# Scalability, Random Surfaces, and Synchronized Computing Networks

# Scalability, Random Surfaces, and Synchronized Computing Networks

In most cases, it is impossible to describe and understand complex system dynamics via analytical methods. The density of problems that are rigorously solvable with analytic tools is vanishingly small in the set of all problems, and often the only way one can reliably obtain a system-level understanding of such problems is through direct simulation. This chapter broadens the discussion on the relationship between complexity and statistical physics by exploring how the computational scalability of parallelized simulation can be analyzed using a physical model of surface growth. Specifically, the systems considered here are made up of a large number of interacting individual elements with a finite number of attributes, or local state variables, each assuming a countable number (typically finite) of values. The dynamics of the local state variables are discrete events occurring in continuous time. Between two consecutive updates, the local variables stay unchanged. Another important assumption we make is that the interactions in the underlying system to be simulated have finite range. Examples of such systems include: magnetic systems (spin states and spin flip dynamics); surface growth via molecular beam epitaxy (height of the surface, molecular deposition, and diffusion dynamics); epidemiology (health of an individual, the dynamics of infection and recovery); financial markets (wealth state, buy/sell dynamics); and wireless communications or queueing systems (number of jobs, job arrival dynamics). Often—as in the case we study here—the dynamics of such systems are inherently stochastic and asynchronous. The simulation of such systems is nontrivial, and in most cases the complexity of the problem requires simulations on distributed architectures, defining the field of parallel discrete-event simulations (PDES) [186, 367, 416]. Conceptually, the computational task is divided among n processing elements (PEs), where each processor evolves the dynamics of the allocated piece. Due to the interactions among the individual elements of the simulated system (spins, atoms, packets, calls, etc.) the PEs must coordinate with a subset of other PEs during the simulation. For example, the state of a spin can only be updated if the state of the neighbors is known. However, some neighbors might belong to the computational domain of another PE, thus, message passing will be required in order to preserve causality.

*Keywords:*
biased surface diffusion, coarse graining, dynamic network flows, molecular beam epitaxy model, nonlinear biased diffusion, parallel computers, scalability, time horizon, universality classes

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