# The Giant Component and Explosive Percolation

DevFeed: [The Giant Component and Explosive Percolation](<https://devfeed.tech/articles/the-giant-component-and-explosive-percolation-40375.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/02/02/the-giant-component-and-explosive-percolation/>)

Published: 2015-02-02T09:00:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Graphs](<https://devfeed.tech/topics/graphs.md>), [Data analysis](<https://devfeed.tech/topics/data-analysis.md>)

Tags: [big-o-notation](<https://devfeed.tech/tags/big-o-notation.md>), [erdos-renyi](<https://devfeed.tech/tags/erdos-renyi.md>), [first-moment-method](<https://devfeed.tech/tags/first-moment-method.md>), [giant-component](<https://devfeed.tech/tags/giant-component.md>), [graph](<https://devfeed.tech/tags/graph.md>), [graphs](<https://devfeed.tech/tags/graphs.md>), [inequality](<https://devfeed.tech/tags/inequality.md>), [markov](<https://devfeed.tech/tags/markov.md>), [method-of-moments](<https://devfeed.tech/tags/method-of-moments.md>), [network-science](<https://devfeed.tech/tags/network-science.md>), [percolation](<https://devfeed.tech/tags/percolation.md>), [probability-theory](<https://devfeed.tech/tags/probability-theory.md>), [random-graph](<https://devfeed.tech/tags/random-graph.md>), [random-graphs](<https://devfeed.tech/tags/random-graphs.md>), [random-variables](<https://devfeed.tech/tags/random-variables.md>)

## AI overview

This article develops rigorous results about Erdős-Rényi random graphs, following a conjecture about connectivity at edge probability p = 5/n. It introduces threshold theorems and begins explaining the first moment method, Markov's inequality, and isolated vertices.

## Source excerpt

Last time we left off with a tantalizing conjecture: a random graph with edge probability $ p = 5/n$ is almost surely a connected graph. We arrived at that conjecture from some ad-hoc data analysis, so let's go back and treat it with some more rigorous mathematical techniques. As we do, we'll discover some very interesting "threshold theorems" that essentially say a random graph will either certainly have a property, or it will certainly not have it.