Random-walk based techniques, such as PageRank, encode the structure of the graph in the form of a transition matrix of a stochastic process from which significances of the graph nodes can be inferred. Recommendation systems leverage such node significance measures to rank the objects in the database. Context-aware recommendation techniques complement the data graph with additional data that provide the recommendation context. However, despite their wide-spread use in many graph-based knowledge discovery and recommendation applications, conventional PageRank-based measures have various shortcomings. As we experimentally show in this paper, one such shortcoming is that PageRank scores are tightly coupled with the degrees of the graph nodes, whereas in many applications the relationship between the significance of the node and its degree in the underlying network may not be as implied by PageRank-based measures. In fact, as we also show in the paper, in certain applications, the significance of the node may be negatively correlated with the node degree and in such applications a naive application of PageRank may return poor results. To address these challenges, in this paper, we propose degree decoupled PageRank (D2PR) techniques to improve the effectiveness of PageRank based knowledge discovery and recommendation systems. These suitably penalize or (if needed) boost the transition strength based on the degree of a given node to adapt the node significances based on the network and application characteristics. ∗This work is supported by NSF Grants 1339835 ”E-SDMS: Energy Simulation Data Management System Software”, 1318788 ”Data Management for Real-Time Data Driven Epidemic Spread Simulations”, 1518939 ”RAPID: Understanding the Evolution Patterns of the Ebola Outbreak in West-Africa and Supporting Real-Time Decision Making and Hypothesis Testing through Large Scale Simulations”, and 1430144 ”Fraud Detection via Visual Analytics: An Infrastructure to Support Complex Financial Patterns (CFP) based Real-Time Services Delivery”.
In recent years, there has been considerable interest in measuring the significance of a node in a graph and relatedness between two nodes in the graph, as if measured accurately, these can be used for supporting many knowledge discovery, search, and recommen1.1
PageRank as a Measure of Significance
Since enumerating all paths among the graph nodes would
require time exponential in the size of the graph, random-walk based
techniques encode the structure of the network in the form of a
transition matrix of a stochastic process from which the node
significance can be inferred.PageRank [
~r = αTG~r + (1 − α)~t where TG is a transition matrix corresponding to the graph G, ~t is a teleportation vector (such that ~t[i] = kV1 k ), and α is the residual probability (or equivalently, (1 − α) is the so-called teleportation probability). Unless the graph is weighted, the transition matrix, TG, is constructed such that for a node v with k (outgoing) neighbors, the transition probability from v to each of its (outgoing) neighbors will be 1/k. If the graph is weighted, then the transition probabilities are adjusted in a way to account for the relative weights of the (outgoing) edges. 1.2
Tight Coupling of PageRank Scores of Nodes and their Degrees
Let us consider an undirected graph G(V, E). There are two factors that contribute to the PageRank of a given node, v ∈ V : • Factor 1: Significance of Neighbors: The more significant the neighbors of a node are, the higher its likelihood to be also significant. • Factor 2: Number of Neighbors (Degree of the Node) : Even if the neighbors are not all significant, a large number of Correlation between PageRank and Degree neighbors would imply that the node, v, is well-connected and, thus, likely to be structurally important.
In theory, these two factors should complement each other. In practice, however, the PageRank formulation described above implies that there is a very tight coupling between the degrees of the nodes in the graph and their PageRank scores (see Table 1). 1.2.1
Problem I: When a Large Node Degree Does Not Indicate High Node Significance
In this paper, we highlight (and experimentally show) that, in many applications, node degree and node significance are in fact inversely related and that the tight-coupling between node degrees and PageRank scores might be counter-productive in generating accurate recommendations.
EXAMPLE 1. Consider, for example, a recommendation application where a movie graph, consisting of movie and actor nodes, is used for generating movie recommendations. In this application, the first factor (significance of neighbors) clearly has a positive contribution: a movie with good actors is likely to be a good movie and an actress playing in good movies is likely to be a good actress. On the other hand, the second factor (number of neighbors) may in fact be a negative contributor to node significance: the fact that an actor has played in a large number of movies may be a sign that he is a non-discriminating (’B movie’) actor, whereas an actress with relatively fewer movies may be a more discriminating (’A movie’) actress.
As we see in Section 4, this observation turns out to be true in many applications, where (a) acquiring additional edges has a cost that is correlated with the significance of the neighbor (e.g. the effort one needs to invest to a high quality movie) and (b) each node has a limited budget (e.g. total effort an actor/actress can invest in his/her work). 1.2.2
Problem II: When PageRank Does Not Sufficiently Account for Contributions of Degrees The mismatch between PageRank and node significance is not limited to the cases where node degrees are inversely related to the node significance. As we see in Section 4, there are other scenarios where PageRank may, in fact, fail to sufficiently account for the contribution of the node degrees to their significances. 1.3
PageRank Revisited: De-coupling Node Significance from Node Degrees
As we discussed above, one key shortcoming of the conventional PageRank scores is that they are often tightly coupled with the degrees of the graph nodes and in many applications the relationship between the significance of the node and its degree in the underlying network may not be as implied by PageRank-based measure: in certain applications, the significance of the node may be negatively correlated with the node degree, whereas in others PageRank may not be sufficient in accounting for degree contributions. Naturally, in such applications a naive application of PageRank in generating recommendations may return poor results.
To address these challenges, in this paper, we propose degree decoupled PageRank (D2PR) techniques to improve the effectiveness of PageRank based knowledge discovery and recommendation systems. These techniques suitably penalize or (if needed) boost1 the transition strength based on the degree of a given node to adapt the node significances based on the network and application characteristics. This paper is organized as follows: Next, we discuss the related literature. In Sections 3, we introduce the proposed degreedecoupled PageRank techniques. We evaluate the proposed techniques in Section 4 and conclude in Section 5. 2. 2.1
Path-length based definitions of node relatedness, such as those
proposed by [
In [
Improvements to the PageRank Function
Due to the obvious relationship between ranking and monetary
rewards (e.g. through selling of advertisements on web search
applications), there has been considerable effort in engineering (or
manipulating) graphs in a way to maximize ranking scores of
particular nodes. This is commonly referred to as PageRank
optimization. One way to achieve this goal is carefully adding or removing
certain links: If, for example, one or more colluding webmasters
can add or remove edges, PageRank scores of target web pages
or domains can be increased [
There are some efforts to change the impact of degrees on the
PageRank computation. [
DEGREE DE-COUPLED PAGERANK The key difficulty of de-coupling node degrees from the PageRank scores is that the definition of the PageRank, based on random walk transitions, is inherently dependent on the number of transitions available from one node to the other. As we mentioned above, the more ways there are to reach into a node, the higher will be its PageRank score. 3.1
Therefore, to de-couple the PageRank score from node degrees, we need to modify the transition matrix. In particular, for each node vi in the graph, we would like to be able to control the transition process with a single parameter (p), such that • if p ≪ −1, transitions from node vi are ∼ 100% towards the neighbor with the highest degree, • if p = −1, transition probabilities from node vi are proportional to the degrees of its neighbors, • if p = 0, the transition probabilities mirror the standard PageRank probabilities (assuming undifferentiated neighbors), • if p = 1, transition probabilities from node vi are inversely proportional to the degrees of its neighbors, • if p ≫ 1, transitions from node vi are ∼ 100% towards the neighbor with the lowest degree.
In other words, the transition function should de-couple the transition process from node-degrees and penalize or boost the contributions of node degrees in the transition process, as needed. 3.2
Degree De-coupling Transition Matrix In this subsection, we will consider degree de-coupling of the transition matrix as implied by the above desideratum. 3.2.1
Let G = (V, E) be an undirected and unweighted graph. Let α also be a given residual probability parameter, and deg(v) be a function which returns the number of edges on the node v. We represent degree de-coupled PageRank (D2PR) scores in the form of a vector
d~ = αTDd~ + (1 − α)~t, where ~t is the teleportation vector, such that ~t[i] = kV k for all i 1 and TD is a degree de-coupled transition matrix,
TD(j, i) =
deg(vj )−p Pvk∈neighbor(vi) deg(vk)−p , (1) where • TD(j, i) denotes the degree de-coupled transition probability from node vi to node vj over an edge eij = [vi → vj ] when there exists at least one edge between two nodes, • neighbor(vi) is the set of all neighbors of the source node, vi, and A C D
F (a) A sample graph
Intuitively, the numerator term, deg(vj )−p, ensures that the edge incoming to vj is weighted by its degree: if p > 0, then its degree negatively impacts (reduces) transition probabilities into vj , if p < 0 then its degree positively impacts (boosts2) transition probabilities into vj , and if p = 0, we obtain the standard PageRank formulation without degree de-coupling. In other words, the transition function satisfies our desideratum of de-coupling the transition process from node-degrees and penalizing or boosting the contributions of node degrees on-demand. Note that, since all transitions from the node vi are degree de-coupled individually based on the degrees of their destinations, the denominator term, Pvk∈neighbor(vi) deg(vk)−p, ensures that the transition probabilities from node vi add up to 1.0. Note also that when there is no edge between node vi and vj , TD(j, i) = 0 and, consequently, the term TD(j, i) is not affected by the degree de-coupling process.
EXAMPLE 2. Figure 1 shows how the random walk probabilities are differentiated in a degree de-coupled transition matrix on a sample graph where a node A has three neighbors, B (with degree 2), C (with degree 3), and D (with degree 1). In conventional PageRank, the transition probabilities from node A to all its neighbor nodes are equal to 0.33. In degree de-coupled PageRank (D2PR), however, the value of p is used for explicitly accounting for the impact of node degree on the transition probabilities: When p = 2, the transition probabilities from A to its neighbors are 0.18, 0.08, and 0.74, which penalizes nodes which have larger degrees, whereas when p = −2, D2PR boosts the transition probabilities to large degree nodes leading to transition probabilities 0.29, 0.64, and 0.07, respectively. ⋄
This example shows that, in degree de-coupled PageRank
(D2PR), as we also see in Table 2, the value of p can be used to
penalize (p > 0) or boost (p < 0) transition probabilities based on
the degree of the destination, vj .
2In fact, a similar function was used in [
The semantics of degree de-coupling is slightly different in directed graphs. In particular, edges incoming to vi often do not require a particular effort from vi to establish and hence are often out of the control of vi, but indicate a certain degree of interestingness, usefulness, or authority as perceived by others. The same is not true for edges outgoing from vi; in particular, a vertex with a large number of outgoing edges may either indicate a potential hub or simply indicate a non-discerning connection maker. The distinction between these two situations gains importance especially in applications where establishing a new connection has a non-negligible cost to the source node and, thus, a large number of outgoing edges may indicate either (a) a very strong participant to the network or (b) a very poor participant with a large number of weak linkages.
Let G = (V, E) be a directed graph and for the simplicity of the discussion, without any loss of generality, let us assume that G is unweighted. Let us also be given a residual probability parameter, α and let outdeg(v) be a function which returns the number of outgoing edges from the node v. The degree de-coupled PageRank (D2PR) scores can be represented in the form of a vector d~, d~ = αTD d~ + (1 − α)~t, where ~t is the teleportation vector, such that ~t[i] = kV k for all i and
1 where TD(j, i) denotes the degree de-coupled transition probability from node vi to node vj over an edge eij = [vi → vj ], out_edges(vi) is the set of out-going edges from the source node, vi, and p ∈ R is a degree de-coupling weight.
EXAMPLE 3. Figure 2 (a) in Section 4 provides an example illustrating the correlations between the degree de-coupled PageRank (D2PR) scores and external evidence for different values of p for some application: here, the higher the correlation, the better resulting ranking reflects the application semantics. As we see in this example, which we will investigate in greater detail in Section 4, the optimal de-coupling weight is not always p = 0 as implied by the conventional PageRank measure. In this particular case, for example, the correlation between D2PR and external evidence of significance is maximized when the de-coupling weight, p, is equal to 0.5, implying that in this application a moderate degree of penalization based on the node degrees is needed to align PageRank scores and application semantics. ⋄ 3.2.3
Once again, the semantics of degree de-coupling need to be reconsidered for weighted graphs. Let G = (V, E, w) be a directed, weighted graph, where w(e) is a function which returns the weight of the edge associated with edge e. It is important to note that, in such a graph, the weight of an edge can 1) indicate the strength of the connection between two nodes (thus positively contributing to the significance of the destination node); and at the same time and 2) contribute to the degree of a node as a multiplier (thus positively or negatively contributing to the node significance depending on the degree-sensitivity of the application). In other words, given an edge eij = [vi → vj ], from node vi to node vj , the transition probability from vi to vj can be written as
T(j, i) = βTconn_strength(j, i) + (1 − β)TD(j, i), where
Tconn_strength(j, i) = w(vi → vj ) P[vi→vh]∈out_edges(vi) w(vi → vh) accounts for the connection strength (as in the conventional PageRank) whereas TD is a degree de-coupled transition matrix, TD(j, i) =
Θ(vj )−p
P[vi→vk]∈out_edges(vi) Θ(vk)−p , such that, TD(j, i) denotes the degree de-coupled transition probability from node vi to node vj over an edge eij = [vi → vj ], p ∈ R is a degree de-coupling weight, and Θ(v) =
X [v→vh]∈out_edges(v) w(v → vh).
Note that, above, β controls whether accounting for the connection strength or degree de-coupling is more critical in a given application. In Section 4, we will study the impact of degree de-coupling in weighted graphs for different scenarios. 4.
In this section, we present case studies assessing the effectiveness of the degree de-coupling process and the relationship between the degree de-coupling weight p and recommendation accuracy for different data graphs.
For all experiments, the degree de-coupling weight, p, is varied between -4 and 4 with increments of 0.5. The residual probability, α, is varied between 0.5 and 0.9, with default value chosen as 0.85. We also varied the β parameter, which controls whether accounting for the connection strength or degree de-coupling is more critical in a given application, between 0.0 and 1.0, with the default value set to 0 (indicating full decoupling). 4.1.1
Four real data sets are used for the experiments. Each data set
is used to create two distinct data graphs and corresponding ratings
data. Table 3 provides further details about the various graphs
created using these four data sets. These recommendation tasks based
on these data graphs are detailed below:
• For the IMDB [
In this section, our goal is to observe the impact of different D2PR degree de-coupling weights on the relationship between D2PR rankings and application specific significance measures for the above data sets3. We also aim to verify whether de-coupling weights can also be used to improve recommendation accuracies.
In order to measure the relationship between the degree decoupled PageRank (D2PR) scores and the application-specific node significance, we used Spearman’s rank correlation,
Pi(xi − x¯)(yi − y¯) pPi(xi − x¯)2 Pi(yi − y¯)2 , which measures the agreement between the D2PR ranks of the nodes in the graph and their application-specific significances. Here, x are rankings by D2PR and y are significances for an application and x¯ and y¯ are averages of two values. 4.3
Impact of De-Coupling in Different Applications (Unweighted Graphs)
In this subsection, we present results that aim to assess D2PR under the settings described above. For these experiments, the residual probability, α, and the parameter, β, are set to the default values, 0.85 and 0, respectively. In these experiments, we consider only unweighted graphs (we will study the weighted graphs and the impact of parameter β later in Section 4.5).
Figures 2 through 4 include charts showing the Spearman’s correlations between the D2PR ranks and application specific node significances for different values of p and for different data graphs. These figures clearly illustrate that different data graphs require different degrees of de-coupling4 to best match the application specific node significance criterion. 4.3.1
Application Group A: When Degree Penalization Helps
The actor-actor (based on common movies) and commentercommenter (based on common products) graphs have highest correlation at p = 0.5, with the correlations dropping significantly 3In this paper, we are not proposing a new PageRank computation mechanism. Because of this (and since the focus is not improving scalability of PR), we do not report execution times and compare our results with other PageRank computation mechanisms. 4Degree penalization or degree-based boosting when the degrees are over-penalized (i.e., when p ≫ 0.5). The Epinions product-product graph (based on common commenters, Figure 2(c)) also provides the highest correlations with p > 0, but behaves somewhat differently from the other two cases: the correlations stabilize and do not deteriorate significantly when degrees are over-penalized, indicating that the need for degree penalization is especially critical in this case: this is due to the fact that, the larger the number of comments a product has, the more likely it is that the comments are negative (Figure 5). In fact, we see that, among the three graphs, this is the only graph where the traditional PageRank (with p = 0) leads to negative correlations between node ranks and node significances.
These results indicate that actors who have had many co-actors, commenters who commented on products also commented by many others, or products which received comments from individuals who also commented on many other products are not good candidates for transition during random walk. This aligns with our expectation that, in applications where each new movie role or comment requires additional effort, high degree may indicate lower per-movie or per-comment effort and, hence, lower significance.
Figure 3 shows that, for movie-movie (based on common actors) and author-author (based on common articles) graphs, the peak correlation is at p = 0 indicating that the conventional PageRank which gives positive weight to node degree, is appropriate.
This perhaps indicates that movies with a lot of actors tend to be big-budget products and that authors with a large number of coauthors tend to be experts with whom others want to collaborate. Note that, in these applications, additional boosting, with p < 0, negatively affects the correlation, indicating that the relationship between node degree and significance is not very strong (Figure 5). The quick change when p < 0 is because, as we see in Table 3, median standard deviations of neighbors’ degrees are low; i.e., degrees of neighbors of a node are comparable: there is no dominant contributor to TD(j, i) in Equation 1 (Section 3) and, thus, the transition probabilities are sensitive to changes in p, when p < 0. 4.3.3
Application Group C: When Degree Boosting Helps
Figure 4 shows that there are scenarios where additional boosting based node degrees provides some benefits. The article-article (based on common authors), listener-listener (based on common artists), and artist-artist (based on common listeners) graphs reach their peaks around p ∼ −1, indicating that these also benefit from large node degrees though improvements over p = 0 are slight.
A significant difference between applications in Group B and Group C is that, for p < 0, the correlation curve is more or less stable. This is because, as we see in Table 3, in these graphs median standard deviations of neighbors’ degrees are high: in other words, for each node, there is a dominant neighbor with a high degree and this neighbor has the highest contribution to TD(j, i); thus, the rankings are not very sensitive to p, when p < 0. **(-%%%)!$$$##+'"',,.&!&% !'#(% !'% !)#(% !)% !*#(% /!#*0%%##!"*/!#"""(####%"""12&"&,%%%3""%-4'0"*#5(%#406*7%*8"9**#(% )% )#(% '% '#(% &% 4.3.4 Summary: Correlations between Node Degrees and Application Specific Significances The experiments reported above show that degree de-coupling is important as different applications, even on the same data set, may associate different semantics to node degrees and the conventional PageRank scores are too tightly coupled with node degrees to be effective in all scenarios. Figure 5, which plots correlations between node degrees and application specific significances for different data graphs, re-confirms that the ideal value of the p is related to the usefulness of the node degree in capturing the application specific definition of node significance. 4.4 Relationship between α and p
In Figures 6 through 8, we investigate the relationship between the value α and the degree de-coupling parameter p for different application types. Here we use the default value, 0, for the parameter β and present the results for unweighted graphs (the results for the weighted graphs are similar).
First thing to notice in these figures is that the grouping of the applications (into those where, respectively, p > 0, p = 0, or p < 0 is useful) is preserved when different values of α are considered.
Figure 6 studies the impact of the value of α in application group A, where degree penalization helps (p > 0). As we see here, for the IMDB actor-actor (Figure 6(a)) and Epinions commentercommenter (Figure 6(b)) graphs, having a lower value of α (i.e., lower probability of forward movement during the random walk) provides the highest possible correlations between D2PR ranks and node significance (with the optimal value of p being ∼ 0.5 independent of the value of α). This indicates that in these graphs, it is not necessary to traverse far during the random walk. Interestingly, though, when degrees are over-penalized (i.e., p ≫ 0), smaller values of α start leading to worse correlations, indicating that (while not being optimal) severe penalization of node degrees helps make random traversals more useful than random jumps. As we have already observed in Figure 2(c), the Epinions productproduct graph (Figure 6(c)) behaves somewhat differently from the other two cases where degree penalization (p > 0) leads to larger correlations: in this case, unlike the other two graphs, the highest possible correlations between D2PR ranks and node significance are obtained for large values of α, indicating that this application benefits from longer random walks (though the differences among the correlations for different α values are very small).
Figure 7 shows that the pattern is different for application group B, where conventional PageRank is ideal (p = 0): in this case, having a larger value of α (i.e., larger probability of forward movement during the random walk) provides the highest correlations between ranks and significance. Interestingly, in these applications, when p ≪ 0 or p ≫ 0, higher probabilities of random walk traversal (i.e., larger α) stop being beneficial and lower values of α lead to larger correlations. This re-confirms that, for these applications, p ∼ 0 leverages the random walk traversal the best.
As we see in Figure 8, in application group C, where degree boosting helps (p < 0), it is also the case that larger values of α (i.e., larger probabilities of forward transitions during the random walk) provides the highest correlations between node ranks and significance. On the other hand, in these applications, p ∼ 0.5 serves as a balance point where the value of α stops being relevant; in fact, for p > 0.5 the higher values of α stops being beneficial and lower values of α lead to larger correlations. This re-confirms that smaller values of p (which provides degree boosting) help leverage the random walk traversal the best. 4.5 Relationship between β and p in
Weighted Graphs
Finally, in Figures 9 through 11, we investigate the relationship between the value β (which controls whether accounting for the connection strength or degree de-coupling is more critical in a given application) and the degree de-coupling parameter p for different application types. Here we use the default value, 0.85, for the parameter α and present the results for weighted graphs:
Figure 9 depicts the impact of the value of the parameter β in application group A, where degree penalization helps (p > 0). As we see here, for all three weighted graphs, performing degree penalization (i.e., β < 1.0) provides better rank-significance correlation than relying solely on the connection strength (i.e., β = 1.0). Note that the value of β impacts the optimal value of degree penalization parameter p: the more weight is given to connection strength (i.e., the greater β is), the larger is the optimal value of p.
Figure 10 shows that, for applications in group B, where p ∼ 0 is ideal, when the connection strength is given significantly more weight than degree de-coupling (i.e., β ∼ 0), we observe high ranksignificance correlations. Interestingly however, for the moviemovie graph (where the edge weights denote common actors) the highest correlations are obtained not with p = 0, but with p = 0.5 and β = 0.75, indicating that degree penalization is actually beneficial in this case: movies that share large numbers of actors with other movies are likely to be B-movies, which are not good candidates for transitions during the random walk.
Figure 11 shows that in application group C, where degree boosting (p < 0) helps, giving more weight to connection strength (i.e., β ∼ 1.0) is a good, but not necessarily the best strategy. In fact, in these graphs, the highest overall correlations are obtained with β = 0 or β = 0.25, indicating that degree de-coupling is beneficial also in these cases. Interestingly, (unlike the case with the unweighted listener-listener graph, where the best correlation was obtained when p < 0) for the weighted version of the listenerlistener graph (where edge weights denote the number of shared friends), when β = 0 through 0.5, p = 0 provides the highest correlation and when β = 0.75, p = 0.5 provides the highest correlation – these indicate that listeners who have large numbers of shared friends with others are good candidates for random walk.
Note that a key observation from the above results is that the conventional PageRank, based on connection strength (i.e., β = 1.0), is not always the best strategy for the applications considered.
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CLMFN*
CLMFON* CLP* ship between the significance of the node and its degree in the underlying network may not be as strong (or as weak) as implied by PageRank-based measures. We proposed degree de-coupled PageRank (D2PR) to improve the effectiveness of PageRank based knowledge discovery and recommendation tasks. Evaluations on different data graphs and recommendation tasks have confirmed that degree de-coupling would be an effective way to match application specific node significances and improve recommendation accuracies using PageRank based approaches.