To Randomize or Not To Randomize: Space Optimal Summaries for Hyperlink Analysis

# ABSTRACT

Personalized PageRank expresses link-based page quality around user selected pages. The only previous personalized PageRank algorithm that can serve on-line queries for an unrestricted choice of pages on large graphs is our Monte Carlo algorithm [WAW 2004]. In this paper we achieve unrestricted personalization by combining rounding and randomized sketching techniques in the dynamic programming algorithm of Jeh and Widom [WWW 2003]. We evaluate the precision of approximation experimentally on large scale real-world data and find significant improvement over previous results. As a key theoretical contribution we show that our algorithms use an optimal amount of space by also improving earlier asymptotic worst-case lower bounds. Our lower bounds and algorithms apply to SimRank as well; of independent interest is the reduction of the SimRank computation to personalized PageRank.

## Categories & Subject Descriptors

H.3.3 [Information Storage and Retrieval]: Information Search and Retrieval
G.2.2 [Discrete Mathematics]: Graph Theory - Graph algorithms
G.3 [Mathematics of Computing]: Probability and Statistics - Probabilistic algorithms

## General Terms

Algorithms, Theory, Experimentation

## Keywords

link-analysis, similarity search, scalability, data streams

# 1 Introduction

The idea of using hyperlink mining algorithms in Web search engines appears since the beginning of the success of Google's PageRank [24]. Hyperlink based methods are based on the assumption that a hyperlink implies that page votes for as a quality page. In this paper we address the computational issues [13,17,11,12] of personalized PageRank [24] and SimRank [16].

Personalized PageRank (PPR) [24] enters user preferences by assigning more importance to the neighborhood of pages at the user's selection. Jeh and Widom [16] introduced SimRank, the multi-step link-based similarity function with the recursive idea that two pages are similar if pointed to by similar pages. Notice that both measures are hard to compute over massive graphs: naive personalization would require on the fly power iteration over the entire graph for a user query; naive SimRank computation would require power iteration over all pairs of vertices.

We give algorithms with provable performance guarantees based on computation with sketches [7] as well as simple deterministic summaries; see Table 1 for a comparison of our methods with previous approaches. We may personalize to any single page from which arbitrary page set personalization follows by linearity [13]. Similarly, by our SimRank algorithm we may compute the similarity of any two pages or the similarity top list of any single page. Motivated by search engine applications, we give two-phase algorithms that first compute a compact database from which value or top list queries can be answered with a low number of accesses. Our key results are summarized as follows:

We give practical methods for serving unrestricted on-line personalized PageRank (Section 2.1) as well as SimRank queries with space a reasonable constant per vertex (Section 3). The methods are based on deterministic rounding.
We give a theoretically optimal algorithm for personalized PageRank value queries (Section 2.2) based on randomized sketching. Given an additive error and the probability of an incorrect result, we improve the disk usage bound from [11,12] to .
We give theoretically optimal algorithms for SimRank value and top list queries (Section 3.1) by a nontrivial reduction of SimRank to personalized PageRank.
We improve the communication complexity based lower
bounds of [11,12] for the size of the database (Section 4); our bounds are matched by our algorithms. Our sketch-based algorithms use optimal space; surprisingly for top list queries deterministic rounding is already optimal in itself.
In Section 5 we experimentally analyze the precision of approximation over the Stanford WebBase graph and conclude that our summaries provide better approximation for the top personalized PageRank scores than previous methods.

Table 1: Comparison of personalized PageRank algorithms for graphs of vertices, additive error and error probability .

## 1.1 Related Results

The scalable computation of personalized PageRank was addressed by several papers [13,18,17] that gradually increase the choice for personalization. By Haveliwala's method [13] we may personalize to the combination of 16 topics extracted from the Open Directory Project. The BlockRank algorithm of Kamvar et al. [18] speeds up personalization to the combination of hosts. The state of the art Hub Decomposition algorithm of Jeh and Widom [17] computed and encoded personalization vectors for approximately 100K personalization pages.

To the best of our knowledge, the only scalable personalized PageRank algorithm that supports the unrestricted choice of the teleportation vector is the Monte Carlo method of [11]. This algorithm samples the personalized PageRank distribution of each page simultaneously during the precomputation phase, and estimates the personalized PageRank scores from the samples at query time. The drawback of the sampling approach is that approximate scores are returned, where the error of approximation depends on the random choice. In addition the bounds involve the unknown variance, which can in theory be as large as , and hence we need random samples. Indeed a matching sampling complexity lower bound for telling binomial distributions with means apart [1] indicates that one can not reduce the number of samples when approximating personalized PageRank. Similar findings of the superiority of summarization or sketching over sampling is described in [5]. The algorithms presented in Section 2 outperform the Monte Carlo method by significantly reducing the error.

We also address the computational issues of SimRank, a link-based similarity function introduced by Jeh and Widom [16]. The power iteration SimRank algorithm of [16] is not scalable since it iterates on a quadratic number of values, one for each pair of Web pages; in [16] experiments on graphs with no more than 300K vertices are reported. Analogously to personalized PageRank, the scalable computation of SimRank was first achieved by sampling [12]. Our new SimRank approximation algorithms presented in Section 3 improve the precision of computation.

The key idea of our algorithms is that we use lossy representation of large vectors either by rounding or sketching. Sketches are compact randomized data structures that enable approximate computation in low dimension. To be more precise, we adapt the Count-Min Sketch of Cormode and Muthukrishnan [7], which was primarily introduced for data stream computation. We use sketches for small space computation; in the same spirit Palmer et al. [25] apply probabilistic counting sketches to approximate the sizes of neighborhoods of vertices in large graphs. Further sketching techniques for data streams are surveyed in [23]. Lastly we mention that Count-Min Sketch and the historically first sketch, the Bloom filter [2] stem from the same idea; we refer to the detailed survey [4] for further variations and applications.

Surprisingly, it turns out that sketches do not help if the top highest ranked or most similar nodes are queried; the deterministic version of our algorithms show the same performance as the randomized without even allowing a small probability of returning a value beyond the error bound. Here the novelty is the optimal performance of the deterministic method; the top problem is known to cause difficulties in sketch-based methods and always increases sketch sizes by a factor of . By using times larger space we may use a binary search structure or we may use sketches accessed times per query [7]. Note that queries require an error probability of that again increase sketch sizes by a factor of .

In Section 4 we show that our algorithms build optimal sized databases. To obtain lower bounds on the database size, we apply communication complexity techniques that are commonly used for space lower bounds [21]. Our reductions are somewhat analogous to those applied by Henzinger et al. [14] for space lower bounds on stream graph computation.

## 1.2 Preliminaries

We briefly introduce notation, and recall definitions and basic facts about PageRank, SimRank and the Count-Min sketch.

## Personalized PageRank

Let us consider the web as a graph. Let denote the number of vertices and the number edges. Let and denote the number of edges leaving and entering , respectively. Details of handling nodes with and are omitted.

In [24] the PageRank vector , ..., is defined as the solution of the following equation , where , ..., is the teleportation vector and is the teleportation probability with a typical value of . If is uniform, i.e. for all , then is the PageRank. For non-uniform the solution is called personalized PageRank; we denote it by PPR. Since PPR is linear in [13,17], it can be computed by linear combination of personalization to single points , i.e. to vectors consisting of all 0 except for node where . Let PPRPPR.

An alternative characterization of PPR [10,17] is based on the probability that a length random walk starting at node ends in node . We obtain PPR by choosing random according to the geometric distribution:

 PPR (1)
the summation is along walks starting at and ending in . Thus
 PPR PPR (2)

Similarly we get PPR if we sum up only to instead of . An equivalent reformulation of the path summing formula (2) is the Decomposition Theorem proved by Jeh and Widom [17]:
 PPR PPR (3)
The Decomposition Theorem immediately gives rise to the Dynamic Programming approach [17] to compute personalized PageRank that performs iterations for , 2, ...with PPR:
 PPR PPR (4)

## SimRank

Jeh and Widom [16] define SimRank by the following equation very similar to the PageRank power iteration such that Sim and

 Sim (5)
where summation is for .

## Count-Min Sketch

The Count-Min Sketch [7] is a compact randomized approximate representation of non-negative vector , ..., such that a single value can be queried with a fixed additive error and a probability of returning a value out of this bound. The representation is a table of depth and width . One row of the table is computed with a random hash function . The th entry of the row is defined as . Then the Count-Min sketch table of consists of such rows with hash functions chosen uniformly at random from a pairwise-independent family.

Theorem 1 (Cormode, Muthukrishnan [7]) Let where the minimum is taken over the rows of the table. Then and Prob hold.

Count-Min sketches are based on the principle that any randomized approximate computation with one sided error and bias can be turned into an algorithm that has guaranteed error at most with probability by running parallel copies and taking the minimum. The proof simply follows from Markov's inequality and is described for the special cases of sketch value and inner product in the proofs of Theorems 1 and 2 of [7], respectively.

# 2 Personalized PageRank

We give two efficient realizations of the dynamic programming algorithm of Jeh and Widom [17]. Our algorithms are based on the idea that if we use an approximation for the partial values in certain iteration, the error will not aggregate when summing over out-edges, instead the error of previous iterations will decay with the power of . Our first algorithm in Section 2.1 uses certain deterministic rounding optimized for smallest runtime for a given error, while our second algorithm in Section 2.2 is based on Count-Min sketches [7].

The original implementation of dynamic programming [17] relies on the observation that in the first iterations of dynamic programming only vertices within distance have non-zero value. However, the rapid expansion of the -neighborhoods increases disk requirement close to after a few iterations, which limits the usability of this approach2. Furthermore, an external memory implementation would require significant additional disk space.

Figure 1: A simple example showing the superiority of dynamic programming over power iterations for small space computations.

We may justify why dynamic programming is the right choice for small-space computation by comparing dynamic programming to power iteration over the graph of Fig. 1. When computing PPR, power iteration moves top-down, starting at , stepping into its neighbors and finally adding up all their values at . Hence when approximating, we accumulate all error when entering the large in-degree node and in particular we must compute PPR values fairly exact. Dynamic programming, in contrast, moves bottom up by computing the trivial PPR vector, then all the PPR, then finally averages all of them into PPR. Because of averaging we do not amplify error at large in-degrees; even better by looking at (4) we notice that the effect of earlier steps diminishes exponentially in . In particular even if there are edges entering from further nodes, we may safely discard all the small PPR values for further computations, thus saving space over power iteration where we require the majority of these values in order to compute PPR with little error.

We measure the performance of our algorithms in the sense of intermediate disk space usage. Notice that our algorithms are two-phase in that they preprocess the graph to a compact database from which value and top list queries can be served real-time; preprocessing space and time is hence crucial for a search engine application. Surprisingly, in this sense rounding in itself yields an optimal algorithm for top list queries as shown by giving a matching lower bound in Section 4. The sketching algorithm further improves space usage by a factor of and is hence optimal for single value queries. For finding top lists, however, we need additional techniques such as binary searching as in [7] that loose the factor gain and use asymptotically the same amount of space as the deterministic algorithm. Since the deterministic rounding involves no probability of giving an incorrect answer, that algorithm is superior for top list queries.

The key to the efficiency of our algorithms is the use of small size approximate values obtained either by rounding and handling sparse vectors or by computing over sketches. In order to perform the update step of Algorithm 1 we must access all vectors; the algorithm proceeds as if we were multiplying the weighted adjacency matrix for with the vector parallel for all values of . We may use (semi)external memory algorithms [27]; efficiency will depend on the size of the description of the vectors.

The original algorithm of Jeh and Widom defined by equation (4) uses two vectors in the implementation. We remark that a single vector suffices since by using updated values within an iteration we only speed convergence up. A similar argument is given by McSherry [22] for the power iteration, however there the resulting sequential update procedure still requires two vectors.

## 2.1 Rounding

In Algorithm 1 we compute the steps of the dynamic programming personalized PageRank algorithm (4) by rounding all values down to a multiple of the prescribed error value . As the sum of PPR for all equals one, the rounded non-zeroes can be stored in small space since there may be at most of them.

We improve on the trivial observation that there are at most rounded non-zero values in two ways as described in the next two theorems. First, we observe that the effect of early iterations decays as the power of in the iterations, allowing us to similarly increase the approximation error for early iterations . We prove correctness in Theorem 2; later in Theorem 4 it turns out that this choice also weakens the dependency of the running time on the number of iterations. Second, we show that the size of the non-zeroes can be efficiently bit-encoded in small space; while this observation is less relevant for a practical implementation, this is key in giving an algorithm that matches the lower bound of Section 4.

Theorem 2 Algorithm 1 returns values between PPR and PPR.
Proof. By induction on iteration of Algorithm 1 we show a bound that is tighter for than that of the Theorem:
PPR
By the choice of and we have , thus the case is immediate since PPR.

Since we use a single vector in the implementation, we may update a value by values that have themselves already been updated in iteration . Nevertheless since and hence decreases in , values that have earlier been updated in the current iteration in fact incur an error smaller than required on the right hand side of the update step of Algorithm 1. In order to distinguish values before and after a single step of the update, let us use to denote values on the right hand side. To prove, notice that by the Decomposition Theorem (3)

 PPR
As introduces at most error, by the triangle inequality
 PPR

Using the inductive hypothesis this leads us to
 PPR
completing the proof.

Next we show that multiples of that sum up to 1 can be stored in bit space. For the exact result we need to select careful but simple encoding methods given in the trivial lemma below.

Lemma 3 Let non-negative values be given, each a multiple of that sum up to at most 1. If we unary encode the values as multiples of and use a termination symbol, the encoding uses space bits. If we combine the same encoding with sparse vector storage by recording the position of non-zeroes in space each, we may encode the sequence by bits.
Theorem 4 Algorithm 1 runs in time of bit operations3and builds a database of size bits. In order to return the approximate value of PPR and the largest elements of PPR, we may binary search and sequentially access bits, respectively.
Proof. We determine the running time by noticing that in each iteration for each edge we perform addition with the sparse vector PPR. We may use a linked list of non-zeroes for performing the addition, thus requiring bit operations for each non-zero of the vector. Since in iteration we store all values of a vector with norm at most one rounded down to a multiple of , we require space to store at most non-zeroes of PPR by Lemma 3. By the total running time becomes .

## 2.2 Sketching

Next we give a sketch version of Algorithm 1 that improves the space requirement of the rounding based version by a factor of , thus matches the lower bound of Section 4 for value queries. First we give a basic algorithm that uses uniform error bound in all iterations and is not optimized for storage size in bits. Then we show how to gradually decrease approximation error to speed up earlier iterations with less effect on final error; finally we obtain the space optimal algorithm by the bit encoding of Lemma 3.

The key idea is that we replace each PPR vector with its constant size Count-Min sketch in the dynamic programming iteration (4). Let denote the sketching operator that replaces a vector by the table as in Section 1.2 and let us perform the iterations of (4) with SPPR and . Since the sketching operator is trivially linear, in iteration we obtain the sketch of the next temporary vector SPPR from the sketches SPPR.

To illustrate the main ingredients, we give the simplest form of a sketch-based algorithm with error, space and time analysis. Let us perform the iterations of (4) with wide and deep sketches times; then by Theorem 1 and the linearity of sketching we can estimate PPR for all from SPPR with additive error and error probability . The personalized PageRank database consists of sketch tables SPPR for all . The data occupies machine words, since we have to store tables of reals. An update for node takes time by averaging tables of size and adding , each in time. Altogether the required iterations run in time.

Next we weaken the dependence of the running time on the number of iterations by gradually decreasing error as in Section 2.1. When decreasing the error in sketches, we face the problem of increasing hash table sizes as the iterations proceed. Since there is no way to efficiently rehash data into larger tables, we approximate personalized PageRank slightly differently by representing the end distribution of length walks, PPR, with their rounded sketches in the path-summing formula (2):

 (6)
where denotes the -th hash function of the -th iteration. By (6) we need to calculate efficiently in small space. Notice that unlike in the dynamic programming where we gradually increase the precision of as grows, we may compute less precise with growing since its values are scaled down by . Hence we obtain our algorithm by using wide hash tables in the -th iteration and replacing the last line of Algorithm 1 with
 (7)
where is the recoding function shrinking hash tables from width to . To be more precise, we round the width of each sketch up to the nearest power of two; thus we maintain the error bound, increase space usage by less than a factor of two, and use the recoding function that halves the table when necessary.
Theorem 5 Let us run the sketch version of the dynamic programming Algorithm 1 with sketching to width and rounding all table entries to multiples of in iteration . The algorithm runs in time ; builds a database of size bits; and returns a value such that Prob.
Proof. As still holds, along the same lines as Theorem 4 we immediately get the running time; space follows by Lemma 3.

We err for three reasons: we do not run the iteration infinitely; in iteration we round values down by at most , causing a deterministic negative error; and finally the Count-Min Sketch uses hashing, causing a random positive error. For bounding these errors, imagine running iteration (7) without the rounding function but still with wide and deep sketches and denote its results by SPPR and define

First we upper bound to obtain the last claim of the Theorem. Since , we need Prob. By the Count-Min principle it is sufficient to show that for a fixed row , the expected overestimation of SPPR is not greater than . Since the bias of each sketch row SPPR is , the bias of their exponentially weighted sum is bounded by .

Finally we lower bound ; the bound is deterministic. The loss due to rounding down in iteration affects all subsequent iterations, and hence

And since we sum up to instead of infinity, underestimates PPR by at most , proving the Theorem.

# 3 SimRank

In this section first we give a simpler algorithm for serving SimRank value and top-list queries that combines rounding with the empirical fact that there are relatively few large values in the similarity matrix. Then in Section 3.1 we give an algorithm for SimRank values that uses optimal storage in the sense of the lower bounds of Section 4. Of independent interest is the main component of the algorithm that reduces SimRank to the computation of values similar to personalized PageRank.

SimRank and personalized PageRank are similar in that they both fill an matrix when the exact values are computed. Another similarity is that practical queries may ask for the maximal elements within a row. Unlike personalized PageRank however, when rows can be easily sketched and iteratively computed over approximate values, the matrix structure is lost within the iterations for Sim as we may have to access values of arbitrary Sim. Even worse PPR while

Sim
can in theory be as large as ; an -size sketch may hence store relevant information about personalized PageRank but could not even contain values below 1 for SimRank. An example of a sparse graph with is an -node star where Sim for all pairs other than the center.

In practice is expected be a reasonable constant times . Hence first we present a simple direct algorithm that finds the largest values within the entire Sim table. In order to give a rounded implementation of the iterative SimRank equation (5), we need to give an efficient algorithm to compute a single iteration. The naive implementation requires time for each edge pair with a common source vertex that may add up to . Instead for we will compute the next iteration with the help of an intermediate step when edges out of only one of the two vertices are considered:

 ASim Sim (8) Sim ASim (9)

Along the same line as the proof of Theorems 2 we prove that (i) by rounding values in iterations (8-9) we approximate values with small additive error; (ii) the output of the algorithm occupies small space; and (iii) approximate top lists can be efficiently answered from the output. The proof is omitted due to space limitations. We remark here that (8-9) can be implemented by 4 external memory sorts per iteration, in two of which the internal space usage can in theory grow arbitrary large even compared to . This is due to the fact that we may round only once after each iteration; hence if for some large out-degree node a value Sim is above the rounding threshold or ASim becomes positive, then we have to temporarily store positive values for all out-neighbors, most of which will be discarded when rounding.

Theorem 6 Let us iterate (8-9) times by rounding values in iteration down to multiples of for (9) and for (8).

(i)
The algorithm returns approximate values for with
Sim.

(ii)
The space used by the values is bits where Sim.

(iii)
Top list queries can be answered after positive values are sorted for each in time.

## 3.1 Reduction of SimRank to PPR

Now we describe a SimRank algorithm that uses a database of size matching the corresponding lower bound of Section 4 by taking advantage of the fact that large values of similarity appear in blocks of the similarity table. The blocking nature can be captured by observing the similarity of Sim to the product PPRPPR of vectors PPR and PPR.

We use the independent result of [10,17,16] that PageRank type values can be expressed by summing over endpoints of walks as in equation (1). First we express SimRank by walk pair sums, then we show how SimRank can be reduced to personalized PageRank by considering pairs of walks as products. Finally we give sketching and rounding algorithms for value and top queries based on this reduction.

In order to capture pairs of walks of equal length we define reversed'' PPR by using walks of length exactly by modifying (1):

 RP (10)
where , ..., is a walk from to on the transposed graph. Similarly [16] shows that Sim equals the total weight of pairs
with length that both end at and one of them comes from while the other one from . The weight of the pair of walks is the expected meeting distance as defined in [16]:
 (11)
Importantly the walks satisfy two properties: they have (i) equal length and (ii) no common vertex at the same distance from start, i.e. for . Except for the last two requirements Sim has a form similar to the inner product of PPR and PPR on the reversed graph by (1).

Next we formalize the relation and give an efficient algorithm that reduces SimRank to PPR on the reversed graph. As a step 0 try'' we consider

 SimRPRP (12)
with form similar to SimRank with exact length walks except that these walks may have common vertices unlike stated in (ii).

In order to exclude pairs of walks that meet before ending, we use the principle of inclusion and exclusion. We count pairs of walks that have at least meeting points after start as follows. Since after their first meeting point the walks proceed as if computing the similarity of to itself, we introduce a self-similarity measure by counting weighted pairs of walks that start at and terminate at the same vertex by extending (12):

 SSim RP RP (13)
Now we may recursively define a value that counts all pairs of walks with at least inner points where the walks meet; the values below in fact count each pair times that meet at exactly inner points. First we define self-similarity as
 SSimRP RP SSim (14)
and then similarity with at least inner meeting points as
 SimRP RP SSim (15)
By the principle of inclusion and exclusion
 Sim Sim Sim Sim (16)
where Sim is defined in (12) as the weighted number of walk pairs that are unrestricted in the number of meeting times. Induction on shows that Sim, thus the infinite series (16) is (absolute) convergent if . By changing the order of summation (16) becomes
 Sim RP RP SSim where SSim SSim (17)

The proof of the main theorems below are omitted due to space limitations.

Theorem 7 If Algorithm 2 uses bits to store and values; the factor can be replaced by by sketching in width . The algorithm computes these values in time measured in RAM operations and approximates in bit operations time. For given the set of non-negative values can be computed in bit operations time .
Theorem 8 If the above algorithm gives ; when sketching RP, this holds with probability at least .

# 4 Lower bounds

In this section we will prove lower bounds on the database size of approximate PPR algorithms that achieve personalization over a subset of vertices. More precisely we will consider two-phase algorithms: in the first phase the algorithm has access to the edge set of the graph and has to compute a database; in the second phase the algorithm gets a query and has to answer by accessing the database, i.e. the algorithm cannot access the graph during query-time. A worst case lower bound on the database size holds, if for any two-phase algorithm there exists a personalization input such that a database of size bits is built in the first phase.

We will consider the following queries for :

- value approximation: given the vertices approximate with such that
top query: given the vertex , with probability compute the set of vertices which have personalized PPR values according to vertex greater than . Precisely we require the following:

As Theorem 6 of [11] shows, any two-phase PPR algorithm solving the exact ( ) PPR value problem requires an bit database. Our tool towards the lower bounds will be the asymmetric communication complexity game bit-vector probing [14]: there are two players and ; player has a vector of bits; player has a number ; and they have to compute the function , i.e., the output is the th bit of the input vector . To compute the proper output they have to communicate, and communication is restricted in the direction . The one-way communication complexity [21] of this function is the number of transferred bits in the worst case by the best protocol.

Theorem 9 ([14]) Any protocol that outputs the correct answer to the bit-vector probing problem with probability at least must transmit at least bits.

Now we are ready to state and prove our lower bounds, which match the performance of the algorithms presented in Sections 2 and 3.1, hence showing that they are space optimal.

Theorem 10 Any two-phase - PPR value approximation algorithm with builds a database of bits in worst case, when the graph has at least nodes.