ar X iv : p hy si cs / 9 80 20 35 v 1 [ m at h - ph ] 1 8 Fe b 19 98 THEORY OF HIERARCHICAL COUPLING

Recursion relation between intensity of hierarchical objects at neighbouring levels of a hierarchical tree, strength of coupling between them and level distribution of nodes of the hierarchical tree is proposed. Regular (including Fibonacci), degenerate and irregular trees are considered. It is shown that the strength of hierarchical coupling is a exponentially, logarithmically or power law decreasing function of distance from a common ancestor, respectively. 1 Formulation of the problem Despite widespread occurence of hierarchy in social life and recognizing of its importance to other systems [1], the theory of hierarchically subordinated ensembles has been mainly evolved as a necessary part needed to understand dynamics of spin glasses [2], [3]. The key point is that the hierarchically subordinated objects form ultrametric space. Geometrically, the latter can be conceived of as a Cayley tree (see Fig.1). Degree of hierarchical coupling between objects, w, corresponding to the nodes of given level depends on the distance between them defined by the number of steps m to a common ancestor, so that the ultrametric space is equipped with metrics, ζ ∝ m (ζ is the distance). The primary goal of this work is to show how the function w(ζ) can be derived for different types of the hierarchical trees. Let z k be an intensity of a hierarchical object at the level k, assuming that the intensity z k increases by going from the level k to the nearest higher level k − 1 (it looks like climbing the career ladder). Mathematically, it can be expressed in terms of the simplest recursion relation z k−1 = z k + N −1 k w(z k), (1) where N k is the number of nodes at level k and w(z k) is the required function of hierarchical coupling. In the case of regular tree, shown in Fig.1a, we have the exponentional dependence 1