ISSN 20700482, Mathematical Models and Computer Simulations, 2010, Vol. 2, No. 2, pp. 200–210. © Pleiades Publishing, Ltd., 2010. Original Russian Text © A.P. Mikhailov, D.F. Lankin, 2009, published in Matematicheskoe Modelirovanie, 2009, Vol. 21, No. 8, pp. 108–120.
The Structures of Power Hierarchies A. P. Mikhailov and D. F. Lankin Institute of Mathematical Modeling, Russian Academy of Sciences, Moscow, Russia Moscow State University, Moscow, Russia Received January 24, 2007
Abstract—On the basis of analytical research of the base model of treelike hierarchies, various prob lems of reforming power structures are considered. With the help of corresponding computing exper iments which were made for the solution of some problems of optimization of power systems, optimal (in a certain sense) structures of power hierarchies were defined and the consequences of applying a particular reforming strategy were estimated. Key words: power hierarchies, power distribution, “Power and Society” System, bureaucracy, power defect. DOI: 10.1134/S2070048210020079
1. THE BASE MODEL OF A TREELIKE HIERARCHY The problem of optimizing the power hierarchy is of great importance for the participants of the “Power and Society” System [1–5]. The reformation of power systems (reduction or expansion of the bureaucracy body, the transformation of official power authorities, the formation of new power institu tions or the abolition of old ones, etc.) influence both the functioning of the power system as a whole and the public response as regards to separate (or all) levels. By having an idea of the consequences of applying one or other reformation strategy, one can consider the problem of the selection of a conclusive strategy, which provides certain results. In the paper, some approaches to solve this problem are covered and the results of the conducted numerical experiments are presented. In [4, 5], a dynamic equation of power distribution in treelike hierarchies was obtained and examined. Within the framework of the developed theory, it is possible to raise the question of the optimization of the power hierarchy, which consists in the selection of a specific hierarchy, answering postulated conditions (the set of postulated conditions will be denoted by Θ), from a set of the “accepted” power hierarchies (i.e., answering the same conditions Θ). The construction of a set of the “accepted” power hierarchies, which we will denote by Ω is naturally close to the optimization problem. The structure of a set Ω is determined by the conditions, which are imposed on the hierarchy, i.e., by the set Θ. The postulated conditions obviously should have a clear sociologic interpretation and provide the existence of the solution of the optimization problem solution (i.e., Ω = Ω(Θ) ≠ ∅), meeting the com mon requirements for the solution of the “Power and Society” System (see [1–5]). Thus, we do not restrict the set Θ. In this paper we will consider and solve two different optimization problems (i.e., we will define a set of imposed conditions Θ and under them we will build a set of the “accepted” power hierarchies Ω = Ω(Θ)). Let us proceed to the problem of the optimization of the power hierarchy. We mention that the basic dynamic equation of power distribution (see [5]) is written as follows: ∂p ∂ ∂p ∂p n ( x ) = ⎛ x, t, p, , p 1, p 2, …⎞ n ( x ) + n ( x )F ( x, t, p, p 1, p 2, … ), ⎠ ∂t ∂x ⎝ ∂x ∂x
(1)
with boundary conditions ( x, t, p, … )n ( x ) ∂p ∂x
x=0
∂p = ( x, t, p, … )n ( x ) ∂x 200
= 0. x=l
(2)
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It is also essential to set initial power distribution p ( x, t 0 ) = p 0 ( x ) ≥ 0,
0 ≤ x ≤ l.
(3)
The model (1)–(3) is mathematically closed and wellformed; i.e., it uniquely determines a solution—a smooth nonnegative function p(x, t) for all 0 ≤ x ≤ l and t ≥ t0 (if some restrictions to the input data are unessential for generality). We note the content of variables, which model (1)–(3) includes. The function p(x, t) (solution) describes the spacetime dynamics of the power distribution in the hierarchy, i.e., the place (x coordinate) and time t dependence of a variable (level), which is objectively exercised by the power source. The rate of the p(x, t) function (the left side of equation (1)) is defined by the following variables: (1) The difference of power streams obtained by the mechanism of closerange interaction from the closest neighbors in the hierarchy and given back to them (the first, differential component on the right side (1)); (2) The system response, i.e., the function F(x, t, p1, p2, …), and also by the following prescribed char acteristics of the “Power and Society” System; (3) The minimum and maximum power authorization in the hierarchy—steadily decreasing in x func tions p1(x, t) > 0 and p2(x, t) > 0; ∂p, p , p , …⎞ > 0; (4) The inner behavioral properties of the hierarchy structure—the function ⎛ x, t, p, 1 2 ⎝ ⎠ ∂x (5) The number of bureaucrats in the hierarchical section—the function n(x); (6) The initial power distribution in the structure—p0(x, t) ≥ 0. Thus, the model (1)–(3) is the mathematical implementation of the general pattern of the “Power and Society” System, corresponding to a selforganizing object with different feedforwards and feedbacks. The hierarchical power distribution described by it is set up not voluntarily, but as a result of the interaction of all system components. It should be also noticed that when constructing the model (1)–(3) assumption 1 was essentially used (and also other conditions (for details refer to [1–5])). Assumption 1: the hierarchy under consideration is strongly regular; i.e., it can be considered that all heads related to a single power section have almost the same number of subordinates. One of the most naturally formalized “Power and Society” systems is a legal system. The “Power and Society” System is called legal, if its response to the actions of any source in the hier archy always focuses on retaining the power distribution within the framework of imposed powers (it means that minimum and maximum powers in the hierarchal structure correspond to law in the ordinary way of this conception). Such a response type conforms to the legal public conscience. The previously formulated designation is implemented by the definition of the corresponding function of the system response F(x, t, p, p1, p2, …). We will make an analysis of the model (1)–(3) in the case of a legal system under the following assump tions: (1) The function , corresponding the mechanism of handing over power on command, is regular, i.e., = 0 = const > 0; (2) The public response is a linear function of the departure from the perfect power level and does not depend on t, i.e., F(p, x) = k1(p0(x) – p), where k1 > 0 characterizes the response amplitude; The variable p0(x, t) can be called the “perfect” (“average”) power distribution to mean that in the pro cess of such distribution the public response to the hierarchy activities would be always and everywhere equal to zero (the function p0(x, t) is considered to be monotone to x). (3) The perfect power distribution p and power authorization p1 and p2 do not change in time and lin early decrease with a raise in coordinate x, i.e., p0 = H – kx,
H > 0, p1 = (1– α)p0,
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Here, k > 0 is a decreasing power of the function p0 in x, 0 < α < 1 is the relative difference between p0 and p1, p2 (where p0 = (p1 + p2)/2 is an arithmetic average of the minimum and maximum power authori zation). As a result of the simplifications (1)–(3), we obtain the general model of a treelike hierarchy of the legal system: ⎧ ∂p = ∂ ⎛ n ( x ) ∂p⎞ + n ( x ) ( p 0 ( x ) – p ), ⎪ n ( x ) k1 0 ⎪ ∂t ∂x ⎝ ∂x⎠ ⎪ ⎨ n ( x ) ∂p ∂p = 0 n ( x ) = 0, ⎪ 0 ∂x ∂x x = l x = 0 ⎪ ⎪ p ( x, t 0 ) = p 0 ( x ) ≥ 0, ⎩
0 < x < l,
t > t0 , (4)
t ≥ t0 , 0 ≤ x ≤ l.
We will give a definition of the functional Z(t), characterizing the absolute divergence of the actual power profile p(x, t) from the perfect distribution p0(x, t), taking account of the branching of hierarchy n(x): x
*
∫
l 0
∫
0
Z ( t ) = 2 p ( x, t ) – p ( x, t ) n ( x ) dx = 2 p ( x, t ) – p ( x, t ) n ( x ) dx > 0. 0
x
(5)
*
The functional Z(t) is called a power defect. The point x∗ ∈ (0, l) is a cross point of the perfect and actual power profiles. The equality and positivity of the two integrals on the right side (5) result from the properties of the solutions of parabolic equation (4). It follows from assumption (2) that the public response is oriented to decrease the power defect. Thus, it is found that the greater the power defect the more actively a society responds to the actions of the authorities (it means the overall public response). It may be said that the power defect determines the polit ical tension in society (in the case of the legal power system). The base model of a treelike hierarchy is called the general model of the treelike hierarchy of the legal system (4), when n(x) = eβx. ⎧ βx βx ⎪ e βx ∂p = ∂ ⎛ 0 e ∂p ⎞ + k 1 e ( H – kx – p ), ⎝ ⎠ ⎪ ∂t ∂x ∂x ⎪ βx ⎨ e βx ∂p = 0 e ∂p = 0, ⎪ 0 ∂x ∂x x = 0 x = l ⎪ ⎪ p ( x, t 0 ) = p 0 ( x ) ≥ 0, ⎩
0 < x < l,
t > t0 , (6)
t ≥ t0 , 0 ≤ x ≤ l.
The optimization problem is raised in terms of a base model in a treelike hierarchy (6); its system of equa tions presents the second boundary problem for a quasilinear parabolic equation. The selection of the base model naturally involves a reduction of the great number of hierarchies under consideration (as the base model corresponds to the exponential function of the number of bureaucrats in the section n(x) = exp(βx)), but it does not affect the strong restriction of generality, because most of the real hierarchies are described considerably well in terms of the base model in tree like hierarchies. By varying the parameter β, one can define the hierarchy of the different degree of branching (includ ing the chain hierarchy), whereby the class of “base” hierarchies (with an exponential function of the number of bureaucrats) proves to be wide enough to approximate an actual power hierarchy (with a steadily increasing function n(x)) by a component of this class. Besides, the application of the base hierarchy model enables us in some cases to obtain an analyt ical solution of the optimization problem, which is much easier to examine in contrast to a numerical solution. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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p
1.5 4 2.0 1.0
2
4
1
2 1.0
0.5
1 3
3
0
0.5
x
0.5
0
Fig. 1.
x
Fig. 2.
2. THE BASE MODEL EXAMINATION The stationary solution of problem (6) is found by the formula (a nondimensionalized solution is given here as in [1]) c
–c–q
–c+q
( – c + q )x ( – c – q )x be 1–e 1–e p ( x ) = e + e – bx + 1 – 2abc, – c + q c + q 2 sinh q
0 kl βl where p = p , x = x , 0 < x < 1, a = 2 , b = , c = , q = H l H 2 k1 l
(7)
1 c +. a 2
As is clear, it represents a linear combination of two sign exponents of different signs and a firstdegree polynomial and characterizes a “specific” power distribution. The complete power is a product of “specific” power by the distribution function n(x) of the number of bureaucrats. Quantitative solution (7) (means “specific” power, i.e., the power of one bureaucrat in a hierarchical section) resembles the power distribution in a chain hierarchy. This is shown in Fig. 1. In Fig. 1, line 1 shows a diagram of a specific power distribution in a treelike hierarchy at a given set of parameters: β = 2,
0 = 0.01,
H = l = 1,
k 1 = 0.5,
Line 2 is the perfect power distribution p0(x) = H – kx = 1 – 0.9x, where lines 3 and 4 are the minimum and maximum power authorization p1 = (1 – α)p0, p2 = (1 + α)p0. The complete power diagram is presented in Fig. 2. As is obvious, exceeding of the power authorization takes place at low levels; as a whole, the stationary solution is in the legal environment. Let us compare the power distribution in a chain hierar chy with the power distribution in a treelike one. In Fig. 1, there is a power distribution diagram in n(x) = e2x (a treelike hierarchy), and in Fig. 3 there is a power distribution dia gram in n(x) ≡ 1. This diagram corresponds to a chain hier archy with the same set of parameters (8) (naturally, except ing β = 0). MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
k = 0.9,
α = 0.75.
(8)
p 1.5 4
1.0 2 1 0.5 3 0.5
0
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The diagrams, which are shown in Figs. 1 and 3 are similar to each other; the power distribution func tion both in a treelike hierarchy and a chain one is a steadily decreasing one in x, but there are also some differences: —Firstly, the power distribution in a treelike hierarchy for middle levels is more widely deviated from the perfect power profile and when the exponent β grows, the deviation only increases; —Secondly, the “equilibrium” point (it is a cross point of the perfect and actual power distribution) in the case of a treelike hierarchy is shifted to the right; moreover, as the value β increases, so does the size of the shift. 3. NUMERICAL EXPERIMENTS WITH A BASE MODEL Let us formalize two concepts of a bureaucrat’s power level: (1) The perfect power is one, which a bureaucrat must have in the public opinion (the perfect power is determined by the function p0(x)). (2) The actual power is one, which a bureaucrat really has (the actual power is a solution of the power distribution equation, which includes the public response function F(x, t, p, (x, t), …), the bureaucrat’s behavioral property κ, and the distribution function of the number of bureaucrats n(x); i.e., the actual power depends on all the parameters of the “Power and Society” System. For example, the actual power of the head of a hierarchy is always less than the perfect one and the actual power of the lower bureaucrats is always more (this results from the properties of the solution of the power distribution equation). 3.1. The First Goal of Optimization Formulation. The main goal of the experiment is to compare two different hierarchies exercising the same power authority (by any criterion). We will assume that one of the hierarchies is given and let, for instance, it be a chain one (this is an original hierarchy). Let us solve some possible optimization problem: the hierarch is going to reduce a certain number of levels in the original hierarchy (it is not essential what the reasons are: to express public opinion, the hier arch’s own view, the initiative of other authorities, etc.), and at the same time retain his own perfect and actual power. How should he distribute the bureaucrats for it and how should their perfect power authori zation be defined? To solve this problem one needs to fulfill the following conditions for the hierarchies under examina tion: Condition 1: The complete power of the bureaucracy in the chain (original) hierarchy is equal to the complete power of the bureaucracy in the desired hierarchy. Condition 2: The perfect power of a high official in the original hierarchy is equal to the perfect power of a high official in the desired hierarchy. Conditions 1 and 2 are quite natural: they result from the problem statement and determine a set of power hierarchies, from which a selection of the desired hierarchy will be made. The selection is mainly defined by the following condition: Condition 3: The actual power of a high official in the original hierarchy is equal to the actual power of a high official in the desired hierarchy. Furthermore, we assume that condition 4, which is close to condition 3, is met: Condition 4: The behavioral response of a bureaucrat κ and the public response coefficient k1 are con stant. Thus, conditions 1–4 determine a set of postulated conditions Θ. The set Θ of the “accepted” power hierarchies Ω = Ω(Θ) remains to be built. Mathematical formalization. We will follow the basic symbols of hierarchy parameters. The parameters of the desired hierarchy are primed. Let us write condition 1 mathematically: l
l'
∫ p ( x ) dx = ∫ p ' ( x )n' ( x ) dx. 0
0
0
(9)
0
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Table 1 l'
k'
β'
Z'
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.9 0.897 0.891 0.883 0.87 0.855 0.836 0.84 0.915 1.362
0 0.07 0.2 0.42 0.78 1.39 2.47 4.63 9.83 29.4
0.018 0.018 0.019 0.019 0.020 0.021 0.021 0.021 0.019 0.015
Let us write condition 2 mathematically: H = H'. (10) We will transform condition 1 to find a relation among the parameters of the desired hierarchy, using con dition 2: l
l
∫
0
p ( x ) dx =
0
l'
∫ 0
∫
2
l ( H – kx ) dx = Hl – k , 2
0
l' 0
p ' ( x )n' ( x ) dx =
∫ 0
β'l'
β'x β'l' k' H β'l' e 1 ( H – k'x )e dx = ( e – 1 ) – l'e – + . β' β' β' β'
So, the relation between k', l', and β' was obtained: β'l'
2
β'l' β'l' e k' 1 Hl – k l = H ( e – 1 ) – l'e – + . β' 2 β' β' β'
(11)
The k', characterizing the “slope” of the powers can be determined by the explicit formula: 2
l H β'l' ( e – 1 ) – Hl + k 2 β' k' = . β'l' β'l' e 1 1 l'e – + β' β' β'
(12)
The fulfillment of condition 3 is equivalent to the requirement of the maintenance of the deviation of the specific power for x = 0 (for the head of a hierarchy). With the use of a base model, the specific power deviation both for the original hierarchy and for the desired one may be calculated analytically (under certain parameters). Results. The problem under consideration may be solved, as we know variable l' (a characteristic of the number of levels in the desired hierarchy), we can determine uniquely, using conditions (1)–(4) (retaining complete power, realized by the whole bureau cracy, retaining the perfect and actual power of high officials, and when the behavioral response of the bureaucracy κ' and the public response coefficient k '1 are known), unambiguously determine the value β', which is responsible for the number of bureaucrats in the desired hierarchy and the value k', characterizing the differen tial of ideal powers among bureaucrats in the desired hierarchy. So, the set of the “accepted” power hierarchies Ω consists of at most one component (because it is necessary in addition to establish if the feasible solution has a meaning and delete solu MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
β'
20
10
0.8
1.0
1.2 Fig. 4.
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MIKHAILOV, LANKIN β'
k'
20
1.0
10
0.5
0
0.2
0.4
0.6
0.8 l'
0
0.2
Fig. 5.
0.4
0.6
0.8 l'
Fig. 6.
tions not satisfying model restrictions), which will be determined later for the specific values Θ, estimated by different values of the variable l'. Let us review the results of the numerical experiment for the optimization problem described above. As input data, the following values of parameters were given: the parameters of the chain (original) hierarchy: l = 1, p0(x) = H – kx = 1 – 0.9x, = 0.01, k1 = 1; the parameters of the treelike (desired) hierarchy: ' = = 0.01, k '1 = k1 = 1, H' = H = 1. In Table 1, the results of values k', β', and Z' (power defect), depending on the value l' are presented. Using the data isoline in k', the β'plane can be built (see Fig. 4). Along the isoline, the actual power of the head of the hierarchy is retained. There takes place a clockwise motion that corresponds to the loss in the value l'; the isoline starts at (0.9; 0), fitting the chain hierarchy. The graph of the dependence of the dependence of the value β' on the treelike hierarchy length l' and the graph of the dependence of k' on l' are given. In Fig. 5, a strictly monotone function is shown (the smaller l' the greater β'). The function, whose diagram is shown in Fig. 6, is not a monotone one: when l' decreases, first the value k' decreases weakly and, then, from l' ≈ 0.3, increases rapidly. The interpretation of the results. If the hierarchy’s length decreases, the parameter β' increases rather quickly from l' ≈ 0.3. That is, if a high official wants to reduce the hierarchy more than threefold, they should distribute a great number of bureaucrats at low levels. The number of bureaucrats at a low level is denoted as N'. It will be calculated in the formula N' = exp(l'β'). One can determine how the number of bureaucrats in the treelike (desired) hierarchy changes in com parison with the chain (original) hierarchy. We will consider the number of bureaucrats to be known and to be determined by the length of the lhierarchy. So, the length of the chain hierarchy is a measure of the number of bureaucrats in the chain hierarchy. The measure will be denoted as μ. In this case, l = 1 and thus μ = 1. In the treelike hierarchy, μ' will be obviously calculated by the following formula: l'
μ' =
∫ exp ( β'x ) dx = 0
N' – 1 . β'
(13)
Then, comparing μ with μ', we can determine how the number of bureaucrats in the desired hierarchy changes as against the original hierarchy. In Table 2, we have listed the number of lowlevel bureaucrats and a measure of the number of bureau crats in relation to the length of the treelike hierarchy. Table 2 shows that if l' decreases, the measure μ' becomes smaller as well. In other words, cutting down the levels in a hierarchy involves a “staff reduction.” MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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Table 2 l'
N'
μ'
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
1.0 1.065 1.174 1.342 1.598 2.004 2.686 4.011 7.142 18.916
1.0 0.929 0.87 0.814 0.767 0.722 0.683 0.65 0.625 0.609
For instance, if we decided to reduce levels by half, we have to terminate slightly more than half the bureaucrats. This can explain the small values N' when l' > 0.5 (as bureaucrats have been fired!). It should be noted that N' is the optimal (resulting from retaining the actual power of high officials), but not the maximum possible number of bureaucrats at the low level. For example, if l' = 0.1 (the number of levels were decreased tenfold), the greatest permissible value of β' is equal to β* = 49.4. Then, we obtain N* = 139.77, i.e., the number of low bureaucrats in comparison with the optimal case can be sevenfold greater (naturally one should realize that the average perfect power N* – 1 of the low bureaucrat will be minimal here (nearly zero)). We also note that μ* = = 2.809, i.e., the β* higher bureaucrat has to increase staff nearly three times. In addition their actual power will decrease rap idly (compared with the chain hierarchy) and will relate to the actual power of a low bureaucrat. First, the k'parameter drops weakly; it means that the average perfect power of the bureaucrat in a treelike hierarchy is slightly more than the perfect power of the corresponding level in the chain hierarchy. Then, the k'parameter takes on a greater value than k at l' < 0.3. It means that the average perfect power of a bureaucrat in the treelike hierarchy is less than the perfect power of the corresponding level in the chain hierarchy. As is clear from Table 1, the power defect changes weakly, when the number of levels in the hierarchy varies in terms of the postulated conditions. So, political tension can be regarded to be practically at the same level in the obtained hierarchies. 3.2. The Second Optimization Problem Formulation. Now, we will take not a chain hierarchy but rather a ramified treelike one as the original. Let us solve the following optimization problem: it is needed (it is not important for what reasons) to reduce the number of bureaucrats in a hierarchy (i.e., we have a known value β' beforehand), maintaining, at the same time, the number of levels in the hierarchy (i.e., l' = l). In addition, the perfect and actual power of the low bureaucracy is to remain the same (as well as the complete power exercised by all levels). How should the bureaucrats be distributed and their perfect power authorization be determined in this case? We require the fulfillment of the following conditions for the hierarchies under review: Condition 1: The complete power of the bureaucrats in the hierarchy is retained. Condition 2: The complete perfect power of the low bureaucrats in the hierarchy is retained. Condition 3: The number of levels in the hierarchy remains the same. We will consider the behavioral characteristics of the bureaucrats do not change (i.e., κ' = κ), and we will calculate the public response coefficient k '1 from condition 4: Condition 4: The complete actual power of low bureaucrats in the hierarchy is retained. Conditions 1–4 determine the set of postulated conditions Θ. Now, we define the set of the “accepted” power hierarchies Θ by the set Ω = Ω(Θ). MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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MIKHAILOV, LANKIN H'
H'
2
2
1
1
0
0.5
1.0
k'
0
Fig. 7.
1
β'
2 Fig. 8.
Here, it should be mentioned that within the framework of the “Power and Society” System, we see it is difficult to raise the question of forecasting public behavior, when the system parameters change (for this purpose it is necessary at least to analyze an inverse problem that, in itself, is not simple). Using condition 4, we calculate not the actual but the required public response coefficient (if the actual public response coefficient is the same as the calculated (required) k '1 , condition 4 will be met). Mathematical formalization. We present conditions 1–3 mathematically, following the notation system of experiment 1. l
l'
∫ p ( x )n ( x ) dx = ∫ p ' ( x )n' ( x ) dx, 0
0
0
(14)
0
0
0
p ( l ) exp ( βl ) = p ' ( l' ) exp ( β'l' ),
(15)
l = l'. (16) Having obviously transformed relations (14)–(16), we come to the system of linear equations concerning the parameters k' and H'. Having solved this system, we obtained the following values for the parameters: Π – G, k' = M–F
(17)
where βl
βl
H ( e – 1 ) – kle k ( exp ( βl ) – 1 ) Π = + , 2 β β G = D ( H – kl ) exp [ l ( β – β' ) ],
l exp ( β'l ) 1 – exp ( β'l ) F = + , 2 β' β'
exp ( β'l ) – 1 D = ; β'
M = lD, (18)
H' = k'l + ( H – kl ) exp [ l ( β – β' ) ].
The fulfillment of condition 4 can be checked by directly substituting different values of k '1 in formula (6) (it should be remembered that the complete power of the bureaucracy at a particular level is the product of specific power by the number of bureaucracy at that level, which is defined by the function n(x) = exp(βx)). Thus, the set of the “accepted” power hierarchies Ω consists of at most one component (refer to 3.1), which will be later defined for particular values Θ, determined by different values of β'. The results. We solve the problem under examination, using formulas (14)–(18), and give the results of the numerical experiment. As input data, the following values of parameters were given: the parameters of the original hierarchy: β = 3, l = 1, p0(x) = H – kx = 1 – 0.9x, = 0.01, k1 = 1; the parameters of the desired hierarchy: ' = = 0.01. In Table 3, the results of values k', H', k '1 , and Z', depending on the value β' are presented. For conve nience, Table 3 includes the value N', which is the number of low bureaucrats in the desired hierarchy and the value μ', which is a measure of the number of bureaucrats. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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Table 3 N'
μ'
β'
H'
k'
k '1
Z'
20 18 16 14 12 10 8 6 4 3.32 2.72 2 1.65 1
6.33 5.88 5.41 4.92 4.43 3.91 3.37 2.79 2.16 1.93 1.72 1.44 1.3 1.0
3.0 2.89 2.775 2.64 2.485 2.303 2.08 1.793 1.388 1.2 1.0 0.695 0.5 0.0
1.0 1.052 1.109 1.177 1.258 1.356 1.481 1.645 1.88 1.987 2.096 2.25 2.335 1.481
0.9 0.941 0.983 1.034 1.09 1.155 1.23 1.311 1.379 1.382 1.358 1.247 1.116 0.472
1.0 0.867 0.745 0.619 0.497 0.379 0.265 0.158 0.0588 – – – – –
0.167 0.177 0.189 0.203 0.222 0.247 0.284 0.342 0.447 – – – – –
As seen in the table, the required (to meet condition 4) public response coefficient decreases, when β' decreases, and if the number of bureaucrats in the hierarchy drops rapidly (when N ' < 4 and μ' < 2.16, i.e., the number of bureaucrats decreases more than threefold), one cannot find the value of k '1 , when the actual power of lowlevel bureaucrats is retained (it will be less than the actual power of lowlevel bureaucrats in the original hierarchy). In other words Ω = ∅ at N' < 4. Here, we have to use other mech anisms to meet condition 4 (e.g., to vary the value of '). Using the data from Table 3, the isoline in the (k', H')plane can be built. In the isoline, the perfect power of lowlevel bureaucrats in the desired hierarchy is retained (see Fig. 7). A counterclockwise motion takes place, which corresponds to the loss in the value β'; the isoline starts at (0.9;1.0), fitting β' = 3.0, and ends at (0.472; 2.481), fitting β' = 0 (the chain hierarchy). The diagrams of the H'value (the perfect power of the head of the hierarchy) against the treelike hier archy branching β' (see Fig. 8) and of the k'value (the difference of power authorization) against the tree like hierarchy branching β' (see Fig. 9) are given. The function shown in Fig. 8 is a strictly monotone one: the smaller β', the greater H'. The dependence of k' on β' (see Fig. 9) is more complicated: with the reduction β', the value k' initially increases and then decreases rather quickly from β' ≈ 1.2. The interpretation of the results. H' (the perfect power of high officials) increases practically linearly with the reduction of the number of bureaucrats. If, for instance, it is required to reduce the number of bureaucrats by half (when N' = 8, μ' = 3.37), the perfect power of high official has to be increased almost by half as much again (this is “the payment” for retaining the com plete power of bureaucrats in the hierarchy). First, the k'parameter increases; it means that the relative differ ence of perfect power authorization of a bureaucrat at the level of the desired hierarchy is larger than the corresponding difference at the level of the original hierarchy. Later at β' <1.2, the k'parameter begins to decrease and at β' ≈ 0.4, the value of k' becomes equal to k. Further, the k'parameter begins to decrease rapidly, i.e., the relative difference of the perfect power authorization of the bureaucrat at the level of the desired hierarchy becomes smaller than the corre sponding difference at the level of the original hierarchy.
k' 1.5
1.0
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The parameter k '1 diminishes with a reduction of the number of bureaucrats, i.e., the society should be more loyal to the activities of the powers, in order for the actual power of lowlevel bureaucrats to remain the same. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS
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In addition, the power defect, as seen from Table 3, increases when the number of bureaucrats in the hierarchy changes in terms of the postulated conditions Θ. Therefore, it is arguable that the political ten sion in the obtained hierarchies grows with the reduction of the number of bureaucrats. So, for example, at N' = 8 (the number of bureaucrats was reduced by 3.3 times), the tension in the obtained hierarchy increases twofold, compared with the original hierarchy. It must be emphasized that the requirement to retain the actual power of lowlevel bureaucrats in the hierarchy, owing to a change of the public response coefficient, cannot be performed if the number of bureaucrats (more than by three times) reduces rapidly, though the perfect power of lowlevel bureaucrats will be maintained. CONCLUSIONS The results of the conducted experiments with the base model of a treelike hierarchy show that the optimization problem of a power hierarchy can have a solution in a correct setting. The answers to the fol lowing questions are given: how a society should behave and how a hierarchy should be reorganized to achieve particular goals in the operation of the power system. In spite of the rather simple applied mathematical apparatus, the nontrivial dependences between parameters of the “Power and Society” System that are of some interest for further investigations were obtained. During the experimentations, it is found that in the case of a sudden change (more than by three times) of both the number of levels and the number of bureaucrats, “resonance” effects were formed in the system (a rapid growth of the number of lowerlevel bureaucrats in the first instance and a sharp fall of the relative difference of the power authorization between levels in the second one). In the present paper, we limited ourselves to the base model of a treelike hierarchy and represented the results of two numerical experiments. This extends further the true potential of the possible settings for optimization problems; the main point is that they should be meaningful. The main conclusion is that the problem of the search of optimal structures of power hierarchies can be solved by means of mathematical modeling methods, whose application enables us to estimate, apart from the definition of the hierarchy’s structure itself, a number of consequences arising, when the power system is “modernized”. REFERENCES 1. A. P. Mikhailov, “Mathematical Simulation of Power Distribution in Hierarchical Structures,” Mat. Modelir. 6 (6), 108–138 (1994). 2. A. A. Samarskii and A. P. Mikhailov, Principles of Mathematical Modeling, Ideas, Methods, Examples (Nauka, Moscow, 1997) [in Russian]. 3. A. A. Samarskii and A. P. Mikhailov, Principles of Mathematical Modeling, Ideas, Methods, Examples (Taylor and Francis, 2002). 4. A. P. Mikhailov and A. V. Savel’ev, “Models Ground for Power Hierarchies via Their Microdescription,” Mat. Modelir. 13 (4), 19–34 (2001). 5. A. P. Mikhailov, “Simulation of the System “Power–Society” (Nauka–Fizmatlit, Moscow, 2006) [in Russian].
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