Geodesic
Equilibrium Resource Distribution in a Model of Group Interaction
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 11 (2011) no. 3, pp. 61-76

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We consider a distributed system represented by weighted bipartite graph $G=(I\cup J, \mathcal{E})$. Each vertex $i\in I$ (agent $i$) possesses a certain amount of resource and distributes it among adjacent vertices $j\in J$ (fields of interaction). Agent $i$ evaluates the efficiency of allocation of its resource in the field $j$ according to value of given function $c_{ij}(x_{ij},\hat{X}_{j})$. Here $x_{ij}$ is the quantity of resource assigned to $j$ by $i$ and $\hat{X}_j$ is the total amount of resources allocated in $j$ by all the adjacent agents. A feasible distribution of resources is called equilibrium distribution, if the following condition is satisfied: $c_{ij}(x_{ij}, \hat{X}_j)=c_i$ for each $(i,j)\in\mathcal{E}$. In this paper we consider the problem of existence of equilibrium resource distributions in systems with linear functions $c_{ij}$ and represented by different kinds of graphs. We formulate sufficient conditions for the existence of equilibriums and obtain explicit expressions to compute these distributions.
Mots-clés : group interaction
Keywords: equilibrium, distributed network. 
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     author = {S. N. Astrakov and I. I. Takhonov},
     title = {Equilibrium {Resource} {Distribution} in a  {Model} of {Group} {Interaction}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {61--76},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2011_11_3_a3/}
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S. N. Astrakov; I. I. Takhonov. Equilibrium Resource Distribution in a  Model of Group Interaction. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 11 (2011) no. 3, pp. 61-76. http://geodesic.mathdoc.fr/item/VNGU_2011_11_3_a3/