=Paper=
{{Paper
|id=Vol-2654/paper10
|storemode=property
|title=Cyber Object State Maximal Probability Timing Obtained Through Multi-Optional Technique
|pdfUrl=https://ceur-ws.org/Vol-2654/paper10.pdf
|volume=Vol-2654
|authors=Andriy Goncharenko
|dblpUrl=https://dblp.org/rec/conf/cybhyg/Goncharenko19
}}
==Cyber Object State Maximal Probability Timing Obtained Through Multi-Optional Technique==
https://ceur-ws.org/Vol-2654/paper10.pdf
Cyber Object State Maximal Probability Timing
Obtained Through Multi-Optional Technique
Andriy Goncharenko [0000-0002-6846-9660]
National Aviation University, Kyiv, Ukraine
andygoncharenco@yahoo.com
Abstract. In this publication a Doctrine for the Conditional Extremization of
the Hybrid-Optional Effectiveness Functions Entropy is discussed as a tool for
the Cyber Object State Maximal Probability Assessments. Traditionally, most
of the problems having been dealt with in this area must relate with the proba-
bilistic problem settings. Regularly, the optimal solutions are obtained through
the probability extremizations. It is shown a possibility of the optimal solutions
“derivation”, with the help of a model implementing a variational principle
which takes into account objectively existing parameters and components of the
Markovian process. The presence of an extremum of the objective state proba-
bility is observed and determined on the basis of the proposed Doctrine with
taking into account the measure of uncertainty of the hybrid-optional effective-
ness functions in the view of their entropy. Such approach resembles the well
known Jaynes’ Entropy Maximum Principle from theoretical statistical physics
adopted in subjective analysis of active systems as the subjective entropy max-
imum principle postulating the subjective entropy conditional optimization. The
developed herewith Doctrine implies objective characteristics of the process ra-
ther than subjective individual’s preferences or choices, as well as the states
probabilities maximums are being found without solving a system of ordinary
linear differential equations of the first order by Erlang corresponding to the
graph of the process.
Keywords: cyber hygiene, conflict management, global information networks,
effectiveness functions entropy, hybrid-optional effectiveness, multi-optionality,
optimal distribution, variational principle, entropy maximum principle.
1 Introduction
1.1 Literature Survey
Cyber hygiene and conflict management in global information networks can be con-
sidered from the point of view of the theoretical developments for reliability [1].
The analogy of the hygiene to the maintenance procedures is very good. Therefore,
the apparatus of theoretical physics related with the uncertainty measures [2-4] is
quite applicable here. Thus, in the field of the Social Networking Services it is critical
to take into considerations subjective entropy of preferences [5, 6]. The similar to the
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0). CybHyg-2019: International Workshop on
Cyber Hygiene, Kyiv, Ukraine, November 30, 2019.
aircraft maintenance and repair approaches [7], combinations with the rationality of
the choice behavior [8], [9], inspire global science and social science entropy research
[10]. In addition, economic issues [11] in respect of risk [12], like in aviation [13], are
complicated with the group decision making [14, 15].
All this initiated search for a new explanation of the described process. The pre-
sented doctrine, like developed in [16-30], is to demonstrate the possibilities of the
entropy paradigm use to the variety of the problems solutions, for example discussed
in works [31-39]. Mathematical means intended to be used are of the regular calculus
[40]. Also, adjacent and similar formalism scientific areas, let us say mentioned in
publications of [41-49], can implement the presented doctrine results.
1.2 The Problem Statement
Management of cyber incidents, warfare and conflicts are considered in terms of the
mass service theory [37-39].
The considered cyber object (space) can change its states. Illustration of that is in
the simplified graph (see Fig. 1).
0
01 02
10 20
12
1 2
21
Fig. 1. A graph of three states of a cyber space.
Here, in Fig. 1, “0”, “1”, and “2” designate the states of the cyber object. The corre-
sponding values of the rates ij and ji will determine the process going on in the
system.
The problem is to find the timing for the maximal values of the states probabilities,
for instance of P1 t , analytically and in an easier than the traditional way. The pro-
posed is the multi-optional way.
2 Main Content
2.1 Traditional Methods
Even for the simplified (partial to Fig. 1) case, although implying the possible return
of the system from the state “D” into the state of “A” without the transition into the
state “F” (this transition is carried out with the parameter of 1 illustrated on the
graph, see Fig. 2) the procedure is quite challenging analytically.
1
2
A D F
1
Fig. 2. A simplified graph of three states of a cyber space.
The corresponding, to the graph of Fig. 2, system of differential equations by Erlang
will have the view of
dPA
1PA 1PD ;
dt
1PA 2 1 PD ;
dPD
(1)
dt
dPF
2 PD .
dt
The characteristic equation for system (1) will be similarly [40]:
1 k 1 0
1 2 1 k 0 0. (2)
0 2 0k
Determinant (2) yields
1 k 2 1 k 0 k 1 2 0 1 0 0
2 1 k 0 0 11 0 k 1 k 2 0 0 . (3)
Which means
1 k 2 1 k k 11k 0 . (4)
k 11 1 k 2 1 k 0 . (5)
Thus, we have already known at least one root:
k1 0 . (6)
Then, for finding two other roots from Eq. (5)
11 12 11 1k k2 k1 k 2 0 . (7)
Reducing Eq. (7) and cancelling the similar members 11 and 11 ,
k 2 k 1 2 1 1 2 0 . (8)
The sought roots are
b b 2 4ac b b 2 4ac
k2 ; k3 ; (9)
2a 2a
where a 1 , b 1 2 1 , c 1 2 are corresponding coefficients of (8).
For each root ki of Eq. (2)-(5), (7), (8), namely k1 , k 2 , k 3 Eq. (6) and (9) we will
write down the system of linear algebraic equations for i , i , i [40]:
1 2 3
1 k 1 1 2 0 3 0;
11 2 1 k 2 0 3 0; (10)
0 1 2 2 0 k 3 0.
The system of Eq. (10) derives from an assumption of a partial solution existence
PA 1ekt ; PD 2ekt ; PF 3ekt ; (11)
for the system of Eq. (1).
Since having three roots in the stated problem setting [40], the solution of (1):
PA1 11ek1t ; PD1 21ek1t ; PF1 31ek1t ; (12)
PA2 12ek2t ; PD2 22ek2t ; PF2 32ek2t ; (13)
PA3 13e k3t ; PD3 23e k3t ; PF3 33e k3t . (14)
In the way of direct substitution of partial solutions (12)-(14) into equations, one can
be convinced that the system of functions, similarly to [40]:
PA C1PA1 C2 PA2 C3 PA3 C111e k1t C212 e k2t C313e k3t ;
PD C1PD1 C2 PD2 C3 PD3 C121ek1t C222 e k2t C323e k3t ; (15)
PF C1PF1 C2 PF2 C3 PF3 C131e k1t C232 ek2t C333e k3t ;
where C1 ; C 2 ; C3 are arbitrary constants; also is the solution of the differential
equations system (1). This is the general solution of the differential equations sys-
tem (1), [40].
Satisfying the condition of Eq. (6) for root k1 0 from the system of Eq. (10)
1 k1 11 1 21 0 31 0;
111 2 1 k1 21 0 31 0; (16)
0 1 2 21 0 k1 31 0.
1
111 1 21 0 31 0;
111 2 1 21 0 31 0; (17)
0 2 21 0 31 0.
From where, immediately the coefficients are
21 0 ; 11 0 ; 31 1 ; (18)
since 31 is an arbitrary number, supposedly 31 1 , [40].
For the Eq. (8) roots of k 2 and k 3 , Eq. (9), the system of Eq. (10) analogous to the
system of Eq. (16) it yields
1 k2,3 12,3 1 22,3 0 32,3 0;
2, 3
1 1
2 1 k2,3 2 2 ,3 0 2,3 0;
3 (19)
0 2, 3 2, 3 0 k2,3 3 0.
2,3
1 2 2
The system of Eq. (19) can be solved for unknown sought coefficients.
Since one of the alpha coefficients can be chosen arbitrary, [40], let us assume
22,3 1 . (20)
Then, from the first equation of system (19)
1
1 k2,3 12,3 1 0 ; 12,3 . (21)
1 k2,3
Or from the second equation
2 1 k2,3
112,3 2 1 k2,3 0 ; 12,3 . (22)
1
Or summing the first and second equations
2 k 2,3
k2,312,3 2 k2,3 22,3 0 ; 12,3
k 2,3
. (23)
All three expressions for 12,3 , i.e. Eq. (21)-(23) are equivalent because all of them
use the roots k 2 and k 3 , Eq. (9) of the initial quadratic equation Eq. (8).
Indeed. Equalizing Eq. (21) and (22) we get
11 1 2 11 1k2,3 2k2,3 1k2,3 k22,3 . (24)
And cancelling for 11 in both parts of Eq. (24) it yields Eq. (8):
1 2 1 2 1 k2,3 k22,3 0 . (25)
The same result is obtained if make equal Eq. (21) and (23):
1k2,3 1 2 1k2,3 2k2,3 k22,3 ; k22,3 1 2 1 k2,3 1 2 0 . (26)
When equalling Eq. (22) and (23) it gives the same. Indeed:
2 1 k2,3 k22,3 1 2 1k2,3 ; k 22,3 1 2 1 k 2,3 1 2 0 . (27)
For coefficient 32,3 , from the third equation of Eq. (19) and condition (20),
2
2 22,3 k 2,3 32,3 0 ; 32,3 . (28)
k 2,3
Thus, turning back to the system of Eq. (15), we determine the unknown coefficients
of the general solution of the differential equations system (1), [40], satisfying the
initial conditions: t0 0 ; PA t t 1 ; PD t t 0 ; PF t t 0 ; and have already
0 0 0
known the coefficients of alpha; i.e. Eq. (18); (20); (21); (28):
1 1
PA C111e k1t C212 e k2t C313e k3t ; 1 0 C2 k C3 k ;
1 2 1 3
PD C1 21e k1t C2 22 e k2t C3 23e k3t ; 0 0 C2 C3 ; (29)
1 k1t 2 k 2t 3 k3t 2
PF C13 e C23 e C33 e ;
t0 0 0 C1 C2 C3 2 .
k2 k3
From the second equation of the system of Eq. (29) it yields
C2 C3 . (30)
Substituting the values of Eq. (30) for the corresponding members into the first equa-
tion of the system of Eq. (29) we get
1 1 1
1 C3 ; C3 . (31)
1 k3 1 k 2 1 1
1 k3 1 k 2
In order to make the notations shorter let us put down the indications with the alpha
symbolizations:
1 C312 C313 C3 13 12 ; C3 3
1
1 12
. (32)
1
C2 3 . (33)
1 12
From the third equation of the system of Eq. (29) we obtain
C1 C232 C333 . (34)
Now, all coefficients are expressed through the given values, hence, the system of
Eq. (1) is successfully solved. The Laplace integral transformation methods give the
same results. For the general case described with the graph shown in Figure 1
k ek1t k2ek2t ek1t ek2t
P0 t 1 a1
k1 k2 k1 k2
b1 b1 b b1 k2t
1 ek1t e . (35)
k1k2 k2 k2 k1 k1k2
2 2 1
k k k
e k1t e k2t c k 2t
P1 t 01
c c1 c1
1 1 e k1t e . (36)
k1 k2 k1k2 k2 k2 k1 k1k2 k2 k2 k1
e k1t e k2t d k 2t
P2 t 02
d d1 d1
1 1 e k1t e . (37)
k1 k2 k1k2 k2 k2 k1 k1k2 k2 k2 k1
The values of the parameters in (35) - (37) have the mathematical expressions corre-
sponding to the general case (see Fig. 1). Then, it has to be found the possible extreme
values of the probabilities. For distinctness, let it be P1 t .
dP1 t
01
k1e k1t k2e k2t k1
c1 c
1 e k1t k2
c1 k 2t
e . (38)
dt k1 k2 k2 k2 k1 k1k2 k2 k2 k1
After equalizing (38) to zero, the needed timing is
ln 01k1 c1 ln 01k2 c1
t *p . (39)
k 2 k1
2.2 The Proposed Approach
Herein it is suggested to formulate the own concept (idea, problem, hypotheses).
In such respect [1-40], the considered example may be given an attention to in re-
gards with the Multi-Optional Hybrid-Effectiveness Functions Uncertainty Measure
Conditional Optimization Doctrine (method, approach, concept) applicable (used,
implemented) to the cyber object state maximal probability timing determination [17,
20, 22, 25].
The values can be obtained not only in the entire probabilistic way, but also in a
hybrid partially probabilistic partially optional way [17, 20, 22, 25].
The essence of the doctrine (method, idea, approach, concept) is to consider the
process developing in the system from the position of some hybrid optional functions
distribution optimality.
Consider the options essential to the general view three state system (see Fig. 1).
Objective functional, like proposed in references [17, 20, 22, 25], is as follows:
xF ln xF xF M xF 1 ,
3
t *p 3 3
h 1
i
1
i
1
i i
12 1
i
i 1 01 i 1 i 1
i ki 01 c1
F1i
M12
,
M p p 2 pe b c d
1 1 1 1
i k c ,
M12 i 01 1
M p p 2 pe1 b1 c1 d1 , (40)
where x is an unknown parameter; hi xF1i is the multi-optional hybrid functions
depending upon the options effectiveness functions of F i ; t * is the intrinsic 1 p 01
parameter of the system and the process, which is the ratio of the timing (delivering
the sought maximal value to the probability) t *p , it is unknown yet for such problem
formulation and the time of t *p is going to be determined as a solution, i.e. it is not the
Eq. (39) so far, however it will be, that is why the indication is the same, to the flow
intensity 01 ; M12i is the algebraic addition of the initial elementary intensities ma-
trix M formed in the style likewise from the Erlang’s system, Eq. (1), element of
m12 ; is the parameter, coefficient, function (uncertain Lagrange multiplier, weight
coefficient) for the normalizing condition.
Consider an extremum existence necessary conditions for the objective functional
of (40), [17, 20, 22, 25]:
h h
0, i 1, 3 . (41)
hi
xF1i
t k c 1 ln xF t k c .
* *
ln xF11
p 2 p
01 1 1 1 01 2 1 (42)
01 01
From where
t k c ln xF t k c .
* *
ln xF11
p 2 p
01 1 1 1 01 2 1 (43)
01 01
After that, we have got the law of subjective conservatism on one hand and on the
other hand
t k c k c .
*
ln xF11 ln xF12
p
01 2 1 01 1 1 (44)
01
c c
ln xF11 ln xF12 t *p k2 1 k1 1 .
01 01
(45)
After that likewise Eq. (39)
t *p
ln F11 ln F12
.
(46)
k 2 k1
And finally equivalent with Eq. (39) with taking into account the roots, i.e. the sec-
ond, third, and fourth expressions of the Eq. (40)
k1 01 c1 k2 01 c1
ln
.
ln
p p pe1 b1 c1 d1
2
p p pe1 b1 c1 d1
2
t *p (47)
k2 k1
ln k1 01 c1 ln k 2 01 c1
t *p . (48)
k 2 k1
3 Discussion
Thus, the result of Eq. (39) is obtained in absolutely not probabilistic rather in the
Multi-Optional Hybrid-Effectiveness Functions Uncertainty Measure Conditional
Optimization Doctrine way [17, 20, 22, 25].
The same approach is applicable to F2i with yielding the parallel to the Eq. (39)
and (48) results.
Now we ought to say that for the situation when the probability of P2 t undergoes
the extremum instead of the probability of P1 t , the problem, due to the symmetry,
has a symmetrical solution:
ln 02 k1 d1 ln 02 k 2 d1
t *p . (49)
k 2 k1
4 Conclusions
That is the system according to the developing stationary Poison flow process has the
possible states optimal options related with either the system of parameters
ki , 02 , d1 or ki , 01, c1 (50)
values for the initial moment probability of the state “0” being equaled to “1”. The
proposed optional method is more compact and applicable for a cyber object state
maximal probability timing determination.
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