Distribution of optimized parameter vector

Figure 5-1 shows the distribution of the 2-tuple (𝜃, 𝛼 ) per user optimized parameter vectors, where 𝜃 stands for the ratio of the target user to the tag selected as a query and 𝛼 stands for the restart probability. Most of the optimized parameter vectors prefer small value of 𝜃. We could explain that most of users prefer to use more explicit tags to describe items. When the query is polysemic, we may concern the behavior of the user to find what the query means for the user. Thus, if a user usually uses ambiguous tags for tagging, the random walk-based recommendation system may not get sufficient information by almost only concerning about the query (i.e., 𝜃 is small). However, a user usually has different interests, and may tag what he or she doesn’t like, and therefore user behaviors are divergent.

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Figure 5-1 The distribution of the 2-tuple per user optimized parameter vector.

Notice that the distribution of these per user optimized parameter vectors do not concentrate tightly. It is impossible to find a parameter vector to fit all users.

Therefore, to obtain more suitable recommendation, we would not only use a single optimized parameter vector but also multiple parameter vectors. We divide the optimized parameter vectors by clustering and then select the representative vector for each cluster. Each cluster could represent a sort of user behavior. In our experiments, there are two clustering methods to cluster these per user optimized parameter vectors.

For each cluster, we compute the mean vector where each element is the mean of the elements in all optimized parameter vectors in the cluster. Figure 5-2 and 5-3 shows the clustering result by K-Means and DBSCAN, respectively.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

Ratio of User to Query (θ)

Restart Probability (α)

Distribution

41 into two sets, and then we could give the initial set of K-Means where the values of 𝜃 are separated. On the other hand, for DBSCAN, either the initial means of clusters or the number of clusters we don’t need to give. DBSCAN requires two parameters: the

0

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search radius (eps) and the minimum number of points required to form a cluster (minPts). Both of the parameters are easier to be tuned. From our experiments, the result by these two clustering methods are similar. One cluster locates at the area where the ratio of user (i.e., 𝜃) is quite small (i.e., 0 ~ 0.2), and the other could be taken as the outliers. Because the outliers are too many to be neglected, the mean vector of all outliers is still used as a parameter vector for recommendation. Thus, our random walk model using this parameter vector would fit the users who are taken as outliers because of their linking behaviors.

Figure 5-4 shows the distribution of the 3-tuple (𝜃, 𝛼, 𝛾) per user optimized parameter vectors, where the three parameters stand for the ratio of user to query, the restart probability and the self-transition probability, respectively. Like the distribution of the 2-tuple vectors, most of the optimized parameter vectors prefer small 𝜃. We still explain that most of users prefer to use more explicit tags to describe items. Moreover, from the projection of 𝛼-𝛾 plane, there are three clusters: one locates along the line 𝛼+𝛾=0.8, another locates near the y-axis (i.e., 𝛼 ≈ 0) and the other locates on the x-axis (i.e., 𝛾 ≈ 0). The first cluster infers that, the walk goes forward with the probability of 0.2 whatever the ratio between 𝛼 and 𝛾 is. Most of vectors are located in this cluster. The second cluster shows that the Supervised FolkRank (SFR) could be reduced to the self-transition model that M. Clements et al.

propose [5]. As shown in Figure 5-5, the distribution of PageRank-based model always assign the highest value to the nodes closest to the starting position, while in the lazy random walk model the distant nodes are more relevant [6].

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Figure 5-4 The distribution of the 3-tuple (𝜃, 𝛼, 𝛾) per user optimized parameter vector. (a) The projection of 𝜃 − 𝛼 plane. (b) The projection of 𝛾 − 𝛼 plane. (c) The projection of 𝛾 − 𝜃 plane. (d) The distribution of 𝜃, 𝛼 and 𝛾.

Figure 5-5 The Probability mass function (PMF) of the walk distance after a fixed number of steps through the social graph, for restart probability and self-transition probability are both 0.8.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 PQ(q)

Walk distance (q)

PageRank Model

Lazy Random Walk Model

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Unlike the distribution of the 2-dimension relation graph, the distribution of 3-dimension graph cannot be figured out easily. Moreover, the initial set of K-Means would affect the clustering. Figure 5-6 and Figure 5-7 show that the result of clustering by K-Means and DBSCAN. Among a multi-dimension relation graph, it needs other algorithm to find an initial set of K-Means. Without further information about the data distribution, the number of clusters is unknown. Moreover, the K-Means divides nodes into clusters according to distance and we know that the distance-based clustering methods do not adapt to certain data sets, whose clusters are not circle-like, very well. From Figure 5-4, the distribution could not be divided into circle-like clusters. For DBSCAN, three clusters are obtained without assigning the number of clusters. The clustering result is similar to our discussion stated above. If we use the centers of clusters obtained by DBSCAN as the initial set of K-Means, the result by K-Means is similar. Again, the mean vector of all outliers is used. The search radius (eps) and the minimum number of points required to form a cluster (minPts) could be predicted according to the range of each dimension.

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Figure 5-6 The clustering result by K-Means. The per user optimized parameter vectors are 3-tuple (𝜃, 𝛼, 𝛾). (a) The projection of 𝜃 − 𝛼 plane. (b) The projection of 𝛾 − 𝛼 plane. (c) The projection of 𝛾 − 𝜃 plane. (d) The distribution of 𝜃, 𝛼 and 𝛾.

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Figure 5-7 The clustering result by DBSCAN. The per user optimized parameter vectors are 3-tuple (𝜃, 𝛼, 𝛾). (a) The projection of 𝜃 − 𝛼 plane. (b) The projection of 𝛾 − 𝛼 plane. (c) The projection of 𝛾 − 𝜃 plane. (d) The distribution of 𝜃, 𝛼 and 𝛾.