Define $G$ to be random graph, with the following constructiion: The network begins with an initial node. New nodes are added to the network one at a time. Each new node is connected to the existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability ${p_{i}} $ that the new node is connected to node $i$ is $p_{i}={\frac {k_{i}}{\sum _{j}k_{j}}}$, where $ k_{i}$ is the degree of node $i$ and the sum is made over all pre-existing nodes $j$ (i.e. the denominator results in twice the current number of edges in the network). Let's say this graph has $n$ nodes.
Modification of Random Graph: Given the graph $G$, we modify it by adding extra edges to it. For each node pair $i, j$ that are not connected, with probability $q$, we put an edge between them. Call the modified graph $G'$.
Question: what is the probability that there is a path between vertices $i$ and $j$ in G', assuming that there is no path between $i$ and $j$ in G.
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