One way to rank the relative importance of vertices on a directed graph (network) is to use the Pagerank algorithm. The Pagerank algorithm was developed by Page and Brin to address the conundrum between quantity versus quality in the context of webpages [2], as a webpage could be considered influential either because it is linked to by many sites or by a few, high-quality sites. For instance, many space-enthusiasts may link to a NASA, indicating its authority. In turn, NASA may link to a revered astrophysicist on its own, indicating its endorsement.

Pagerank and similar network algorithms have been developed for other contexts. In the of a sports league, the graph could have teams be the vertices and the directed edges represent a team losing against the other team. Previous work has utilized this idea of to rank NCAA College Football teams where it is often the case that some teams perform very well but with weaker opponents [3]. In social media, the graph would have users of vertices and the directed edges represent one user following or mentioning one another. In [6], the PageRank algorithm is used to provide an alternative measure of influence of Twitter users beyond simply counting the number of retweets. That is, who retweets our message is just as important as how many times it is retweeted.

Consider a directed graph, $G$. For our purposes and throughout, $G$ may have loops and bidirectional edges (i.e., there is an arc from $i$ to $j$ and another arc from $j$ to $i$). However, there will be no duplicate arcs (i.e., there is at most one arc from $i$ to $j$). The Pagerank algorithm produces a vector of length $n$ (the number of vertices in $G$), denoted $\pi$, where each entry measures on the proportion of the time we would encounter each vertex by simply wandering around the graph randomly and occasionally jump around. Hence, the Pagerank vector is a stochastic vector.

To begin calculating the Pagerank vector, consider a directed graph $G$ and choose an $\alpha$ to be your jumping parameter, $0<\alpha<1$. The parameter $\alpha$ will determine the proportion of the time we will follow an arc in our random process. Intuitively, $\alpha$ balances the impact of arcs from high-quality vertices and high quantities of arcs. Once $\alpha$ is chosen, randomly pick a vertex of $G$ uniformly at random to start, denoted $v$. Then, with probability $\alpha$, choose an out-neighbor of $v$, uniformly at random, and walk to that vertex; and with probability $1-\alpha$, jump to an another random vertex chosen uniformly at random. If $v$ has no out neighbors, then simply jump to a random vertex, chosen uniformly at random. Take the vertex we are on and declare that the new $v$ and repeat. The Pagerank vector is the limiting probability of finding ourselves at each vertex after an arbitrarily large number of steps. Intuitively, the more often we visit a vertex, the more important it is and the higher the (Page)rank it has!

For a given directed graph $G$ and $\alpha$, we define the Pagerank vector with jumping parameter $\alpha$

for each vertex $i$.

The Pagerank can be interpreted using linear algebra in the context of Markov chains where the probability transition matrix for a directed graph $G$, $\mathbf{R}$, is given by

Where $deg_{j}^{out}$ is the number of out neighbors of j. In which case, the Pagerank vector $\pi$ is a right eigenvector of $\mathbf{R}$ with eigenvalue 1

and is the stationary distribution of the random walk.

The existence and uniqueness of $\pi$ guaranteed by the Perron-Frobenius Theorem for all $0\leq\alpha<1$ (see, for example, [4]). For $\alpha=1$, $\pi_{\alpha}$ is well-defined provided $G$

• •

  is aperiodic (i.e., the greatest common divisor of the length of all cycles is 1),

• •

  is weakly connected and

• •

  there is a unique maximal induced subgraph that is strongly connected (perhaps with just a vertex with a single loop).

For simplicity, we will typically drop one of the subscripts of the Pagerank vector. We note that in this paper we will use subscripts to both the value of $\alpha$ and which vertex is being spoken of. For instance, we write $\pi_{\alpha=1}$ to refer to the entire Pagerank vector when $alpha=1$ but also we write $\pi_{A}$ to denote to entry of the Pagerank vector corresponding to the vertex $A$; in the later case, the context of $\alpha$ will be clear.

Even on the same graph, choosing different values for $\alpha$ can result in substantially different Pagerank vectors. For example, when $\alpha=0$, the walker always jumps, in which case $\pi_{\alpha}$ is the uniform vector: $(1/n,1/n,\ldots,1/n)$ where $n$ is the number of vertices in $G$. On the other extreme, when $\alpha=1$, the walker always follows the arcs (unless they are stuck), so naturally, vertices with more in-arcs will be visited more often. In this situation, it is possible that $\pi_{\alpha}$ is concentrated at one vertex. This leads to the question: For the same graph, how different can these rankings be? Our main problem is to determine the maximum difference between Pagerank vector with one $\alpha$ and a Pagerank with a different $\alpha$ but on the same graph. Formally we ask:

We focus on the 2-norm as it is the traditional norm, though we briefly discuss the problem for other norms in Section 3 which we leave as an open problem for the reader.

As we will discover, this problem is difficult because it requires carefully constructed examples with precisely chosen parameters $\alpha_{1}$ and $\alpha_{2}$. While we do not answer this problem explicitly, we provide a construction that provides an extremal value of $\sqrt{\frac{67}{50}}$. Surprisingly, to achieve this, we use $\alpha_{1}=1$ and $\alpha_{2}$ very close to 1. We believe $\sqrt{\frac{67}{50}}$ to be best possible.

Based on the linear-algebraic interpretation of Pagerank mentioned previously, this problem is fundamentally one about the principal eigenvector of graph-theoretic matrices. Indeed, problems of a similar nature have been considered before. For an irreducible nonnegative matrix, the principal ratio is the ratio between the maximum and minimum entries of the principal eigenvector. For Markov transition matrices of simple directed graphs, Askoy, Chung, and Peng show that the principal ratio of a random walk can be superexponentially small [1]. For the adjacency matrix of undirected graphs, Tait proved that the “kite graph” (a complete graph with a long path) acheives the maximum principal ratio [9].

To get our feet wet, here are some important properties of Pagerank vectors we use.

Where $n$ is the number of vertices in the graph. What this says is that, in practice, one should choose an $\alpha$ away from 0 so that the resulting values vertices are more differentiated. Indeed, the original $\alpha$ for the Pagerank algorithm was $\alpha=0.85$ [2].

In the context of maximizing $\|\pi_{\alpha=1}-\pi_{\alpha=x}\|_{2}$, this suggests that we would want to choose $\alpha$’s away from 0. Our the extremal choices for $\alpha$ are closer to 1.

Also, in practice, people usually avoid choosing an $\alpha$ close to 1 because accurately computing $\pi_{\alpha\approx 1}$ numerically is challenging using methods such as the power method [7]. Our graphical plots fail to show the true behavior near $\alpha=1$ because when $k$ is large, these numerical methods are not accurate enough! See for example figure 2

This last bound is important because it provides an upper bound for how high $\|\pi_{\alpha=1}-\pi_{\alpha=x}\|_{2}$ can be. While we are unable to achieve $\sqrt{2}\approx 1.4142$, we are not too far from it: $\sqrt{\frac{67}{50}}\approx 1.1576$.

Our construction is a long “ladder” with $k+3$ vertices: $C_{1},C_{2},B_{1},\ldots,B_{k},$ and $A$. It is illustrated in graph Figure 1 and is constructed as follows. Vertex $C_{1}$ has an arc to itself and to $C_{2}$. $C_{2}$ has an arc back to $C_{1}$, a loop to itself and a path to $B_{1}$. $B_{1}$ has an arc back to $C_{1}$ and $C_{2}$, and an arc to $B_{2}$. Likewise, for all $i<k$, $B_{i}$, has arcs back to both $C_{1}$ and $C_{2}$ and an arc to $B_{i+1}$. Then, $B_{k}$ which has an arc back to $C_{1}$ and $C_{2}$ and to a vertex $A$. However, vertex $A$ only has a loop to itself.

This construction will end up concentrating the random walkers either in $A$ or in $C_{1}$ and $C_{2}$, depending on the value of the jumping constant $\alpha$.

When $\alpha=1$, the random walker will always follow the arcs. The walker will climb up and fall down “the ladder”, but inevitably, they will eventually make it to $A$ and become stuck. Therefore, when $\alpha=1$ we have $\pi_{A}=1$.

On the other hand, for other, carefully chosen, values of $\alpha$ close to 1, the random walker will frequently “fall down” to $C_{1}$ and $C_{2}$. In which case, it is next to impossible for the walker to get to $A$. And even if the walker gets to $A$, since $\alpha<1$, the walker is able to avoid becoming stuck. Within the proof of Theorem 2.1, we show that $\pi_{A}\to 0$ as $k\to\infty$ for $\alpha=1-1/k$.

Our proof is outlined as follows. We will start by explicitly computing $\pi_{\alpha_{1}=1}$ for $\Gamma(k)$ for any $k>0$. We then compute $\pi_{\alpha_{2}=1-1/k}$ and show that $\pi_{C_{1}}$ and $\pi_{C_{2}}\to\frac{2}{5}$ as $k\to\infty$. We then analyze $\pi_{B_{i}}$ and show that $\sum_{i}\pi_{B_{i}}^{2}$ can be approximated by a geometric series in terms of $\pi_{C_{2}}$. As a result, we will have that as $k\to\infty$, $\|\pi_{\alpha_{1}=1}-\pi_{\alpha_{2}=1-1/k}\|_{2}^{2}\to 67/50$.