Geodesic
Equilibrium resource distribution in a~network model
Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 3, pp. 58-68.

Voir la notice de l'article provenant de la source Math-Net.Ru

A network is modeled by a weighted undirected graph $G$. Some certain time invariable resource is assigned to each node and is distributed among the incident edges at each time (time is assumed to be discrete). A state of the network corresponds to a distribution of resources of all nodes among the edges of $G$. At each time a vertex $i$ evaluates its relationship with an adjacent vertex $j$ according to a given function $c_{ij}(x_{ij},x_{ji})$ of the resources $x_{ij}$ and $x_{ji}$ provided by the nodes $i$ and $j$ to the edge $(i,j)$. Since resources of the nodes are redistributed at every time, the state of the system varies in time. Some sufficient conditions are found for the existence of the limit and equilibrium states of the model; and precise formulas are given to compute these states in the case of a special function $c_{ij}$ for an arbitrary graph $G$. 
@article{SJIM_2005_8_3_a7,
     author = {A. I. Erzin and I. I. Tfkhanov},
     title = {Equilibrium resource distribution in a~network model},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {58--68},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a7/}
}
TY  - JOUR
AU  - A. I. Erzin
AU  - I. I. Tfkhanov
TI  - Equilibrium resource distribution in a~network model
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2005
SP  - 58
EP  - 68
VL  - 8
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a7/
LA  - ru
ID  - SJIM_2005_8_3_a7
ER  - 
%0 Journal Article
%A A. I. Erzin
%A I. I. Tfkhanov
%T Equilibrium resource distribution in a~network model
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2005
%P 58-68
%V 8
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a7/
%G ru
%F SJIM_2005_8_3_a7
A. I. Erzin; I. I. Tfkhanov. Equilibrium resource distribution in a~network model. Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 3, pp. 58-68. http://geodesic.mathdoc.fr/item/SJIM_2005_8_3_a7/

[1] Erzin A. I., Astrakov S. N., Takhonov I. I., Gadyatskaya O. A., “Odna zadacha funktsionirovaniya raspredelennoi seti”, Materialy Mezhdunar. seminara “Vychislitelnye metody i reshenie optimizatsionnykh zadach”, Bishkek, 2004, 77–82

[2] Makeev S. P., “O realizuemosti vzveshennykh grafov s zadannymi vesami vershin”, Upravlyaemye sistemy, 1993, no. 13, 40–52

[3] Khakimi S. L., “O realizuemosti mnozhestva tselykh chisel stepenyami vershin grafa”, Kibernetika. Novaya seriya, 2 (1966), 40–53

[4] Adamidou E. A., Kornhauser A. L., Koskosidis Y. A., “A game theoretic/network equilibrium solution approach for the railroad freight car management problem”, Transportation Research Part B: Methodological, 1993, no. 27(3), 237–252 | DOI

[5] Smolyakov E. R., Ravnovesnye modeli pri nesovpadayuschikh interesakh uchastnikov, Nauka, M., 1986 | MR

[6] Fudenberg D., Tirole J., Game Theory, MIT Press, Massachusetts, 1991 | MR

[7] Gibbons R., Game Theory for Applied Economists, Univ. Press, Princeton, 1992

[8] Cobb J. A., Gouda M. G., Musunuri R., “A stabilizing solution to the stable path problem”, Proc. 6 Internat. Symp. “Self-Stabilizing Systems”, San Francisco, 2003, 169–183 | Zbl

[9] Dolev S., Schiller E., “Self-stabilizing group communication in directed networks”, Proc. 6 Intern. Symp. “Self-Stabilizing Systems”, San Francisco, 2003, 61–76 | MR | Zbl

[10] Astrakov S. N., Erzin A. I., “Odna model samoreguliruyuscheisya sistemy”, Matematicheskie struktury i modelirovanie, 13, 2004, 30–38 | MR | Zbl

[11] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR

[12] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966 | MR