SWhat changed
Add edges at random to a graph with n labelled vertices, giving every graph with the same number N of edges equal probability. Erdős and Rényi identify a change as N/n approaches a fixed density c. For fixed 0<c<1/2 the largest connected component has logarithmic order. For fixed c>1/2, one giant component contains a positive fraction of all vertices with probability tending to one. This is an asymptotic transition in a specified random model, rather than a rule for every finite network.
Scientific scope
Add edges at random to a graph with n labelled vertices, giving every graph with the same number N of edges equal probability. Erdős and Rényi identify a change as N/n approaches a fixed density c. For fixed 0<c<1/2 the largest connected component has logarithmic order. For fixed c>1/2, one giant component contains a positive fraction of all vertices with probability tending to one. This is an asymptotic transition in a specified random model, rather than a rule for every finite network. This entry concerns the off-critical fixed-density asymptotics of the original uniform fixed-edge model. It is not the earlier 1959 connectivity threshold, a deterministic guarantee for every graph, or a literal finite-n discontinuity. No complete modern critical-window scaling law, exact limiting distribution, universal statement for all random graph models, or independent certification of every printed theorem is claimed.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This entry concerns the off-critical fixed-density asymptotics of the original uniform fixed-edge model. It is not the earlier 1959 connectivity threshold, a deterministic guarantee for every graph, or a literal finite-n discontinuity. No complete modern critical-window scaling law, exact limiting distribution, universal statement for all random graph models, or independent certification of every printed theorem is claimed.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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