five classes of models. The further development ekonofizicheskogo the approach allows to put a problem of management of distribution of innovations.

## More on topic 1.3 Mathematical models of distribution of innovations:

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