Construction Features and Analysis of Warfare Information
Model with Impulse Perturbations under Poisson
Approximation Conditions
Ihor Samoilenkoa, Anatolii Nikitinb,c, Ganna Verowkinaa, and Tetiana Nikitinad
a
  Taras Shevchenko National University of Kyiv, 60 Volodymyrska str., Kyiv, 01033, Ukraine
b
  National University of Life and Environmental Science, 15 Heroiv Oborony str., Kyiv, 03041, Ukraine
c
  Jan Kochanowski University of Kielce, 5 Zeromskiego str., Kielce, 25-369, Poland
d
  Dmytro Motornyi Tavria State Agrotechnological University, 18b Khmelnytsky ave., Melitopol, 32312,
  Ukraine

                Abstract
                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 supporters 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.

                Keywords 1
                Random evolution, information warfare model, Markov switch, Lotka-Volterra equations,
                Poison approximation scheme

1. Introduction
   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 𝑁0 , 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 𝑁1 (𝑡), 𝑁2 (𝑡) are the numbers of “adherents”
depending on time 𝑡 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.

2. Classical Information Warfare Model
     The main model assumptions are:
1. Both information threats are distributed among the community through the two information
   channels:

Cybersecurity Providing in Information and Telecommunication Systems, January 28, 2021, Kyiv, Ukraine
EMAIL: isamoil@i.ua (A.1); nikitin2505@univ.kiev.ua (B.2); avv@univ.kiev.ua (A.3); nikitina0217@gmail.com (D.4)
ORCID: 0000-0001-6858-8224 (A.1); 0000-0001-5137-0114 (B.2); — (A.3); 0000-0002-6091-0597 (D.4)
             ©️ 2021 Copyright for this paper by its authors.
             Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
             CEUR Workshop Proceedings (CEUR-WS.org)




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       The first one is “external” in relation to the community, for example, advertising media
campaigns. Its intensity is characterized by the parameters 𝛼1 > 0 and 𝛼2 > 0 respectively, both are
considered to be independent of time.
       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 𝛽1 > 0 and 𝛽2 > 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 𝑁1 (𝑡)), make
their contribution to the recruitment process by affecting non-recruited members (their number is
equal to the value of 𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡). The same is for the adherents of the second idea.
2. The rate of change of the number of adherents 𝑁1 (𝑡) and 𝑁2 (𝑡) (that is, the number recruited into
   the unit time) consists of:
       External recruitment rate (it is proportional to the product of the intensities
𝛼1 and 𝛼2 and on the number of individuals who are not yet recruited 𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡)), that is,
𝛼1 (𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡)) and 𝛼2 (𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡)) respectively.
        Internal recruitment rate (it is proportional to the product of intensities 𝛽1 and 𝛽2 , on the
corresponding number of active adherents 𝑁1 (𝑡), 𝑁2 (𝑡) and on the number of non-recruited 𝑁0 −
𝑁1 (𝑡) − 𝑁2 (𝑡)), that is, 𝛽1 𝑁1 (𝑡)(𝑁 − 𝑁1 (𝑡) − 𝑁2 (𝑡)) and 𝛽2 𝑁2 (𝑡)(𝑁 − 𝑁1 (𝑡) − 𝑁2 (𝑡)) respectively.
                                      0                                  0

   Consequently, the model is described by Lotka-Volterra-type equations [4]:

                            𝑑𝑁1 (𝑡)/𝑑𝑡 = (𝛼1 + 𝛽1 𝑁1 (𝑡))(𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡)),
                            𝑑𝑁2 (𝑡)/𝑑𝑡 = (𝛼2 + 𝛽2 𝑁2 (𝑡))(𝑁0 − 𝑁1 (𝑡) − 𝑁2 (𝑡)), 𝑡 > 0.

3. Information warfare model with impulsive influence
   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:
                              𝑑𝑁 𝜀 (𝑡) = 𝐶(𝑁 𝜀 (𝑡), 𝑥(𝑡/𝜀 2 ))𝑑𝑡 + 𝑑𝜂 𝜀 (𝑡),                    (1)
where
                                                          𝑡
                                          𝐶 (𝑁 𝜀 (𝑡), 𝑥 ( 2 )) =
                                                         𝜀
        −𝛼1 (𝑥) + 𝛽1 (𝑥)𝑁0 (𝑥) − 𝛽2 (𝑥)𝑁1𝜀 (𝑡)              −𝛼1 (𝑥) − 𝛽1 (𝑥)𝑁1𝜀 (𝑡)         𝑁 𝜀 (𝑡)
    =(                             𝜀                                                  𝜀  ) ( 1𝜀 ), (2)
                −𝛼2 (𝑥) − 𝛽2 (𝑥)𝑁2 (𝑡)             −𝛼2 (𝑥) + 𝛽2 (𝑥)𝑁0 (𝑥) − 𝛽1 (𝑥)𝑁2 (𝑡) 𝑁2 (𝑡)
                                  𝜀
𝜀 is a small series parameter; 𝑁 (𝑡) is a two-dimensional vector of solutions, components of which
are the quantities of the adherents of different ideas; 𝑥(𝑡/𝜀 2 ) is uniformly ergodic Markov process in
standard phase space (𝑋, 𝑿), is defined by the generator [1], [2], [6], [13]
                               𝑸𝜑(𝑥) = 𝑞(𝑥) ∫ 𝑃(𝑥, 𝑑𝑦)[𝜑(𝑦) − 𝜑(𝑥)]
                                                𝑋
on the Banach space 𝐵(𝑋) of real-valued bounded functions  ( x) with the supremum norm
                                         ||𝜑|| = max|𝜑(𝑥)|.
                                                     𝑥∈𝑋
   The stochastic kernel 𝑃(𝑥, 𝐵), 𝑥 ∈ 𝑋, 𝐵 ∈ 𝑿, uniformly ergodic embedded Markov chain
                                           𝑥𝑛 = 𝑥(𝜏𝑛 ), 𝑛 ≥ 0,
with stationary distribution 𝜌(𝐵), 𝐵 ∈ 𝑿. Stationary distribution 𝜋(𝐵), B  X , of the Markov process
𝑥(𝑡), 𝑡 ≥ 0 is defined by the relation
                                         𝜋(𝑑𝑥)𝑞(𝑥) = 𝑞𝜌(𝑑𝑥),
where




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                                            𝑞 = ∫ 𝜋(𝑑𝑥)𝑞(𝑥).
                                                     𝑋

     Denote by 𝑅0 the potential operator of the generator 𝑸, which is defined by the equality
                                             𝑅0 = 𝛱 − (𝛱 + 𝑸)−1,
where 𝛱𝜑(𝑥) = ∫𝑋 𝜋(𝑑𝑦)𝜑(𝑦) 𝟏(𝑥) is the projector of zeroes of generator 𝑸 onto the subspace
                                               𝑁𝑄 = {𝜑: 𝑸𝜑 = 0};
𝜂 𝜀 (𝑡) is the impulse perturbation process [1], [5], [6], [7], [16] defined by the relation
                                                   𝑡
                                                                   𝑠
                                       𝜂 𝜀 (𝑡) = ∫ 𝜂 𝜀 (𝑑𝑠, 𝑥 ( 2 )),
                                                  0               𝜀
where the family of processes with independent increments
                       𝜞𝜀 (𝑥)𝜑(𝜔) = 𝜀 −2 ∫ (𝜑(𝜔 + 𝑣) − 𝜑(𝜔)) 𝛤 𝜀 (𝑑 𝑣, 𝑥), 𝑥 ∈ 𝑋
                                            𝑅

and satisfies the properties of Poisson approximation;
   P1. The approximation of averages
                           ∫𝑅 𝑣𝛤 𝜀 (𝑑𝑣, 𝑥) = 𝜀(𝑎(𝑥) + 𝜃𝑎 (𝑥)), 𝜃𝑎 (𝑥) → 0, 𝜀 → 0,

and
                          ∫𝑅 𝑣 2 𝛤 𝜀 (𝑑 𝑣, 𝑥) = 𝜀 2 (𝑏(𝑥) + 𝜃𝑏 (𝑥)), 𝜃𝑏 (𝑥) → 0, 𝜀 → 0.
     P2. The condition imposed on the distribution function
                        ∫𝑅 𝑔(𝑣) 𝛤 𝜀 (𝑑 𝑣, 𝑥) = 𝜀 2 (𝛤𝑔 (𝑥) + 𝜃𝑔 (𝑥)), 𝜃𝑔 (𝑥) → 0, 𝜀 → 0
for all g (v)  C3 ( R) , and C3 ( R) is the space of real-valued bounded functions such that
                                               𝑔(𝑣) /| 𝑣 |2 → 0, | 𝑣 | → 0,
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):
                                𝛤𝑔 (𝑥) = ∫𝑅 𝑔(𝑣)𝛤0 (𝑑𝑣, 𝑥), 𝑔(𝑣) ∈ 𝐶3 (𝑅).
     P3. The uniform quadratic integrability

                                       lim ∫         𝑣 2 𝛤0 (𝑑 𝑣, 𝑥) = 0.
                                       𝑐→∞ | 𝑣 |>𝑐
     Р4: Absence of diffusion component

                                            𝑏(𝑥) = ∫𝑅 𝑣 2 𝛤0 (𝑑𝑣, 𝑥).

     We give a simple example of a random variable 𝜉 that satisfies the conditions of the Poisson
approximation:
                                                𝑃{𝜉 = 𝑏} = 𝜀𝑝,
                                             𝑃{𝜉 = 𝜀𝑎} = 1 − 𝜀𝑝.
     The relations for the moments of this random variable are as follows:
                                         𝐸 𝜉 = 𝜀(𝑎 + 𝑏𝑝) + 𝑜(𝜀),
                                          𝐸 𝜉 2 = 𝜀(𝑏 2 𝑝) + 𝑜(𝜀).




14
4. Asymptotic analysis of the model
   Firstly, we consider the asymptotic properties of the perturbation process.
   Theorem 1. Under conditions P1 – P4, weak convergence
                                         𝜂 𝜀 (𝑡) → 𝜂 0 (𝑡), 𝜀 → 0.
holds true for the impulse perturbation process.
   The limit process 𝜂 0 (𝑡) is defined by the generator
                             𝛤𝜑(𝑤) = 𝑎̂𝜑′(𝑤) + ∫𝑅[𝜑(𝑤 + 𝑣) − 𝜑(𝑣)]𝛤̂0 (𝑑𝑣),
where
                                          𝑎̂ = ∫ 𝜋 (𝑑𝑥)𝑎(𝑥),
                                               𝑋

                                      𝛤̂0 (𝑣) = ∫ 𝜋 (𝑑𝑥)𝛤0 (𝑣, 𝑥).
                                                𝑋
   Thus, we deal in a limit with a random process, which has deterministic drift and Poisson jumping
component.
   Further, we investigate the asymptotic properties of the original evolutionary system (1), in
particular, using the approaches proposed in [14], [15].
   Theorem 2. If conditions P1 – P4 are satisfied, the weak convergence in the sense of generators
convergence
                                 (𝑢𝜀 (𝑡), 𝜂 𝜀 (𝑡)) → (𝑢0 (𝑡), 𝜂 0 (𝑡)), 𝜀 → 0.
holds true.
   The limiting coupled process is defined by the generator
                                 𝑳𝜑(𝑤, 𝑣) = 𝐶̂ (𝑢)𝜑′ (𝑤,∙) + 𝛤𝑤 𝜑(𝑤,∙),
where generator 𝛤𝑤 are the same as defined in Theorem 1, but acting on argument 𝑤 of the vector-
function 𝜑(𝑤, 𝑣), corresponding to a coupled process.
   The averaged function has a form
                                       𝐶̂ (𝑢) = ∫ 𝜋(𝑑𝑥)𝐶(𝑢, 𝑥).
                                                𝑋
    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
of the switching Markov process.

5. Investigation methods and results
    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
                            𝜑𝜀 (𝑢, 𝑤, 𝑣) = 𝜑(𝑢, 𝑤) + 𝜀𝜑1 (𝑢, 𝑤, 𝑣) + 𝜀 2 𝜑2 (𝑢, 𝑤, 𝑣).
    The following resumes should be made:
    1. The weak convergence of the processes
                                               𝑢𝜀 (𝑡) ⇒ 𝑢0 (𝑡),   0 .
follows from the convergence of respective generators when compactness of the prelimiting set of
processes 𝑢𝜀 (𝑡) 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.



                                                                                                  15
         2. The limit process 𝑢0 (𝑡) can be given by stochastic differential equation
                               𝑑𝑢̂(𝑡) = [𝐶̂ (𝑢̂(𝑡)) + 𝑎̂]𝑑𝑡 + ∫𝑅 𝑣𝜈̃ (𝑑𝑡, 𝑑 𝑣),
where
                                              𝑬𝜈̃(𝑑𝑡, 𝑑𝑣) = 𝑑𝑡𝛤̃0 (𝑑𝑣).
                               0
       3. The limit process 𝑢 (𝑡) has two components. The deterministic drift is defined by the
         solution of the differential equation
                                        𝑑𝑢̂𝑑 (𝑡) = [𝐶̂ (𝑢̂𝑑 (𝑡)) + 𝑎̂]𝑑𝑡,
where the additional term 𝑎̂ appears due to accumulation with the normalized time 𝑡/𝜀 2 , 𝜀 → 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 𝛤̃0 (𝑑𝑣) by the generator

                           Γ𝜑(𝑤) = ∫ [𝜑(𝑤 + 𝑣) − 𝜑(𝑤) − 𝑣𝜑′(𝑤)]𝛤̃0 (𝑑𝑣)
                                      𝑅


6. Conclusions
     The following results of our investigation were obtained:
       A model that is more general than the classical one was proposed.
       The limit generator of the dynamical system was constructed.
       The behavior of the limit process in terms of its components was analyzed.

7. Interpretation
   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. We suppose that such a model could be more
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.

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