Comparison of methods of complex system evaluation

Full text

Cybernetics and Systems Analysis, Vol. 56, No.2, March, 2020
VULNERABILITY OF COMPLEX NETWORK
STRUCTURES AND SYSTEMS
O. D. Polishchuk UDC 519.6
Abstract. Structural and functional approaches to the determination of vulnerability of complex
network structures and systems to negative internal and external influences are considered.
The concept of parameters of influence and betweenness of system elements is introduced, which
allows us to identify the most important from the functional point of view nodes and edges of the
network and develop scenarios for identifying those components of the system whose blocking
can cause greatest losses in the process of its functioning, and also quantify these losses.
The sensitivity of the system to small variations in the volume of flow movement, which are close to
the critical loading of its components, is analyzed. The obtained results can be used to improve the
available methods and develop new ones to protect real network systems from various natural and
artificial damages.
Keywords: complex network, network system, flow, stability, influence, betweenness.
INTRODUCTION
One of the areas of system research that has started to develop rapidly over the past decades [1, 2] is the study of
complex network systems (CNS). Network structures exist in the micro- and macrocosm [3, 4], biological systems
(neural, protein, metabolic, food, environmental networks, etc.), and human society (economic, social, financial, political,
religious, professional, family, and many others) [5–7]. The subject of research in the theory of complex networks (TCN)
is the formulation of universal models of network structures, determination of statistical properties that characterize their
behavior, and prediction of the behavior of networks when their structural properties change. The similarity of many
natural and artificial networks helps with the development of universal methods for studying such structures, but it does
not always help to study the processes of functioning of such systems. One of the defining features of substantially
functioning CNS is the movement of flows in them. In some cases, ensuring the movement of flows is the main goal of
creation and functioning of such systems (transport networks and systems for the supply of resources; trade and social
networks, etc.), in others, it is the process that ensures their vital activity (the movement of blood, lymph, and nerve
impulses in the human body). Discontinuation of such flows can lead to the termination of such systems. Therefore,
the flow movement can be attributed to the main or one of the main functions that is exercised by CNS [8, 9].
Among the complex networks (CN) of various types, the most interesting from an applied point of view are the
so-called scale-free networks [10]. A distinctive feature of these networks is the distribution of node degrees, which leads
to a small number of nodes that have high degree, and a huge number of nodes with a low degree (number of cities in
each country is small compared to the total number of settlements, but their importance in the life of the country cannot
be overemphasized). One of the main problems that is being investigated in the TCN is the vulnerability of the network
to accidental or targeted internal and external influences on its nodes [11–13]. This is due to the risk of consequences of
312 1060-0396/20/5602-0312 ©2020 Springer Science+Business Media, LLC
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of
Ukraine, Lviv, Ukraine, od_polishchuk@ukr.net. Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April,
2020, pp. 166–176. Original article submitted March 26, 2019.
DOI 10.1007/s10559-020-00247-4

the deployment of such processes, which can lead to the destabilization of transport systems and the Internet, financial
crises and failures in the operation of energy supply systems, etc. It turns out that scale-free networks are quite resistant
to accidental attacks and are very vulnerable to targeted attacks [12]. A number of structural approaches to the study of
such network phenomena has been developed within the TCN [13, 14]. However, the system may be vulnerable not only
to attacks on its structure, but also to the process of functioning. For example, global computerization makes almost all
areas of human activity sensitive to cyber attacks. During 2014–2018 hacker attacks and computer viruses infected and
destabilized the work of a number of government agencies, security services, military departments, transport and energy
systems in many countries of the world. Losses from such attacks amounted to tens of billions of dollars [15]. In Ukraine,
at the same time, hacker attacks have repeatedly blocked the work of the State Treasury of Ukraine, Ministry of Finance
of Ukraine and the Pension Fund, dozens of large banks, railway and aviation systems, energy supply systems of certain
regions, etc. It is obvious that in the future such threats will only increase, and the damage from them will grow as well.
Therefore, the identification of the most functionally important, and therefore the most attractive elements for attacks of
real CNS will help to improve existing and build new, much more effective protection systems.
The purpose of the article is to define criteria for the functional importance of elements of complex network
systems and develop scenarios for negative impacts on them in order to prevent or minimize the consequences of
potential defeats.
STRUCTURAL APPROACHES TO DETERMINING THE VULNERABILITY
OF COMPLEX NETWORKS
The structure and a number of characteristics of its elements are completely determined by the adjacency
matrix [1]. These characteristics include the degree of a node, which value for binary undirected networks is determined
by the number of its connections to adjacent nodes. For scale-free networks of this type, the following scenarios of
targeted attacks are investigated in the TCN [12, 13]:
(i) a list of network nodes is prepared in descending order of their degrees, and nodes from the beginning of this
list are sequentially removed from the structure until the percolation threshold is reached;
(ii) after extracting the next node generated in the first scenario. the list of nodes is rewritten according to the
same principle and the attack is performed on the first node of the modified list.
The second scenario of a targeted attack on a non-scale CN, which takes into account the possible change after the
next withdrawal of the network connection structure, was much more dangerous than the first [12]. In particular, it was
found that as a result of applying this scenario, after removing 4% of the nodes with the highest degree from the Internet,
this network is divided into disconnected components.
The degree of a node is its local characteristic in the network. One of the global characteristics of a node is the
centrality of mediation, which is determined by the number of all shortest paths in the network that pass through it [16].
The centrality of mediation allows one to create more effective scenarios for attacks on non-scale networks, in particular:
(i) a list of network nodes is prepared in descending order of their mediation centrality values, and nodes are
sequentially removed from the structure from the beginning of this list until the percolation threshold is reached;
(ii) after extracting the next node, the list of nodes is formed according to the previous scenario, and the attack is
carried out on the first node from the modified list.
The last two scenarios are more dangerous than the previous ones, i.e., the nodes with higher central mediation are
more important for the network than the nodes with a high degree. This was confirmed, in particular, by the analysis of
a number of global aviation networks [14].
A node may have a high degree or centrality of mediation in the network structure, but this does not always
determine its real importance in the process of functioning of the system. In many countries, there are regions that, after
periods of intensive development, went into a state of depression (exhaustion of mineral deposits that were extracted in
the region; a decrease in demand for products that the region specialized in producing, etc.). In such regions, there
is usually a well-developed infrastructure, in particular a dense transport network and energy supply network, but the
volume of flows to/from it is significantly reduced. This means that despite the high degree of nodes that are located in
the corresponding parts of the network, their functional value in the system may be small [8]. The degree of cities such as
Korosten, Kupyansk, Ternopil, and Kyiv in the railway network of Ukraine is equal to five. Obviously, this does not
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correspond to the functional importance of these cities in the life of the country. In addition, the structural approach to
analyzing CNS vulnerability does not provide a clear answer to at least three questions:
(i) attacks on the functioning of system elements can further destabilize its operation even if the structure
is not affected;
(ii) which part of the system can be affected by this destabilization process;
(iii) what quantitative losses are expected by the system and its individual components as a result of the defeat.
FUNCTIONAL APPROACHES TO DETERMINING THE VULNERABILITY
OF NETWORK SYSTEMS
The problem of a system stability is much deeper and more complex than the problem of the stability of its
structure. Of course, the destruction of the structure, and, for example, its division into unrelated components, will
inevitably lead to the destabilization of the CNS, but failures in the system can also occur when the structure is unstable.
Moreover, the number of system elements where targeted attacks on their operation can lead to CNS failures is usually
significantly higher than the number of nodes in the structure, which removal leads to crossing the percolation threshold.
In order to confirm this, M. Newman’s approach can be used [17], who mapped a weighted network with integer weights
to a multigraph with the same adjacency matrix V==
{}Vij ij
N
,1
, whereVij is the weight of the edge that combines nodes ni
and nj,ij N,,=1,Nis the number of network nodes. If we assume that the weights are reduced to integer values of the
volume of flows that pass through the edges of the network for a certain period of time [, ]0T, then, by this approach,
even multigraph transit nodes can have a high degree of centrality of mediation on the routes of heavy flows (Fig. 1).
This means that attacks on them can lead to serious system failures.
The functional importance of a node in CNS is defined as follows. Let ukij
nn
out (, )
stand for the amount of flows
generated in the node niand accepted in the node nj, that passed by the path pnn
ki j
(, )
over the period [, ]0T,Kij is
a number of all possible paths connecting nodes niand nj,kK
ij
=1, ,ij N,,=1. Let us denote by
Vnn nn
ij kij
k
Kij
out out
(, ) (, )=
=
åu
1
total amount of flows generated in the node niand used for acceptance within the node njby all possible means
for the period [, ]0T. Parameter Vnn
ij
out (, )
determines the actual impact of the node nion the node njbased on
the total volume of flows that flow from it to the node njover the period Ò,ij N,,=1. A clear definition of the
ways and volumes of flows is typical for most industrial, financial, transport, trade networks, resource supply
systems, public administration, etc. For example, it is easy to determine how much of the railway tickets are sold
using the Internet, what percentage of utility payments or pensions are paid through the offices of “Ukrposhta”
(Ukrainian Postal Service), what percentage of public procurement is carried out using the “Prozorro” system (public
e-procurement system), etc. These data makes it possible to quantify the losses that may be inflicted on the relevant
systems, for example, as a result of cyber attacks on their computer networks.
Let Rjj
iii
L
out {}=1, ..., be the set of node numbers, the end receivers of flows generated in the node ni, and Lbe
a number of elements of the set Ri
out . Parameter
xx
i
jR
ij i
i
Vnns
out out out
out
=Î
Î
å(, )/(), [,]V01,
determines the impact force of the node niapplied on CNS in general, iN=1, . Here, s()Vstands for the sum of
the elements of the matrix V, which is equal to the total volume of flows in the network for the period [, ]0T.
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The power of the node nithat is applied on the system is defined using the parameter pLNp
ii
out out
=Î/, [,]01 ,
and a set Ri
out ,iN=1, , is called the area of influence of this node on CNS (Fig. 2).
For example, the area of influence of local governments or regional media is usually limited to the relevant region
of the country, while national governments and national media are limited to the entire state. Parameters xi
out ,pi
out , and
Ri
out are called the output force, power, and area, or the initial parameters of the node’s ni,iN=1, , influence on CNS,
respectively. In the simplest case, the initial area of influence of each CNS node is limited to adjacent nodes, and in the
most complex case, it forms a complete graph. Area Ri
out and parameter pi
out allow us to assess, which part of the CNS
is affected by the consequences of failures in the operation of the node ni, and the value of xi
out determines what losses it
cause in terms of under-delivery or delay in delivery of the corresponding volume of flows.
Let ukji
nn
in (,)
be the amount of flows generated in the node njand accepted in the node nithat passed by the
path pnn
kj
i
(, )
over the period [, ]0T, and Kji be the number of all possible paths connecting nodes njand ni,
kK
ji
=1, ,ij N,,=1. Let us denote
Vnn nn
ji kji
k
Kij
in in
(,) (,)=
=
åu
1
total amount of flows generated in the node njand used for acceptance within the node niby all possible means
for the period [, ]0T. Parameter Vnn
ji
in (,)
determines the actual impact of the node njon the node niaccording to
the total volumes of flows that were accepted in the node niover the period Ò,ij N,,=1. Let Gjj
iii
M
in {}=1,..., be
a set of node numbers that generate flows that are used for acceptance to the node niand Mbe the number of
elements of the set Gi
in . Parameter
xi
jG
ji
i
Vnns
in in
in
=
Î
å(,)/()V,xi
in Î[, ]01,
315
Fig. 1. Fragment of an integer weighted network: the value of the
weights is six (a); the corresponding multigraph (b).
ab
Fig. 2. The area of the input (vertical stripes Gi
in )
and output (horizontal stripes Ri
out ) influence are
reflected by the square of the network system node.

determines the impact of CNS on the node ni,iN=1, . Power of the system’s impact on the node niwe
determine using the parameter pMNp
ii
in in
=Î/, [,]01, and a set Gi
in we will call the area of influence of CNS
on the node ni. Parameters xi
in ,pi
in , and Gi
in we will call the input force, power, and area (or input parameters of
the CNS effect on the node ni), respectively. In the simplest case, the initial area of influence of each CNS node is
limited to adjacent nodes, and in the most complex case, it forms a complete graph.
The parameters of the input and output influence of CNS nodes make it possible to give at least partial answers to
the questions formulated above regarding the most functionally important elements for destabilizing the system.
However, there is another aspect of the protection problem. It consists in timely detection and blocking of those CNS
nodes that contain a potential or real threat and can destabilize the system (hacker and terrorist groups, sources of
dangerous infectious diseases, etc.). In social online services, so-called bot networks [18] are often found, with the help
of which one person can create the illusion of a common/shared thought of many people, massively spread
disinformation, organize DDoS attacks, etc. Such botnets are often created during election campaigns and can distort the
will of citizens. They are a powerful tool for unfair competition, advertising low-quality products or, conversely,
anti-advertising of new products. Detection of generator nodes of such botnets and their blocking allows one to prevent
many negative social and economic phenomena. The input and output parameters of the CNS node influence make it
possible to accurately identify the botnet generators. Usually, the botnet generator does not require and does not receive
a return response when sending commands to the bots it created (information about the target and the content of the
attack), i.e., the following inequality is true for such entities:
x
x
i
i
in
out << 1.
From these considerations, it also follows that the area and power of the output impact of such nodes is quite large
(botnets that number more than 350 thousand nodes were found in the Twitter network [19]), and the area and power of
the input impact are small, moreover, RG
ii
inout ǻ0.
In real CNS, there are almost no nodes that are exclusively generators or receivers of flows. Indeed, for the
production of certain products, it is necessary to supply raw materials and components, mining can not be carried out
without appropriate mining equipment, etc. Let us denote RGicombining the input and output influence areas of a node
ni, i.e., RG R G
ii i
=È
out in .
The interaction strength of node niwith CNS is defined by using the parameter xxx
iii
=+()/
in out 2and the power
of this interaction by using the parameter of piequal to the number of elements of the set RGi.
The interaction parameters allow us to define the following attack scenarios on CNS:
(i) a list of network nodes is prepared in order of decreasing the values of the forces of their interaction with the
system and nodes from the beginning of this list are sequentially removed from the structure until a predetermined level
of critical losses is reached;
(ii) after extracting the next node, the list of nodes is formed according to the previous scenario, and the attack is
carried out on the first node from the modified list.
The second scenario takes into account the need to replace blocked generator nodes and receiver nodes and the
corresponding redistribution of traffic flows by the network. Depending on how to counter potential threats, the last two scenarios
can be generated separately for generator nodes (for example, searching for initiators of DDoS attacks) and flow receiver nodes
(searching for the most likely targets of DDoS attacks). It should also be taken into account that in reality, the behavior of the
parameters of the influence of CNS nodes can be much more complex. A node that directed a flow to all adjacent nodes can
become a receiver again, and adjacent nodes from receivers can turn into generators that direct this flow further. This is how
infectious disease epidemics are deployed according to the so-called SIS scenario [20]. In addition, the parameters of the
influence of CNS nodes in general are dynamic characteristics, the value of which can change significantly over time.
As mentioned above, one of the most used methods for determining the importance of network nodes is the degree
of centrality of mediation. Perhaps the term “mediation” is the best way to define the participation of the CNS element in
the process of joint functioning and interaction of all nodes of the network or a certain part of it. Therefore, to determine
316

the functional importance of a node or edge in the system, we will use the term “mediation”. Let us denote
Pp
ij
K
ij
k
k
K
ij ij
==
{}
1a set of paths that connect generator nodes and receiver nodes of CNS flow and contain the following
edge as an element: (, )nn
ij
,ij N,,=1. Let uij
kbe the volume of flows that have passed through the path of pij
kfrom the
generator node to the receiver node, and, hence, through the edge (, )nn
ij
over the period [, ]0T. Then the value
Vij
K
ij
k
k
K
ij
ij
=
=
åu
1
defines the total amount of flows that have passed through a set of paths Pij
Kij , and, therefore, through an edge
(, )nn
ij
, for this period of time. Parameter Fij ij
K
Vs
ij
=/( )V, which determines the specific weight of flows passing
through an edge (, )nn
ij
over the period [, ]0T, we will call as the measure of mediation of this edge in the process
of CNS functioning. A set Lij of all network nodes that lie on paths from the aggregate Pij
Kij , we will call the
mediation area, and the number hij of these nodes are called the edge mediation function (, )nn
ij
,ij N,,=1
(Fig. 3).
Parameters of the degree, area, and power of edge mediation (, )nn
ij
,ij N,,=1, are global characteristics of its
importance in the process of CNS functioning. They, in particular, determine how blocking of this edge will affect the
work in the mediation area, the size of this area, and, as a result, the loss of the entire system.
Let us denote Pp
i
K
i
k
k
K
ii
==
{}
1a set of paths that connect generator nodes and receiver nodes of CNS flows and
that pass through the node ni,iN=1, . Let ui
kbe the volume of flows that have passed through the path of pi
kfrom the
generator node to the receiver node, and therefore through the node niover the period [, ]0T. Then the value
Vi
K
i
k
k
K
i
i
=
=
åu
1
defines the total amount of flows that have passed through a set of paths Pi
Ki, and hence through the node ni, for
this very period of time. Parameter Fii
K
Vs
i
=/( )V, which determines the specific weight of flows passing through
the node niover the period [, ]0T, we will call as measure of mediation of this node in the process of CNS
functioning. A set Miof all CNS nodes that lie on paths from the aggregate Pi
Ki, we will call the mediation area,
and the number hiof these nodes is called the node mediation function ni. Parameters of the degree, area, and
power of node mediation ni,iN=1, , are global characteristics of its importance in the process of CNS functioning.
They, in particular, determine how blocking of this node will affect the work in the mediation area, the size of this
area, and, as a result, the loss of the entire system.
317
Fig. 3. The area of edge mediation (, )nn
ij
in the process of CNS functioning.
ninj

The mediation parameters allow us to define the following attack scenarios on CNS:
(i) a list of network nodes is prepared in order of decreasing the values of the degree of their mediation in the
system, and the nodes from the beginning of this list are sequentially removed from the structure until a predetermined
level of critical losses is reached;
(ii) after extracting the next node, the list of nodes is formed according to the previous scenario, and the attack is
carried out on the first node from the modified list.
The second scenario takes into account the need to replace blocked nodes-generators and receivers of flows and
search for alternative routes for transit flows that passed through the blocked nodes, i.e., the corresponding redistribution
of traffic flows by the network. Similar attack scenarios are formed for edges, since in many cases it is much easier to
remove a network edge from the CNS operation process than to block one of the nodes that this edge connects. The
parameters for mediation of nodes and edges allow us to estimate which part of the CNS will be affected by the
consequences of failures of the corresponding element of the system and what losses this will cause in the sense of
non-delivery of certain volumes of transit flows.
Above, we defined the parameters of node mediation, taking into account only the transit flows that pass through it.
However, the values of mediation parameters can be significantly expanded, given that the node nican be not only a transiter,
but also a generator and a final receiver of flows. Then, a set Pi
Kican be supplemented with the paths of traffic flows that start
(generated) or end (accepted) in the node ni. Let us denote such an augmented set ~
Pi
Ki,iN=1, . Then, the value
~()/FF
iiii
=++xx
in out 3
we will call a generalized node mediation measure niin the process of CNS functioning. Respectively, a set ~
Miof
all CNS nodes that lie on paths from the aggregate ~
Pi
Ki, we will call the generalized mediation area, and the
number ~
hiof these nodes are called the generalized mediation power of the node ni,iN=1, . The generalized
mediation parameters take into account the interaction between all directly and indirectly connected CNS nodes
(generators, receivers, and transiters) and allow you to create the most effective attack scenarios for them.
The principles for building such scenarios are described above.
SENSITIVITY OF NETWORK SYSTEMS TO SMALL CHANGES
Along with targeted attacks, there is another aspect of the system’s stability, which is its sensitivity to small
changes in the structure or process of functioning. Such changes can be caused by both internal and external factors, and
lead to no less negative consequences than targeted attacks. In this case, the stability of the structure is determined by the
sensitivity to small changes in its composition (a set of nodes and connections between them) [12]. The structure is
unstable if such changes can lead to the loss of certain network properties, such as connectivity. The stability of the CNS
operation process is determined by its sensitivity to small changes in the volume of flows. For example, the system is
vulnerable to critical (close to their capacity) congestion of part of its edges or nodes, and in CNS with fully ordered
traffic flows, such as a railway transport system, it is sensitive to small delays in the train schedule. It is obvious that the
stability of the process is more or less related to the stability of the CNS structure. If small changes (blocking several
nodes and edges of the network) lead to the loss of its connectivity, this directly affects the process of functioning of the
system. If the stress on certain elements of the structure is critical, this also creates a threat of blocking them.
We will determine the most vulnerable to critical stress conditions functionally important components of CNS. For
this purpose, we will introduce [21] flowing l-core of the CNS as the largest subnet of the source network for which the
elements of the matrix Vare no less than the value of lÎ[, ]01. The adjacency matrix Vll
==
{}Vij ij
N
,1
of l-core we will
define by the formula
VVV
VijN
ij
ij ij
ij
ll
l
=³
<=
ì
í
î
if
if
,
,, , .01
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Elements of matrix Vlwith the growing importance of ldetermine the functional priority of the corresponding
CNS subsystems.
Let us also define the CNS stress matrix U==
{}Uij ij
N
,1
with the elementsUVU
ij ij ij
=/max ,where
Uij
max is a bandwidth
(maximum allowed amount of flows) of the edge (, )nn
ij
,ij N,,=1, and let us introduce b-core of the CNS stress as the largest
subnet of the source network for which the elements of the matrix Uare no less than the values of bÎ[, ]01.
Let us define by the formula
UuU
UijN
ij
ij ij
ij
bb
b
=³
<=
ì
í
î
if
if
,
,, , ,01
the adjacency matrix Ubb
==
{}Uij ij
N
,1
of b-core.
Elements of matrix Ubwith the growing importance of bdetermine the most stressed components of the system.
Then, the nonzero elements of the matrix
Wlb l b´
=
=´{}VU
ij ij ij
N
,1
for values land b, close to one, determine the most vulnerable of the functionally most important components of the
system belonging to the intersection of l-and b-cores of CNS. A slight increase in the volume of flows in such
components can lead to their blocking in the system, causing it the greatest harm. In Fig. 4a, the main highways of
the central part of the Lviv city, the structure of which was formed long before the advent of automobile and electric
transport, are reflected. This is the main reason for long traffic jams (blocking important components of the road
network) that constantly occur in the center of Lviv. In Fig. 4b and 4c, fragments of the flow 0.8-core and 0.9-core
load of this transport system (TS), are shown, respectively. In Fig. 4d, the most vulnerable to congestion of the road
network section (non-zero elements of the matrix Wlb´), which repeatedly led to the temporary collapse of the TS
center of Lviv. In general, for an arbitrary network system elements of its matrix Wlb´they are the most vulnerable,
i.e., the components that are attractive from the point of view of successful implementation for targeted attacks on it.
It is obvious that blocking of these components creates difficulties for traffic flows on all adjacent roads.
In contrast to targeted defeats, failures that occur as a result of small changes in the volume of traffic flows can
often be prevented. Thus, the analysis of risks associated with critical stress allows us to increase the throughput of CNS
elements in advance, choose alternative or create new flow paths. The disadvantages associated with a low-latency
sensitive flow schedule can be overcome by optimizing this schedule, etc. Since failures of a single element negatively
affect the operation of all directly or indirectly related CNS elements, i.e., on a certain subsystem, and shortcomings in the
functioning of individual subsystems (on CNS in general), it is advisable to use the methods of continuous monitoring and
comprehensive assessment of the functioning of complex systems, described in detail in [22, 23].
319
Fig. 4. Fragments: main roads of town center (a); flow-core of town center (b);
the core workload of the town center (c); the most vulnerable component
of the road of the town center (d).
ab dc

Usually, after blocking a system whose structure is determined by the matrix Wlb´, CNS tries to redirect flows in
other ways. As a result, other flow cores and load cores are formed in the network, the cross section of which creates new
threats for blocking system components. This process can spread over large areas of the CNS and requires rapid analysis
and appropriate decisions to prevent wide propagation of such processes. In the case of a high rate of propagation, they
can develop into so-called cascading phenomena in the network. The most reliable and effective way to counteract such
processes is to reserve alternative flow paths, in other words, to seal the network.
CONCLUSIONS
Determining the most important elements of real CNS with a scale-free structure from a functional point of view is
relevant for improving the means of protecting these systems from targeted attacks and other negative internal and
external influences. The concepts of influence parameters and mediation of CNS elements introduced in the work made it
possible to develop scenarios for identifying those components of the system whose blocking can lead to the greatest
losses in the course of its operation, as well as to quantify these losses. The sensitivity of the system to small changes in
the volume of traffic flows, the values of which are close to the critical load of composite CNS, was analyzed. It is shown
that critical load of system elements can lead to the same consequences as deliberate removal of them from the network
structure or purposeful blocking of the functioning process. The obtained results can be used to improve existing and
develop new methods for protecting real network systems from natural and artificial lesions of various types.
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