Synchronization in interacting Scale Free Networks

We study the fluctuations of the interface, in the steady state, of the Surface Relaxation Model (SRM) in two scale free interacting networks where a fraction $q$ of nodes in both networks interact one to one through external connections. We find that as $q$ increases the fluctuations on both networks decrease and thus the synchronization reaches an improvement of nearly $40\%$ when $q=1$. The decrease of the fluctuations on both networks is due mainly to the diffusion through external connections which allows to reducing the load in nodes by sending their excess mostly to low-degree nodes, which we report have the lowest heights. This effect enhances the matching of the heights of low-and high-degree nodes as $q$ increases reducing the fluctuations. This effect is almost independent of the degree distribution of the networks which means that the interconnection governs the behavior of the process over its topology.Comment: 13 pages, 7 figures. Added a relevant reference.Typos fixe

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

Any Load-Balancing Regimen for Evolving Tree Computations on Circulant Graphs Is Asymptotically Optimal

Abstract. We analyze evolving tree computations on circulant (rings with “regular ” chords) and related graphs. In an evolving α-ary tree computation, a complete tree grows level by level, i. e., every leaf gener-ates α new nodes that become the new leaves. The load balancing task is to spread the new nodes on a network of processors in the moment they were created in such a way that the accumulated number of nodes per processor, i. e., its load, is as close as possible to the average number of nodes per processor. Gao/Rosenberg [2] introduced evolving compu-tations and investigated the growth of complete binary trees on rings of processors. They showed that the so-called ks-regimen behaves opti-mally in the course of long computations. In this paper, we generalize evolving computations to trees of arbitrary degree and we generalize the regimen notion. We show that any regimen behaves optimally. For this purpose, we model the actual load distribution, the generation process, and the distribution regimen by formal infinite polynomials. Then we show that evaluating these polynomials for certain inputs leads to the analysis of these regimens on circulant and related graphs. It is shown that any regimen leads to a close to optimal load distribution.