Kinetic Theory of Random Graphs: from Paths to Cycles}
E. Ben-Naim and P.L. Krapivsky
Structural properties of evolving random graphs are investigated.
Treating linking as a dynamic aggregation process, rate equations
for the distribution of node to node distances (paths) and of cycles
are formulated and solved analytically. At the gelation point, the
typical length of paths and cycles, $l$, scales with the component
size $k$ as $l\sim k^{1/2}$. Dynamic and finite-size scaling laws
for the behavior at and near the gelation point are
obtained. Finite-size scaling laws are verified using numerical
simulations.
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