Seminarium Fizyki Materii Skondensowanej

We consider models of a growing tree with the growth process driven by therules of tournament selection, where a new node is attached to a contestantnode at the best hierarchy level (closest to the tree root). The proposedevolution reflects limited information about the network topology that isavailable for new nodes.Two cases are considered: the constant tournament (CT) model where thenumber of tournament participants is constant throughout the treeevolution, and the proportional tournament (PT) model where it is growsproportionally to the actual tree size. The results of analyticalcalculations based on a rate equation fit well to numerical simulations forboth models.In the CT model all hierarchy levels emerge in the tree but thebirth time of the hierarchy level increases exponentially or faster withlevel number.The number of nodes at the first hierarchy level (just below the root)grows logarithmically in time, while the size of the last, ``worst''hierarchy level oscillates quasi log-periodically. In the PT model theoccupations of the first two hierarchy levels increase linearly but worsehierarchy levels either do not emerge at all or appear only by chanceduring an early stage of tree evolution and stop growing soon afterwards.We observe that information restrains the emergence of new hierarchy levelsand that it is the absolute amount of information, not relative, thatgoverns this behavior.The explanation for these results is that the larger the choice where tojoin, the more likely a new node connects to ''top'' nodes, thus slowingdown the emergence of new hierarchy levels.Ref. PHYSICAL REVIEW E 89, 062810 (2014)