Матрично-графічне моделювання соціальної мережі: ергодичні властивості
We propose mathematical tools for social network simulation to obtain sufficient conditions for network ergodicity, defined as the existence of a steady state as time approaches infinity. The proposed model is linear; the network elements form a two-dimensional array (matrix), where each entry repre...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2025
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System research and information technologies| _version_ | 1866303055890219009 |
|---|---|
| author | Spectorsky, Igor Statkevych, Vitalii Stus, Oleksandr |
| author_facet | Spectorsky, Igor Statkevych, Vitalii Stus, Oleksandr |
| author_sort | Spectorsky, Igor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
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| datestamp_date | 2026-02-02T20:49:24Z |
| description | We propose mathematical tools for social network simulation to obtain sufficient conditions for network ergodicity, defined as the existence of a steady state as time approaches infinity. The proposed model is linear; the network elements form a two-dimensional array (matrix), where each entry represents the state of an element at a specific time.
An impact operator, structured as a four-dimensional array, defines the interactions between elements. This operator is also presented as a directed graph where vertices correspond to network elements, and arcs represent the impact of one element on another. The model incorporates boundary elements that influence the internal states of the network.
Sufficient conditions for network ergodicity are derived from the connectivity properties of the impact graph, which must contain paths between all pairs of vertices and loops for all vertices. These conditions ensure that the operator's spectrum (with the possible exception of the value 1) is located inside the open unit disk. We prove that 1 is an eigenvalue if and only if the boundary is isolated. These spectral properties guarantee that a steady state exists and can be found using an iterative procedure with linear (geometric) convergence. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.4.03 |
| first_indexed | 2026-02-08T08:06:10Z |
| format | Article |
| fulltext |
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus, 2025
38 ISSN 1681–6048 System Research & Information Technologies, 2025, № 4
UDC 519.87, 519.17
DOI: 10.20535/SRIT.2308-8893.2025.4.03
MATRIX-GRAPHIC SIMULATION OF SOCIAL NETWORK:
ERGODIC PROPERTIES
I.Ya. SPECTORSKY, V.M. STATKEVYCH, O.V. STUS
Abstract. We propose mathematical tools for a social network simulation in order to
obtain some sufficient conditions of the network’s ergodicity, that is, the existence
of a steady state as t . The proposed model is linear; the elements of the net-
work form a two-dimensional array (i.e., a matrix) },,2,1{},,2,1{ mn ,
where ]1,0[)(, tA ji is the state of the element ),( ji , 0t is time. An impact
operator T is a four-dimensional array; the element 0,,, lkjiT denotes the impact
of the element ),( ji on the element ),( lk :
n
k
m
l
lklkjiji ATTA
1 1
,,,,,)( . The impact
operator T is also presented as a directed graph TG , whose vertices correspond to
elements ),( ji : a directed edge (an arc) leads from the vertex ),( lk to the
vertex ),( ji if and only if 0,,, lkjiT , and this edge is labelled by the number
lkjiT ,,, . A bound B is introduced in such a way that 0,,, lkjiT for
),( lk , Bji ),( . The state )1( tA at time 1t is defined by the state )(tA
at the current time 0t via equation )()1( tTAtA , where matrix of di-
mension mn defines the states of bound elements Bji ),( ; 0, ji for inter-
nal elements Bji \),( . Some sufficient conditions for the network’s ergodicity
are given in the form of connectivity properties of the impact graph TG . This graph
must contain paths between all pairs of vertices and loops for all vertices. Suggested
conditions provide the spectrum of T (with the possible exception of 1 ) to be
located inside the open unit disk; we prove that 1 is an eigenvalue of T if and
only if the bound B is isolated (no bound element impacts any internal one).
These spectral properties of T provide that the steady state exists and can be found
by the iterative procedure )()1( tTAtA with the given )0(A ; the iterative
process converges linearly (geometrically).
Keywords: social system, simulation, ergodicity, eigenvalue, Jordan normal form.
INTRODUCTION
Social network analysis is currently one of the most important methods for scien-
tific investigation in sociology, social psychology and other areas (see, e. g., [1; 2]).
A social network is defined by the interaction of network elements, or, in other
words, by impact of network elements on other ones.
Various toolkits can be used to simulate a social network. For example, in
[3; 4] graph theory methods are used to visualize network elements interaction, in
[1] matrix analysis gives a more convenient way to analyze network elements in-
teraction.
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 39
Usually, a social network is not a static structure, i.e. the state of network el-
ements changes over time. The social network’ behaviour is currently being in-
tensively investigated (see, e.g., [3; 5; 6]), and steady states are of a special
interest (see, e.g., [3]).
The purpose of this work is to obtain some sufficient conditions for social
network ergodicity (independence of a network’s behaviour from initial condi-
tions in extremely distant time), using matrix and graph methods of social net-
work representation.
REPRESENTATION OF A SOCIAL NETWORK AND ITS DYNAMICS
Suppose that the social network (hereinafter referred to as the network) contains
nm elements, arranged in n rows and m columns ( mn , ), i.e. position of
each element is defined by a pair ),( ji , where },,2,1{},,2,1{ mn
}1 ,1:),{( mjniji is the set (area) of coordinates of network elements.
The current state of the element ),( ji is defined by a number ]1,0[)(, tA ji ,
which can be particularly treated as an attitude of the element ),( ji towards
some problem arisen in the network ( 0 means completely negative attitude, 1
means completely positive one); hereinafter 0t denotes discrete time
( }0{0 ). Therefore, the current state of the network can be represented as
a matrix ]1,0[)( mnMtA of dimension mn with elements ]1,0[)(, tA ji
( ),( ji ).
Let B be the bound of the coordinate area . The states of boundary
elements are described by the boundary condition matrix of dimension mn ,
assuming 0, ji for all Bji \),( . Elements Bji \),( that do not belong
to B are called internal. Hereinafter, assume that B (excluding a trivial case
B ).
In order to simulate network dynamics, introduce a linear impact operator
][][: mnmn MMT , where ][mnM denotes a linear space of mn
matrices with elements from . The operator T is considered as 4-dimensional
mnmn array with elements from , its action on a matrix ][mnMX is
defined point-wise:
n
k
m
l
lklkjiji XTTX
1 1
,,,,,)( , (1)
the element lkjiT ,,, ( },,2,1{},,2,1{},,2,1{},,2,1{),,,( mnmnlkji )
defines impact of the state of the network element ),( lk on the state of the network
element ),( ji . The following conditions are assumed for normalization reasons:
0:},,2,1{},,2,1{},,2,1{},,2,1{),,,( ,,, lkjiTmnmnlkji ; (2)
n
k
m
l
lkjiTBji
1 1
,,, 1:\),( . (3)
Since the states of elements on the bound B are defined by matrix , as-
sume that
0:),( ),( ,,, lkjiTlkBji . (4)
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 40
Given relation (1), this means that 0)( , jiTX for all Bji ),( .
Assume that the network state )1( tA at time 1t ( 0t ) is defined by the
network state )(tA at time 0t according to the equation
)()1( tTAtA , (5)
the network state )0(A at the initial time 0t is assumed to be defined by the
initial condition matrix )0(A of dimension mn with elements jiA ,))0((
]1,0[)0(, jiA ( ),( ji ). Note that the summand )(tTA in relation (5) defines
the states of internal elements Bji \),( , the summand defines the states of
elements Bji ),( on the bound B .
The correspondence of the initial condition )0(A with the boundary condi-
tion (on the bound B ) requires assumption
jijiA ,,))0(( for all Bji ),( . (6)
Lemma 1. Let X be an arbitrary mn matrix with elements from ]1,0[
( ]1,0[mnMX ). Then TX ]1,0[mnM , i.e. the matrix set ]1,0[mnM is closed
with respect to the operatorT .
Proof. Equation (1) immediately implies nonnegativity of elements
jiTX ,)( ( ),( ji ). For upper bounding jiTX ,)( apply relation (1) giv-
en condition (3):
n
k
m
l
lkji
n
k
m
l
lklkjiji TXTTX
1 1
,,,
1 1
,,,,, 1)( ,
which proves the statement of the lemma. □
The operator T can be visualized as a labelled directed impact
graph TG with vertices corresponding to elements ),( ji : a directed edge
leads from the vertex ),( lk to the vertex ),( ji if and only if 0,,, lkjiT ,
this edge is labelled by the number lkjiT ,,, . Note that the operator T in fact
defines adjacency matrix TG , deployed in 4-dimensional array for convenience.
Example 1. Consider the network on the coordinate area },,2,1{ n
},,2,1{ m with the bound }1 ,1:),(),1,(),,(),,1{( mjnimiijnjB , the
impact operator T simulates equal impact on each internal element by its 4
neighbours:
jijiT ,,, for 12 ni and 12 mj ;
)1(25.01,,,1,,,,1,,,1,, jijijijijijijiji TTTT
for 12 ni and 12 mj ;
0,,, lkjiT for 2 ljki ; 0,,, lkjiT for },1{ ni or },1{ mj .
Here a constant )1,0( defines the impact value on the element by its 4 neigh-
bours and by itself. The corresponding impact graph TG for a case of 5n ,
6m , 6.0 is depicted in Fig. 1, vertices of boundary elements are denoted
by ( ) .
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 41
Remark 1. Relations (2) and (3) in the case of B define linear opera-
tors with stochastic (Markov) matrices (see, e.g., [7; 8]), and some its properties
can be extended on a more general matrix types (see, e.g., [9]).
SPECTRUM OF THE OPERATOR T
It is well known that, although ][][: mnmn MMT is the linear operator on
the linear space ][mnM (i.e. on the field of real numbers), eigenvalues and ei-
genvectors of the operator T in a general case are complex. Hereinafter, in the
context of the operator T , usual notion ‘eigenvector’ is used (despite that the ar-
gument of the operator T is matrix ][mnMX ).
Lemma 2. Let be an eigenvalue of the operator T , i.e. ETE for
some nonzero ][mnME . Then 1|| .
Proof. Let ji
ji
ji EE ,
),(
, max
00
, i.e. jiE , reaches its maximal value on
),( 00 ji (obviously, this maximum can be reached at several points). Then, simi-
larly to the proof of Lemma 1, one can obtain:
000000000000 ,
1 1
,,,,
1 1
,,,,
1 1
,,,,,)( ji
n
k
m
l
lkjiji
n
k
m
l
lklkji
n
k
m
l
lklkjiji ETEETETTE
.
However, on the other hand,
0000 ,,)( jiji ETE , thus
0000 ,, jiji EE .
Therefore, since 0max ,
),(
, 00
ji
ji
ji EE , it yields the desired estimate 1 . □
Theorem 1. Let be an eigenvalue of the operator T that belongs to the
unit circle ( 1|| ), and let the impact graph TG satisfy the following conditions:
for any internal elements Bjiji \),(),,( 2211 there exists a directed
path from the vertex ),( 11 ji to ),( 22 ji ;
for any internal element Bji \),( there exists a ‘loop’ (an edge lead-
ing from the vertex ),( ji to the same vertex ),( ji ).
0.1
0.1
0.1
0.1
0.1 0.10.1
0.1
0.1
0.6
0.1
0.10.6
0.1
0.6
0.1
0.6
0.1
0.1 0.10.1
0.10.1 0.10.6 0.6 0.6
0.60.1 0.1
0.1
0.1
0.10.1
0.1
0.1
0.1
0.1 0.10.1
0.10.1 0.10.6 0.6 0.6
0.60.1 0.1
0.1
0.1
0.1
0.1 0.10.10.1
0.1 0.1 0.1
0.1
Fig. 1
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 42
Then 1 , and the corresponding eigenspace is generated by the eigenvec-
tor ][\1 mn
B M
such that
.),(,0
,\),(,1
,
\1
Bji
Bji
ji
B
Proof. Let ][mnME be some eigenvector that corresponds to the eigen-
value on the unit circle ( 1 ). Note that 0, jiE for all Bji ),( due to
relation ETE .
Firstly prove that jiE , is a constant that does not depend on Bji \),( .
Let ji
ji
ji EEc ,
),(
, max
00
, i.e. jiE , reaches its maximal value on ),( 00 ji .
Since any eigenvector is nonzero by definition, constant ji
ji
ji EEc ,
),(
, max
00
is positive, thus Bji \),( 00 . Using relation (1), one can obtain:
n
k
m
l
lklkjijiji ETTEE
1 1
,,,,,, 000000
)( . (7)
Given normalizing conditions (2) and (3), equality (7) implies
000000000000 ,
1 1
,,,,
1 1
,,,,,, ji
n
k
m
l
lkjiji
n
k
m
l
lklkjijiji ETEETEE
.
Therefore, the triangle inequality
n
k
m
l
lklkji
n
k
m
l
lklkjiji ETETE
1 1
,,,,
1 1
,,,,, 000000
turns into equality:
n
k
m
l
lklkji
n
k
m
l
lklkjiji ETETE
1 1
,,,,
1 1
,,,,, 000000
, (8)
which is possible if and only if
00 ,, jilk EE for all Blk \),( such that
0,,, 00
lkjiT . Further, recall that for nonzero 21, zz the equality
2121 zzzz holds if and only if their arguments are equal: 21 argarg zz .
So, relation (7) implies that
00 ,, jilk EE for all ),(),( 00
1 jilk E , where the set
}0:),{(),( ,,,00
1
00
lkjiTlkjiE contains all elements Blk \),( that directly
impact the element ),( 00 ji (there exists an edge from ),( lk to ),( 00 ji ). There-
fore, since ),(),( 00
1
00 jiji E (by the theorem conditions, there is the loop for the
element Bji \),( 00 ), it is easy to see that 1 , and
00 ,, jilk EE for all
),(),( 00
1 jilk E . Repeating these considerations for each ),()
~
,
~
( 1 lklk E ,
),(),( 00
1 jilk E , one can obtain that ),()
~
,
~
( 00 jilk E , where the set ),( 00 jiE
contains all elements Blk \),( that directly or indirectly impact the element
),( 00 ji (there exists a path from ),( lk to ),( 00 ji ). Finally, the theorem condi-
tions provide that for any Bjiji \),(),,( 2211 there exists a directed path from
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 43
),( 11 ji to ),( 22 ji , thus Bji \),( 00 E , B
jiEE \
, 1
00
, which completes the
proof of the theorem.□
Corollary. Under the conditions of Theorem 1, the eigenvalue 1 of the
operator T is a simple one (a single root of its characteristic polynomial).
Proof. Assume that the eigenvalue 1 is not simple, i.e. it is a root of the
characteristic polynomial of multiplicity 2 or more. Then, by Theorem 1, the
eigenvalue 1 corresponds to a one-dimensional eigenspace, and for 1
there exists (see, e.g., [10, 11]) a generalized eigenvector ][\1
~
mn
B M
:
BBBBBT \\\\\ 11
~
11
~
1
~
. So, for an arbitrary t one can
obtain: BBBt tT \\\ 11
~
1
~
, which contradicts to Lemma 1 for sufficiently
large t . This contradiction proves that the eigenvalue 1 is indeed simple. □
Theorem 1 proves that, under the given conditions, the unit circle can con-
tain at most one eigenvalue of the operator T , namely 1 . However, the theo-
rem does not exclude the case when the unit circle does not contain any eigen-
value of the operator T (by Lemma 2, in this case all eigenvalues of T are
located inside the open unit disk). It is easy to derive from the proof of Theorem 1
that 1 is an eigenvalue of the operator T if and only if 1
\),(
,,,
Blk
lkjiT for all
Bji \),( , which, given conditions (2) and (3), is equivalent to the following
condition:
0:),( \),( ,,, lkjiTBlkBji . (9)
Obviously, condition (9) means that the network bound is isolated from the
rest of the network: no element Blk ),( can impact any element Bji \),( . If
condition (9) does not hold, there is at least one element Blk ),( that impacts
some element Bji \),( .
To simplify further analysis, consider a block structure for matrices on with
respect to the partition BB )\( . For an arbitrary matrix ][mnMA con-
sider a block BA \ of elements from B\ . Although the rectangle structure for
area B\ may be distorted (see, e.g., Fig. 3), one can treat ][\ BM (as well as
][\ BM if necessary) as a linear space of real (complex) ‘vectors’, whose en-
tries are numbered by coordinate pairs Bji \),( . So, for the network in Fig. 3,
one can obtain the following block ][\\ BB MA (vertices of boundary
elements are denoted by « »):
4,33,32,31,3
4,21,2
4,13,12,11,1
\
AAAA
AA
AAAA
A B
Similarly, (provided B ) consider the linear space ][BM and the block
][BB MA .
Further, define ][][: \\\,\ BBBB MMT as a linear operator on the
block space ][\ BM :
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 44
.)()()(
;)(:\),( \),(
\),(
,\,,,\,\,\\,\
,,,,,,\,\
Blk
lkBlkjiBBjiBBB
lkjilkjiBB
ATAT
TTBlkBji
Similarly (provided B ) one can define a linear operator :,\ BBT
][][ \ BB MM :
.)()()(
;)(:),( \),(
),(
,,,,,\,,\
,,,,,,,\
Blk
lkBlkjiBBjiBBB
lkjilkjiBB
ATAT
TTBlkBji
Note that one can in a similar way define linear operators :\, BBT
][][\ BB MM and ][][:, BBBB MMT ; however, according to
definition of the bound B (condition (4)), these linear operators are zero.
Now equation (5) can be rewritten as a system:
.)1(
;)()()1( \,\\\,\\
BB
BBBBBBBB
tA
tATtATtA
Note that, by definition of the network bound, 0, ji in equation (5) for
Bji \),( , that is 0\ B , so (for 0t )
.)1(
);()()1( ,\\\,\\
BB
BBBBBBB
tA
tATtATtA
Further, the second equation implies BB tA )( for all 1t , and for 0t
the initial condition corresponds to the boundary one by relation (6), so the ob-
tained system can be written for any 1t as
,)1(
;)()1( ,\\\,\\
BB
BBBBBBB
tA
TtATtA
(10)
where )0(BA is defined by the initial conditions.
Consider system (10) in two cases: when condition (9) holds (isolated
bound) and when it does not hold (non-isolated bound).
A. Suppose that condition (9) holds. Along with condition (4) it means that in
the impact graph TG all boundary elements are isolated (see, e.g., Fig. 2 and 3),
vertices of boundary elements are denoted by « ».
0.1
0.1 0.1 0.1
0.8
0.1
0.8
0.1
0.8
0.1
0.8
0.8
0.8 0.1
0.1
0.1
0.1 0.1 0.1
0.1 0.10.8
0.8 0.8
0.8 0.1 0.1
0.1
0.1
0.1
0.1
0.1 0.10.1
0.8
0.1
0.7
0.1
0.7
0.1
0.8
0.1 0.10.1
0.1 0.1 0.7 0.6 0.6
0.70.1 0.1
0.1
0.1
0.1 0.1
0.1
0.1
0.10.1
0.1 0.1 0.8 0.7 0.7
0.80.1 0.1
0.1
0.1
0.1
0.1 0.1 0.1 0.1
0.1
Fig. 2 Fig. 3
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 45
In the case of isolated bound, the structure of the operator T is similar to a
block-diagonal matrix: 0,,, lkjiT if either Bji \),( and Blk ),( , or
Bji ),( and Blk \),( . The operator BBT ,\ is zero in the case of isolated
bound, thus system (10) for 0t takes a form
.)1(
);()1( \\,\\
BB
BBBB
tA
tATtA
(11)
Given normalizing conditions (2) and (3), the linear operator BBT \,\ can be
treated as an operator with stochastic matrix, which has the eigenvalue 1 with the
corresponding eigenvector ][\\1 BB M : 1)( ,\1 jiB for all Bji \),( .
B. Suppose that condition (9) does not hold. It means that in the impact
graph TG at least one boundary element is not isolated (see Fig. 1 in Example 1).
According to condition (4), 0,,, lkjiT for any Bji ),( and ),( lk , so
for the operator T it is worth considering a block structure similar to one consid-
ered in case A; in the case of non-isolated bound this structure of course is not
block-diagonal.
In the case of non-isolated bound equation (5) can also be written in a general
form like system (10) (but not like system (11), since the operator BBT ,\ is nonzero).
Remark 2. For irreducible matrix Perron–Frobenius theorem is well known
(see, e.g., [10; 12]). This theorem is similar to Theorem 1, but does not require
positivity for the diagonal elements (existence of loops on the corresponding im-
pact graph), which makes it possible for several eigenvalues to have the maximal
absolute value (in the context of Theorem 1 it means that the unit circle can con-
tain several eigenvalues of the operator T ). Perron–Frobenius theorem is also
applied for the Analytic Hierarchy Process, developed by T. Saaty, particularly in
practice problems of economic, industrial, administrative and psychological
kinds, in problems of conflict analysis and in other areas [12].
SUFFICIENT CONDITIONS FOR THE NETWORK ERGODICITY
Jordan normal form of matrix: existence of limit t
t
Q
lim
To analyze the network’s behaviour as t , it is essentially important to
know the spectral properties of the impact operator T , and these properties can
be effectively explored via Jordan normal form of the corresponding matrix (see,
e.g., [10; 11]). For referring convenience, consider some statements related to Jor-
dan normal form, which are known or can be easily proven.
It is known (see, e.g., [10; 11]) that for any matrix ][NNMQ there ex-
ists the nondegenerate transition matrix ][NNMV such that 1VJVQ ,
where ][NNMJ is the following block-diagonal matrix:
pJ
J
J
J
00
00
00
2
1
; [
0000
1000
0100
0010
0001
ss NN
s
s
s
s
s
s MJ
],
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 46
s , ps 1 .
Matrix J is called Jordan, each matrix sJ ( ps 1 ) is called a Jordan
block of dimension sN corresponding to an eigenvalue s ; by this construc-
tion, NNNN p 21 . Note (see, e.g., [10; 11]) that the columns of the
transition matrix V are eigenvectors and generalized eigenvectors of matrix Q ,
and they form so called Jordan basis in N .
To compute Jordan matrix, it is convenient to use the following well–known
formula:
t
p
t
t
t
J
J
J
J
)(
)(
)(
00
00
00
2
1
;
t
st
t
st
t
st
t
st
t
st
s
t
t
t
st
t
st
C
CC
CC
CCC
t
sJ
0
110
110
0110
000
00
0
0
0
)(
( ps 1 , 1t ). (12)
(see, e.g., [10] for approaches to defining polynomial and even analytic functions
of a matrix).
Given the transition equation 1 VVJQ tt , formula (12) provides conven-
ient tools to analyze tQ for different t , particularly as t .
Hereinafter in the space N the norm j
Nj
vv
1
max ( Nv ) is used, in
the space ][NMM the corresponding matrix norm is used:
N
k
ki
Nivv
RRvR
n 1
,
11 :
maxsup
, (13)
the norm of a linear operator is assumed to be defined by the operator norm of the
corresponding matrix by formula (13).
The convergence RRt
t
lim of the matrix sequence ][NMt MR
( t ) to matrix ][NMMR with respect to norm (13) is equivalent to the
entrywise convergence: jijit
t
RR ,,lim
for all Mi 1 , Nj 1 ; the con-
vergence of the sequence of the linear operators is treated as the convergence of
the sequence of corresponding matrices.
Lemma 3. Let sJ be a Jordan block corresponding to an eigenvalue s
such that 1s . Then:
ss NN
t
s
t
J
0)(lim , where
ss NN 0 is a zero matrix of dimension
ss NN ;
The convergence
ss NN
t
s
t
J
0)(lim is linear, i.e. there exist constants
0sC , )1,0[sq and a number st such that
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 47
t
sss qCJ )( for all stt . (14)
Proof. It is sufficient to prove that each entry of the matrix sJ converges to
zero: 00
!
t
it
si
tit
s
i
t
i
C , which implies the required convergence
0lim
it
s
i
t
t
C for all sNi 0 (assuming 0i
tC for ti ). To prove esti-
mate (14), one can choose sufficiently large st so that for
it
s
i
tCtH
)( the
following estimate holds: 1
1
)(
)1(
s
N
t
t
tH
tH
s
s
s
s
s
sq . □
Corollary. Let the matrix Q have the simple eigenvalue 1
0
s with an ei-
genvector N
sv
0
, and let any other eigenvalue s of Q belong to the open
unit disk ( 1s for 0ss ). Then:
There is the convergence ][ˆlim NN
t
t
MQQ
with
0
ˆ
scvvQ for
any vector Nv , where the constant c is defined by the vector Nv ;
The convergence QQt
t
ˆlim
is linear, i.e. there exist constants 00 C ,
)1,0[0 q and a number 0t such that
tt qCQQ 00
ˆ
for all 0tt . (15)
Proof. The convergence ][ˆlim NN
t
t
MQQ
is implied by formula
(12) and Lemma 3; moreover, Lemma 3 and the condition of the corollary yield
equality 1ˆˆ VJVQ , where
000
010
000
ˆ
J
(the only nonzero element of the matrix Ĵ corresponds to the block
)1(
00
t
ss JJ for all 1t ). So, 1ˆrank J , whence, due to nondegeneracy of the
transition matrix V , 1ˆrankˆrank JQ . Therefore, the matrix Q̂ defines the
linear mapping with one-dimensional image generated by the vector
0sv , so
equality
0
ˆ
scvvQ holds for some constant c . Finally,
JJVVVJJVQQ ttt ˆ)ˆ(ˆ 11
))((max)(max
00
11 t
ss
ss
t
s
ss
qCVVJVV
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 48
for all s
ss
ttt
0
max0
, that proves estimate (15) and thereby completes the proof
of the corollary. □
Sufficient conditions for the network ergodicity: isolated bound
In the case of isolated bound, the network’s behaviour is completely described by
system (11).
Theorem 2. Let the impact graph TG satisfy the following conditions:
for any internal elements Bjiji \),(),,( 2211 there exists a directed
path from the vertex ),( 11 ji to ),( 22 ji ;
for any internal element Bji \),( there is a loop (an edge leading
from the vertex ),( ji to the same vertex ),( ji );
all boundary elements Bji ),( are isolated vertices.
Then:
BtB ctA \\ 1)( , where the constant )1,0[c is defined by the
block ]1,0[)0( \\ BB MA representing the states of internal elements at the ini-
tial time 0t , ]1,0[\\1 BB M , 1
,
\1
ji
B for all Bji \),( ;
The convergence BtB ctA \\ 1)( is linear, i.e. there exist con-
stants 00 C , )1,0[0 q and a number 0t such that
t
BB qCctA )()( 00\\ 1
for all 0tt .
Proof. The statement of the theorem is implied by Theorem 1 and corollary
of Lemma 3. □
Theorem 2 states that (under the given conditions) for the network with
isolated bound there is a set of steady states Bc \1 ( )1,0[c ), where B\1
under the given conditions (see also Theorem 1) is an eigenvector of the operator
BBT \,\ corresponding to the eigenvalue 1 . Recall that the linear operator
BBT \,\ under the given conditions can be treated as an operator with stochastic
matrix, which always has the eigenvalue 1 with the eigenvector
][\\1 BB M (since the initial and boundary conditions are located inside the
line segment ]1,0[ , one can choose ]1,0[\\1 BB M ).
Remark 3. Under the fixed initial conditions ]1,0[)0( MA (or equiva-
lently, ]1,0[)0( \\ BB MA ), computation of the steady state Bc \1 (in fact, it
means computation of the constant )1,0[c ) can be reduced through decomposi-
tion of )0(\BA by the Jordan basis of the operator BBT \,\ . However, computa-
tion of the Jordan basis for real-world networks can become significantly more
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 49
complicated due to the large size of the set B\ and (consequently) the dimen-
sion of the space ][\ BM . Therefore, practically reasonable approach is to ap-
proximate (numerically) the eigenvector Bc \1 using the iterative procedure
described by system (11).
Obviously, in the case of isolated bound the block of internal elements of the
network can be considered as a network with empty bound ( B ), and this
network’s behaviour is described by the first equation of system (11).
Example 2. Consider the network on the coordinate area },,2,1{ n
},,2,1{ m with the bound B , the impact operator T simulates equal im-
pact on internal elements by its 4 neighbours:
jijiT ,,, for all ni 1 and mj 1 ;
)1(25.0,1,, jijiT , if 12 ni and mj 1 ;
)1(25.01,,, jijiT , if ni 1 and 12 mj ;
)1(5.0,1,,,2,,1 jnjnjj TT , if mj 1 ;
)1(5.01,,,2,,1, mimiii TT , if ni 1 ;
0,,, lkjiT , if 2 ljki ,
where )1,0( is a fixed constant. The corresponding impact graph TG for the
case of 3n , 4m , 6.0 is depicted in Fig. 4.
Rewrite equation for transition of state )(tA to the next time moment:
jijiji tAtAtA ,1,, ))((()1(25.0))(())1((
)))(())(())(( 1,1,,1 jijiji tAtAtA for 12 ni and 12 mj ;
1,1,1,1 ))((()1(25.0))(())1(( jjj tAtAtA
jj tAtA ,21,1 ))()(1(5.0)))(( for 12 mj ;
1,,, ))((()1(25.0))(())1(( jnjnjn tAtAtA
0.1
0.2
0.10.1
0.6
0.1
0.6
0.1
0.6
0.2
0.6
0.2
0.10.1
0.1 0.10.6 0.6 0.6
0.60.2 0.2
0.2
0.2
0.10.1
0.2
0.1
0.2
0.10.1
0.1 0.10.6 0.6 0.6
0.60.1 0.1
0.1
0.1
0.2
0.2 0.2
0.2
0.2
Fig. 4
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 50
jnjn tAtA ,11, ))()(1(5.0)))(( for 12 mj ;
)))(())((()1(25.0))(())1(( 1,11,11,1, iiii tAtAtAtA
2,))()(1(5.0 itA for 12 ni ; nini tAtA ,, ))(())1((
1,,1,1 ))(()1(5.0)))(())((()1(25.0 ninini tAtAtA for 12 ni ;
)))(())((()1(5.0))(())1(( 1,22,11,11,1 tAtAtAtA ;
)))(())((()1(5.0))(())1(( 1,12,1,1, nnnn tAtAtAtA ;
)))(())((()1(5.0))(())1(( ,21,1,1,1 mmmm tAtAtAtA ;
)))(())((()1(5.0))(())1(( ,11,,, mnmnmnmn tAtAtAtA .
Considering coefficients of jitA ,))(( for the different pairs ),( ji , one
can construct a function ]1,0[]1,0[: MSw as a ‘weighted’ sum of jitA ,))(( ,
which is a constant value for all 0t :
1
2
,
1
2
1,
1
2
,
1
2
,1
1
2
1
2
, 5.0)(
m
i
mi
m
i
i
m
j
jn
m
j
j
n
i
m
j
jiw XXXXXXS
)(25.0 ,,11,1,1 mnmn XXXX for ]1,0[MX .
For )1( tAX one can obtain:
1
2
1
2
,))1(())1((
n
i
m
j
jiw tAtAS
1
2
,1,
1
2
,,1 )))1(())1((()))1(())1(((5.0
n
i
mii
m
j
jnj tAtAtAtA
)))1(())1(())1(())1(((25.0 ,,11,1,1 mnmn tAtAtAtA .
Simplify separately three summands in the right-hand side of the obtained
relation:
1
2
1
2
,))1((
n
i
m
j
jitA
1
2
1
2
1,1,,1,1, ))))(())(())(())()((1(25.0))(((
n
i
m
j
jijijijiji tAtAtAtAtA
1
2
1
2
,))((
n
i
m
j
jitA
1
2
1
2
1,1,,1,1 )))(())(())(())((()1(25.0
n
i
m
j
jijijiji tAtAtAtA
)1(25.0))((
1
2
1
2
,
n
i
m
j
jitA
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 51
1
2 3
,
1
2
2
1
,
3
1
2
,
2
1
1
2
, ))(())(())(())((
n
i
m
j
ji
n
i
m
j
ji
n
i
m
j
ji
n
i
m
j
ji tAtAtAtA
1
2
1
2
,
1
2
1
2
,
1
2
1
2
, ))(())(()1(5.0))((
n
i
m
j
ji
n
i
m
j
ji
n
i
m
j
ji tAtAtA
1
2
,
1
2
,2
1
2
,1
1
2
,1 ))(())(())(())(()1(25.0
m
j
jn
m
j
j
m
j
jn
m
j
j tAtAtAtA
1
2
,
1
2
2,
1
2
1,
1
2
1, ))(())(())(())(()1(25.0
n
i
mi
n
i
i
n
i
mi
n
i
i tAtAtAtA
1
2
,2,
1
2
,1,1
1
2
1
2
, )))(())((()))(())((()1(25.0))((
m
j
jjn
m
j
jnj
n
i
m
j
ji tAtAtAtAtA
;)))(())((()))(())((()1(25.0
1
2
2,,
1
2
1,1,
n
i
imi
n
i
mii tAtAtAtA
1
2
,
1
2
1,
1
2
,
1
2
,1 ))1(())1(())1(())1((5.0
n
i
mi
n
i
i
m
j
jn
m
j
j tAtAtAtA
1
2
,1,
1
2
,,1 )))(())((()))(())(((5.0
n
i
mii
m
j
jnj tAtAtAtA
1
2
1,1,1,11,1 )))(())(())(())((()1(125.0
m
j
jnjnjj tAtAtAtA
1
2
,1,11,11,1 ))))(())((())(())((()1(125.0
n
i
mimiii tAtAtAtA
1
2
1,2,
1
2
,1,2 )))(())((()))(())((()1(25.0
n
i
mii
m
j
jnj tAtAtAtA
1
2
,1,
1
2
,,1 )))(())((()))(())(((5.0
n
i
mii
m
j
jnj tAtAtAtA
1
2
,1,
1
2
,,1 )))(())((()))(())((()1(25.0
n
i
mii
m
j
jnj tAtAtAtA
)))(())(())(())()((1(125.0 ,12,11,11,1 mm tAtAtAtA
)))(())(())(())()((1(125.0 ,2,1,1, mnnmnn tAtAtAtA
)))(())(())(())()((1(125.0 1,1,21,11,1 nn tAtAtAtA
)))(())(())(())()((1(125.0 ,,2,1,1 mnmmnm tAtAtAtA
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 52
1
2
1,2,
1
2
,1,2 )))(())((()))(())((()1(25.0
n
i
mii
m
j
jnj tAtAtAtA
1
2
1
2
,1,,,1 ))())((()))(())(((25.0
m
j
n
i
miijnj tAtAtAtA
1
2
,1,
1
2
,,1 )))(())((()))(())(((25.0
n
i
mii
m
j
jnj tAtAtAtA
1
2
1,2,
1
2
,1,2 )))(())((()))(())((()1(25.0
n
i
mii
m
j
jnj tAtAtAtA
)))(())(())(())()((1(125.0 ,12,11,11,1 mm tAtAtAtA
)))(())(())(())()((1(125.0 ,2,1,1, mnnmnn tAtAtAtA
)))(())(())(())()((1(125.0 1,1,21,11,1 nn tAtAtAtA
;)))(())(())(())()((1(125.0 ,,2,1,1 mnmmnm tAtAtAtA
)))1(())1(())1(())1(((25.0 ,,11,1,1 mnmn tAtAtAtA
)))(())(())(())(((25.0 ,,11,1,1 mnmn tAtAtAtA
)))(())(())(())()((1(125.0 2,1,11,22,1 nn tAtAtAtA
.)))(())(())(())()((1(125.0 1,,11,1,2 mnmnmm tAtAtAtA
Collecting these summands together, one can obtain:
))1(( tASw
1
2
1
2
,1,,,1
1
2
1
2
, )))(())((()))(())(((5.0))((
m
j
n
i
miijnj
n
i
m
j
ji tAtAtAtAtA
)),(()))(())(())(())(((25.0 ,,11,1,1 tAStAtAtAtA wmnmn
whence
BwBw
t
ww SccStASAS \\ 11)(lim))0((
),1)(1()425.0)22(25.0)2)(2(( mncmnmnc
i.e.
)1)(1(
))0((
с
mn
ASw . Particularly, for 20n , 10m , 8.0 , jiA ,))0((
,otherwise,0
,1,1 ji
one can obtain 00146.0с
)110)(120(
25.0 . All data are written
with precision up to 0.00001 which corresponds to relative error 01.0
00146.0
00001.0 ,
the convergence by the iterative procedure (10) with such precision is achieved
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 53
approximately for 5000t . It is interesting to note that for this impact operator
T the steady state Bc \1 does not depend on )1,0( ; however, the value
)1,0( affects on the convergence rate (for 6.0 precision of 0.00001 is
achieved approximately for 2500t ).
Sufficient conditions for the network ergodicity: non-isolated bound
In the case of non-isolated bound, the network’s behaviour is completely de-
scribed by system (10).
Theorem 3. Let the impact graph TG satisfy the following conditions:
for any internal elements Bjiji \),(),,( 2211 there exists a directed
path from the vertex ),( 11 ji to ),( 22 ji ;
for any internal element Bji \),( there is a loop (an edge leading
from the vertex ),( ji to the same vertex ),( ji );
at least one boundary element Bji ),( is not an isolated vertex.
Then there exists the unique vector ][ˆ
\\ BB MA such that:
BtB AtA \\
ˆ)( ;
the convergence BtB AtA \\
ˆ)( is linear, i.e. there exist con-
stants 00 C , )1,0[0 q and a number 0t such that
tBB qCAtA 00\\
ˆ)(
for all 0tt .
Proof. System (10) yields the explicit form for )(\ tA B ( 1t ):
1
0
,\\,\\\,\\ )()0()()(
t
s
BBB
s
BBB
t
BBB TTATtA (16)
Due to Theorem 1, all eigenvalues of the operator BBT \,\ are located in-
side the unit disk, so by virtue of Lemma 3 there exist constants 0C , )1,0[q
and a number 0t such that t
BB qCT
\,\ for all 0tt , which yields the
required convergence:
0
,\\,\\\
ˆ)(
t
BBB
t
BBBtB TTAtA .
Finally,
q
q
BBB
ts
BBB
s
BBBB
t
TCTTAtA
1,\
1
,\\,\\\
1
)(ˆ)( ,
so the convergence BtB AtA \\
ˆ)( is indeed linear. □
Theorem 3 proves that, under the given conditions, for the case of non-
isolated bound there exists the unique steady state ][ˆ
\\ BB MA (moreover:
]1,0[ˆ
\\ BB MA , since the initial and boundary conditions are located inside the
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 54
line segment ]1,0[ ). The equation for the state BA \
ˆ
can be obtained from the first
equation of system (10) as t :
BBBBBBB TATA ,\\\,\\
ˆˆ . (17)
Remark 4. Since all eigenvalues of the operator BBT \,\ under the given
conditions are located inside the open unit disk, equation (17) under these condi-
tions has the unique solution. However, direct solving equation (17) for real-
world networks usually becomes significantly more complicated due to the large
size of the set B\ and (consequently) the large dimension of the space
][\ BM . Therefore, practically reasonable approach is to approximate (numeri-
cally) BA \
ˆ
using the iterative procedure described by system (10).
Example 3. The network with the impact operator T from Example 1 is ob-
viously the network with non-isolated bound. The conditions of Theorem 3 hold,
so the network has the unique steady state ]1,0[ˆ
\\ BB MA . Equation (17) for
this network takes the form:
jiBjiB AA ,1\,\ )ˆ)((1(25.0)ˆ(
jiBjiBjiBjiB AAAA ,\1,\1,\,1\ )ˆ())ˆ()ˆ()ˆ(
for all internal Bji \),( (i.e., for all 12 ni and 12 mj ). Thus,
given 1 , one can obtain:
))ˆ()ˆ()ˆ()ˆ(()ˆ( 1,\1,\,1\,1\4
1
,\ jiBjiBjiBjiBjiB AAAAA .
Assume that the block BBB AA ˆ representing the states of boundary el-
ements of the network is defined by four arithmetic progressions:
;1 ,)1()()( 1
)()(
1,1,1
1,1,1 mjj mBjB
BmB
;1 ,)1()()( 1
)()(
1,,
1,, mjj mnBjnB
nBmnB
;1 ,)1()()( 1
)()(
1,11,
1,11, nii nBiB
BnB
,1 ,)1()()( 1
)()(
,1,
,1, nii nmBmiB
mBmnB
where the states of the corner elements 1,1left top, )( Bb , mBb ,1right top, )( ,
1,left bottom, )( nBb , 1,right bottom, )( nBb are the given constants from ]1,0[ . It is
easy to see that all elements of matrix ]1,0[ˆ
MA (the steady state of the net-
work) also form arithmetic progressions by each row and each column:
1left top,,
left top,left bottom,)1()ˆ(
n
bb
ji ibA
1
)1()(
1
)left top,right top,()left bottom,right bottom,(
left top,right top,
)1(
m
ibb
n
bbbb
j
)()( left top,right top,1
1
left top,left bottom,1
1
left top, bbbbb
m
j
n
i
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 55
))()(( left top,right top,left bottom,right bottom,11
11 bbbbmn
ji
(18)
The table contains the steady state  of the given network for 20n ,
10m , 8.0 , 3.0left top, b , 5.0right top, b , 8.0left bottom, b , right bottom,b
9.0 ; all data are written with precision up to 0.01. Precision 0.01 is achieved
by the iterative procedure (10) approximately for 650t . It is interesting to note
that for this impact operator T the steady state  does not depend on )1,0( ;
however, the value )1,0( affects on the convergence rate (for 6.0 preci-
sion of 0.01 is achieved approximately for 320t ).
T a b l e
j
i
1 2 3 4 5 6 7 8 9 10
1 0.30 0.32 0.34 0.37 0.39 0.41 0.43 0.46 0.48 0.50
2 0.33 0.35 0.37 0.39 0.41 0.43 0.46 0.48 0.50 0.52
3 0.35 0.37 0.39 0.42 0.44 0.46 0.48 0.50 0.52 0.54
4 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56
5 0.41 0.43 0.45 0.46 0.48 0.50 0.52 0.54 0.56 0.58
6 0.43 0.45 0.47 0.49 0.51 0.53 0.55 0.57 0.59 0.61
7 0.46 0.48 0.50 0.51 0.53 0.55 0.57 0.59 0.61 0.63
8 0.48 0.50 0.52 0.54 0.56 0.57 0.59 0.61 0.63 0.65
9 0.51 0.53 0.55 0.56 0.58 0.60 0.62 0.63 0.65 0.67
10 0.54 0.55 0.57 0.59 0.60 0.62 0.64 0.66 0.67 0.69
11 0.56 0.58 0.60 0.61 0.63 0.65 0.66 0.68 0.69 0.71
12 0.59 0.61 0.62 0.64 0.65 0.67 0.68 0.70 0.72 0.73
13 0.62 0.63 0.65 0.66 0.68 0.69 0.71 0.72 0.74 0.75
14 0.64 0.66 0.67 0.69 0.70 0.72 0.73 0.74 0.76 0.77
15 0.67 0.68 0.70 0.71 0.72 0.74 0.75 0.77 0.78 0.79
16 0.69 0.71 0.72 0.74 0.75 0.76 0.78 0.79 0.80 0.82
17 0.72 0.73 0.75 0.76 0.77 0.79 0.80 0.81 0.82 0.84
18 0.75 0.76 0.77 0.78 0.80 0.81 0.82 0.83 0.85 0.86
19 0.77 0.79 0.80 0.81 0.82 0.83 0.84 0.86 0.87 0.88
20 0.80 0.81 0.82 0.83 0.84 0.86 0.87 0.88 0.89 0.90
Remark 5. In Examples 2 and 3 it is possible to compute analytically the
steady state as t . However, as it is mentioned in Remark 4, direct solving
equation (17) for real-world networks usually becomes significantly more com-
plicated due to the large dimension of the space ][\ BM . Therefore, practically
reasonable is to apply the iterative procedure described by system (10). For more
details about analytical solving recurrent relations with multiple indices (with in-
dices ),( ji in the given case of two-dimensional network) see, e.g., [13].
CONCLUSIONS
Matrix model for social network is proposed, mutual impact of network
elements is represented by the linear impact operator T and the corresponding
labelled directed impact graph TG .
I.Ya. Spectorsky, V.M. Statkevych, O.V. Stus
ISSN 1681–6048 System Research & Information Technologies, 2025, № 4 56
Sufficient conditions for the network ergodicity are given in terms of ex-
istence of a steady state, which defines the network’s behaviour as t .
For the proposed model, spectral properties of the operator T are explored.
Sufficient conditions for the network ergodicity are given in the form of
existing eigenvalues of the operator T on the unit circle, and in the form of
strong connectivity of the impact graph TG .
REFERENCES
1. Alan R. Wagner, “Creating and Using Matrix Representations of Social Interaction,”
HRI09: International Conference on Human Robot Interaction, 09 March 2009, La
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Received 02.08.2024
INFORMATION ON THE ARTICLE
Igor Ya. Spectorsky, ORCID: 0000-0003-4863-7986, Educational and Research Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: i.spectorsky@gmail.com
Vitalii M. Statkevych, ORCID: 0000-0001-5210-9890, Educational and Research
Institute for Applied System Analysis of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: mstatckevich@yahoo.com
Matrix-graphic simulation of social network: ergodic properties
Системні дослідження та інформаційні технології, 2025, № 4 57
Oleksandr V. Stus, ORCID: 0000-0003-3426-5093, Educational and Research Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor Sikorsky
Kyiv Polytechnic Institute”, Ukraine, e-mail: o.stus@kpi.ua
МАТРИЧНО-ГРАФІЧНЕ МОДЕЛЮВАННЯ СОЦІАЛЬНОЇ МЕРЕЖІ:
ЕРГОДИЧНІ ВЛАСТИВОСТІ / І.Я. Спекторський, В.М. Статкевич, О.В. Стусь
Анотація. Запропоновано математичний апарат моделювання соціальних ме-
реж, який дозволяє отримати достатні умови ергодичності мережі, тобто існу-
вання граничного стаціонарного стану при t . Запропонована модель є
лінійною: елементи мережі утворюють двовимірний масив (матрицю), елемен-
том матриці в момент часу 0t є стан ]1,0[)(, tA ji
елемента з координа-
тами },,2,1{},,2,1{),( mnji . Взаємний вплив між елементами зада-
но оператором впливу T — чотиривимірним масивом, елементи 0,,, lkjiT
якого позначають вплив елемента ),( lk на елемент ),( ji :
n
k
m
l
lklkjiji ATTA
1 1
,,,,,)( . Для T запропоновано зображення у вигляді графу
TG , вершини якого відповідають елементам ),( ji : орієнтоване ребро
(дуга) з міткою lkjiT ,,, веде від вершини ),( lk до вершини ),( ji
тоді й тільки тоді, коли 0,,, lkjiT . На виділено край B : 0,,, lkjiT
для ),( lk , Bji ),( . Стан )1( tA мережі у момент часу 1t визначаєть-
ся станом )(tA мережі у момент часу 0t згідно з рівнянням
)()1( tTAtA , де матриця розмірності mn визначає стан крайо-
вих елементів мережі; 0, ji для внутрішніх елементів Bji \),( . До-
статні умови ергодичності мережі надано у термінах властивостей зв’язності
графу впливу TG : мають існувати шляхи між довільними вершинами та усі
петлі. Наведені умови забезпечують розташування спектра оператора T все-
редині одиничного круга за винятком, можливо, 1 ; доведено, що 1 є
власним числом T лише у випадку ізольованого краю (жоден крайовий еле-
мент не впливає на жоден внутрішній елемент мережі). Наведені спектральні
властивості T забезпечують існування стаціонарного стану, який можна зна-
ходити ітераційною процедурою )()1( tTAtA за заданим )0(A з гео-
метричною (лінійною) швидкістю збіжності.
Ключові слова: соціальна мережа, моделювання, ергодичність, власне число,
жорданова нормальна форма.
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| id | journaliasakpiua-article-351418 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-02-08T08:06:10Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/86/7d5e5af11d9774caf5925b3956da6b86.pdf |
| spelling | journaliasakpiua-article-3514182026-02-02T20:49:24Z Matrix-graphic simulation of social network: ergodic properties Матрично-графічне моделювання соціальної мережі: ергодичні властивості Spectorsky, Igor Statkevych, Vitalii Stus, Oleksandr social system simulation ergodicity eigenvalue Jordan normal form соціальна мережа моделювання ергодичність власне число жорданова нормальна форма We propose mathematical tools for social network simulation to obtain sufficient conditions for network ergodicity, defined as the existence of a steady state as time approaches infinity. The proposed model is linear; the network elements form a two-dimensional array (matrix), where each entry represents the state of an element at a specific time. An impact operator, structured as a four-dimensional array, defines the interactions between elements. This operator is also presented as a directed graph where vertices correspond to network elements, and arcs represent the impact of one element on another. The model incorporates boundary elements that influence the internal states of the network. Sufficient conditions for network ergodicity are derived from the connectivity properties of the impact graph, which must contain paths between all pairs of vertices and loops for all vertices. These conditions ensure that the operator's spectrum (with the possible exception of the value 1) is located inside the open unit disk. We prove that 1 is an eigenvalue if and only if the boundary is isolated. These spectral properties guarantee that a steady state exists and can be found using an iterative procedure with linear (geometric) convergence. Запропоновано математичний апарат моделювання соціальних мереж, який дозволяє отримати достатні умови ергодичності мережі, тобто існування граничного стаціонарного стану при прямуванні часу до нескінченності. Запропонована модель є лінійною: елементи мережі утворюють двовимірний масив (матрицю), де кожен елемент відображає стан учасника мережі у певний момент часу. Взаємний вплив між елементами задано оператором впливу — чотиривимірним масивом, для якого запропоновано візуалізацію у вигляді графу, де вершини відповідають елементам мережі, а орієнтовані ребра вказують на наявність та силу впливу. У моделі виділено крайові елементи, стани яких визначають зовнішній вплив на систему. Достатні умови ергодичності надано у термінах властивостей зв’язності графу впливу: він має містити шляхи між довільними парами вершин та петлі для всіх вершин. Доведено, що ці умови забезпечують розташування спектра оператора (за винятком, можливо, одиниці) всередині одиничного круга. Одиниця є власним числом лише у випадку ізольованого краю, коли жоден крайовий елемент не впливає на внутрішні. Визначені властивості гарантують існування стаціонарного стану, який можна знайти ітераційною процедурою з геометричною швидкістю збіжності. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-12-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/351418 10.20535/SRIT.2308-8893.2025.4.03 System research and information technologies; No. 4 (2025); 38-57 Системные исследования и информационные технологии; № 4 (2025); 38-57 Системні дослідження та інформаційні технології; № 4 (2025); 38-57 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/351418/338430 |
| spellingShingle | соціальна мережа моделювання ергодичність власне число жорданова нормальна форма Spectorsky, Igor Statkevych, Vitalii Stus, Oleksandr Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title | Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title_alt | Matrix-graphic simulation of social network: ergodic properties |
| title_full | Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title_fullStr | Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title_full_unstemmed | Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title_short | Матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| title_sort | матрично-графічне моделювання соціальної мережі: ергодичні властивості |
| topic | соціальна мережа моделювання ергодичність власне число жорданова нормальна форма |
| topic_facet | social system simulation ergodicity eigenvalue Jordan normal form соціальна мережа моделювання ергодичність власне число жорданова нормальна форма |
| url | https://journal.iasa.kpi.ua/article/view/351418 |
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