Analytical and numerical studies of creation probabilities of hierarchical trees
We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure is studied. We argue that a consistent probabilistic picture...
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Olemskoi, A.I. Borysov, S.S. Shuda, I.A. 2017-06-10T14:08:23Z 2017-06-10T14:08:23Z 2011 Analytical and numerical studies of creation probabilities of hierarchical trees / A.I. Olemskoi, S.S. Borysov, I.A. Shuda // Condensed Matter Physics. — 2011. — Т. 14, № 1. — С. 14001: 1-6. — Бібліогр.: 14 назв. — англ. 1607-324X PACS: 02.50.-r, 89.75.-k, 89.75.Fb DOI:10.5488/CMP.14.14001 arXiv:1106.3578 https://nasplib.isofts.kiev.ua/handle/123456789/119973 We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure is studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied analytically, while the creation probabilities of Fibonacci, scale-free and arbitrary trees are determined numerically. Розглянуто аналiтично i чисельно умови утворення рiзних iєрархiчних дерев. Дослiджено зв’язок мiж ймовiрностями утворення iєрархiчних рiвнiв та ймовiрностi об’єднання цих рiвнiв у єдину структуру. Показано,що побудова послiдовної ймовiрнiсної картини вимагає використання деформованої алгебри. Даний розгляд заснований на дослiдженнi основних типiв iєрархiчних дерев,серед яких регулярне i вироджене дослiдженi аналiтично, тодi як ймовiрностi утворення дерева Фiбоначчi, безмасштабного та довiльного дерева визначенi чисельно. . en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Analytical and numerical studies of creation probabilities of hierarchical trees Аналiтичне i чисельне дослiдження ймовiрностi утворення iєрархiчних дерев Article published earlier |
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Analytical and numerical studies of creation probabilities of hierarchical trees |
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Analytical and numerical studies of creation probabilities of hierarchical trees Olemskoi, A.I. Borysov, S.S. Shuda, I.A. |
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Analytical and numerical studies of creation probabilities of hierarchical trees |
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Analytical and numerical studies of creation probabilities of hierarchical trees |
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Analytical and numerical studies of creation probabilities of hierarchical trees |
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Analytical and numerical studies of creation probabilities of hierarchical trees |
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analytical and numerical studies of creation probabilities of hierarchical trees |
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Olemskoi, A.I. Borysov, S.S. Shuda, I.A. |
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Olemskoi, A.I. Borysov, S.S. Shuda, I.A. |
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Аналiтичне i чисельне дослiдження ймовiрностi утворення iєрархiчних дерев |
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We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure is studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied analytically, while the creation probabilities of Fibonacci, scale-free and arbitrary trees are determined numerically.
Розглянуто аналiтично i чисельно умови утворення рiзних iєрархiчних дерев. Дослiджено зв’язок мiж ймовiрностями утворення iєрархiчних рiвнiв та ймовiрностi об’єднання цих рiвнiв у єдину структуру. Показано,що побудова послiдовної ймовiрнiсної картини вимагає використання деформованої алгебри. Даний розгляд заснований на дослiдженнi основних типiв iєрархiчних дерев,серед яких регулярне
i вироджене дослiдженi аналiтично, тодi як ймовiрностi утворення дерева Фiбоначчi, безмасштабного та довiльного дерева визначенi чисельно.
.
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1607-324X |
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https://nasplib.isofts.kiev.ua/handle/123456789/119973 |
| citation_txt |
Analytical and numerical studies of creation probabilities of hierarchical trees / A.I. Olemskoi, S.S. Borysov, I.A. Shuda // Condensed Matter Physics. — 2011. — Т. 14, № 1. — С. 14001: 1-6. — Бібліогр.: 14 назв. — англ. |
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2025-11-24T02:24:29Z |
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Condensed Matter Physics 2011, Vol. 14, No 1, 14001: 1–6
DOI:10.5488/CMP.14.14001
http://www.icmp.lviv.ua/journal
Rapid Communication
Analytical and numerical studies of creation
probabilities of hierarchical trees
A.I. Olemskoi1,2, S.S. Borysov2, I.A. Shuda2
1 Institute of Applied Physics, Nat. Acad. Sci. of Ukraine, 58 Petropavlivs’ka Str., 40030 Sumy, Ukraine
2 Sumy State University, 2 Rimskii-Korsakov Str., 40007 Sumy, Ukraine
Received February 18, 2011
We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connec-
tion between the probabilities to create hierarchical levels and the probability to associate these levels into
a united structure is studied. We argue that a consistent probabilistic picture requires the use of deformed
algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both
regular and degenerate ones are studied analytically, while the creation probabilities of Fibonacci, scale-free
and arbitrary trees are determined numerically.
Key words: probability, hierarchical tree, deformation
PACS: 02.50.-r, 89.75.-k, 89.75.Fb
1. Introduction
As it is shown for diverse systems, ranging from the World Wide Web [1] to biological [2]
and social [3] networks, real networks are governed by strict organizing principles displaying the
following properties: i) most networks have a high degree of clustering; ii) many networks have
been found to be scale-free [4] which means that the probability distribution over node degrees,
being the set of the numbers of links with neighbors, follows the power law.
A formal basis of the theory of hierarchical structures is represented by the fact that hierar-
chically constrained objects are related to an ultrametric space whose geometrical image is the
Cayley tree with nodes and branches corresponding to elementary cells and their links [5]. One of
the first theoretical pictures [6] has been devoted to the diffusion process on either uniformly or
randomly multifurcating trees. The consequent study of hierarchical structures has shown [7] that
their evolution is reduced to an anomalous diffusion process in ultrametric space that arrives at a
steady-state distribution over hierarchical levels, which represents the Tsallis power law inherent to
non-extensive systems [8]. The principal peculiarity of the Tsallis statistics is known to be governed
by a deformed algebra [9].
This paper briefly represents the results of our study of creation conditions of a vast variety of
hierarchical trees on the basis of methods initially developed within the quantum calculus [10]. An
extended version of our analysis is published elsewhere [11]. The outline of the paper is as follows.
In section 2, we discuss the statistical peculiarities of the picture of hierarchical structure creation
to demonstrate that effective energies of hierarchical levels remain to be additive values, while the
set of corresponding probabilities becomes both non-additive and non-multiplicative due to the
coupling between different levels. Further consideration is based on an analytical and numerical
study of the main types of hierarchical trees in section 3. Section 4 is devoted to the discussion of
obtained results.
c© A.I. Olemskoi, S.S. Borysov, I.A. Shuda 14001-1
http://dx.doi.org/10.5488/CMP.14.14001
http://www.icmp.lviv.ua/journal
2. Statistical peculiarities of hierarchical ensembles
As pointed out above, the stationary creation probability of the l-th hierarchical level takes the
Tsallis form
pl = p0 expq
(
−
εl
∆
)
, expq(x) := [1 + (1 − q)x]
1
1−q
+ , [y]+ ≡ max(0, y). (2.1)
Here, p0 is the top-level probability fixed by the normalization condition, q > 0 is a deformation
parameter, εl is an effective energy of the l number, ∆ is an effective temperature. Although the
energy is a key concept of the network optimization theory, it is not always possible to match
its value to a given graph. However, basing on heuristic ideas, it is always possible to attach an
effective value of energy to some phenomenological parameter. Also, for our purpose it is convenient
to consider the nodes of the hierarchical tree as particles of a statistical ensemble, while its edges
represent couplings between these particles.
In contrast to the statistical theory of complex networks [12], the hierarchical systems under
our consideration cannot simultaneously display the properties of additivity of effective energies
and multiplicativity of related probabilities. The cornerstone of our approach is that the creation
of a hierarchical structure does not break the law of the energy conservation, so that the energies
εl remain to be additive values:
ǫn :=
n
∑
l=0
εl. (2.2)
Within the statistical theory of random networks [12], effective energies εl are reduced to a constant
for microcanonical ensemble and are fixed by the set of particular probabilities pl according to the
relation εl = −∆ ln(pl)+ const for both canonical and grand canonical ensembles with an effective
temperature ∆. On the other hand, due to the coupling between different levels, the hierarchy
essentially deforms the corresponding probabilities pl, which become non-multiplicative. Indeed,
the probability Pn to create an n-level hierarchical structure is connected with total energy ǫn by
means of the relation ǫn = −∆ lnq(Pn) with the deformed logarithm lnq(x) :=
(
x1−q − 1
)
/(1− q).
Then, the condition (2.2) leads to the additivity of these logarithms: lnq(Pn) =
∑n
l=0
lnq(pl), and
one obtains the probability relation
Pn := p0 ⊗q p1 ⊗q p2 ⊗q · · · ⊗q pn, (2.3)
where the deformed product is defined as x ⊗q y :=
[
x1−q + y1−q − 1
]
1
1−q
+
. Thus, in contrast to
ordinary statistical systems, the creation probability Pn of a hierarchical structure is equal to the
deformed production of specific probabilities pl related to the levels l = 0, 1, . . . , n.
The above law of the deformed multiplicativity determines the probabilities pl to create a set
of hierarchical levels simultaneously. Another problem emerges when we consider the connection
between the creation probability of a given hierarchical level l and the same for each node at this
level. For simplicity let us consider a regular tree, whose nodes multifurcate to generate a set of
the Nl nodes determined with inherent probabilities π = p0/Nl where p0 is their top magnitude
being a normalization constant. If one permits additivity of the node probabilities, we arrive at
the total probability of the l level realization to be independent of their numbers: pl := Nlπ = p0.
Since the probability pl to create a hierarchical level decays with the level number l, we are forced
again to replace the trivial additive connection of the level probability pl with the node value π by
a deformed sum.
Finally, since the creation probabilities of the hierarchical levels go beyond probabilities related
to non-hierarchical structures, the standard normalization condition based on the use of the usual
sum is broken as well. With the growth of the difference |q − 1|, the probability (2.1) increases at
arbitrary values of the energy εl with respect to the non-deformed value related to the parameter
q = 1. On the other hand, the deformed sum x⊕q y := x+ y+(1− q)xy decreases with the growth
of the parameter q > 1. As a result, one can anticipate that a self-consistent probabilistic picture
of hierarchical ensembles is reached if one proposes the normalization condition
p0 ⊕q p1 ⊕q · · · ⊕q pn = 1, q > 1 (2.4)
14001-2
Creation probabilities of hierarchical trees
that is deformed to fix the top level probability p0.
Taking into account the above statements, one obtains an explicit form of the creation proba-
bility of a hierarchical structure [11]
Pn = expq
[
∑n
l=0
p1−q
l − (n+ 1)
1− q
]
=
(
n
∑
l=0
p1−q
l − n
)
1
1−q
+
. (2.5)
The relations (2.5) mean the decrease of the creation probability with the growth of the hierarchical
tree in accordance with the difference equation
P 1−q
n−1 − P 1−q
n = 1− p1−q
n . (2.6)
In the non-deformed limit q → 1, relations (2.3) and (2.5) are reduced to the ordinary rule
Pn =
∏n
l=0
pl (respectively, equation (2.6) reads Pn/Pn−1 = pn), while at q = 2 the creation
probability (2.5) takes a maximal value.
According to equation (2.5) the subsequent step in the definition of the creation probability Pn
of a hierarchical structure is the determination of a set of probabilities {pl}
n
0 related to different
hierarchical levels.
3. Level probabilities for different hierarchical trees
First, we consider a regular tree whose nodes multifurcate at the level l with constant branching
index b > 1 to generate a set of the Nl = bl nodes determined with inherent probabilities π =
p0/Nl = p0b
−l. Within naive proposition, one could permit additivity of the node probabilities to
arrive at the total probability of the l level realization to be pl := Nlπ = p0. Thus, within the
condition of additivity of the node probabilities, the related values pl = p0 = (n+1)−1 for all levels
appear to be independent of their numbers l = 0, 1, . . . , n.
To avoid this trivial situation, we propose to replace the common additive connection of the
level probability with the node value π by the deformed one. Such deformation leads to the required
level distribution in the binomial form [11]
pl =
[1 + (1 − q)b−l]b
l
+ − 1
1− q
p0 . (3.1)
This probability increases with the growing number l of hierarchical level at q < 1 and decays at
q > 1. From the physical point of view, the creation probability of a lower hierarchical level should
be less than for higher levels, so that one ought to conclude that only the case q > 1 is meaningful.
Characteristically, the form of this distribution very weakly depends on both deformation parame-
ter q and branching index b excluding the domain 2− q ≪ 1, where the probability does not decay
that fast at small values of the branching index b. With large growth of the parameter b ≫ 1 or
level number l, the dependence pl decreases faster to exponentially reach the minimum value
p∞ =
e1−q − 1
1− q
p0 = p0 lnq e. (3.2)
There is a distinctive feature in the behavior of the regular hierarchical tree near the limit
value q = 2 where the dependence (3.1) has no singularity. This feature is corroborated with the
dependence of the top level probability p0 on the deformation parameter. This probability increases
monotonously with the q-growth to reach sharply the limit value p0 = 1 in the point q = 2.
Obviously, this means an anomalous increase of probabilities pl for the whole set of hierarchical
levels. Though, within the domain 2 − q ≪ 1, the ordinary normalization condition
∑n
l=0
pl = 1
is violated appreciably, the definition of the deformed sum shows that the deformed normalization
condition (2.4) can be recovered with the q parameter growing. However, beyond the border q =
2, this condition is not satisfied at all. As a result, we arrive at the conclusion that physically
meaningful values of the deformation parameter are concentrated within the domain q ∈ [1, 2].
14001-3
The difference between regular and degenerate trees is that all nodes multifurcate at each level
in the former case, while the only one node branches in the latter. In this sense, the degenerate
tree can be considered as an antipode of the regular one to be studied analytically. Taking into
account this peculiarity, the creation probability of the lth hierarchical level takes the form [11]
pl =
[
1 + (1− q)b−l
]
∏l
m=1
[
1 + (1 − q)b−m
]b−1
−1
1− q
p0 . (3.3)
Similarly to the case of the regular tree, this distribution decays exponentially fast to the limit
probability p∞ determined by equation (3.2).
Above, we have considered two conceptual examples of hierarchical trees with self-similar struc-
ture, i.e., regular and degenerate trees. By contrast, a scale-free tree has rather random structure,
but the probability distribution over hierarchical levels tends to the power-law form inherent in self-
similar statistical systems. In this case, the probability distribution over tree levels is determined
by the discrete difference equation [7]
pl+1 − pl = −pql /∆, l = 0, 1, . . . , n (3.4)
accompanied with the deformed normalization condition (2.4) (∆ being a distribution dispersion).
In figure 1 we compare the probability distributions over hierarchical levels of scale-free, regular,
(a) (b)
Figure 1. Probability distributions over hierarchical levels for scale-free, regular, Fibonacci and
degenerate trees (curves 1-4, respectively) at ∆ = 2, b = 2, n = 10 and q = 1.9 (a) and
q = 1.9999 (b).
degenerate and Fibonacci trees at different values of the deformation parameter. As it is seen, at all
q-values, that the forms of these distributions are actually similar for all the above trees excluding
the scale-free one. In the latter case, the level probability decays to zero as a power law, whereas
there is a limit non-zero value (3.2) for the regular trees. In accordance with such a behavior, the
creation probabilities depicted in figure 2 decay faster for the scale-free tree than in the case of
the regular and degenerate ones. Characteristically, this difference appears only within the domain
2− q ≪ 1 of the deformation parameter variation.
Finally, let us consider two examples of arbitrary trees, among which the former concerns the
Fibonacci tree (number of nodes at its each level is equal to Fibonacci number), while the latter
relates to the schematic evolution tree shown in figure 3a (in the latter case, the nodes identify
substantial stages in the evolution of life, e.g., human is situated on the 24th level). Using the
approach developed for the node and level probabilities, obeying the normalization condition,
we show that the probability distributions of the Fibonacci tree depicted in figures 1, 2 do not
actually differ from the related dependencies for both regular and degenerate trees. As concerns
the evolution tree, its probability distributions (figure 3b) show that the presence of the stopped
14001-4
Creation probabilities of hierarchical trees
(a) (b)
Figure 2. Creation probabilities of scale-free, regular, Fibonacci and degenerate hierarchical
trees (curves 1-4, respectively) as function of the whole level number at ∆ = 2, b = 2 and
q = 1.9 (a) and q = 1.9999 (b).
(a) (b)
Figure 3. (a) Schematic representation of evolution tree (from Ref. [13]). (b) Creation probability
of the evolution tree vs. the level number at: q = 1.0001, 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.9, 1.9999
(curves 1-9, respectively).
branches (type of two rightmost ones in figure 3a) considerably decreases the creation probability of
new hierarchical levels. Particularly, the probability of human appearance takes the values greater
than 10−4 only at the deformation parameter q = 1.9999.
4. Concluding remarks
To avoid ambiguities it is worthwhile to stress that our consideration concerns rather the prob-
abilistic picture of creation of the hierarchical trees themselves than hierarchical phenomena and
processes evolving on these trees (for example, hierarchically constrained statistical ensembles [14],
diffusion processes on multifurcating trees [6], et cetera).
The principal peculiarity of the probabilistic picture elaborated is a distinction between the de-
formed and non-deformed quantities. Thus, effective energies of hierarchical levels in equation (2.2)
are non-deformed quantities because the creation of hierarchical structures does not break the
conservation law of the energy being an additive value. Moreover, the node probabilities are de-
termined using the non-deformed relations because these probabilities relate to the configuration
of the hierarchical tree itself (in other words, they are determined for geometrical, rather than for
probabilistic reasons). At the same time, the hierarchy appearance essentially deforms the probabil-
14001-5
ity relations (2.3)–(2.5) due to the coupling level probabilities pl . Similarly, the definition of these
probabilities through corresponding node values is based on the use of a deformed summation.
Making use of the deformed algebra leads to an increase of probabilities pl for the whole
set of hierarchical levels assuming an anomalous character near the point q = 2. The deformed
normalization condition (2.4) is fulfilled only at q 6 2, while it is broken beyond the limit q = 2.
As a result, taking into account the fact that the creation probability of a lower hierarchical level
should be less than the one for higher levels, the physically meaningful values of the deformation
parameter belong to the domain q ∈ [1, 2].
References
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Аналiтичне i чисельне дослiдження ймовiрностi утворення
iєрархiчних дерев
О.I. Олємской1,2, С.С. Борисов2, I.О. Шуда2
1 Iнститут прикладної фiзики НАН України, вул. Петропавлiвська, 58, 40030 Суми, Україна
2 Сумський державний унiверситет, вул. Римського-Корсакова, 2, 40030 Суми, Україна
Розглянуто аналiтично i чисельно умови утворення рiзних iєрархiчних дерев. Дослiджено зв’язок мiж
ймовiрностями утворення iєрархiчних рiвнiв та ймовiрностi об’єднання цих рiвнiв у єдину структуру.
Показано, що побудова послiдовної ймовiрнiсної картини вимагає використання деформованої
алгебри. Даний розгляд заснований на дослiдженнi основних типiв iєрархiчних дерев, серед яких
регулярне i вироджене дослiдженi аналiтично, тодi як ймовiрностi утворення дерева Фiбоначчi,
безмасштабного та довiльного дерева визначенi чисельно.
Ключовi слова: ймовiрнiсть, iєрархiчне дерево, деформацiя
14001-6
http://dx.doi.org/10.1038/43601
http://dx.doi.org/10.1038/35036627
http://dx.doi.org/10.1073/pnas.021544898
http://dx.doi.org/10.1126/science.286.5439.509
http://dx.doi.org/10.1103/RevModPhys.58.765
http://dx.doi.org/10.1103/PhysRevLett.57.1965
http://dx.doi.org/10.1134/1.568335
http://dx.doi.org/10.1007/978-0-387-85359-8
http://dx.doi.org/10.1016/j.physa.2004.03.082
http://dx.doi.org/10.1137/S003614450342480
http://dx.doi.org/10.1126/science.300.5626.1691
http://dx.doi.org/10.1016/j.physa.2008.11.019
Introduction
Statistical peculiarities of hierarchical ensembles
Level probabilities for different hierarchical trees
Concluding remarks
|