We study a continuous model that describes the conflict interaction for two complex systems. External conflict interaction is modeled by the additional influence of chanc e. The dynamics of internal conflict are similar to the Lotka-Volterra model. We interpret the new model of information warfare as the influence of rare events that rapidly change certain ideas of a large number of people. As a result, the number of suppor ters of different ideas makes stochastic jumps that we can see using the Poison approximation scheme. We suggest that such a model could be more natural, as important news now has a quick and powerful impact on audiences through television and the Internet .
The information warfare models with an additional interaction between them may be interpreted as some kind of correlation between the habitants of different regions. In general, the Lotka-Volterra model of prey-predator interaction is one of the main models for simulation of similar processes in applied mathematics, social sciences, and economics [3], [8–11], [17]. Application of this approach to the information warfare model was proposed in [4]. Authors regard some social community of quantity 0, potentially exposed some information threat of two types, that is, for example, the threat of a negative change in its state by transmitting some information relevant to this group by information two different channels. The values 1( ), 2( ) are the numbers of “adherents” depending on time who accepted the new information, ideas, norms, etc. of type 1 and 2 respectively. These are the main current characteristics of the degree of prevalence of information threats. We propose some results where the generator of the limit process is constructed in explicit form. We also give some interpretations of our model.
considered to be independent of time. campaigns. Its intensity is characterized by the parameters 1 > 0 and 2 > 0 respectively, both are
The second, “internal” channel is interpersonal communication between members of the social community (its intensity, that is, the number of equivalent informational contacts, characterized by the parameters 1 > 0 and 2 > 0 respectively, that are also independent of time). As a result, the adherents of the first idea that has been already “recruited” (their number is equal to 1( )), make their contribution to the recruitment process by affecting non-recruited members (their number is equal to the value of 0 − 1( )− 2( ). The same is for the adherents of the second idea. 2. The rate of change of the number of adherents 1( )and 2( ) (that is, the number recruited into the unit time) consists of:
(it is proportional to the product of the intensities 1 and 2 and on the number of individuals who are not yet recruited 0 − 1( )− 2( )), that is, 1( 0 − 1( )− 2( ))and 2( 0 − 1( )− 2( ))respectively.
Internal recruitment rate (it is proportional to the product of intensities 1 and 2, on the corresponding number of active adherents 1( ), 2( ) and on the number of non-recruited 0 − 1( )− 2( )), that is, 1 1( )( 0 − 1( )− 2( ))and 2 2( )( 0 − 1( )− 2( ))respectively.
1( )/ = ( 1 + 1 1( ))( 0 − 1( )− 2( )), 2( )/ = ( 2 + 2 2( ))( 0 − 1( )− 2( )), > 0.
As we know, the outside world speaks with us the language of probability theory, so the deterministic model is only part of the real situation. That is why we are building a model that describes a model of information warfare that has not yet been explored, that is, a model based on contingencies, and contingencies of different types: ( )= ( ( ), ( / 2)) + ( ), (1) where ( ( ), ( 2)) =
= (− 1( )+ 1( ) 0( )− 2( ) 1 ( ) − 2( )− 2( ) 2( )
− 1( )− 1( ) 1 ( ) − 2( )+ 2( ) 0( )− 1( ) 2( ) )( 1 ( ) 2( )), (2) is a small series parameter; ( )is a two-dimensional vector of solutions, components of which are the quantities of the adherents of different ideas; ( / 2)is uniformly ergodic Markov process in standard phase space ( , ), is defined by the generator [1], [2], [6], [13]
( )= ( )∫ ( , )[ ( )− ( )] on the Banach space ( )of real-valued bounded functions (x) with the supremum norm || || = max| ( )|.
∈ = ( ), ≥ 0, (
) ( )= ( ),
with stationary distribution ( ), ∈ . Stationary distribution ( ), BX , of the Markov process ( ), ≥ 0 is defined by the relation where
Denote by 0 the potential operator of the generator , which is defined by the equality where ( )= ∫ (
) ( ) ( )is the projector of zeroes of generator onto the subspace ( )is the impulse perturbation process [1], [5], [6], [7], [16] defined by the relation = ∫ (
) ( ). ( ) ( )= −2 ∫( ( + )− ( )) ( , ), ∈
and satisfies the properties of Poisson approximation;
∫ ( , )= ( ( )+ ( )), ( )→ 0, → 0, and
P2. The condition imposed on the distribution function ∫ 2 ( , )= 2( ( )+ ( )), ( )→ 0, → 0.
∫ ( ) ( , )= 2(
( )+ ( )), ( )→ 0, → 0 for all g(v)C3(R) , and C3(R) is the space of real-valued bounded functions such that where measure g(x) is bounded for all g(v)C3(R) and is defined by the relation (functions from the space C3(R) separate the measures):
( )/| |2 → 0,| | → 0, ( )= ∫ ( ) 0( , ), ( )∈ 3( ). approximation:
P3. The uniform quadratic integrability Р4: Absence of diffusion component lim ∫ →∞ | |>
2 0 ( , )= 0. ( )= ∫ 2 0( , ).
{ = } = , { = } = 1 − . = ( + )+ ( ), 2 = ( 2 )+ ( ).
We give a simple example of a random variable that satisfies the conditions of the Poisson
Theorem 1. Under conditions P1 – P4, weak convergence
( ) → 0( ), → 0. holds true for the impulse perturbation process.
The limit process 0( ) is defined by the generator where component. convergence holds true.
̂ = ∫ (
) ( ), ̂0( ) = ∫ ( ) 0( , ). particular, using the approaches proposed in [14], [15].
Thus, we deal in a limit with a random process, which has deterministic drift and Poisson jumping Further, we investigate the asymptotic properties of the original evolutionary system (1), in Theorem 2. If conditions P1 – P4 are satisfied, the weak convergence in the sense of generators The limiting coupled process is defined by the generator where generator function ( , ), corresponding to a coupled process.
The averaged function has a form ( ( ), ( )) → ( 0( ), 0( )), → 0. ( , ) = ̂( ) ′( ,∙)+ ( ,∙), ̂( ) = ∫ ( ) ( , ).
The last correlation means that to obtain the limit characteristics that describe the information warfare model, all the functions in (2) that depend on x should be averaged by the stationary measure are the same as defined in Theorem 1, but acting on argument of the vectorof the switching Markov process.
We apply the approaches to the construction and analysis of complex systems proposed in the works of Korolyuk V.S. [1], [2] and his followers, in particular, we apply the following scheme: 1. Construction of the generator of the Markov additive process. 2. Asymptotic form of the generator acting on some special type of test functions. 3. Solving of a singular perturbation problem on test functions in a form ( , , ) = ( , )+ 1( , , )+ 2 2( , , ).
( ) ⇒ 0( ), 0 . follows from the convergence of respective generators when compactness of the prelimiting set of processes ( ) holds true. Weak convergence of stochastic processes is usually proved by checking the two conditions: tightness of the distributions of the converging processes which ensures the existence of a converging subsequence and uniqueness of the weak limit. The passage to the limit can be done on the semigroups which correspond to the converging processes as well as on appropriate generators. While proving convergence of generators a natural question arises concerning the uniqueness of a limit semigroup. It can be answered by representing the process in focus as a unique solution to a martingale problem which is formulated with the help of the limit generator. 2. The limit process 0( ) can be given by stochastic differential equation ̂ ( ) = [ ̂( ̂ ( ))+ ̂ ] + ∫ ̃ ( , ), where ̃( , ) = ̃0( ). 3. The limit process 0( ) has two components. The deterministic drift is defined by the solution of the differential equation
̂ ( ) = [ ̂( ̂ ( ))+ ̂ ] , where the additional term ̂ appears due to accumulation with the normalized time / 2, → 0 of small jumps of the impulse process that happen with probability, close to one. The second component is rare big jumps that take place with nearly zero probability and are defined in terms of an averaged measure of jumps ̃0( ) by the generator Γ ( ) = ∫ [ ( + )− ( )− ′( )] ̃0( )
As we can see in many works on mathematical biology and economics the modeling of population dynamics or economical processes is based on Lotka-Volterra type equations.
We propose a new model of information warfare with an additional influence of chance. That may be interpreted as some kind of rare event that rapidly changes some beliefs of large quantities of people. As a result, the quantities of adherents of different ideas make stochastic jumps, which we may see applying the Poisson approximation scheme.
essential, as soon as now breaking news produce a quick and astonishing influence on the audience through TV and Internet.
The behavior of our model could not be analyzed obviously for any fixed moment as it was done in a classical case. But, as it is usual for stochastic models, we may obtain functional limit theorems that present the behavior on large time intervals. Thus, we have averaged limit characteristics of the process and may use them to construct obvious solutions. We hope to obtain recommendations for prevalence strategies in information warfare fights in the future. 8. References 2005.
723–728. [1] V. S. Korolyuk, N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, [2] V. S. Korolyuk, N. Limnios, I. V. Samoilenko, Lévy and Poisson approximations of switched stochastic systems by a semimartingale approach, Comptes Rendus Mathématique 354 (2016) [3] A. J. Lotka, Relation between birth rates and death rates, Science 26 (1907) 21–22. [4] А. P. Mikhailov, N. A. Marevtseva, Models of information warfare. Mathematical Models and [5] I. V. Samoilenko, A. V. Nikitin, Differential Equations with Small Stochastic Terms Under the Lévy
[6] A.V. Nikitin, Asymptotic Dissipativity of Stochastic Processes with Impulsive Perturbation in the Levy Approximation Scheme, JAI(S) 50 (2018). doi:10.1615/JAutomatInfScien.v50.i4.50. [7] Y. M. Chabanyuk, A.V. Nikitin, U. T. Khimka, Asymptotic properties of the impulse perturbation process under Levy approximation conditions with the point of equilibrium of the quality criterion, Matematychni Studii 52 (2019). doi:10.30970/ms.52.1.96-104. [8] L. Stone, R. Olinky, Phenomena in ecological systems, in: Experimental Chaos: 6th
[9] K. I. Takahashi, K. Salam, M. Md., Mathematical model of conflict with non-annihilating multiopponent, J. Interdisciplinary Math 9 (2006) 459–473. doi:10.1080/09720502.2006.10700457. [10] J. Tufto, Effects of releasing maladapted individuals: a demographic evolutionary model, The
American Naturalist 158 (2001) 331–340. doi:10.1086/321987. [11] P. P. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondence mathematique et physique publiee par A. Quetelet 10 (1838) 113–121. [12] V. Volterra, Sui tentativi di applicazione della matematiche alle scienze biologiche e sociali,
[13] A.V. Nikitin, Asymptotic Properties of a Stochastic Diffusion Transfer Process with an Equilibrium Point of a Quality Criterion, Cybernetics and System analysis 51 (2015) 650–656. doi:10.1007/s10559-015-9756-3. [14] M. B. Nevelson, R. Z. Khas'minskii, Stochastic approximation and recursive estimation,
Moscow, 1972. [15] G. Papanicolaou, D. Stroock, S.R.S. Varadhan, Martingale approach to some limit theorems, in: Duke turbulence conference, Durham, NC, April 23-25, 1976, Duke University Mathematics Series III, New York, 1977, 120 p. [16] I. V. Samoilenko, A. V. Nikitin, B. V. Dovhai, Asymptotic Dissipativity for Merged Stochastic Evolutionary Systems with Markov Switchings and Impulse Perturbations under Conditions of Lévy Approximation, Cybernetics and System analysis 56 (2020) 392–400. doi:10.1007/s10559020-00255-4. [17] Zh. Hu, V. Buriachok, V. Sokolov, Implementation of Social Engineering Attack at Institution of Higher Education, in: 1th International Workshop on Cyber Hygiene & Conflict Management in