---- > [!definition] Definition. ([[Price's DAG growing model]]) > **Price's model for growing a [[network|directed]] [[acyclic network|acyclic graph]]** with a [[power law]] [[degree distribution]] works as follows: > 1. Nodes are created one-by-one. Each new node cites $c$ previously created nodes on average. (i.e., has average out-degree $c$.) > 2. The $c$ nodes to cite are chosen randomly, in proportion to how many citations they already have (+ a constant $a \in \mathbb{N}$ to get things started). That is, a new node cites a previous node $i$ with probability $\frac{q_{i}+a}{\sum_{i=1}^{n}(q_{i}+a)}=\frac{q_{i}+a}{n \langle q \rangle + na }= \frac{q_{i}+a}{n(c+a)}$ > where $q_{i}$ denotes the [[degree|in-degree]] of node $i$ and the second equation uses the fact that in [[mean degree|average in-degree equals average out-degree]] in [[network|directed networks]]. > > [!basicproperties] > - [[in-degree distribution of Price's model is a power law]] > [!basicexample] > - (Newman 13.2) [[Price's model plug-and-chug example]] > - (Newman 13.4) [[Price's model when c=a=1]] > - (Newman 13.6) [[one must have both growth and preferential attachment to get a power law degree distribution]] > - (Newman 13.7) [[one must have both growth and preferential attachment to get a power law degree distribution]] (same as above) > - (Newman 13.8) ---- #### ---- #### 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 > ```