1 Introduction
For a directed graph , representing e.g. the web graph, a ranking function maps elements of to a numeric value, so that subsets of can be sorted. Link spam is the manipulation of the edges to game the ranking function.
Spamfighting is the subject of intense research (see Section 2). In 1998, Brin and Page [12] introduced PageRank as a graphtheoretic spamresistant ranking function. However, as we discuss below, the standard formulation of PageRank is easy to spam.
In this paper, we show how to make PageRank [12] spam resistant.
We begin by giving a simple model of link spam. The following discussion relies on the rather mundane observations that if you own a web page, you can edit its links, but acquiring a link from a page you do not own requires effort (or money).
1.1 A cost model for spamming
Define the web graph to be , where is a partition over that defines ownership, that is, if two nodes are in the same partition of , then they have the same owner. Each partition in is called the property of its owner. is a set of trusted sites that are known not to be owned by spammers. The ranking function is only responsible for ranking nodes reachable from some node in .
An owner can perform the following operations for free:

Create a node in its own property;

Change the outlinks of the nodes in its property.
An owner can perform the following operation on , but at a cost:

Purchase a link from a node outside its property to a node in its property;

Purchase an existing node (with its inlinks) from the current owner.
For any property , we define the boundary to be the nodes in that have links from other properties and the interior to be those that do not.
Define the cost of a node to be , where . Normalize ranking function so that . We say that is spam resistant if there is a way to set so that, for every property , , that is, the amount of ranking function accumulated in the interior of a property is at most proportional to the cost of acquiring a boundary.
Notice that we say a ranking function is spam resistant if any pricing scheme shows that you must pay for your rank value. It is too much to require all pricing schemes to yield a payforrank bound. For example, suppose that all but one node cost 0, and one node costs 1. Then any ranking scheme would be free to spam. So our notion of spam resistance is a necessary condition: there exists a market value for nodes so that a spammer is forced to pay for what they get. The more they have to pay, the more resistant.
What about the cost of buying an edge? There is no restriction that the cost function assign nonzero costs only to nodes outside of a partition. If the spammer chooses to buy edges, they have targets inside the property. We can assign the cost of edges to their target node. Thus it is sufficient to assign costs to nodes to capture the cost of both buying nodes (as when a spammer buys an existing domain), or acquiring edges (via all kinds of techniques, including making a highquality page that people choose to link to, market places for buying edges, etc).
Our (unspecified) cost function allows us to combine costs in terms of money, effort, etc. What about setting the cost of a node to be its rank value? As we will see below (see Theorem 5), this is too restrictive. Ranking schemes can have nonlocal effects that must be accounted for in the cost scheme.
This definition makes the simplification that a cabal, in which different agents cooperate, say, by exchanging links, is a single owner.
We can assume that a spammer begins disconnected from the main web graph (or even with an empty partition). The spammer must therefore buy one node to connect to the network, and there is always a node that costs . Therefore, the smallest resistance a ranking function can have is . Surprisingly, some ranking functions can be spammed for just this cost. Other ranking functions are spam resistant. The most resistance a function can have is , since it could simply by all the ranking (and receive benefit ), by buying all the nodes (for cost ). Here we give examples that range from resistance to resistance.
In 1999, Kleinberg [26] introduced the notion of hub and authority scores and the HITS algorithm to compute them. Intuitively, a page receives a high authority score if it is pointed to by many highquality hubs. A page receives a high hub score if it points to many highquality authorities. It suffices to note that hub scores depend on outlinks and is therefore free to spam. But once an agent owns many pages with high hub scores, it can create pages with high authority scores, once again for free. Such considerations are far from hypothetical. Assano et. al [5] report that HITS was unusable by 2007, due to its spammability. This vulnerability was already well understood in 2004 [19]. In the parlance of this paper, HITS is spam resistant.
PageRank was introduced to be harder to spam. The idea is that if many highquality nodes point to a node, then that is evidence for the high quality of the target. The stationary distribution of a random walk on a graph has a similar behavior: the more inedges from highprobability nodes one has, the higher one’s own probability in the random walk.
However, not every directed graph has a well defined stationary distribution. Therefore Brin and Page introduced the notion of a reset. Let be the reset probability. Consider a random walk which, with probability , traverses an out edge from the current node, selected uniformly at random. With probability , it resets to a node chosen according to , the
reset probability distribution
, orreset vector
, over . We refer to such a random walk model as . The PageRank of is the stationary distribution of this process.But this definition presents a problem: it might be possible to linkspam itself, and thus spam the PageRank! Indeed, the reset vector
is usually taken to be the uniform distribution, as it was in the original paper. Spamming PageRank with a uniform reset vector is basically free. Once a single node is purchased, the spammer can create an arbitrarily large subgraph (for free), which can gather and focus PageRank. Indeed, this method can focus an arbitrarily large fraction of the total PageRank into the spam region, yielding a spam resistance of
.In fact, both of these ranking function ignore the trusted sites and suffer from the lowest possible spam resistance. This conjunction of features is not accidental:
Lemma 1.
Any ranking that gives the same values to all nodes, no matter what the trusted sites are, is spam resistant.
Proof.
The spammer can make a copy of the rest of the graph for free. If the ranking function ignores the trusted sites, it has no way of knowing which copy is the “real” graph. Thus the spammer can accumulate half the ranking for the cost of one edge. ∎
Therefore, we need to consider ranking functions that use the trusted sites, if we want to have any hope of achieving spam resistance. Consider a PageRank where all the reset goes to a single trusted node, . Such a PageRank is called a personalized PageRank [24] of center node , which we denote . Personalized PageRanks are not susceptible to the direct link spam attack described above, because a spammer can only acquire PageRank by gaining more and more nodes from the “legitimate” web. Furthermore, they are not susceptible to manipulations of the reset vector, since the interior of any property that does not include receives no reset. To see how spammable this ranking function is, set the cost of a node to be its Personalized PageRank. The total PageRank in the interior of property is a constant times the amount that flows across the boundary, as long as is a constant. Thus, this ranking function is spam resistant.
However, Personalized PageRanks are not very useful for ranking pages, since any page that’s downstream from and near to the center will have a huge PageRank. Such a PageRank is too biased to serve as a reasonable ranking function.
1.2 Our proposed linkspamfighting method
We would like to keep the spam resistance of Personalized PageRanks, while reducing the large PageRanks downstream from the center node.
Consider a set of trusted nodes and their corresponding Personalized PageRanks, . We define the MinPPR method to be the componentwise min of these PageRanks, that is, the th component of the MinPPR is the min of the th component of all the (perhaps normalized so that the value of the MinPPR sum to ).
Now, the node does not receive a huge PageRank, as it would in its own Personalized PageRank. Instead, it gets a value that is set by one of the other Personalized PageRanks. Thus, intuitively, MinPPR should avoid the local bias inherent in Personalized PageRanks. We discuss the bias of MinPPR in the conclusion (see Section LABEL:sec:conclusion).
On the spamfighting side, componentwise min combinations of personalized PageRanks would appear to inherit the spam resistance of personalized PageRanks, since, in order to spam one’s way to a high min value, one must engineer being in the downstream neighborhood of all the nodes used for each constituent .
In the following, we show that this simple analysis of spam resistance is, in fact, too simple. Nonetheless, we show that MinPPR is spam resistant, when is a constant.
1.3 Our Results
In this paper, we explore the algebra of PageRank. Recall that the space of PageRanks is the set of stationary distributions of , over all choices of and . Results include:

We show necessary and sufficient conditions for a vector to constitute a PageRank for a graph. We show how to compute the source reset vector and reset probability for a putative PageRank vector.

We demonstrate a class of functions which, when applied componentwise to a set of PageRanks and rescaled, yields another PageRank. That is, this class of functions is closed over the set of PageRanks. Most notably, we establish that componentwise min with rescaling is closed for PageRanks.
We use the machinery so developed to propose a spamfighting method:

We propose a method – called the MinPPR– for fighting link spam. This method consists of taking the componentwise min of a set of personalized PageRanks centered on trusted sites.

We show that MinPPR is spam resistant, when is a constant.
2 Related Work
Originally, and most famously, PageRank was used by Google as a ranking function for web pages [2]
, but since then, it has been used to analyze networks of neurons
[14], Twitter recommendation systems [17], protein networks [23], etc. (See [16] for a survey of nonweb uses). Here, we focus on its application to Web link analysis.As noted above, PageRank is susceptible to link spam. Thus, other ranking functions have been proposed [29, 11, 20]. TrustRank [20] for example is based on assigning higher reputation to a subset of pages curated by an expert, and the assumption that pages linked from these reputable pages are reputable as well. A similar method can be applied for low reputation pages, which is called AntiTrust Rank [28]. In both, reliability lowers as distance from the reference pages increases.
Other work is geared towards modifications of the PageRank mechanism. For instance, Global Hitting Time [22] was designed as a transformation of PageRank to counter crossreference link spam, where nodes link each other to increase their rank, but it still suffers if the number of spammers is large. Variants include Personalized Hitting Time [32].
Despite the progress on other ranking mechanisms, PageRank still stands as the most popular [37] ranking function, and therefore the most attractive for linkspammers.
Google discouraged PageRank manipulation through the buying of highly ranked links by announcing that pages discovered to participate in such activity will be left out of the PageRank calculation (hence, their rank lowered) and encouraging the public to notify Google about such pages [1].
Other research has focused on linkspam detection [18] and quantifying the rank increase obtained by creating Sybil pages [13].
Linkspam detection may be useful for excluding pages from the PageRank calculation, but better than building a fortress is to make the attack futile, that is, to develop techniques that yield PageRank spam resistance. Towards that end, some work limits or assign reset probability selectively [15, 20]. These approaches are generalizations of Personalized PageRank [24].
As expected, personalized PageRank is biased towards the vicinity of the trusted node. This undesired effect can be compensated for to some extent by concentrating reset probability on a subset of nodes rather than one (as in [15, 20]). Indeed, the approach has been successful for particular areas where the search space is relatively small (e.g. in Linguistic Knowledge Builder graph [3], Social Networks [8, 25], and Protein Interaction Networks [23]). But the scale of the web graph may require a large set of trusted pages for a general purpose PageRank.
The question of how to compute PageRank fast enough in practice has attracted a lot of attention, yielding a variety of theoretical and experimental results. The literature includes exact as well as approximation algorithms.
Exact computations involve the application of standard iterative methods, such as the Jacobi, Krylov subspace and power methods. A survey was published in [30]. If the computation is parallel, the web graph is first partitioned in blocks (cf. [27] and the references therein).
Approximations include a variety of techniques, either heuristic or with theoretical guarantees
[4, 9, 33, 35]. A common approach is to start multiple random walks from different nodes, aiming to approximate the rank with the number of visits to each node.Less related work includes distributed implementations [36], stochastic methods [7]
, and bounds on the second Eigenvalue of the web hyperlink matrix. The second Eigenvalue is related to convergence time to compute PageRank by iterated methods
[21].3 Preliminaries
In this section, we introduce notation and define a somewhat extended notion of PageRank.
Let be a graph. For any node with no outgoing links, we assume it has a self loop, in order to simplify definitions and lemmas. For any edge , let be called an outneighbor of and be called an inneighbor of . For any , let and denote the set of inneighbors and outneighbors of respectively. For , let . Note that for all , because of the self loops. The definitions of IN and OUT are extended to subsets of nodes in a straightforward manner.
Let be the ndimensional nonnegative unit sphere denoting all possible dimensional probability vectors.
Note: throughout the rest of the paper, if is not specified, we will take it to be a default value. In the literature, this value is almost always .
The transition probability matrix, , of the random walk is
(1) 
where and denote matrices as follows:
For instance, for the case of a uniform reset vector, is the all matrix.
The random walk is a Markov chain defined by the sequence of moves of a particle between the nodes of
, where the location of the particle at any given time is the state of the system, and the onestep transition matrix is . This random walk on is well behaved, as summarized in the following.Theorem 1.
The random walk on , as defined above, has a unique stationary distribution for any graph , reset vector , and reset probability .
Proof.
Let be the set of nodes with positive probability in . Notice that these nodes belong to a strongly connected component in consisting of the nodes reachable from . These nodes form a unique essential communicating class in the Markov chain of the random walk on . By Proposition 1.26 in [31], such a Markov chain has a unique stationary distribution. ∎
This stationary distribution, , has weight 0 on all nodes not reachable from a node in . On all nodes reachable from a node in , they have PageRank defined by the standard random walk algorithm. It is not necessary that all nodes in form a communicating class, as is often claimed [30, 10, 6, 34].
We call this stationary distribution the PageRank defined by a graph, a reset vector and a reset probability. Theorem 1 states that the PageRank is well defined for any graph, reset vector and reset probability. When it is clear from the context what , and are, we simply write instead of . We denote the set of all possible PageRanks for with reset probability as , and the set of all possible PageRanks for with any reset probability as .
Let be the PageRank of . Then, is the unique vector that satisfies:
(2) 
To see why, notice that, being a stationary distribution, must satisfy
Then, from Equation 2, we can obtain the following expression for the th component of a reset vector .
(3) 
Equation 3 establishes a test to see if vector .
Lemma 2.
Let be a graph and let . Then belongs to if and only if there exists an such that the obtained by Equation 3 belongs to .
Next, we show how to compute a valid from a valid PageRank and reset vector .
Lemma 3.
Let be a graph and let for some . Let and define for each
Then, one of the following holds:

If , then there is a (unique) such that

If , then for every , .
Proof.
The proof of the first item is as follows. If , Equation 3 holds. Rearranging it, we get that for any
Rewriting in terms of ,
Given that for all , we can write
Yielding a unique value as claimed. The other direction of the implication is immediate by reversing the algebra above.
The proof of the second item is as follows. Recall that . Substituting for (see Equation 1), and setting , we get:
Thus, any suffices. ∎
The second case happens when or .
From Equation 3, we can also compute the th component of a PageRank vector obtained from an arbitrary reset vector as follows.
(4) 
Finally, it will sometimes be convenient in the following to consider scaled PageRanks, defined as a positive scalar times a PageRank. For any scaled PageRank, , normalizing yields (standard) PageRank .
4 Closed Operators for PageRanks
In this section, we show that PageRank is closed under a class of functions. It is straightforward to show that the (linear) convex combination of any two PageRank vectors is, itself, a PageRank. Here we show that a more general class of operators is also closed for PageRanks.
Let denote an operator whose domain is tuples of nonnegative reals and whose range is the nonnegative reals. For example, might be ‘’ or ‘’. We abuse the notation by extending to vectors, where the result is the componentwise application of . If is ’‘, then applying to vectors is simply vector addition.
For clarity in the analysis that follows, we define the following notation. We denote the second term in Equation 4 as
Definition 1.
A function is SemiDualCommutative (SDC) if
What kind of functions are semidualcommutative?
If is monotoneincreasing and , then is SDC, since for all :
where in the first inequality we used the monotonicity of and in the second line we used the SDC property. Both ‘’ and ‘’ are SDC, but ‘median’ is not. For instance, consider the following counterexample (order of the addends is important, “med” means median).
Moreover, this counterexample can be adjusted (up to normalization) so that a negative reset is needed in order to compensate the flow adjustments of the ‘median’.
We now show that SDC functions are closed over the class of PageRanks.
Theorem 2.
Let be PageRanks, for some , and let be SemiDualCommutative. Then, if is defined, then .
Proof.
We obtain the components of each from Equation 4 as follows.
Rearranging, we have that
Applying for PageRanks on the righthand side of the latter, gives us:
Normalizing,
Replacing Equation 4 and the definition of , we have
Given that is SDC, we observe that the latter is nonnegative for each . Furthermore, summing over all yields . Thus, by Lemma 2, is a PageRank. ∎
5 The Cost of Spamming MinPpr
In this section, we describe our method for addressing link spam. We show that this method spam resistant.
Recall that is a set of trusted sites. We define MinPPR
where the implicit is fixed for all Personalized PageRanks. In certain circumstances we will want to consider the normalized version of , but using the unnormalized version will make the algebra in this section more transparent.
Let be the reset vector of and let . That is, is the (scaled) reset vector derived by applying Equation 3 to .
Our naive analysis in the introduction suggested that the only way to get PageRank from a MinPPR system was to find (and acquire) nodes that are in the neighborhoods of all the trusted centers of the Personalized PageRanks. This analysis is incomplete. It is also possible for a property to acquire PageRank by engineering , the reset vector. The main result of this section is that the total in the interior of a property is bounded by a function of the behavior of the on the boundary. Thus, in order to spam MinPPR, it is necessary to spend money/effort on the boundary itself. We conclude by showing that MinPPR is spam resistant by exhibiting an appropriate cost function for nodes.
We begin by building up some intuition about , before proving our main theorem and exploring some consequences.
5.1 The Structure of
Notice that induces a natural coloring of nodes in , as follows. Suppose node derives its value in from , that is, . Then we set the color of to be the minimum such (in case of ties). We call a node homogeneous if so that , for all We begin by showing that homogeneous nodes receive no reset.
Lemma 4.
Node is homogeneous iff , that is, receives no reset.
Proof.
Whether is homogenous or not,
If is homogeneous, then for all . Thus,
Otherwise, there exists some in such that . Therefore,
In other words, in order to garner more reset, an owner needs to have a mixture of colors in its nodes, and, as we will see, at its boundary. We next show that this effort has limited efficacy.
5.2 Combining Two PageRanks
Consider two (possibly scaled) PageRanks, which, for ease of reference we call the yellow PageRank, , and the blue PageRank, , with (correspondingly scaled) reset vectors and , respectively. Let the green PageRank , and let be ’s scaled reset vector.
In this section we bound the total reset over the interior of any property, as a function of and at the boundary of the property, in addition to the resets and , themselves, over those nodes. The main theorem of this section serves as the technical workhorse for the results on spam resistance of MinPPR, established in subsequent sections.
As with above, we say a node is yellow (resp. blue) if it achieves its minimum in the yellow (resp. blue) computation.
For any edge , define
and similarly for . That is, in the flow interpretation of PageRank, is the amount of yellow flow that receives from . Define
and similarly for edges.
Let be the reset of a node in this twocolor MinPPR. We can bound the reset as follows.
Lemma 5.
For any MinPPR and any node ,
Proof.
Let be a blue node. By using MinPPR, the reset of , that is , increases (above ) by the difference between the yellow and blue values of ’s yellow parents, To see why, notice that the value of corresponds to the blue PageRank, and its value is the sum of two contributions: reset and fractions of parents values in the blue PageRank (cf. Equation 4). Therefore, if some of ’s parents are yellow instead of blue, they are contributing less to ’s value. But ’s value is fixed to the blue PageRank. Hence, it has to gain on reset as stated.
Notice that
and
Subtracting, we get
Thus,
If is a yellow node, then the are negated, but the increase in the resets comes out the same, but the is replaced by . ∎
Now that we have a bound on the reset at any node, we can sum over the nodes in the interior of a region and prove that the increase in the reset of the interior of a property depends only on Personalized PageRanks at the boundary. This will form the basis for showing that MinPPR is spam resistant, since getting more reset requires buying a bigger boundary.
Theorem 3 (The Reset Theorem).
For any two (possibly scaled) PageRanks, and , over , and for any with boundary and interior ,
Proof.
We begin by showing that
(5) 
as follows. Notice that
Then, replacing in Lemma 5, we know that
From the definition of , we know that only has inlinks from nodes in the same property, but some of them may be in the boundary and some in the interior. Thus, . On the other hand, may have outlinks to nodes in the interior as well as any other nodes. Hence, it is . Therefore, we know that
Substituting, we get
(6) 
By symmetry, we have that
and given that nodes in the interior only have links from the same property, it is
so, it is
and we can introduce the latter in Equation 6 getting
and rearranging Equation 5 follows.
The theorem follows directly by noting that . ∎
5.3 Implications of the Reset Theorem
Theorem 3 serves as the base case for a bound on a MinPPR for properties that do not contain a trusted site in the interior. We begin with some notation. Let , and let . Then, we get the following.
Theorem 4.
For the MinPPR on any permutation of the PageRank trusted sites in , and for any property with boundary and interior , such that , the following holds.
Proof.
Consider and . Given that , and that and are both Personalized PageRanks, the reset on is on both. Then, we can apply Theorem 3. Subsequently, consider adding the th Personalized PageRank to the min of the first (which is a scaled PageRank, by the closure of min). The second term has reset computed inductively, and the first has zero reset. So, once again by Theorem 3, the inductive step holds, thus establishing the theorem. ∎
Corollary 1.
If every node in the boundary of a property has the same color, then the interior receives no reset.
Proof.
Lemma 6.
In Theorem 4, we can ignore any color that does not occur on the boundary.
Proof.
Consider a color that does not occur on the boundary. Put it last in the permutation order in Theorem 4. Then , for every . Thus the two summations for ’s contribution to the reset cancel, and this color adds no reset. We an extend this to any number of colors by placing them at the end of the permutation. ∎
5.4 Spam Resistance
Here we show that MinPPR is spam resistant by assigning a cost to each node, so that the PageRank in the interior of a property is bounded by the cost of the boundary.
We conclude with the following.
Theorem 5.
If is a constant, then MinPPR is spam resistant.
Proof.
Fix the cost of a node to be at least its PageRank, plus the amount by which it contributes to the reset in the interior of a property.
We can derive the amount that any unit of added PageRank contributes to the interior as it flows from node to node by noticing that only a fraction gets propagated. Then the PageRank multiplier is
Then, for any property with boundary and interior , by Theorem 4, we know that
(7) 
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