Ranking is an important problem in machine learning. Given a set of documents, the goal is to sort them in a specific order based on certain criteria. Ranking is widely used in information retrieval systems to sort search results or in recommender systems to filter content will potentially be interesting to a particular user.
Based on a given problem and objective, there exist an abundance of ranking algorithms. The one we are going to study in this article is named PageRank. Its main objective is to rank a set of documents (web pages) by using the information about their connectivity. The rank assigned to each web page indicates its importance: the higher the rank is, the higher the importance is. The algorithm is based on two assumptions which we are going to look at in the next section.
We can define the term “importance” of a web page by making two assumptions.
The importance of a web page is high if there are many other web pages pointing to it.
Imagine we have a popular research paper and many other articles linking to it by using quotes or results from it. Primarily, it makes sense to give this article a large importance. On the other hand, if there is an unknown web page with no links to it from other resources, it seems logical to assign low importance to the page.
In reality, we should also care about the quality of the incoming links.
The importance of a web page is proportional to the importance of the web pages pointing to it.
If a page is originally cited by a high-quality article on Wikipedia, then such a link should have a larger weight. Conversely, when an absolutely unknown resource points to another web page, it should not normally have high importance.
Let us say that the importance of a node is equal to the sum of the weights of incoming links.
Imagine a node i with importance rᵢ having k outcoming links. How can we determine the weight of each link? The most straightforward approach is to take the node’s importance and divide it equally between all the outcoming links. This way, each outcoming link will receive the weight of rᵢ / k.
Given a graph of n web pages, we can create a system of n linear equations to find the weights of the graph. However, such a system can have an infinite number of solutions. That is why we should add another constraint that will impose a unique solution. By the way, PageRank adds the normalized condition that the sum of all node importance is equal to 1.
We have come up with a solution but it is not scalable. Even with Gaussian elimination, we end up with O(n³) complexity. Keeping in mind that the number of analyzed web pages n can reach billions, we need to come up with a more efficient approach.
First of all, let us simplify the notation. For this, we introduce the adjacency square matrix G which will contain link weights for every pair of linked web pages (if two web pages are not linked, we will put 0 in a corresponding matrix element). More formally:
Matrix G is called stochastic because each of its columns sums up to 1.
Next, we define the rank vector r whose i-th element is equal to the rank (importance) of page i. The sum of all elements of this vector also equals 1. Our ultimate goal is to find values of this vector r.
Let us see what will happen if we multiply matrix G by vector r. Based on the example with the graph from the section above, we can see that it results in the same vector r!
Why does it happen? Is it just a coincidence? Remember that the i-th row of matrix G contains weights of all input links to the page i. When we multiply the j-th element of the i-th row by r[j], we actually get the component rj / d[j]out — the importance which flows from node j to i. If there is no link between nodes i and j, then the respective component is set to 0. Logically, the final result of the multiplication of the i-th row by the vector r will be equal to the sum of all importances which flow from any connected node of the graph to node i. By definition, this value equals the rank of the node i. In general, we can write the following equation:
Therefore, our goal is to find such a vector r which being multiplied by the input matrix G will remain the same.
We can find the solution to the equation above by revising the theory on eigenvectors from linear algebra. Given a matrix A, the vector v is called the eigenvector if there exists such a number α which satisfies the following equation:
The number α is called the eigenvalue. We can notice that the PageRank equation corresponds to the eigenvalue equation where A = G, v = r and α = 1. Normally, any square matrix has several eigenvalues and eigenvectors but since our matrix G is stochastic, the theory claims that its largest eigenvalue is equal to 1.
One of the most popular ways of finding matrix eigenvectors is the Power iteration method. It consists of initializing an initial vector r with some values (we will use 1 / n where n is the number of web pages), then constantly computing the value of G * r and assigning this value to r again. If on any iteration the distance between vectors r and G * r is less than a certain threshold ε, we stop the algorithm as it has converged successfully.
In the example above we can see that by setting ε to 0.0005 the algorithm correctly converges just in 9 iterations:
Obviously, this is only a toy example but in practice, this method works very well for a larger number of variables as well.
Imagine a surfer (walker) being at any node of the graph at time t. Let us denote by p(t) the vector whose i-th component equals the probability that at time t the surfer is present at node i. Then the surfer randomly (with equal probabilities) chooses another linked node to the current one and moves there at time t + 1. Ultimately, we want to find the distribution vector p(t + 1) at the moment t + 1.
We can notice that the probability of the surfer appearing at a node i at the moment t + 1 is the sum of probabilities (over all linked nodes to i) that the surfer was previously at an adjacent node j multiplied by the probability of moving from node j to i.
- We already know the probability of the surfer appearing at node j at moment t: p(t)[j].
- The probability of moving from node j to i is equal to G[j][i].
By summing up these probabilities, we get the value for p(t + 1)[i]. For finding the value of p(t + 1) for all the graph nodes, we can write the same equation in the matrix form:
This equation has absolutely the same form as what we have obtained for the PageRank before! This means these two problems have the same solution and interpretation.
At some point, the distribution vector p(t) will converge: p(t + 1) = M * p(t) = p(t). The converged vector p(t) in such case is called the stationary distribution. At all the following moments of time, the probability of residing at any given node does not change.
The PageRank score of a node equals the probability that the surfer will be located at this node in the future by randomly walking through the graph.
The described process of walking throughout the graph is often referred to as “Markov chains”. There exists a theorem in Markov chains theory which states that:
Under certain conditions on the graph structure, the stationary distribution is unique and can be reached with any initial probability distribution at the moment t = 0.
In the following section, we will go more in-depth into the conditions that need to be satisfied for the unique convergence. It turns out that for not all the graphs the unique convergence can be achieved.
Principally, there exist 2 kinds of cases that we want to avoid at all costs.
Nodes that do not have out links are called dead ends. The problem with such kind of nodes is that because of them the total importance leaks out of the network. Here is an example:
A group of nodes form a spider trap if they do not have out links to other nodes outside of this group. Basically, once there, it is impossible to get outside of this group of nodes. Spider traps lead to the two following problems:
- The algorithm never converges.
- The group of nodes forming a spider trap absorbs all the graph importance. As a result, these nodes have very high importance while other nodes have importance being equal to 0.
The first problem is illustrated in the figure below:
The absorption of importance is demonstrated in the next figure. Though it might not look like a serious problem in the toy example below, imagine a web network with millions of web pages where several of them form a spider trap. As a consequence, these several pages will distribute all of the available importance while the importance of all other web pages will be set to 0. Obviously, this is not what we normally want in real life.
One of the solutions proposed by Google is to add the following condition before each move of the surfer:
- With probability β, move to another linked node.
- With probability (1 — β), move to a random node (through a teleport).
The parameter β is called the dumping factor. Authors of the original PageRank algorithm recommend choosing the value for β = 0.85 meaning that on average after 5 transitions the surfer will randomly jump to another node. The idea is that if the surfer falls into a spider trap, then after some time it will eventually get out of there through a teleport.
The diagram below shows how teleports can help to deal with the spider trap problem. If the surfer walks into the node c, then it will stay there forever. Introducing teleports (blue lines) helps eliminating this problem guaranteeing that after some time the surfer will have to walk to another random node.
However, for dead-end nodes, we need to slightly modify the approach. From one of the examples above, we know that dead ends lead to importance leakage in a graph. This phenomenon can be observed during the power iteration method, when the rank vector becomes full of zeros because of a corresponding zero column in the initial matrix G. Ultimately, what we can do is the following:
Whenever the surfer lands on a dead-end node, then it should immediately jump to a random node (with an equal probability) of the graph.
In fact, we can modify the initial matrix G to satisfy this statement: we just need to replace zeros to 1 / n in place of all the elements of the columns of all dead-end nodes of matrix G. The example below demonstrates this principle.
The node c is a dead-end node with a corresponding column of zeros in the matrix G. Adding n = 3 teleports from c to all of the nodes of the graph imposes equal probability p = 1 / 3 of moving from c to any node. To account for this, we fill the column of the matrix G corresponding to node c with values of 1 / 3.
We can notice that after adding teleports the sum of all matrix columns is now equal to 1. In other words, the matrix G becomes stochastic. This is an essential property which we will be used later.
There exists a crucial theorem from the theory of Markov chains that defines the convergence condition.
For any start vector, the transition matrix G converges to a unique positive stationary distribution vector r if the chain corresponding to G is stochastic, aperiodic and irreducible.
Let us remind the last three properties in this theorem and check if introduced teleports solve the problems above.
A chain G is called stochastic if sum of its each column equals to 1.
As we observed above, adding teleports to dead-end nodes eliminates zero columns in the matrix and makes the sum of all its columns equal to 1. The condition is already satisfied.
A chain G is called periodic if there exists a number k > 1 that the path length between any pair of nodes is always a multiple of k. Otherwise, the chain is called aperiodic.
This condition means that any return to the same state must occur in multiple of k times. In the case of aperiodicity, the return occurs at irregular times. Basically, the condition refers to the spider trap problem. Since we have already dealt with spider traps by adding teleports, the chain G is aperiodic.
A chain G is called irreducible if the probability of transitioning from any one node to any another node is always greater than 0.
This condition implies that there always exists a link between any two nodes, so it is impossible to stuck at any node. In other words, the matrix G needs to consist of all non-zero elements. We are going to see in the next section below how this condition will be satisfied by connecting all the nodes of the graph.
Modifying the algorithm
PageRank algorithm proposed by Google takes the initial matrix G and adjusts it by adding teleports from dead ends to other nodes. This ensures stochasticity. To guarantee aperiodicity and irreducibility it then adds the condition described before to each node:
- With probability β, move to another linked node.
- With probability (1 — β), move to a random node.
Mathematically, it results in the following rank equation for every node:
We can transform this equation into the matrix form:
Let us draw the modified graph and the corresponding transition matrix R from on of the examples above:
The only problem left for us is how to store the transition matrix R. Remember that R is a square matrix of size n x n where n is the number of web pages. Currently, Google has more than 25 billion web pages! The matrix R does not have any zeros, so it is dense which means we have to fully store it. Let us assume that every matrix element requires 4 bytes to be stored. The total memory size required to store the matrix R equals (25 * 10⁹)² * 4 (bytes) ~ 3 * 10²¹ (bytes). This is a gigantic memory size! We need to come up with another approach to reduce at least by several orders.
Firstly, we can simply notice that adding teleports is equivalent to reducing initial matrix G elements by (1 — β)% and distributing them evenly across every node. Keeping this in mind we can transform the matrix equation of PageRank into another format:
Let us look at the last obtained equation. G is the initial link matrix with most of the elements being equal to 0. Why is it so? In reality, if you take any web page, it will probably contain at most a few dozen links to other web pages. Keeping in mind that are more than 25 billion web pages we get that the relative number of total links compared to the number of web pages is extremely small. Therefore, there are a lot of zeros in G, so G is sparse.
Storing sparse matrices requires much less memory than dense ones. Let us assume that each web page links on average to other 40 pages. The total number of bytes required to store the matrix G now becomes 25 * 10⁹ * 40 (bytes) = 10¹² (bytes) = 1 (TB). It turns out we only need 1 terabyte to store G. Compared to what we had previously, this is a fabulous improvement!
In fact, at each iteration, we only need to compute the multiplication of matrix G by vector r, multiply it by β and add a constant (1 — β) / n to each element of the resulting vector.
Also keep in mind that if the initial chain G contains dead-end nodes, then the sum of vector r at each iteration will be less than 1. To deal with this, it is enough to renormalise it, so all the vector components sum up to 1.
In the figure below we can see the full version of the PageRank algorithm. At each iteration, the update of ranks proceeds in 2 stages. The first stage includes only update according to the initial matrix G. Then we sum up the components of the rank vector into the variable s. This way, the value of (1 — s) is the value by which the total input rank of a single node was reduced. To compensate for this, in the second stage, we account for teleports and add them from a node to all the nodes with the equal value of (1 — s) / n.
In this article, we have looked through different formulations of the PageRank algorithm to ultimately come up with its optimised version. Despite the existence and evolution of other methods for ranking search results, PageRank remains the most efficient algorithm among others which works under the hood of Google’s search engine.
The logical structure of this article is based on the lecture from Stanford University on large graphs.
All images unless otherwise noted are by the author