## Parametric controllability of the personalized PageRank: classic model vs biplex approach.(English)Zbl 1432.91094

Summary: Measures of centrality in networks defined by means of matrix algebra, like PageRank-type centralities, have been used for over 70 years. Recently, new extensions of PageRank have been formulated and may include a personalization (or teleportation) vector. It is accepted that one of the key issues for any centrality measure formulation is to what extent someone can control its variability. In this paper, we compare the limits of variability of two centrality measures for complex networks that we call classic PageRank (PR) and biplex approach PageRank (BPR). Both centrality measures depend on the so-called damping parameter $$\alpha$$ that controls the quantity of teleportation. Our first result is that the intersection of the intervals of variation of both centrality measures is always a nonempty set. Our second result is that when $$\alpha$$ is lower that $$0.48$$ (and, therefore, the ranking is highly affected by teleportation effects) then the upper limits of PR are more controllable than the upper limits of BPR; on the contrary, when $$\alpha$$ is greater than $$0.5$$ (and we recall that the usual PageRank algorithm uses the value $$0.85)$$, then the upper limits of PR are less controllable than the upper limits of BPR, provided certain mild assumptions on the local structure of the graph. Regarding the lower limits of variability, we give a result for small values of $$\alpha$$. We illustrate the results with some analytical networks and also with a real Facebook network.

### MSC:

 91D30 Social networks; opinion dynamics 05C82 Small world graphs, complex networks (graph-theoretic aspects) 15B51 Stochastic matrices

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