Cliques and duplication-divergence network growth.

A population of complete subgraphs or cliques in a protein network model is studied. The network evolves via duplication and divergence supplemented with linking a certain fraction of target-replica vertex pairs. We derive a clique population distribution, which scales linearly with the size of the network and is in a perfect agreement with numerical simulations. Fixing both parameters of the model so that the number of links and abundance of triangles are equal to those observed in the fruitfly protein-binding network, we precisely predict the 4- and 5-clique abundance. In addition, we show that such features as fat-tail degree distribution, various rates of average degree growth and nonaveraging, revealed recently for a particular case of a completely asymmetric divergence, are present in a general case of arbitrary divergence.

Phase transition in a traffic model with passing

We investigate a traffic model in which cars either move freely with quenched intrinsic velocities or belong to clusters formed behind slower cars. In each cluster, the next-to-leading car is allowed to pass and resume free motion. The model undergoes a phase transition from a disordered phase for the high passing rate to a jammed phase for the low rate. In the disordered phase, the cluster size distribution decays exponentially in the large size limit. In the jammed phase, the distribution of finite clusters is independent on the passing rate, but it accounts only for a fraction of all cars; the "excessive" cars form an infinite cluster moving with the smallest velocity. Mean-field equations, describing the model in the framework of Maxwell approximation, correctly predict the existence of phase transition and adequately describe the disordered phase; properties of the jammed phase are studied numerically.

Multiscaling in inelastic collisions

We study relaxation properties of two-body collisions on the mean-field level. We show that this process exhibits multiscaling asymptotic behavior as the underlying distribution is characterized by an infinite set of nontrivial exponents. These nonequilibrium relaxation characteristics are found to be closely related to the steady state properties of the system.

Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0

Extremal paths on a random cayley tree

We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution rho(l). Besides, the number of branches from any node is also random. Exact results are derived for arbitrary distribution rho(l). In particular, for the binary 0,1 distribution rho(l)=pdelta(l,1)+(1-p)delta(l, 0), we show that as p increases, the minimal length undergoes an unbinding transition from a "localized" phase to a "moving" phase at the critical value, p=p(c)=1-b(-1), where b is the average branch number of the tree. As the height n of the tree increases, the minimal length saturates to a finite constant in the localized phase (pp(c)) where the velocity v(min)(p) is determined via a front selection mechanism. At p=p(c), the minimal length grows with n in an extremely slow double-logarithmic fashion. The length of the maximal path, on the other hand, increases linearly as v(max)(p)n for all p. The maximal and minimal velocities satisfy a general duality relation, v(min)(p)+v(max)(1-p)=1, which is also valid for directed paths on finite-dimensional lattices.