Scaling personalized web search

Glen JehJennifer Widom

article2003WWW1,422 citationsBest Paper Award

Develops a scalable framework for personalized PageRank by decomposing personalized views into precomputed, shared partial vectors that enable fast, query-time customization for arbitrary user preference sets.

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Modern web search engines struggle to return highly relevant results using text matching alone due to the vast scale of the web. While algorithms like PageRank address this by computing a universal importance score for each page based on link structure, users increasingly require personalized search tailored to individual interests or bookmarks. However, delivering personalized importance scores has historically been computationally intractable. Computing personalized rankings across millions of web pages at query time causes unacceptable latency, while precomputing and storing individual rankings for every possible user preference combination is impossible due to prohibitive storage and processing demands.

To resolve this bottleneck, the article evaluates a scalable graph-theoretic framework that decomposes personalized views into reusable mathematical components. The primary objective is to demonstrate that personalized PageRank vectors can be efficiently precomputed, stored in compact partial forms, and assembled in real time at query response time.

To evaluate this framework, the authors implemented dynamic programming algorithms and tested them on a real-world web crawl containing 80 million non-leaf pages from Stanford's WebBase repository. The methodology relies on decomposing basis importance vectors into compact "partial vectors"—which stop traversing the web graph once they hit a predefined set of high-importance "hub" pages—and a compact "hubs skeleton" that tracks the structural relationships among these hubs. Computing workloads were partitioned across disk and memory to simulate large-scale operational environments.

Key findings show that this approach substantially reduces computational and storage overhead while scaling effectively with larger hub sets. First, partial vectors shrink in size as the number of hub pages increases, creating significant storage savings over full vectors. Second, precomputing partial vectors required only about 0.33 seconds per vector at 50,000 hubs, compared to roughly 2.8 seconds per vector for full calculations—an approximate 88% reduction in computation time. Third, at query time, a personalized ranking vector containing 14 million non-zero entries was assembled from partial components in just 6 seconds with an error rate of about 16%, outperforming standard iterative benchmarks.

These findings prove that search engines can move beyond static global rankings and provide fine-grained personalization without linear cost scaling. By sharing partial vectors across multiple user profiles, infrastructure costs and memory footprints are minimized. Furthermore, the ability to assemble rankings incrementally allows systems to balance latency and precision dynamically based on server load.

Organizations operating large-scale search, discovery, or recommendation platforms should consider adopting this hub-decomposition architecture. Engineering teams should select pages with high global PageRank or curated directory topics as hubs, as high-centrality hubs stop graph traversal faster and yield the smallest partial vectors. Future work should explore extending this technique via "web skeletons" to support personalization over arbitrary non-hub pages.

The reported benchmarks are limited to a fixed experimental setup running six iterative passes on an 80-million-page static snapshot. While confidence in the underlying linear and graph-theoretic formulations is very high, practitioners should validate these memory-partitioning and real-time assembly methods against continuously evolving live web graphs and variable real-world query loads.

Stanford University linkage analysis theoretical result viewpoint personalized web search PageRank algorithm PageRank vector

Jeh et al (2003).pdf

Paper: Topic-sensitive PageRank, Taher H. Haveliwala (2002). This paper introduces Topic-Sensitive PageRank, establishing the foundational framework of precomputing topic-biased ranking vectors that the source paper directly builds upon and optimizes for full personalization.
Paper: The PageRank Citation Ranking : Bringing Order to the Web, Lawrence M. Page et al. (1999). This seminal paper introduces the core PageRank algorithm and random-surfer model that provide the fundamental mathematical basis for personalized ranking calculations.
Paper: The anatomy of a large-scale hypertextual Web search engine, Sergey Brin et al. (1998). This foundational work details the large-scale web search engine architecture and link-based scoring mechanisms that the source paper seeks to scale and personalize.
Paper: SimRank: a measure of structural-context similarity, Glen Jeh et al. (2002). This paper presents structural-context similarity algorithms on graphs, offering essential background on iterative random-walk methods across web link topologies.

Paper: TextRank: Bringing Order into Text, Rada Mihalcea et al. (2004). This paper adapts graph-based random-walk ranking principles like PageRank to natural language processing tasks such as keyword and sentence extraction.
Paper: Google news personalization: scalable online collaborative filtering, Abhinandan Das et al. (2007). This work addresses practical system architectures for scaling real-time personalization across millions of users in large web environments.
Paper: PathSim, Yizhou Sun et al. (2011). This paper extends graph-based similarity and ranking mechanisms to heterogeneous information networks by incorporating meta-paths.

Cover for Scaling personalized web search

Abstract

Recent web search techniques augment traditional text matching with a global notion of “importance” based on the linkage structure of the web, such as in Google’s PageRank algorithm. For more refined searches, this global notion of importance can be specialized to create personalized views of importance—for example, importance scores can be biased according to a user-specified set of initially-interesting pages. Computing and storing all possible personalized views in advance is impractical, as is computing personalized views at query time, since the computation of each view requires an iterative computation over the web graph. We present new graph-theoretical results, and a new technique based on these results, that encode personalized views as partial vectors. Partial vectors are shared across multiple personalized views, and their computation and storage costs scale well with the number of views. Our approach enables incremental computation, so that the construction of personalized views from partial vectors is practical at query time. We present efficient dynamic programming algorithms for computing partial vectors, an algorithm for constructing personalized views from partial vectors, and experimental results demonstrating the effectiveness and scalability of our techniques.

Topics

linkage analysis theoretical result viewpoint web graphs web searches multiple views search process

Keywords

personalized web search PageRank algorithm PageRank vector global PageRank selective expansion algorithms web graph preference vectors iterative computation

1 Introduction and Motivation
2 Preliminaries
3 Basis Vectors
4 Decomposition of Basis Vectors
4.1 Inverse P-distance
4.2 Partial Vectors
4.3 Hubs Skeleton
4.4 Discussion
4.4.1 Summary
4.4.2 Choice of H
4.4.3 Web Skeleton
5 Computation
5.1 Decomposition Theorem
5.2 Algorithms for Computing Basis Vectors
5.2.1 Basic Dynamic Programming Algorithm
5.2.2 Selective Expansion Algorithm
5.2.3 Repeated Squaring Algorithm
5.3 Computing Partial Quantities
5.3.1 Partial Vectors
5.3.2 Hubs Skeleton
5.3.3 Web Skeleton
5.4 Construction of PPV's
6 Experiments
6.1 Computing Partial Vectors
6.2 Computing the Hubs Skeleton
6.3 Constructing Hub Vectors from Partial Vectors
7 Related Work
8 Summary
9 Acknowledgment
References
APPENDIX
A Proof: Linearity Theorem
B Proof: Decomposition Theorem
C Inverse P-distance
C.1 Relation to Personalized PageRank
C.2 Loop Factor
D Proof: Hubs Theorem
E Proof: Basic Dynamic Programming Algorithm
F Proof: Selective Expansion Algorithm
G Proof: Repeated Squaring Algorithm
H Proof: Computation of Partial Vectors
I Proof: Computation of the Hubs Skeleton

Knowls

Knowl 1 — Hubs Theorem and Hubs Equation for Basis Vector Reconstruction
theoretical result
For a web graph $G = (V, E)$ , let $H \subseteq V$ denote a set of designated hub pages, $c \in (0, 1)$ the teleportation constant, and $x_p \in \mathbb{R}^{|V|}$ the unit basis vector having value $1$ at index $p$ and $0$ elsewhere. Let $r_p$ be the personalized PageRank basis vector for page $p$ , and let $r^H_p$ denote the restricted component of $r_p$ corresponding to random walk paths passing through at least one hub page in $H$ (excluding endpoints).

The Hubs Theorem states that for any page $p \in V$ and hub set $H \subseteq V$ :

$r^H_p = \frac{1}{c} \sum_{h \in H} (r_p(h) - c x_p(h)) \left(r_h - r^H_h - c x_h\right)$

Substituting this result into $r_p = (r_p - r^H_p) + r^H_p$ yields the Hubs Equation, which constructs the full basis vector $r_p$ from partial vectors $(r_h - r^H_h)$ and the hubs skeleton entries $r_p(H) = \{ (h, r_p(h)) \mid h \in H \}$ :

$r_p = (r_p - r^H_p) + \frac{1}{c} \sum_{h \in H} (r_p(h) - c x_p(h)) \left[(r_h - r^H_h) - c x_h\right]$

A dual formulation based on the first occurrence of a hub node on a path also holds:

$r^H_p = \frac{1}{c} \sum_{h \in H} \left(r_p(h) - r^H_p(h) - c x_p(h)\right) (r_h - c x_h)$
Knowl 2 — Query-Time Construction of Personalized PageRank Vectors from Partial Vectors
algorithm
Let $H \subseteq V$ be a preselected hub set. Given a user preference vector $u = \sum_{i=1}^z \alpha_i x_{p_i}$ with $\sum_{i=1}^z \alpha_i = 1$ and preferences restricted to hub nodes $p_i \in H$ , the full personalized PageRank vector (PPV) $v$ is constructed directly from precomputed partial vectors $(r_h - r^H_h)$ and the hubs skeleton $S = \{r_p(H) \mid p \in H\}$ .
Input: Preference weights and hubs $\{(\alpha_i, p_i)\}_{i=1}^z$ where $p_i \in H$ ; precomputed partial vectors $(r_h - r^H_h)$ for $h \in H$ ; hubs skeleton $S = \{r_p(H) \mid p \in H\}$ ; active hub subset $Q \subseteq H$ ; teleportation constant $c$ .
Output: Personalized PageRank Vector $v$ .
for each $h \in Q$ do
$r_u(h) \leftarrow \sum_{i=1}^z \alpha_i (r_{p_i}(h) - c x_{p_i}(h))$
end for
$v \leftarrow \sum_{i=1}^z \alpha_i (r_{p_i} - r^H_{p_i})$
for each $h \in Q$ such that $r_u(h) > 0$ do
$v \leftarrow v + \frac{1}{c} r_u(h) [ (r_h - r^H_h) - c x_h ]$
end for
return $v$
Setting $Q = H$ produces an exact construction of $v$ . For faster query-time evaluation, $Q$ can be truncated to the $m$ hubs $h \in H$ with the highest values of $r_u(h)$ .
Knowl 3 — Decomposition Theorem for Personalized PageRank Basis Vectors
theoretical result
Let $G = (V, E)$ be a directed graph where each node $p \in V$ has out-degree $|O(p)| > 0$ with out-neighbors $O_1(p), \dots, O_{|O(p)|}(p)$ . Let $c \in (0, 1)$ be the teleportation constant, and $x_p \in \mathbb{R}^{|V|}$ the unit basis vector for node $p$ . Let $r_p$ denote the personalized PageRank basis vector corresponding to preference vector $u = x_p$ .

The basis vector $r_p$ satisfies the linear decomposition:

$r_p = \frac{1 - c}{|O(p)|} \sum_{i=1}^{|O(p)|} r_{O_i(p)} + c x_p$

Thus, node $p$ 's personalized view of the graph is the average of the personalized views of its out-neighbors scaled by $(1-c)$ , plus a teleportation compensation component $c x_p$ assigned to $p$ itself.
Knowl 4 — Linearity Theorem of Personalized PageRank Vectors
theoretical result
Let $A$ be the transition matrix of a web graph $G = (V, E)$ , where $A_{ij} = \frac{1}{|O(j)|}$ if directed edge $(j, i) \in E$ and $A_{ij} = 0$ otherwise. For teleportation constant $c \in (0, 1)$ and preference distribution vector $u$ ( $\|u\|_1 = 1$ ), the personalized PageRank vector (PPV) $v$ is the unique solution to $v = (1 - c)Av + cu$ .

The mapping from preference vectors to PPVs is linear: for any preference vectors $u_1, u_2$ with corresponding PPVs $v_1, v_2$ , and any non-negative weights $\alpha_1, \alpha_2 \ge 0$ such that $\alpha_1 + \alpha_2 = 1$ :

$\alpha_1 v_1 + \alpha_2 v_2 = (1 - c)A(\alpha_1 v_1 + \alpha_2 v_2) + c(\alpha_1 u_1 + \alpha_2 u_2)$

Consequently, for any general preference vector $u = \sum_{i=1}^n \alpha_i x_i$ (where $x_i$ is the unit vector for page $i$ ), the resulting PPV $v$ is given by $v = \sum_{i=1}^n \alpha_i r_i$ , where $r_i$ is the basis vector for page $i$ .
Knowl 5 — Equivalence of Personalized PageRank Scores and Inverse P-Distance
theoretical result
For a directed graph $G = (V, E)$ with teleportation constant $c \in (0, 1)$ , let a tour $t = \langle w_1, w_2, \dots, w_k \rangle$ from $p = w_1$ to $q = w_k$ have length $l(t) = k - 1$ and probability $P[t] = \prod_{i=1}^{k-1} \frac{1}{|O(w_i)|}$ (with $P[t] = 1$ when $l(t) = 0$ ). The inverse P-distance from $p$ to $q$ is defined as:

$r'_p(q) = \sum_{t: p \leadsto q} P[t] c (1 - c)^{l(t)}$

summed over all directed walks $t$ from $p$ to $q$ , allowing node repetitions and cycles. For every pair of nodes $p, q \in V$ , the inverse P-distance is identically equal to the personalized PageRank score:

$r_p(q) = r'_p(q)$

where $r_p(q)$ is the $q$ -th entry of basis vector $r_p$ . Furthermore, restricting the summation to tours $t: p \xrightarrow{H} q$ of length $l(t) \ge 2$ that pass through at least one hub node $h \in H$ at an intermediate step defines the restricted score:

$r^H_p(q) = \sum_{t: p \xrightarrow{H} q} P[t] c (1 - c)^{l(t)}$
Knowl 6 — Computation of Partial Vectors via Selective Expansion
algorithm
Partial vectors $(r_p - r^H_p)$ capture the random walk paths originating at hub $p \in H$ that terminate without traversing intermediate hub nodes in $H$ . They are computed by restricting the expansion set of the selective expansion algorithm after the initial step.
Input: Web graph adjacency matrix $A$ with out-degrees $|O(q)|$ ; hub set $H$ ; teleportation constant $c$ ; iteration count $K$ .
Output: Approximated partial vectors $(r_p - r^H_p) \approx D_K[p] + c E_K[p]$ for each $p \in H$ .
for each $p \in H$ do
     $D_0[p] \leftarrow 0$
     $E_0[p] \leftarrow x_p$
end for
for $k = 0, 1, \dots, K-1$ do
    if $k = 0$ then
         $Q_k \leftarrow V$
    else
         $Q_k \leftarrow V \setminus H$
    end if

    for each $p \in H$ do
         $D_{k+1}[p] \leftarrow D_k[p] + \sum_{q \in Q_k} c E_k[p](q) x_q$
         $E_{k+1}[p] \leftarrow E_k[p] - \sum_{q \in Q_k} E_k[p](q) x_q + \sum_{q \in Q_k} \frac{1-c}{|O(q)|} \sum_{i=1}^{|O(q)|} E_k[p](q) x_{O_i(q)}$
    end for
end for
return $\{ D_K[p] + c E_K[p] \mid p \in H \}$
Because hub nodes in $H$ are excluded from expansion for all $k > 0$ , paths passing through $H$ are eliminated, and $D_k[p] + c E_k[p]$ converges to $(r_p - r^H_p)$ as $k \to \infty$ .
Knowl 7 — Computation of the Hubs Skeleton via Repeated Squaring
algorithm
The hubs skeleton $S = \{ r_p(H) \mid p \in H \}$ contains hub-to-hub PageRank scores $r_p(h)$ for $p, h \in H$ . It is computed using repeated squaring initialized from the intermediate states $(D_k[p], E_k[p])$ generated during partial vector computation.
Input: Intermediate results $(D_0[p], E_0[p])$ for $p \in H$ from partial vector computation satisfying non-hub error $\sum_{q \notin H} E_0[p](q) < \epsilon$ ; hub set $H$ ; number of squaring iterations $I$ .
Output: Hubs skeleton estimates $D_I[p](H)$ for all $p \in H$ .
for each $p \in H$ do
    Drop all entries $q \notin H$ from $D_0[p]$ and $E_0[p]$
end for
for iteration $i = 0, 1, \dots, I-1$ do
    for each $p \in H$ do
         $D_{i+1}[p] \leftarrow D_i[p] + \sum_{q \in H} E_i[p](q) D_i[q]$
         $E_{i+1}[p] \leftarrow \sum_{q \in H} E_i[p](q) E_i[q]$
    end for
end for
return $\{ D_I[p](H) \mid p \in H \}$
After $i$ iterations of repeated squaring restricted to $Q_k(p) = H$ , total error $|E_i[p]|$ is bounded by $(1 - c)^{2^i} + \frac{\epsilon}{c}$ . Non-hub entries are discarded, making time and storage depend only on $|H|$ rather than $|V|$ .
Knowl 8 — Invariant Formulation and Fixed-Point Algorithms for PageRank Basis Vectors
model/method
Iterative algorithms for computing a basis vector $r_p$ maintain a lower-approximation vector $D_k[p]$ and an error projection vector $E_k[p]$ satisfying the invariant for all $k \ge 0$ and $p \in V$ :

$D_k[p] + \sum_{q \in V} E_k[p](q) r_q = r_p$

Initialized with $D_0[p] = 0$ and $E_0[p] = x_p$ , intermediate states are refined via two algorithms:
1. Selective Expansion: For a chosen expansion set $Q_k(p) \subseteq V$ : $D_{k+1}[p] = D_k[p] + \sum_{q \in Q_k(p)} c E_k[p](q) x_q$ $E_{k+1}[p] = E_k[p] - \sum_{q \in Q_k(p)} E_k[p](q) x_q + \sum_{q \in Q_k(p)} \frac{1-c}{|O(q)|} \sum_{i=1}^{|O(q)|} E_k[p](q) x_{O_i(q)}$ When $Q_k(p) = V$ , error $|E_k[p]|$ decays by a factor of $(1-c)$ per step.
2. Repeated Squaring: For a chosen expansion set $Q_k(p) \subseteq V$ : $D_{2k}[p] = D_k[p] + \sum_{q \in Q_k(p)} E_k[p](q) D_k[q]$ $E_{2k}[p] = E_k[p] - \sum_{q \in Q_k(p)} E_k[p](q) x_q + \sum_{q \in Q_k(p)} E_k[p](q) E_k[q]$ When $Q_k(p) = V$ , $E_{2k}[p] = \sum_{q \in V} E_k[p](q) E_k[q]$ , and the error squares each iteration ( $|E_{2k}[p]| = |E_k[p]|^2$ ).
Knowl 9 — Sparsity and Scalability of Partial Vectors Across Hub Set Sizes
empirical result
Experiments conducted on an 80-million-page crawl (Stanford WebBase with leaf pages removed) using teleportation parameter $c = 0.15$ and hub sets $H$ comprising top-PageRank pages demonstrated that partial vectors shrink as $|H|$ increases, whereas full hub vectors do not:
- Full hub vectors maintained an average size (number of non-zero entries) of 90,000 to 96,000 across $|H| \in [1000, 50000]$ after 6 iterations (taking $\approx 2.8$ seconds per vector at $|H| = 50,000$ ).
- Partial vectors $(r_p - r^H_p)$ $(r_{p} - r_{p}^{H})$ decreased in average size monotonically as $|H|$ $∣ H ∣$ grew:
  - $|H| = 1,000$ : $\approx 58,000$ nonzeros
  - $|H| = 2,000$ : $\approx 60,000$ nonzeros
  - $|H| = 5,000$ : $\approx 54,000$ nonzeros
  - $|H| = 10,000$ : $\approx 49,000$ nonzeros
  - $|H| = 20,000$ : $\approx 44,000$ nonzeros
  - $|H| = 50,000$ : $\approx 34,000$ nonzeros (taking $\approx 0.33$ seconds per vector)
  - $|H| = 100,000$ : $\approx 26,000$ nonzeros
Choosing hub pages from high-PageRank nodes yielded substantially smaller partial vectors than selecting random hub pages because high-PageRank nodes are closer in inverse P-distance to other nodes, causing path expansions from $p$ to encounter $H$ sooner.
Knowl 10 — Trade-offs in Hub Vector Construction Time and Sparsity vs Hubs Skeleton Size
empirical result
Evaluating hub vector construction on the 80-million-page WebBase graph ( $|H| = 10,000$ , $c = 0.15$ ) by assembling 6-iteration partial vectors using the top- $m$ entries of the hubs skeleton $r_p(H)$ showed:
- Reconstruction error remained at $\approx 16\%$ ( $0.166$ at $m = 100$ to $0.163$ at $m = 10,000$ ), much lower than the $38\%$ error $(1-c)^6$ of 6-iteration full hub vectors.
- Average constructed non-zero vector size and in-memory construction time scaled with $m$ $m$ as follows:
  - $m = 100$ : $\approx 2.5 \times 10^6$ nonzeros, $\approx 0.5$ seconds
  - $m = 200$ : $\approx 4.0 \times 10^6$ nonzeros, $\approx 1.0$ seconds
  - $m = 500$ : $\approx 8.5 \times 10^6$ nonzeros, $\approx 2.5$ seconds
  - $m = 1,000$ : $\approx 14.0 \times 10^6$ nonzeros, $\approx 6.0$ seconds
  - $m = 2,000$ : $\approx 24.0 \times 10^6$ nonzeros, $\approx 14.0$ seconds
  - $m = 5,000$ : $\approx 34.0 \times 10^6$ nonzeros, $\approx 38.0$ seconds
  - $m = 10,000$ : $\approx 40.0 \times 10^6$ nonzeros, $\approx 60.0$ seconds
By contrast, a directly computed 6-iteration full hub vector contained only 93,993 nonzeros (reaching nodes at most 6 links away), whereas assembling with $m = 1,000$ yielded 14 million nonzeros in 6 seconds.

Coverage note — Omitted the web skeleton extension for arbitrary non-hub personalization and the loop factor self-influence analysis as they are minor conceptual discussions and secondary extensions.

References

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4.Taher H. Haveliwala. Efficient computation of PageRank. Technical report, Stanford University Database Group, 1999. http://dbpubs.stanford.edu/pub/1999-31.
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6.Jun Hirai, Sriram Raghavan, Andreas Paepcke, and Hector Garcia-Molina. WebBase: A repository of web pages. In Proceedings of the Ninth International World Wide Web Conference, Amsterdam, Netherlands, May 2000. http://www-diglib.stanford.edu/~testbed/doc2/WebBase/.
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8.Jon M. Kleinberg. Authoritative sources in a hyperlinked environment. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, January 1998.
9.Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, United Kingdom, 1995.
10.Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The PageRank citation ranking: Bringing order to the Web. Technical report, Stanford University Database Group, 1998. http://citeseer.nj.nec.com/368196.html.
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Citation

MLA

Jeh, G., and J. Widom. “Scaling Personalized Web Search”. Proceedings of the Twelfth International Conference on World Wide Web - WWW '03, 2003, p. 271, https://doi.org/10.1145/775152.775191.

APA

Jeh, G., & Widom, J. (2003). Scaling personalized web search. Proceedings of the Twelfth International Conference on World Wide Web - WWW '03, 271. https://doi.org/10.1145/775152.775191

Chicago

Jeh, G., and J. Widom. 2003. “Scaling Personalized Web Search”. Proceedings of the Twelfth International Conference on World Wide Web - WWW '03, 271. https://doi.org/10.1145/775152.775191.

Harvard

Jeh, G. and Widom, J. (2003) “Scaling personalized web search”, Proceedings of the twelfth international conference on World Wide Web - WWW '03. ACM Press, p. 271. Available at: https://doi.org/10.1145/775152.775191.

Vancouver

1. Jeh G, Widom J (2003) Scaling personalized web search. In: Proceedings of the twelfth international conference on World Wide Web - WWW '03. ACM Press, p 271

BibTeX

@inproceedings{Jeh_2003, series={WWW ’03}, title={Scaling personalized web search}, url={http://dx.doi.org/10.1145/775152.775191}, DOI={10.1145/775152.775191}, booktitle={Proceedings of the twelfth international conference on World Wide Web  - WWW ’03}, publisher={ACM Press}, author={Jeh, Glen and Widom, Jennifer}, year={2003}, pages={271}, collection={WWW ’03} }

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Scaling personalized web search

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Hubs Theorem and Hubs Equation for Basis Vector Reconstruction

Knowl 2 — Query-Time Construction of Personalized PageRank Vectors from Partial Vectors

Knowl 3 — Decomposition Theorem for Personalized PageRank Basis Vectors

Knowl 4 — Linearity Theorem of Personalized PageRank Vectors

Knowl 5 — Equivalence of Personalized PageRank Scores and Inverse P-Distance

Knowl 6 — Computation of Partial Vectors via Selective Expansion

Knowl 7 — Computation of the Hubs Skeleton via Repeated Squaring

Knowl 8 — Invariant Formulation and Fixed-Point Algorithms for PageRank Basis Vectors

Knowl 9 — Sparsity and Scalability of Partial Vectors Across Hub Set Sizes

Knowl 10 — Trade-offs in Hub Vector Construction Time and Sparsity vs Hubs Skeleton Size

References

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