=Paper=
{{Paper
|id=Vol-3241/paper2
|storemode=property
|title=Relaxation Time as Unique Characteristic for Networks Clustering
|pdfUrl=https://ceur-ws.org/Vol-3241/paper2.pdf
|volume=Vol-3241
|authors=Andrii Snarskii,Dmytro Lande,Oleh Dmytrenko
|dblpUrl=https://dblp.org/rec/conf/its2/SnarskiiLD21
}}
==Relaxation Time as Unique Characteristic for Networks Clustering==
https://ceur-ws.org/Vol-3241/paper2.pdf
Relaxation Time as Unique Characteristic for Networks
Clustering
Andrii Snarskii 1,2, Dmytro Lande 1,2 and Oleh Dmytrenko 1,2
1
National Technical University «Igor Sikorsky Kyiv Polytechnic Institute», 37, Prosp. Peremohy, Kyiv, 03056,
Ukraine
2
Institute for Information Recording of National Academy of Sciences of Ukraine, 2, Mykoly Shpaka Street, Kyiv,
03113, Ukraine
Abstract
This paper researches new unique characteristics of networks – a network relaxation time and
an individual node relaxation time, which characterize the stability of a complex network and,
accordingly, each node separately to external perturbations. While researching of the complex
networks, it is assumed that relaxation time is the number of iterative steps of the corresponding
algorithm required to achieve the initial equilibrium numerical values of a certain characteristic
after some external perturbation. In other words, the network relaxation time for each node
characterize the resistance of a complex network and the individual node relaxation time
characterize the resistance of each node to external perturbations, accordingly. In this work, to
compute the relaxation time, the decelerated iterative HITS algorithm is used. It is shown, that
these characteristics are unique numerical characteristic of network nodes, and they can be
used to find the centroids of clusters and combine nodes into groups according to these
characteristics – for complex networks clustering. The approbation of the presented
characteristics of the relaxation time and the individual relaxation time was carried out on the
example of clustering of random networks with clearly expressed clusters. In particular, a
randomly generated matrix with dimension 30×30 and 3 clusters and a matrix with dimension
100×100 and 4 clusters were researched.
Keywords 1
Complex Network, Network Relaxation Time, Individual Node Relaxation Time, HITS,
PageRank, Clustering, Centroids of Clusters
1. Introduction
Complex networks are widespread in nature and technology. For example, networks such as the World
Wide Web, peer-to-peer networks [1] and others are complex [2, 3]. They have non-trivial topological
properties and, therefore, are of great interest to research.
Despite the fact that various networks, such as electrical, transport, and information networks, fall
within the scope of the consideration of the theory of complex networks, the greatest contribution to
the development of this theory is made by research of social networks [4], where binary relationships
between people in a group can be represented in the form of a network, so these networks also play an
important role. In these networks, each object is a node, and connections between nodes are an edge or
link [5, 6].
An analysis of complex networks that includes the study of statistical and dynamical structural
properties changes that characterise their behaviour, as well as the prediction of the evolution of such
networks, are important directions [7].
Division of a set of network nodes (objects) into subsets (clusters), within which these objects are
strongly connected, and the connection between individual clusters is relatively small, i.e. partitioning
into clusters and, for example, detecting communities in social networks [8, 9] is an urgent task.
XXI International Scientific and Practical Conference "Information Technologies and Security" (ITS-2021), December 9, 2021, Kyiv, Ukraine
EMAIL: asnarskii@gmail.com (A. Snarskii); dwlande@gmail.com (D. Lande); dmytrenko.o@gmail.com (O. Dmytrenko)
ORCID: 0000-0002-4468-4542 (A. Snarskii); 0000-0003-3945-1178 (D. Lande); 0000-0001-8501-5313 (O. Dmytrenko)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
13
There are many methods of clustering graphs of complex networks [10, 11], which differ in the idea
of combining similar nodes. Since different models use different algorithms, different clustering models
are distinguished, in particular, such as connectivity models, centroid, statistical, group, neural models
and others. For example, there are models based on connectivity. In these models, objects that are closer
in space are more similar (related). And also, there are models based on finding the centroid. In these
models, clusters are represented in the form of a central vector.
In this work, we propose to use a new characteristic for clustering complex networks – the relaxation
time [12, 13].
2. Cluster network analysis
Cluster network analysis, as a rule, solves the problem of two-criteria optimization, namely:
1. Within each cluster 𝐾, elements (nodes) should be connected as much as possible, that is
𝑎 → 𝑚𝑎𝑥.
(1)
∈
∈
Hereinafter, 𝑎 is some estimate of the relationship between the elements with indices 𝑖 and 𝑗, which
are included in the cluster 𝐾. 𝑎 can take non-negative values. This estimate can be calculated in various
ways, for example, as the shortest path between nodes 𝑖 and 𝑗 [14].
2. The connectivity between any separate different clusters, for example, 𝐾 and 𝐾 should be
minimal, that is
𝑎 → 𝑚𝑎𝑥 .
∈
∈
(2)
Often, in the general case, the sums for all indices are estimated (𝑁 is a number of clusters):
𝑎 → 𝑚𝑎𝑥 , 𝑎 → 𝑚𝑖𝑛 .
∈ ∈ (3)
∈ ∈
By association, in our case, the first factor in the formula for determining the weight 𝑤 corresponds
to the first requirement, if it is of great importance, then it is connected by strong ties with a certain
number of nodes (including from its own group). If the parameter 𝑥 is of great importance, then the
bonds are concentrated within their own group (cluster) – this corresponds to requirement 2.
We can assume that the individual nodes with the highest discriminant weight (the number chosen
in advance) will constitute the centroids for clustering, or the basis for formation of clusters that will
form the basis of the clusters.
There are numerous clustering algorithms such as K-means [15], LSA/LSI [4], most of which are
used for networks. Recently, the algorithm based on the modularity property has been widely used due
to the fact that it is built into the Gephi system [16].
By definition, modularity is equal to the proportion of the total number of edges that fall into a given
cluster minus the predicted numbers of edges that would fall into the same groups if they were randomly
distributed. The value of modularity lies in the interval [0,1]. For a given partitioning of network nodes
into clusters, modularity shows how many links in clusters in comparison to the links that are randomly
distributed between all nodes without paying attention to clusters. There are various methods for
calculating modularity. In the most commonly accepted version of the concept, edges are randomized
in such a way that the degree of each node is preserved. The difference between the real number of
edges between nodes 𝑖 and 𝑗 and the predicted number of edges between them is
𝑘𝑘
𝑎 , (4)
2𝑚
where 𝑚 is a number of edges in network, 𝑘 and 𝑘 are degrees of the nodes 𝑖 and 𝑗, respectively.
14
The modularity is defined as the sum of all pairs
1 𝑘𝑘
𝑄 𝑎 𝛿 𝑐 ,𝑐 , (5)
2𝑚 2𝑚
,
where 𝛿 is Kronecker delta (shows whether nodes 𝑖 and 𝑗 are in the same cluster).
The matrix formulation of modularity is as follows. By definition, if the node belongs to the cluster
then 𝑆 is equal to 1and else it is equal to 0. Then
𝛿 𝑐 ,𝑐 𝑆 𝑆 , (6)
1 𝑘𝑘 1
𝑄 𝑎 𝛿 𝑐 ,𝑐 𝑇𝑟 𝑆 𝐵𝑆 , (7)
2𝑚 2𝑚 2𝑚
,
where 𝑆 is a matrix, having elements 𝑆 , and 𝐵 is the so-called modularity matrix, which has elements
(4).
The modularity of a network without clusters is equal 0 cause the sum of all columns and rows of
the matrix that corresponds modularity is equal to 0.
Figure 1 shows a fragment of the interface of the Gephi system, which implements the calculation
of the modularity function.
Figure 1: Fragment of the interface for calculating the modularity in the Gephi system
3. Physical meaning of relaxation time
In physics, relaxation is understood as the process of establishing thermodynamic (including stochastic)
equilibrium in a physical system. It is assumed that the system consists of a large number of particles.
The system is characterized by a large number of parameters, and different sets of parameters move
towards equilibrium at different speeds. The parameters characterizing the microstate of the system are
the first to relax. And ultimately, we can talk about local equilibrium when the thermodynamic
parameters relax, such as example, temperature, the concentration of particles or charges, etc.
For such phenomenological parameters responsible for transfer processes in physical systems, the
Le Chatelier-Brown principle was formed [17]. If the physical quantity 𝑓 characterizing the system
15
(temperature, charge density, etc.) has the value 𝑓 і |𝑓 𝑓 |/𝑓 ≪ 1, then in the first approximation
𝑓 that go to equilibrium is given by the kinetic equation
𝑑𝑓 (8)
𝜆 𝑓 𝑓 ,
𝑑𝑡
The solution of this equation is
(9)
𝑓 𝑡 𝑓 𝑓 0 𝑓 𝑒
where 𝑓 0 𝑓 is an initial perturbation, and
1 (10)
𝜏
𝜆
is a relaxation time.
So, for example, for an environment with conductivity 𝜎 and the relative permittivity 𝜀, Maxwell's
equation and Ohm's law𝚥⃗ 𝜎𝐸⃗ are can be written as
𝜕𝜌 (11)
𝑑𝑖𝑣𝜀𝐸⃗ 4𝜋𝜌, 𝑑𝑖𝑣𝚥⃗ 0
𝜕𝑡
where 𝐸⃗ is an electric field strength, 𝚥⃗ is a charge density.
In the simplest case of a homogeneous environment
𝜕𝜌 𝜎 (12)
4𝜋 𝜌 0
𝜕𝑡 𝜀
the next equation immediately follows
(13)
𝜌 𝑡 𝜌 𝑒
where the relaxation time is
𝜀 (14)
𝜏
4𝜋𝜎
i.e., the higher conductivity 𝜎 and the lower relative permittivity 𝜀, the faster the heterogeneity of the
charge distribution is resolved, i.e., the equilibrium is reached faster.
The concept of relaxation is used in the methods for approximate solution of systems of linear equations,
in iterative methods, for example, in methods of coordinate relaxation, block and group relaxation, etc.
These examples can be described in the following terms. Let us consider a system of equations 𝐿⃗𝑓
0, the solution of which is the function 𝑓 . To find solutions, some initial approximation 𝑓 of the
function 𝑓 is chosen and such an iterative procedure is found. In explicit form it is represented as
𝑓 𝑇𝑓 (15)
or in implicit form as
𝐹 𝑓 ,𝑓 0 (16)
that is, there is such an operator 𝑇 that
lim 𝑓 𝑓 (17)
→
Here we assume an approach and terminology that allows us to consider the described methods in
the only way.
Let us consider a system described by a set of parameters (numbers, functions) 𝑓. The equilibrium
state of the system is given by equations (11-12)
𝐿𝑓 0 (18)
is a solution for which 𝑓 defines the equilibrium state of the system. Let's perturbing the system out of
equilibrium by defining some set 𝑓 𝑓 . By choosing some iterative scheme (15), we get the sequence
𝑓 ,𝑓 ,…,𝑓 (19)
16
where 𝑛 is considered as the discrete time.
In the usual case, for a descending sequence|𝑓 𝑓 |, |𝑓 𝑓 |, … let’s define a value △, such that
if
𝑓 𝑓 △ 𝑓 ,𝑓 ,…,𝑓 (20)
that value 𝑛 , which determines (20) will be considered (called) as the relaxation time 𝜏 𝑛 .
So, in the case of researching complex networks, the relaxation time is the number of iterative steps
𝜏 of the corresponding algorithm, which are necessary to reach the initial equilibrium numerical values
of a certain characteristic after its perturbation with some predetermined accuracy △ – the condition of
convergence (usually △=10-4).
When the recovery of the whole network is researched, the number of iteration steps required for
the value of each node to recover (the whole network is recovered) is called the relaxation time of the
network 𝜏 after the perturbation of a certain m-th node or 𝑚𝑎𝑥 𝜏 (k = 1,..,N, where N is a number
of nodes in network). And the number of iterations which are required to recover the particular node
whose numerical value was perturbated, will be called the individual relaxation time indicator of the
node 𝜏 (or simply 𝜏 ). That is, the first characteristic characterizes the node in terms of the stability
of the whole network after an external perturbation, the last one characterizes the individual stability of
the node after its perturbation.
4. Algorithm
For example, let's consider some complex network (a directed graph), for which PageRank [18] (𝑃𝑅 of
each node) can be found. PageRank is one of the algorithms for evaluating the importance and ranking
of web pages by hyperlinks, it was created by Sergey Brin and Larry Page in 1996 at Stanford
University. The principle of calculating the PageRank of nodes is based on the model of "random
wandering" of the user according to the following algorithm: he opens a random node (web page) from
which he goes by a randomly selected link. It then moves to another web page and activates a random
link again, and so on, constantly going from page to page, never coming back. Sometimes, when, with
some probability 1 𝛿, he gets bored with this wandering, or the page does not have links to other
pages, he goes to a random web page again – not by a link, but by manually typing in some URL. It is
assumed that the probability that a user wandering the network will go to a certain web page is its rank.
Obviously, a node's PageRank is higher the more other nodes link to it, and the more popular those
nodes are.
Let's assume, there are n nodes 𝑑 , 𝑑 , . . , 𝑑 that refer to some node (web page 𝐴), and 𝐶 𝐴 is
the total number of links from a node to other nodes. Some fixed value 𝛿 is defined as the probability
that the user, viewing any web page from the set 𝐷, will go to the node 𝐴 by a link, and not by typing
its URL explicitly.
Within the framework of the model, the probability of this user continuing to surf the web from 𝑁
web pages without using links, by manually entering an address (URL) from a random page will be
1 𝛿 (alternative to following links). The PageRank index for a node is considered as the probability
that the user will be at this node at some random moment in time:
1 𝛿 𝑃𝑅 𝑑 (21)
𝑃𝑅 𝐴 𝛿 .
𝑁 𝐶 𝑑
According to this formula, the node rank is calculated by a simple iterative algorithm.
In a complex network of 𝑁 nodes, the 𝑃𝑅 value of each node 𝑃𝑅 , 𝑖 1,2, … , 𝑁 determines the
equilibrium state of this network. After selecting the value of 𝑃𝑅 for each node 𝑖 that deviates from the
value 𝑃𝑅 𝑃𝑅 0, the iterative procedure for finding 𝑃𝑅 is used. The relaxation time of node 𝑗 is
the number of steps 𝑚 for which
𝑃𝑅 𝑃𝑅 △ (22)
17
Note that there are different variants of setting initial values (deviations) for nodes. They can be the
same, they can alternately set the deviations of individual nodes, leaving others at equilibrium values,
etc.
In this work, as a characteristic whose numerical value is perturbated, a characteristic corresponding
to the iterative HITS (Hyperlink Induced Topic Search) rank algorithm was used. This algorithm was
proposed and developed in 1998 by J. Kleinberg [19] to select the best "authors" (primary sources that
refer to other documents) and "hub" (documents that refer to these primary sources) from an array of
documents. For each document j, its authority 𝑎 𝑗 (Authority) and hub ℎ 𝑗 are calculated according
to the formulas:
𝑎 𝑗 ∑→ ℎ 𝑖 , ℎ 𝑗 ∑→ 𝑎 𝑖 , (23)
Let introduce the relaxation time of the k-th node in the complex system – 𝜏 . At the initial stage, a
set of values of nodes 𝑠 ( 𝑠 ⃑ is a vector form) define as the equilibrium state. This state is determined
in accordance with the certain rule, for example, – the HITS or PageRank iterative algorithm, or any
other.
Calculation of 𝑠 ⃑ in accordance with the selected rule (the iterative algorithm) can always be
rewritten in iterative form:
𝑠⃑ 𝑛 1 𝑠⃑ 𝑛 𝐿𝑠⃑ 𝑛 , 𝑛 0,1, … (24)
where the numbers of nodes correspond to the vector elements numbers of 𝑠⃑; 𝐿 is the operator of one
the iterative HITS or PageRank algorithms that are considered in this work; 𝑠⃑ 0 is the initial values of
nodes.
𝑠⃑ 0 𝑙𝑖𝑚 𝑠⃑ 𝑛 , 𝑛 0,1, … (25)
→
and, respectively
𝐿𝑠⃑ 0 0. (26)
The initial values of nodes 𝑠⃑ 0 is considered as the solution – 𝑠 ⃑ (these values are equilibrium).
Next, we propose to make some deviation of the score (called perturbation) of the m-th node, as it is
shown in the next example:
𝑠⃑ 0 𝑠⃑ 𝛼𝛿 𝑠⃑ , (27)
where 𝛼 defines the deviation (perturbation) score of the m-th vector element; 𝛿 is the Kronecker
symbol.
The described above means vector forms of the deviation from the equilibrium state one of the
elements of the vector 𝑠 ⃑. The perturbation of vector 𝑠 ⃑ due to the perturbation of one of its elements
leads a non-equilibrium state of the system.
According to the equation (24), for n=0 it can be rewritten:
𝑠 𝑠 0 ∑ 𝐿 𝑠 0 , (28)
taking into account (27) it looks as follows:
𝑠 1 𝑠 0 ∑ 𝐿 𝑠 𝛼∑ 𝐿 𝛿 𝑠 𝑠 𝛼𝑞 , (29)
where the vector 𝑞 ∑ 𝐿 𝛿 𝑠 can be defined as
𝐿 𝐿 ... 𝐿 ... 𝐿 0 𝐿
𝐿 ... ... 𝐿 ... 𝐿 𝐿
⎛ ⎞⎛ 0 ⎞ ⎛ ⎞
𝑞 ⎜ : : ⎟⎜ : ⎟ ⎜
:
⎟𝑠 . (30)
⎜𝐿 : ⎟ ⎜𝑠 ⎟ ⎜ : ⎟
: : : :
⎝𝐿 𝐿 𝐿 ⎠⎝ 0 ⎠ ⎝𝐿 ⎠
Taking into account the initial condition 𝑠 1 , 𝑠 𝑛 come back to the initial equilibrium state 𝒔 ⃑
after increasing a number of iterations n in accordance to (25).
18
The initial scores 𝑠 0 𝑠 , 𝑘 𝑚. After 𝑛 10 (more than 10 step) the values 𝑠 𝑛 10
decrease and convergence to the value μ.
Figure 2: Shema of values convergence the of the k‐th node
The value 𝜏 is the k-th node relaxation time after the perturbation of the m-th node, when the
following condition is satisfied
𝑠 𝑛 𝜏 𝜇. (31)
In the other words, the maximum value of 𝜏 for ∀ 𝑚𝑎𝑥 𝜏 after the perturbation of the m-th
node is the network relaxation time [12] for the m-th node.
And respectively, the relaxation time of perturbed node called the individual relaxation time of this
node.
5. Deceleration of algorithm
When calculating the relaxation time for many small networks, a problem arises, which is that the
relaxation time (a discrete number of algorithm steps) can be very small, that is, the effect of "small
distributive force" occurs. To solve this problem, it is proposed to apply an artificial deceleration of the
iterative algorithm. That is, the so-called decelerated iterative algorithms for HITS and PageRank can
be used to calculate the relaxation time. Cause the number of connections between the nodes is large,
the process of iterative recalculation of the values of the nodes after perturbation of some node and
achieving their initial values is fast. It means that we need to make only a few iterative steps to achieve
the initial equilibrium state of nodes after perturbation. In other words, a network relaxation is fast. In
order to decelerate the relaxation time, we propose to decelerate the HITS or PageRank algorithm,
respectively. After the deceleration, the process of convergence to the equilibrium initial values of nodes
after their perturbation will be slowed. So, the corresponding decelerated HITS or PageRank algorithm
is applied
𝑃𝑅 𝑗 → 𝑃𝑅 𝑗 → 𝑃𝑅 𝑗 →. . . . → 𝑃𝑅 𝑖 , (32)
𝐴𝑃𝑅 𝑖 𝑃𝑅 𝑖 , 𝐴𝑃𝑅 𝑖 𝑃𝑅 𝑖 , (33)
𝑃𝑅 𝑖 ⎯ 𝛽𝑃𝑅 𝑖 , (34)
where 0 𝛽 1 is a deceleration factor, 𝐴 is the HITS or PageRank algorithm operator.
19
6. Examples of network analysis
Research of complex networks shows that the values of the network characteristics of nodes after
external perturbation and the next recalculation of these characteristics, recover their initial equilibrium
values during some individual time for each node. Comparing the relaxation time with the network
characteristic, the value of which was perturbated, a vague dependence exists, which shows that the
relaxation time is a unique and incomparable numerical characteristic of network nodes [12].
Within this work, the individual relaxation time 𝜏 is researched, i.e., the relaxation time of the
node, which was deviated from the equilibrium state. In this case, we will further use the notation 𝜏 ,
omitting the upper symbol in 𝜏 .
Also in the work, the general relaxation time of the network for the 𝑚-th node is researched –
𝑚𝑎𝑥 𝜏 the highest value of 𝜏 among ∀ when the 𝑚-th node was disturbed.
The approbation of the presented characteristics of the relaxation time and the individual relaxation
time was carried out on the example of clustering of random networks with clearly expressed clusters.
In particular, a randomly generated matrix with dimension 30×30 and 3 clusters and a matrix with
dimension 100×100 and 4 clusters were researched.
For each network described by the corresponding matrix, after perturbing the value of the network
characteristic ℎ 𝑗 using decelerated HITS algorithm, the network relaxation time and the individual
node relaxation time were calculated. For each network, the deceleration factor 𝛽 of the HITS algorithm
and the convergence condition 𝜇 of the numerical values ℎ 𝑗 and 𝑎 𝑗 to the preperturbed values were
chosen individually.
Also, in order to avoid large numerical values that sometimes arise after the using of decelerated
algorithms, all the resulting values of the relaxation time indicators were normalized in the interval
[0,1].
Figures 3 and 4 show a visual representation of networks. In each of the figures, the nodes are colored
according to the modularity class (figures 3(a) and 4(a)) and by the network relaxation time and the
individual node relaxation time (for the presented examples, the normalized numerical values of these
characteristics coincide and are presented in the form of node labels – Figures 3(b) and 4(b)).
The relaxation time of the network, which presented in Figure 1, was obtained as a result of
deceleration the HITS algorithm with a deceleration factor 𝛽 0.9 and convergence condition 𝜇
10 . For the network presented in figure 2 – 𝛽 0.9 та 𝜇 10 .
(a) (b)
Figure 3: An example of a network corresponding to a randomly generated 30x30 matrix with 3
clusters where groups of nodes are united (colored) by a) modularity class and b) relaxation time
(numeric values are represented as node labels)
20
(a) (b)
Figure 4: An example of a network corresponding to a randomly generated 100x100 matrix with 4
clusters where groups of nodes are united (colored) by a) modularity class and b) relaxation time
(numeric values are represented as node labels)
It can be noticed that the groups of nodes united by the modularity class differ from the groups obtained
by the union of nodes by the relaxation time. This means that the network relaxation time and the
individual node relaxation time are unique network node characteristics and can be used to find cluster
centroids and cluster nodes based on these characteristics.
7. Conclusion
This paper researches the proposed network node characteristics – the network relaxation time and the
individual node relaxation time. The approbation of the presented characteristics of the relaxation time
and the individual relaxation time was carried out on the example of clustering of random networks
with clearly expressed clusters. It is shown, that these characteristics are unique numerical characteristic
of network nodes, and they can be used to find the centroids of clusters and combine nodes into groups
according to these characteristics, in other words, it can be used for complex networks clustering.
The proposed and researched in this work numerical network and node characteristics can be used
during research and analysis of the network structure, making it possible to identify the most important
structural elements. Also, results of the research can be used when building personal search interfaces
for users of information and search systems. In turn, it will simplify the process of finding the necessary
information.
The given algorithm, like the well-known LSA algorithm, is a cluster analysis algorithm that uses a
matrix representation of data. The complexity of the algorithm is determined by the complexity of the
PageRank, HITS or any other iterative algorithm, which is used to recalculate node weights.
The novelty of the algorithm is defined by the approach for calculating the weight of nodes (the
values of relaxation time of network or relaxation time of nodes), the most significant of which can be
used as centres for determining clusters (centroids). If it is necessary to improve the quality of the
proposed approach, certain centroids can be passed as input to other well-known algorithms such as K-
means.
21
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