In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of the their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks.
Nature has an intrinsic ability to form complex structures and networks via a process known as . By self-assembly, small components (like e.g. molecules) automatically assemble into large, and usually complex structures (like e.g. a crystal). There is an abundance of such examples in the physical world. Lipid molecules form a cell's membrane, ribosomal proteins and RNA coalesce into functional ribosomes, and bacteriophage virus proteins self-assemble a capsid that allows the virus to invade bacteria @cite_19 . Mixtures of RNA fragments that self-assemble into self-replicating ribozymes spontaneously form cooperative catalytic cycles and networks. Such cooperative networks grow faster than selfish autocatalytic cycles indicating an intrinsic ability of RNA populations to evolve greater complexity through cooperation @cite_12 . Through billions of years of prebiotic molecular selection and evolution, nature has produced a basic set of molecules. By combining these simple elements, natural processes are capable of fashioning an enormously diverse range of fabrication units, which can further self-organize into refined structures, materials and molecular machines that not only have high precision, flexibility and error-correction capacity, but are also self-sustaining and evolving. In fact, nature shows a strong preference for bottom-up design.
@cite_19 @cite_12
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Self-assembly is the process by which small components automatically assemble themselves into large, complex structures. Examples in nature abound: lipids self-assemble a cell's membrane, and bacteriophage virus proteins self-assemble a capsid that allows the virus to invade other bacteria. Even a phenomenon as simple as crystal formation is a process of self-assembly. How could such a process be described as "algorithmic?" The key word in the first sentence is automatically. Algorithms automate a series of simple computational tasks. Algorithmic self-assembly systems automate a series of simple growth tasks, in which the object being grown is simultaneously the machine controlling its own growth. In models of early life it has been suggested that life and evolution would be more easily achieved if RNA molecules could interact, rather than function independently; here an in vitro system is designed with several RNA fragments that can assemble into a ribozyme, showing that cooperative networks formed by these fragments outcompete self-catalytic RNA fragments.