Vladimir Mazalov and Ludmila Truhina
(Institute of Applied Math. Petrozavodsk)
Generating Functions and Cooperation
in Communication Networks
(Petrozavodsk, RuFiDM, Sept. 15-18,
2014)
Betweenness
Myerson vector
Myerson vector is determined by axioms:
А1. For connected component S the sum of values is equal to value of S
(1)
А2.
both players equally obtain or loose the gain of connection
(2)
Determine the characteristic function
The Myerson vector can be calculated as
(3)
where
Characteristic function
(4)
where L – maximal distance between two nodes in the coalition;
ak – number of paths of length k in the coalition.
Characteristic function for grand-coalition
r2
r
0  r 1
For coalition S = {1,2,4,5} L = 2, a1 = 3, a2 = 3
and characteristic function is
Imputation
Step 1. Two directly connected players obtain r. Each of them obtains half, i.e.
r/2. If a player has some connections then he receives r/2 for every connection.
Step 2. Three connected players receive r2 . Each player receives r2/3 for each
path of length 2.
Arguing the same way we obtain imputation:
(5)
— number of paths of length k, contained player i.
Paths, which include player 2:
2
Paths of length 1: {1,2}, {2,4}, {2,5}, i.e.A1  3
A22  4
of length 2: {1,2,4}, {1,2,5}, {4,2,5}, {2,1,3},
A32  3
of length 3: {3,1,2,4}, {3,1,2,5}, {2,1,3,6},
of length 4: {4,2,1,3,6}, {5,2,1,3,6},
A42  2
Then
Generating function for number of paths
We consider a tree gp = (N, E) with root in node p.
Introduce the generating function
— number of paths, included k players (length k −1), which contain the node
p.
Determine in final nodes q of the tree gp
Denote l number of players on maximal path {p, . . . , q}. Consider nodes q,
for which number of players on a path { p, . . . , q} is equal to l − 1. If this node is
not the root p , then we assign
(6)
qi – descendants (offsprings) of node q.
For l = 2 generating function will be determined for all descendants of the
node p. Then we assign
(7)
qi – descendants of the node p.
Proof.
Let q1, . . . , qd — descendants of node q  p. Then each path from the
node q to node s in the tree gq passes through a node qi, and the difference in
the lengths is equal to 1.
If q = p, then to paths from k players with origin in node p, we have to add
the paths from k players, which pass the node p and can be composed from the
and can be composed from the paths included k1 < k players in the tree g q
and
paths included k − k1 players in the tree g q
i j. The number of
these composed paths is determined by the second sum in formula (7).
j
i
x(1+x(1+2x)+x(1+x)+x(1+2x) x(1+x))
x(1+x)
x(1+x+x)
x
We obtain the payoff for player 1
x
x
Chain (linear path)
Maximal distance between two nodes is L = n – 1. The payoff of grandcoalition
i
Evidently Ak1  Akn  1 , An 1  1
Star
Maximal distance between two nodes is L = 2. Characteristic
function is determined by formulas
.
Payoff of player 1, formula (3)
Payoff of any coalition which contains
player 1, depends on the size of coalition.
The number of coalitions from s players
which are not contained player 1 is Cns1
Payoff of player 1, formula (5)
For other players,
formula (3)
formula (5)
Myerson vector for Finland VR
r = 0,9
Tampere
Pieksamaki
Toijala
Juvaskyla
Iisaimi
Orivesi
Seinajoki
Joensuu
Lahti
Varkaus
41,8041
41,1815
34,1785
29,7186
26,2262
26,1892
25,8993
25,6400
24,9033
24,6058
r = 0,5
Tampere
3,6257
Pieksamaki
3,4662
Tikkurila
3,1103
Seinajoki
2,9403
Lahti
2,9142
Jyvaskyla
2,8291
Kouvola
2,8247
Toijala
2,7676
Haapamaki
2,6619
Orivesi
2,6056
r = 0,2
Tampere
Tikkurila
Peiksmaki
Seinajoki
Kouvola
Lahti
Haapamaki
Jyvaskyla
Orivesi
Toijala
0,6172
0,5973
0,5868
0,5753
0,5598
0,4942
0,4749
0,4661
0,4594
0,4527
Main hubs of Finland are Helsinki, Kouvola, Seinajoki, Pieksamaki, Joensuu, Oulu,
Tampere.
r = 0,9
Чжэнчжоу
97,7504
Ухань
87,3205
Сиань
86,7126
Шицзячжуан 81,3611
Чанша
67,3557
Пекин
64,3907
Чэнду
57,5171
Тайюань
56,8764
Цзинань
48,2734
Шэньян
45,6099
r = 0,5
Ухань
8,0690
Чжэнчжоу
7,5028
Чанша
7,2316
Сиань
6,8030
Шицзячжуан 6,4525
Чэнду
5,6921
Гуйян
5,2368
Пекин
5,1592
Тайюань
5,0372
Гуанчжоу
4,9738
r = 0,2
Ухань
0,9669
Чанша
0,9477
Гуйян
0,8255
Чэнду
0,8192
Чжэнчжоу
0,7988
Гуанчжоу
0,7899
Сиань
0,7651
Шицзячжуан 0,7638
Пекин
0,6935
Наньчан
0,6787
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