Power-Law Size Distribution of Supercritical Random Trees

– The probability distribution P (k) of the sizes k of critical trees (branching ratio m = 1) is well known to show a power-law behavior k. Such behavior corresponds to the mean-field approximation for many critical and self-organized critical phenomena. Here we show numerically and analytically that also supercritical trees (branching ration m > 1) are ”critical” in that their size distribution obeys a power-law k. We mention some possible applications of these results. Introduction. – Ideal trees, either regular or random, play a great role in the description of natural phenomena: many systems, from rivers [1, 2] to blood vessels and lungs [3, 4], to real trees can be described by geometrical branched structures. Other phenomena can be described by branched structures in time (i.e. branching processes), with cascades of events generating new events in a multiplicative fashion: examples span from physics (e.g. nuclear chain reactions and directed percolation) [5], to biology (speciation) [6, 7] and many others disciplines. More recently, trees (and branching processes) have been used to model the mean-field approximation of many non-equilibrium, self-organized critical systems such as the Bak-Tang-Wiesenfeld sandpile and the Bak-Sneppen model [8, 9]. The dynamics of these models has a branching structure, where some local events can trigger more events of the same kind, with some rules given by the branching probability distribution (i.e. the probability pn to trigger n new events) and by the geometric constraints induced by the finite spatial dimension d (that is, different branches can interact when coming to the same region). To model branching processes in infinite dimension (mean-field) the usual assumption is that there are no interactions between different branches: the statistical properties of branching, pn, are preserved, but any spatial constraints are lost. In particular, the mean-field limit of the above mentioned critical models is recovered considering critical branching processes [10, 11]. A critical branching process is characterized by an average branching ratio m = ∑ n npn = 1, so that every generation of branching is (on the average) identical to the preceding ones. What people are usually interested in is the probability distribution P (k) of the tree sizes k, that is, the number of sites (or of branching events, in a branching process jargon) making up the trees. It is a well-known result of branching process theory that P (k) ∼ k for critical trees [12]. This power-law behavior is consistent with the assumption of criticality, as common wisdom suggests.