Nonbacktracking PageRank
Abstract
The PageRank algorithm, which has been “bringing order to the web” for more than 20 years, computes the steady state of a classical random walk plus teleporting. Here we consider a variation of PageRank that uses a nonbacktracking random walk. To do this, we first reformulate PageRank in terms of the associated line graph. A nonbacktracking analog then emerges naturally. Comparing the resulting steady states, we find that, even for undirected graphs, nonbacktracking generally leads to a different ranking of the nodes. We then focus on computational issues, deriving an explicit representation of the new algorithm that can exploit structure and sparsity in the underlying network. Finally, we assess effectiveness and efficiency of this approach on some realworld networks.
Keywords
PageRank Nonbacktracking Complex networks CentralityMathematics Subject Classification
05C50 05C82 65F10 65F501 Motivation
The notion of a walk around a graph is both natural and useful. The walker may follow any available edge, with nodes and edges being revisited at any stage. However, in some settings, walks that backtrack—leave a node and then return to it on the next step—are best avoided. The concept of restricting attention to nonbacktracking walks has arisen, essentially independently, in a wide range of seemingly unrelated fields, including spectral graph theory [2, 13, 14], number theory [30], discrete mathematics [4, 28], quantum chaos [26], random matrix theory [27], stochastic analysis [1], applied linear algebra [29] and computer science [24, 33]. Nonbacktracking has more recently been considered in the context of matrix computation, where it has been shown to form the basis of effective algorithms in network science for community detection, centrality and alignment [3, 10, 16, 18, 20, 21, 23, 31]. We note that nonbacktracking walks have typically been studied on undirected networks, but the definition continues to make sense in the directed case.
For network science applications, nonbacktracking can be imposed at very little extra cost and offers quantifiable benefits: centrality measures can avoid localization [3, 10, 20] and discover key influencers [21, 31], and community detection algorithms perform optimally on the stochastic block model [18]. In terms of classical random walks on a graph, it is also known that nonbacktracking enhances mixing [1].
Given that the PageRank algorithm [22] computes the steady state of a particular (teleporting) random walk, it is therefore natural to ask whether there is any scope for designing and analysing a nonbacktracking analog. We emphasize that the PageRank philosophy has been extended well beyond the WWW into a diverse range of areas including bioinformatics, bibliometrics, business, chemistry, neuroscience, social media, and sports [9], where nonbacktracking may be a natural requirement. In particular, PageRanklike measures have been extensively used in transport modelling; see, for example, [6, 25]. In this setting it is reasonable to assume that a vehicle will not immediately head back to the same road junction, train station or bus stop.

What is an appropriate nonbacktracking version of PageRank?

Can the imposition of nonbacktracking lead to a different node ranking?

Is there an efficient way to implement such an algorithm?
The material is organized as follows. In Sect. 2 we describe background material on PageRank. Section 3 gives a transposition of standard PageRank to the dual space and verifies the equivalence of the two formulations. Section 4 defines and analyzes a new nonbacktracking analog. In particular, we show that, even for undirected graphs, nonbacktracking generally leads to a different ranking of the nodes. In Sect. 5 we derive a computable description of the new measure and show how to exploit the structure and sparsity of the Hashimoto matrix. In Sect. 6 we comment on some numerical experiments performed on road networks. Section 7 gives some conclusions.
2 Background
Let \(A = (a_{ij})\in {\mathbb {R}}^{n\times n}\) be the adjacency matrix of an unweighted, weakly connected, directed graph with n nodes and \({\widetilde{m}}\) directed edges. We further assume that there are no multiple edges linking nodes in the network. Under these assumptions, \(a_{ij} = 1\) if there is an edge from node i to node j, otherwise \(a_{ij} = 0\). A node that has no outgoing links is called a dangling node. We use the common approach of removing dangling nodes by modifying the adjacency matrix; allzeros rows in A are replaced by allones rows [15]. This corresponds to adding n edges from each dangling node to all the nodes in the network, itself included.
Let \(G = (V,E)\) be the graph obtained once these edges have been added to the original network in order to remove dangling nodes. In the remainder of the paper, we shall work on this network. Let \(W = A + \varvec{\chi }{\mathbf {1}}^T\) be its adjacency matrix, where \(\varvec{\chi }\) is the indicator vector for the set containing the dangling nodes in the original digraph. Here, \({\mathbf {1}}\in {\mathbb {R}}^n\) denotes the vector of all ones, and we will sometimes write \({\mathbf {1}}_n\) when its dimension may be in doubt. We let k be the number of dangling nodes in the original network, so the new graph G contains \(m:={\widetilde{m}}+kn\) edges among n nodes.
Let \(\mathbf{d}= (d_{i}) = W{\mathbf {1}}\) be the vector of outdegrees for the nodes in the graph represented by W, i.e., let \(d_i\) be the number of edges leaving node i in G. We will denote by D the diagonal matrix whose diagonal entries are \(D_{ii} = d_{i}\) for all i. Since all the nodes have positive outdegree, D is invertible.
Remark 1
If \(W=A\) and the original network contained no dangling nodes, then D would still be invertible, but \(R^TR\) might not be invertible as the network may contain source nodes; that is, nodes with no incoming links. On the other hand, if the original graph contained at least one dangling node, so that \(W \ne A\), then its removal would also remove any source node, thus making \(R^TR\) invertible as well.
As discussed in the original work [22], the PageRank algorithm may be motivated from two different perspectives: (a) the longterm behavior of a web surfer who randomly follows outgoing links and occasionally teleports (jumps) to a random page, and (b) the outcome of an iterative voting procedure where a page votes for pages that it follows, with more important pages given more influence.
Theorem 1
This theorem shows that, in order to compute the PageRank vector \(\mathbf{x}\) one can solve the pseudoPageRank problem and then normalize the solution. Therefore, underlying every pseudoPageRank system there is a true PageRank problem.
In the following section we show that PageRank can be reformulated in the dual space, where edges in G play the role of nodes in the new network. This reformulation then leads to a natural nonbacktracking extension.
3 Moving to the Dual Space
Our immediate goal is to define a PageRank vector in the dual space, i.e., the edge space, as the solution of a linear system involving \(\mathcal{W}\) that, once projected, will allow us to retrieve the PageRank vector \(\mathbf{x}\) defined earlier in the primal space. This forms the basis of the nonbacktracking version that we develop in Sect. 4.
3.1 Formulation of Edge PageRank
Our goal is now to show that \(\mathbf{z}= \mathbf{x}\), thus deriving equivalence between the new formulations of PageRank and (4)–(5). Let us first prove two preliminary results.
Lemma 1
In the above notation, \(\mathcal{D}^{1}RD = R\).
Proof
Lemma 2
In the above notation, \(L^T\mathcal{D}^{1}R = WD^{1}.\)
Proof
Leftmultiplying by \(L^T\) the relation in Lemma 1 and using \(W=L^TR\) it follows that \(L^T\mathcal{D}^{1}RD=W\), thus the conclusion, since D is invertible. \(\square \)
Theorem 2
Proof
The following corollary is then immediate.
Corollary 1
When the network does not contain a source node, the edge PageRank vector in (7) projects to the original PageRank vector in (3); that is, \(R^T\hat{\mathbf{x}} = \mathbf{x}\).
Proof
We now describe in detail what happens when the network contains \(s = nN\ge 1\) source nodes. We will show that the edge and node based systems are not equivalent. It is worth pointing out that we are tacitly assuming that the original network did not contain dangling nodes. If this was not the case, then the removal of the dangling nodes would have also removed the source nodes; see Remark 1.
Theorem 3
Proof
We then have the following general corollary.
Corollary 2
Proof
A proof follows immediately from Theorem 3. \(\square \)
4 The Nonbacktracking Framework
In the following we will assume that there are no dangling edges in the graph; the remainder of this section can be adapted to the case where such edges existed.
Definition 1
Let \(\mathcal{B}\) be the Hashimoto matrix of a directed graph with no dangling nodes or dangling edges. Let \(\mathcal{D}\) be the diagonal matrix whose diagonal entries are the entries of the vector \(\mathcal{B}{\mathbf {1}}\), and let L be the source matrix of the graph. For all \(0<\alpha <1\) we define the nonbacktracking PageRanklike vector as the nonnegative, unit 1norm vector \(\mathbf{y}:=L^T\hat{\mathbf{y}}\), where \(\hat{\mathbf{y}}\ge 0\) with \(\Vert \hat{\mathbf{y}}\Vert _1=1\) solves the the linear system (8), for \({{\varvec{\upsilon }}} = L(L^TL)^{1}{\mathbf {1}}\).
Definition 1 allows us to introduce the nonbacktracking aspect in PageRank. As we will see below, there are situations in which the newly introduced measure and standard PageRank return vectors that are the same or induce the same ranking. However, in applications where backtracking is best avoided, the two measures may differ, as we shall see in Sect. 6.
Let us now briefly discuss the case where the two measures behave similarly. It has been proved in [17] that for an undirected graph with nodes all having at least degree two (so that there are no dangling edges), the steady state of \(P = D^{1}W\) coincides with that of \(\mathcal{P}= \mathcal{D}^{1}\mathcal{B}\), once the latter has been projected in the node space via \(L^T\): so standard PageRank and the new nonbacktracking PageRanklike measures are the same in the absence of teleportation. In other words, in this case where every edge is reciprocated and no teleportation is allowed, this result tells us that the longterm frequency of visits to a particular node during a random walk does not change when nonbacktracking is imposed—the nodes benefit from nonbacktracking precisely in proportion to their original PageRank score. Intuitively, nodes given higher PageRank scores are accessible through many different traversals, and hence are less reliant on backtracking walks than nodes given lower PageRank scores. We now show that this result carries over to the case where we allow teleporting, for a special class of graphs.
Theorem 4
Before proving the result, let us recall that in the case of undirected networks every undirected edge is considered as a pair of directed edges when building the matrices \(\mathcal{W}\) and \(\mathcal{B}\).
Proof
When \(\alpha = 1\) the result follows from [17]. If \(\alpha = 0\), then from the description of P and \(\mathcal{P}\) it follows that \(\mathbf{x}= \frac{(1\alpha )}{n}{\mathbf {1}}\), and \(\hat{\mathbf{y}} = \frac{(1\alpha )}{n}{{\varvec{\upsilon }}}\). The conclusion then follows from the definition of \({{\varvec{\upsilon }}}\) and the fact that \(D = L^TL\).
Before moving on to the numerical tests on realworld networks, we discuss in the next section how to implement the most expensive operation performed by an iterative solver when solving (8): matrixvector multiplications involving \(\mathcal{B}^T\).
5 On the Computation of \(\mathcal{B}^T\mathbf{v}\)
The Hashimoto matrix \(\mathcal{B}\) built from a directed network may be very large, especially if the original graph was large to begin with and contained several dangling nodes. Indeed, to remove such nodes one needs to add n edges to the network for each dangling node. While performing this correction to the graph in the node space corresponds to a low rank modification of the adjacency matrix A, it corresponds to adding n rows and columns to the nonbacktracking edgematrix for each of the dangling nodes originally present in the digraph.

n is the total number of nodes in the digraph;

k is the number of dangling nodes;

\({\widetilde{m}}\) is the number of edges before the correction of the dangling nodes, i.e., the number of edges in the graph associated with A;

\(K\le {\widetilde{m}}\) is the number of edges in the graph represented by A that point to the dangling nodes.
Let us describe the matrices \(S_2\in {\mathbb {R}}^{K\times k(nk)}\) and \(S_4\in {\mathbb {R}}^{k^2\times k^2}\) in more detail. From the structure of \((L_2\otimes {\mathbf {1}}_k^T)\) and \(({\mathbf {1}}_{nk}^T\otimes R_2)\) it can be deduced that the ones in the matrix \(S_2\) are found in position \((e,(i1)k+j)\), where \(e\in \{{\widetilde{m}}K+1,\ldots , {\widetilde{m}}\}\) is the label of the edge \(i\rightarrow nk+j\) that points from a certain node \(i\in \{1,2,\ldots ,nk\}\) to a dangling node \((nk+j)\) for \(j\in \{1,2,\ldots ,k\}\). Similarly, the matrix \(S_4 = ({\mathbf {1}}_k^T\otimes I_k\otimes {\mathbf {1}}_k)\circ ({\mathbf {1}}_k\otimes I_k\otimes {\mathbf {1}}_k^T)\) has a one in position \(((i1)k+j,(j1)k+i)\) for all \(i,j=1,2,\ldots ,k\). Therefore, premultiplying a vector by these matrices corresponds to accessing certain entries of the vector.
6 Numerical Experiments
We now present numerical results to compare the performance of nonbacktracking PageRank in Definition 1 with standard PageRank. We perform our tests on four road networks corresponding to the cities of Birmingham (UK), Philadelphia (USA), Austin (USA), and the Federal State of Hesse (Germany). The data is available in [32]. Here, the edges represent roads and nodes represent road intersections (primal graph).
We computed both the new measure and standard PageRank as the solution of a linear system; see equations (5) and (8). For both problems we used the MATLAB implementation of the linear solver GMRES without restart and we gave as input a function handle that describes the matrix multiplication \(\mathcal{B}^T(\mathcal{D}^{1}\mathbf{v})\) and \(W^T(D^{1}\mathbf{v})\) for vectors \(\mathbf{v}\) of appropriate size. Both these functions were implemented exploiting the structure of the matrices \(\mathcal{B}\) (see Sect. 5) and \(W = A + {\varvec{\chi }}{\mathbf {1}}^T\).
Description of the dataset
n  k  \({\widetilde{m}}\)  K  M  s  \(\delta \)  

FS of Hesse  4660  1  6674  1  245  1  0.19 
Austin  7388  4  18,956  4  405  3  0.88 
Philadelphia  13,389  0  40,003  0  178  0  0.94 
Birmingham  14,639  0  33,937  0  1346  6  0.76 
In Fig. 3 we scatter plot the new measure (NBT PR edge), projected as in (9), against standard PageRank (PR node). The measures show a strong overall linear correlation; the Pearson’s correlation coefficients are \(\rho = 0.94\) for the Federal State of Hesse, \(\rho = 0.90\) for both Austin and Philadelphia, and \(\rho = 0.81\) for Birmingham. However, it is readily seen that the two measures do not completely agree on the rankings. Indeed, the intersection of the sets containing the IDs of the top ten ranked nodes according to the two measures for Birmingham, the Federal State of Hesse, Austin and Philadelphia contain 8, 3, 5 and 6 elements, respectively. It is beyond the scope of this paper to determine which measure is returning the most meaningful results, since there is no ground truth for comparison. However, it may be argued that the nonbacktracking version of the measure is the best fit for road networks since, as we mentioned in the introduction, the concept of nonbacktracking walks is consistent with the typical behavior of a driver or a pedestrian traversing a city. It is notable that for all four scatter plots the lowranked nodes are assigned the same score by nonbacktracking PageRank, whilst they are being assigned different values by the classic version of the algorithm. A further analysis of the data showed that exactly \(M+s\) nodes fall in this class; these are the s source nodes and the M reciprocated leaves (i.e., nodes that have only one incoming link that is the reciprocal of their only outgoing edge) in the original graph associated with A.
PageRank  NBT PageRank  

Time  Iter.  Time  Preproc.  Iter  
FS of Hesse  6.39e−01  38  1.04e−01  6.85e−03  38 
Austin  1.39  31  9.67e−02  9.07e−02  32 
Philadelphia  4.71  28  1.27e−01  4.09e−01  30 
Birmingham  5.18  29  6.46e−02  2.10e−01  31 
Number of iterations required to achieve convergence of GMRES
\(\alpha \)  0.1  0.25  0.3  0.5  0.75  0.85  0.99 

PR node  5  8  9  15  29  45  – 
NBT PR edge  6  9  10  16  31  47  – 
In summary, for the networks studied here, using the ideas from Sects. 4 and 5 we can incorporate nonbacktracking into PageRank in order to obtain a distinct node ranking at reduced computational cost.
7 Conclusions
In this work we showed how to extend the definition of the PageRank algorithm to a nonbacktracking framework by using the dual representation of a network and the Hashimoto matrix. We proved that the elimination of backtracking has no effect on the results in the case of undirected, regular networks, but also showed by example that nonbacktracking can affect the ranking for the generic case of nonregular networks. We described explicitly how to exploit structure and sparsity in order to compute the new PageRank measure and showed through numerical experiments using real world transport networks and a GMRES solver that the computational cost is not increased. Future work will focus on (i) further tests on networks from a range of application fields, (ii) development of customized iterative methods for the structured linear systems that arise, (iii) nonbacktracking personalized PageRank via a nonuniform teleportation distribution vector.
Notes
Acknowledgements
The authors thank the anonymous referees for their valuable suggestions. We also thank Ernesto Estrada for pointing us to the datasets used in the numerical experiments. Francesca Arrigo thanks Mariarosa Mazza and Alessandro Buccini for helpful discussions.
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