----- > [!proposition]+ Proposition. ([[Barabasi-Albert variant]]) > Consider the following variant of the Barabási–Albert model. Nodes are added one by one to a growing undirected network, each node having initial degree $c$. The $c$ edges emanating from a newly added node connect to previously existing nodes $i$ with probability proportional to $k_{i}+a$, where $k_{i}$ is $i$’s (undirected) degree and $a$ is a constant. ^proposition > [!proof]+ Proof. ([[Barabasi-Albert variant]]) > > **a. Given that $c$ edges are added to the network with each node, what is the mean degree of a node in the network in the limit of large network size?** > > There are $nc$ edges in the [[network]], so the [[mean degree]] is $2nc / n=2c$. > > **b. Derive the master equation that gives the fraction of nodes $p_{k}$ having degree $k$ in the limit of large network size. If necessary, give an additional rate equation to cover any special-case value of $k$.** > > The expected number of new connections into a given node $i$ upon introduction of a new node into the [[network]] is $c\frac{k_{i}+a}{\underbrace{\sum_{i=1}^{n}k_{i}}_{\text{=2cn, by (a)} } + a}=c\frac{k_{i}+a}{n(2c+a)}$. And there are $np_{k}(n)$ nodes of degree $k$ in the network so the expected number of connections into nodes of degree $k$ will be $np_{k}(n) \cdot c \frac{k+a}{n(2c+a)}$. Using this we see that when we add a single new node to our network of $n$ nodes, we gain $cp_{k-1}(n) \frac{k-1+a}{c+a}$ nodes of [[degree]] $k$. We lose $cp_{k}(n) \frac{k+a}{c+a}$ nodes of [[degree]] $k$. Thus the [[master equation]] at a given step is > > $(n+1)p_{k}(n+1)=np_{k}(n)+ cp_{k-1}(n) \frac{k-1+a}{2c+a} - cp_{k}(n) \frac{k+a}{2c+a}$ > > for $k > c$. For $k = c$ the middle term does not appear and we have > > $(n+1)p_{0}(n+1)=np_{0}(n) - cp_{0} \frac{k+a}{2c+a}.$ > > The asymptotic form is > > $p_{k} = \begin{cases} > \frac{c}{2c+a}[(k-1+a)p_{k-1} - (k+a)p_{k}] & k \geq c \\ > 1 - c \frac{c+a}{2c+a}p_{c} & k=c. > \end{cases}$ > > The case $k<c$ is impossible so we omit it from our results. > > **c. Show that the fraction of nodes with degree c in the limit of large network size is** $p_{c}=\frac{2c+a}{2c+a+c(c+a)}.$ > > Just rearrange. ^proof #### 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 > ``` #reformatreviseBbatch