Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2011-06-10
Physics
Condensed Matter
Disordered Systems and Neural Networks
5 pages plus SI; Minor corrections in V2
Scientific paper
Contrary to initial beliefs, random graph evolution under an edge competition process with fixed choice (an Achlioptas process) seems to lead to a continuous transition in the thermodynamic limit. Here we show that a simpler model, which examines a single edge at a time, can lead to a strongly discontinuous transition and we derive the underlying mechanism. Starting from a collection of $n$ isolated nodes, potential edges chosen uniformly at random from the complete graph are examined one at a time while a cap, $k$, on the maximum allowed component size is enforced. Edges whose addition would exceed size $k$ can be simply rejected provided the accepted fraction of edges never becomes smaller than a decreasing function, $g(k) = 1/2 + (2k)^{-\beta}$. If the rate of decay is sufficiently small ($\beta < 1$), troublesome edges can always be rejected, and the growth in the largest component is dominated by an overtaking mechanism leading to a strongly discontinuous transition. If $\beta > 1$, once the largest component reaches size $n^{1/\beta}$, troublesome edges must often be accepted, leading to direct growth dominated by stochastic fluctuations and a "weakly" discontinuous transition.
Chen Wei
D'Souza Raissa M.
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