----- > [!proposition] Proposition. ([[no small components in configuration model with nodes of degree greater than 1]]) > Show that in a configuration model network with nodes of degree 2 and greater, but no nodes of degree 0 or 1, there are no small components (or, more properly, the fraction of nodes belonging to such components tends to zero as $n \to \infty$). > > Because [[the configuration model is locally tree-like]], any [[small component]] of $s$ nodes has $s-1$ [[tree iff has n-1 edges|edges]]. But since each node contributes at least $2$ edges to the component this is not possible. > > Can also argue as follows:![[CleanShot 2023-11-07 at 23.21.56.jpg|500]] > > [!proof]- Proof. ([[no small components in configuration model with nodes of degree greater than 1]]) > ~ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```