In the context of rapid advances in information technology, there is an increasing demand for the development of efficient, scalable, and resilient network infrastructures that integrate distributed computing modules or autonomous devices into a cohesive system. Such a need is particularly pronounced in fields that require high levels of automation, fault tolerance, and adaptability - including telecommunications, cloud computing, decision support systems, healthcare information networks, banking infrastructure, and nextgeneration energy grids. When the availability of financial and physical resources is limited, a major challenge arises: not only must the network successfully interconnect all its components, but it must also guarantee maximum functional stability under a constrained total budget and varying costs for establishing individual communication links. This paper proposes a novel method for constructing an optimal node connection topology represented as an undirected graph, with a specific focus on maximizing the network's robustness under cost constraints. The proposed model formalizes functional stability using a functional stability indicator. An optimization algorithm is developed to identify the most resilient network configuration - one that minimizes the risk of fragmentation or failure due to the loss of specific links or nodes. A key component of the model is the integration of edge weights representing individual link costs, alongside a global budget constraint enforced as part of the optimization objective. Furthermore, the study introduces a software prototype that implements the developed algorithm. The tool provides an automated mechanism for generating optimal network topologies, alongside features for visualizing the resulting structure. The results presented in this work may be directly applied in the design and deployment of intelligent distributed systems that demand high resilience and adaptability, ensuring robust performance in realworld operational environments.
The use of various networks is a popular approach for solving tasks in many fields [1-3]. However, to utilize these networks effectively, it is necessary to establish connections between multiple machines (modules, sub-networks, etc.). Naturally, real-world tasks require considering various factors such as connection time, cost, and other constraints. Given the need for highly reliable networks while adhering to budget limitations, it would be beneficial to have an algorithm that generates an optimal connection topology – one that maximizes functional stability [4-6] while staying within a given budget.
This problem can be formulated as a graph enumeration or traversal task [4, 5, 7]. However, many existing graph-based algorithms either provide only partial solutions to this problem or require excessive computation time. This work proposes a new modification of an exhaustive graph search algorithm that fully addresses the described problem while significantly reducing the required computation time.
Optimization of exhaustive graph, described in this work based on two concepts: decreasing of number of traversed graphs an estimation of indicator of functional stability. After description of the algorithm it would be good to see how this algorithm works, For automation of this algorithm authors developed software, what allows to visualize optimal topology of resulting network. Using developed software it will be shown, that described algorithm may give good optimization of finding of topology of connections of machines between each other. 2.1.
Let consider, that we have n machines and we want to connect it between each other by
connection lines. In other hand, let consider, that we know general budget B and we have prizes
matrix
where cij is a cost of laying a communication line between machines vi and v j . For estimation of
functional stability of considered information let use indicator R [4], what may be defined as
integral
(
As we know, selection of topology of the metwork, what is optimal in some sense [8, 9], is equivalent to the task of selection of (mostly, unordered) graph G = G (V , L), what describes a topology of information system, with number of vertices equals the number of machines. If we have criterion of optimality, we may use exhaustive search of graphs to search optimal. But exhaustive ⎛ n(n−1) ⎞ search of graphs in case of graph with n vertices have computational difficulty equals O⎜⎜ 2 2 ⎟⎟ , ⎝ ⎠ what is very bad variant. This factor is a reason to optimization of exhaustive search of graphs.
For optimization of exhaustive search of graphs with n vertices let use two ideas. First idea based on limitation of number of connection lines. To do it let consider sequences: first sequence (let note it as {λu } ) is a sequence of values cij ,i = 2,3,..., n, j = 1, 2,...,i −1, what are sorted in descending order and second sequence (let note it as {λu }) is a sequence of values cij ,i = 2,3,..., n, j = 1, 2,...,i −1, what are sorted in ascending order. Now let consider numbers N and N , what may be calculated by formulas
C = cij i, j=1,2,...,n ,
1 i, j=1,2,...,n ∫0 Rij ( p ) dp, R = min i≠ j
n ⎫ N = max ⎨⎧n : n ∈ • ∧ ∑λu ≤ B⎬, ⎩ u=1 ⎭
n ⎫ N = max ⎨⎧n : n ∈ • ∧ ∑λu ≤ B⎬.
⎩ u=1 ⎭
Using purpose, that there is a sense to work only with connected graphs, and numbers N , N ,
we may formulate interval of permissible number of communication lines formula
⎛ n(n −1) ⎞
max (n −1, min ( N, N )) ≤ L ≤ min ⎜ , max ( N, N )⎟, (
Second idea of optimization based on approximation of indicator R using regression polynomial P%3, what is described in [4] and may be written by formula
R ≈ P%3= P%3(η min ,η max , S, qG ,uG ) =
= 7,01ηmin + 2,57ηmax +1,14S −13,69qG − 3,84uG −14η m2in − 4,56ηminηmax + 5,6ηminS −
−21,83η minqG + 7,54η minuG −1,63η m2ax − 2,11η max S − 6,53η maxqG + 7,05η maxuG − 7,96S 2 +
+32,52SqG + 0,19SuG + 35,09qG2 + 4,38qGuG − 2,89uG2 +12,98η m3in −1,66η m2inηmax − 7,23η m2inS +
+12,31η m2inqG + 4,42η m2inuG + 0,72ηminηmax + 7,89ηminηmaxS + 4,66ηminηmaxqG −1,74ηminηmaxuG −
2
−7,23ηminS2 +17,93ηminSqG −13,87ηminSuG + 6,99ηminqG2 − 4,48ηminqGuG − 0,28ηminuG2 −
−0,49η m3ax + 5,21η m2axS + 3,09η m2axqG − 4,03η m2axuG − 5,78ηmaxS2 − 5,6ηmaxSqG −
−7,64ηmaxSuG + 5,09ηmaxqG2 + 3,35ηmaxqGuG + 7,19ηmaxuG2 + 7,22S3 −1,48S2qG +
+13,25S2uG − 50,9SqG2 − 20,74SqGuG + 0,49SuG2 − 24,57qG3 +11,39qG2uG + 0,47qGuG2 − 3,7uG3, (
In borders of the study a specialized software was realized. This software realizes described algorithm for finding optimal way of connection of machines in one information system. To see its work, let consider two examples.
Let purpose, that it’s needed to connect four servers between each other and we know, that general budget, what is given for connection, equals 10000$ and prizes for laying every connection line may be represented by matrix
If we enter these data into the realized software, we will have result like demonstrated on fig. 1.
As we may see on fig. 1, if we numerate servers using numbers from 0 to 3, the most functionally stable variant of connection, based on given budget, may be demonstrated by built graph.
Now let purpose, that it’s needed to connect five servers between each other and we know, that general budget, what is given for connection, equals 10000$ again, but and prizes for laying every connection line may be represented by matrix
Let compare a number of graphs, what was traversed in the process finding of optimal topology. To do it let use table 1.
As we see on table 1, described method may decrease a number of traversed graphs more, than
30%. It gives excellent optimization in finding of optimal topology of connections of servers. But it
would be interesting to see, for what number of graphs we need do calculate an indicator of
functional stability (
As we see on table 2, condition (
This work demonstrates optimized algorithm of finding of topology of network, optimal in sense of functional stability, considering fixed maximum budget for connection of machines. As examples and additional analysis shows, described algorithm allows to have excellent optimization in context of different parameters of difficulty of calculations.
These results may be used in various fields, for example, in IoT [9, 10], data transmission [11-13] and processing [14-17], development different systems [18-20], etc.
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