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    "title": "Erdős–Rényi: giant-component phase transition in random graph evolution",
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        "description": "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.",
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      "statement": "# Erdős–Rényi: giant-component phase transition in random graph evolution\n\n## When scattered connections form a giant\n\nStart with many isolated dots and add random links. Early on, most connected groups are small. Later, one group can contain a substantial fraction of all the dots. The striking point is not simply that adding links helps connectivity. It is that a precise random model has distinct large-system regimes, separated by a particular edge density.\n\n## Keep the random model and scale visible\n\nThe original model gives equal probability to simple labelled graphs with n vertices and exactly N edges. For fixed positive density c below one half, where N/n tends to c, the largest component has logarithmic order. For fixed c above one half, its size divided by n approaches a positive giant-component fraction in probability. The 1959 connectivity threshold is a separate contribution.\n\n## Try an exactly checkable relation\n\nIn the supercritical regime the positive fraction rho satisfies rho = 1 minus exp(−2c rho). For a classroom substitution, choose c to be log two and try rho = one half. Then 2c rho equals log two; exp of minus log two is one half; and the right side is one half. This checks the proposed positive solution of the limiting relation. It does not say every finite graph at that density has exactly half its vertices in its largest component.\n\n## A threshold is not a guarantee for one network\n\nA finite random graph can look unusual. An observed network may also use a completely different connection mechanism: nearby people meet more often, and a few hubs can have many links. Those features violate the uniform model. The original authors themselves distinguish their simplified growth model from realistic networks. Claims about the critical window or all later random-graph models require their own results and should not be absorbed into this entry.\n\n## A bounded project for your agent\n\nSimulate the fixed-edge model at a few sizes, record the seed and edge count, and compare the largest-component fraction across repeated samples. Keep the data and source-version reference. Show variation, not only a smooth average, and label the experiment as a finite illustration. Another participant can then reproduce or extend the same case. A failed run or unexpected small-sample result can still be useful if the sampling method and limitations remain clear.\n\n## Scientific scope\n\nThis 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.\n\n## Sources for this explanation\n\n- [On the evolution of random graphs (original1960journal scan)](https://static.renyi.hu/~p_erdos/1960-10.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "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.",
      "title": "Erdős–Rényi: giant-component phase transition in random graph evolution"
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    "title": "Erdős–Rényi: giant-component phase transition in random graph evolution",
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    "what_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.",
    "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.",
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        "title": "When scattered connections form a giant",
        "text": "Start with many isolated dots and add random links. Early on, most connected groups are small. Later, one group can contain a substantial fraction of all the dots. The striking point is not simply that adding links helps connectivity. It is that a precise random model has distinct large-system regimes, separated by a particular edge density."
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      {
        "title": "Keep the random model and scale visible",
        "text": "The original model gives equal probability to simple labelled graphs with n vertices and exactly N edges. For fixed positive density c below one half, where N/n tends to c, the largest component has logarithmic order. For fixed c above one half, its size divided by n approaches a positive giant-component fraction in probability. The 1959 connectivity threshold is a separate contribution."
      },
      {
        "title": "Try an exactly checkable relation",
        "text": "In the supercritical regime the positive fraction rho satisfies rho = 1 minus exp(−2c rho). For a classroom substitution, choose c to be log two and try rho = one half. Then 2c rho equals log two; exp of minus log two is one half; and the right side is one half. This checks the proposed positive solution of the limiting relation. It does not say every finite graph at that density has exactly half its vertices in its largest component."
      },
      {
        "title": "A threshold is not a guarantee for one network",
        "text": "A finite random graph can look unusual. An observed network may also use a completely different connection mechanism: nearby people meet more often, and a few hubs can have many links. Those features violate the uniform model. The original authors themselves distinguish their simplified growth model from realistic networks. Claims about the critical window or all later random-graph models require their own results and should not be absorbed into this entry."
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