Foundations of Data Science

1
2 0.85π i
j
p ji
1
0.85π π
2 i
i = 0.85π j p ji + 0.85
i
π i = 1.48π j p ji
0.15π j 0.15π i
2 π i
Figure 5.12: Impact on page rank of adding a self loop
graph strongly connected so that the stationary probabilities exist.
Page rank
The page rank of a vertex in a directed graph is the stationary probability of the vertex,
where we assume a positive restart probability of say r = 0.15. The restart ensures that
the graph is strongly connected. The page rank of a page is the fractional frequency with
which the page will be visited over a long period of time. If the page rank is p, then
the expected time between visits or return time is 1/p. Notice that one can increase the
pagerank of a page by reducing the return time and this can be done by creating short
cycles.
Consider a vertex i with a single edge in from vertex j and a single edge out. The
stationary probability π satisfies πP = π, and thus
π i = π j p ji .
Adding a self-loop at i, results in a new equation
or
π i = π j p ji + 1 2 π i
π i = 2 π j p ji .
Of course, π j would have changed too, but ignoring this for now, pagerank is doubled by
the addition of a self-loop. Adding k self loops, results in the equation
π i = π j p ji +
k
k + 1 π i,
and again ignoring the change in π j , we now have π i = (k + 1)π j p ji . What prevents
one from increasing the page rank of a page arbitrarily? The answer is the restart. We
neglected the 0.15 probability that is taken off for the random restart. With the restart
taken into account, the equation for π i when there is no self-loop is
π i = 0.85π j p ji
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