This document presents mathematical models of information dissemination and warfare. It discusses: 1) Models of information spreading through both vertical (centralized) and horizontal (interpersonal) flows, and how the combination of these determines information dynamics in society. 2) Models of information adoption and forgetting over time, and the effects of incomplete media coverage and two-step perception. 3) Models of information warfare between two information sources, examining the necessary conditions for one to win over the other. 4) Extensions of these models including periodic destabilization, additional factors like forgetting, and a model of individual choice-making during information warfare.

1. A. Mikhailov1
A. Petrov1
O. Proncheva1,2
1Keldysh Institute of Applied Mathematics (RAS)
2Moscow Institute of Physics and Technology (SU)
Mathematical modeling of
information processes
AINL FRUCT: Artificial Intelligence and Natural Language
Conference

2.  The weapon of criticism cannot, of course, replace
criticism of the weapon, material force must be
overthrown by material force; but theory also
becomes a material force as soon as it has gripped
the masses.
Marx, A Contribution to the Critique of Hegel’s Philosophy of
Right (1844)
 A newspaper is not only a collective propagandist and
collective agitator, but also a collective organiser.
Lenin, Where To Begin (1901)
2

3. Information
3
 Information is not entropy, information is not
uncertainty!
Thomas D. Schneider
 Information is a measure of the decrease of
uncertainty.
Claude Shannon
 Information is information, not matter or energy.
Norbert Wiener

4. Society in the information field
4
The vertical
(centralized)
flows of
information
influence
Horizontal
(interpersonal,
networking)
flows of
information
influence
The combination of these flows determines the dynamics of
information dissemination in society
Conceptual framework

5. Sociological argument about information
dissemination through mass-media and
interpersonal communication
5
 The Rwandan genocide, known officially as the genocide
against the Tutsi, was a genocidal mass
slaughter of Tutsi in Rwanda by members of the Hutu majority
government.
Empirical facts:
 The broadcasts increased militia violence not only directly by
influencing behavior in villages with radio reception, but also
indirectly by increasing participation in neighboring villages.
 Spillovers are estimated to have caused more militia violence
than the direct effects.
(D. Yanagizawa-Drott,
Propaganda and conflict: evidence from the Rwandan Genocide» (2014))

6. Basic model of information attack
- the number of spreaders
- external persuasion
- internal persuasion
- size of a group
Information dissemination without
warfare
  0
dN
N N N
dt
   
(Mikhailov A.P., Klusov N.V., 2002)
  0
0N N
  
  
0
0
0 0
exp
exp
N t
N N
N t N
   
   
 

 
6


0N
( )N t
Maximum agiotage: 0
1
2
gN N


 
  
 

7. Information dissemination without
warfare
7
adoption of the information with forgetting   0
dX
X N X X
dt
     
incomplete coverage of society by the media
two-steps information perception
   
  
1
2
dX
X Y N X
dt
dY
X Y N Y
dt
 


   

   

  
 
0 2
dx
X N X x
dt
dX
x X
dt
 
 

   

  


8. Adoption of the information with
forgetting
8
 
2
0 0 04
2
s
N N N
N
      

      

Steady state:
The condition for maximum agiotage
0
0 , gN N N
 


 

9. Incomplete coverage of society by
the media
9
Blue line– use mass-media
Purple line– don’t use mass media

10. Two-steps information perception
10
red line - the number of spreaders
blue line - the number of pre-spreaders
Successive individuals transformation:
ignorants→pre-spreaders→spreaders

11. Information dissemination without
warfare
11
(Mikhailov A.P., Petrov A.P., Marevtseva N.A., Tretiakova I.V., 2014)
   
  
  
 
1
1 2 1 1 1 1 1
2
1 2 2 2 2 2 2
1
1 1 2 1
2
1 2 2 2
2
2
dx
X X N X x x X
dt
dx
X X N X x x X
dt
dX
x X X X
dt
dX
X X x X
dt
   
  
  
 

      

      


    


  


12. Information dissemination under
warfare
The necessary and sufficient condition of the victory I1
over I2
12
1 2
1 0 2 0
1 2
ln(1 ) ln(1 )
2 2
N N
 
 
 

 
  
  
1 1 0
2 2 0
dX
X N X Y
dt
dY
Y N X Y
dt
 
 

   

    

0N
X Y
1 1,  2 2, 
(Mikhailov A.P., Marevtseva N.A., 2011)

13. The model of information warfare
with three additional factors
13
 2
1 2 1 2 1 2
dX
x X X X
dt
   
  1
1 2 2 1 2 2 1
dY
y Y Y Y
dt
     
 2
2 2 1 2 2 2
dY
y Y Y Y
dt
   
   1
1 1 1 2 1 1 1 1 1 1 1 1 12
dx
X X N X Y x y X x
dt
           
  2
1 1 2 2 2 2 2 2 1 2 1 22
dx
X X N X Y x y X x
dt
         
   1
2 2 1 2 1 1 1 1 1 2 1 2 12
dy
Y Y N X Y x y Y x
dt
           
  2
2 1 2 2 2 2 2 2 2 2 2 22
dy
Y Y N X Y x y Y y
dt
         
  1
1 1 1 1 2 1 1
dX
x X X X
dt
     

14. 14
Group 1 (accessible for mass-media ) Group 2 (inaccessible for mass-media)
Strong propaganda VS virality
Intensity of forgetting γ is small
0 1 2 3 4 5
0
20
40
60
80
100
X1(t),Y1(t)
t
0 1 2 3 4 5
0
10
20
30
40
X2(t),Y2(t)
t
Blue line - information source 1 (propaganda)
Red line - information source 2 (virality)

15. Group 1 (accessible for mass-media ) Group 2 (inaccessible for mass-media)
15
Intensity of forgetting γ is large
Blue line - information source 1 (propaganda)
Red line - information source 2 (virality)
0 20 40 60 80 100
0
20
40
60
80
100
X1(t),Y1(t)
t
0 5 10 15 20
0
10
20
30
40
50
X2(t),Y2(t)
t
Strong propaganda VS virality

16. 16
   
  
1
2
dX
t X N X Y
dt
dY
Y N X Y
dt
 
 

   

    

 
 
*
1
1 *
1
, ;
, 0,1,2,3...
, ;
sw
sw
t iT iT t
i
h t iT t iT T



  
 
   
Periodic destabilizing effect on
information warfare
0 5 10 15
0
20
40
60
Y,X
t

17. Periodic destabilizing effect on
information warfare
17
   
  
1
2
dX
t X N X Y X
dt
dY
Y N X Y Y
dt
  
  

    

     

 
 
*
1
1 *
1
, ;
, 0,1,2,3...
, ;
sw
sw
t iT iT t
i
h t iT t iT T



  
 
   
(Mikhailov A.P., Petrov A.P., Proncheva O.G., Pronchev G.B.., Marevtseva N.A., 2016)

18. Rashevsky's neurological scheme
18
• L,B - feed braking elements
• S – external incentives
• R - reaction
• ε, j – generated incentives
• w - internal incentives
(Rashevsky N., 1933)

19. A model of making choices by
individuals during information warfare
in a society
19
(Petrov A.P., Maslov A.I., Tsaplin N.A., 2015)
 X(t) – the number of first party
 ϕ – internal propensity for response selection
 N(ϕ) – distribution of individuals
 N0 – size of society
 ψ - shift of incentives (defined social environment) toward
the first party support
 b1 и b2 - propaganda first and second party, respectively
 
 
   
 
0 1 2
0
2 , 0
t
d
A C N d N b b a X N d
dt  

     
 
 
  
       
  
  
 
   
 t
X t N d

 


 

20. Effect of Social Polarization
20
  0
0,
,
4
0,
d h
N
N d h d h
h
d h

 

 


    

 
Sociological sense:
•society is divided into two groups, each of which has
a propensity to support "their" party
•d parameter characterizes the degree of social
polarization (how radical is each group)
•h parameter characterizes the degree of
fragmentation of each group

21. Equilibrium states
 Consider a case
 There are from 1 to 5 equilibrium states
(depending on initial conditions)
 Stable equilibrium states:
21
o 1 – party 2 wins
o 3 – draw
o 5 – party 1 wins
1 2b b

22. Homogeneous group
22
Case of a slowly polarizing society
 
 0 2
0
2
Q d h hP
h Q

 


 
 0 2
0
2
Q d h hP
h Q

 


 1 2A b b
P
a
 

0A CN
Q
a



23. Medium group
23
 
 0 2
0
2
Q d h hP
h Q

 


 1 2A b b
P
a
 

0A CN
Q
a

  
 0 2
0
2
Q d h hP
h Q

 


Case of a slowly polarizing society

24. Heterogeneous group
24
 1 2A b b
P
a
 

0A CN
Q
a


Case of a slowly polarizing society

25. Thank you for your attention!
Proncheva Olga
Olga.proncheva@gmail.com
25