A Study of YouTube recommendation graph based on measurements and stochastic tools

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3 Table 1 regression R 2 coefficients N = 1 using logarithmic scale N = 1 using linear scale N = 2 using logarithmic scale N = 2 using linear scale N = 3 using logarithmic scale N = 3 using linear scale N = 4 using logarithmic scale N = 4 using linear scale N = 5 using logarithmic scale N = 5 using linear scale N = 8 using logarithmic scale N = 8 using linear scale N = 10 using logarithmic scale N = 10 using linear scale N = 15 using logarithmic scale N = 15 using linear scale Table 1: Regression coefficients and coefficient of determination R-square for different values of N (a) N=3 fig1 randomly picked 1000 videos and we focus on two elements of the data. The first element is the view count of a video and the second element is the average view count of related videos recommended by YouTube recommendation system. We believe that the number of videos that we collected for each experiment is enough to capture all information. First, we investigate the relation between the view count of a video and the average view count of its recommendation list. We consider different values of the number N of videos in the list. In practice, the larger screen is, the larger is N. We shall show later that N plays a crucial role on the properties of the excursions over the recommendation graph. A very small N corresponds to list of recommendations viewed over cellular telephones. In our statistical study, we test two types of model that relate the number x i of views of a video i and the number y i of views of videos in its recommendation list averaged over N: (b) N=3 fig2 N-videos in which we use the linear regression between x i and y i, i {1,...,1000}. N-videos wherein a linear regression is used between log(x i) and log(y i). In figure 3, we plot the view count of one video on the X- axis, and the average view count of videos in its recommendation list. We observe that the coefficient of determination R-squareis small for all valuesofn (see Table 3). This indicates that the regression line cannot explain all the variation of the average of view count of videos in recommendation list of a video by variation in view count of that video. In other hand, in figures 3-3, we use a logarithmic scale to plot the view count of one video on the X-axis, and the average view (c) N=15 fig3 Figure 2: The view count of one video on the X- axis, and the average views of the top N = 3, 15 of its recommendation list

4 count of videos in its recommendation list. Both figures 3-3 show the strong correlation between the view count of a video and its average of view count of top N videos in its recommendation system. The first observation claims that the higher average of view count of related videos in recommendation list of a video, the higher view count of that video. Hence, Youtube recommendation system will prefer to put videos in recommendation list based on the popularity of the current video, located in the same region of the popularity (same region of the long tail of popularity). On the other hand, we characterize a good relationship between the view count x of a video and the average of view count y of videos in its recommendation list. This relationship is y = (e ln(x) ) a +e b. The table 1 shows how well a regression line using logarithm scale, fits a set of data for several values of N. This approximation is more accurate when N is small. Since a walker has more probability to choose a video at top position (Google advertisement [10] showed that the first position in the recommendation list attract 39 times more click that the 10 th position), this relationship is still a very good approximation even if the number of videos in list of recommendation system is high. (a) N=3 fig4 Now we further investigate the correlation between the age of a content and the average of age of videos in its recommendation list. Figures 3-3 show clearly the trend that the higher theaverage of age oftherelated videos, thehigherthe age of the video. This implies that Youtube recommendation will prefer to recommend the videos located in the same region of the age. This means that Youtube s recommendation will not significantly affect the overall videos based on the age of a video and will instead focus more on the same generation of that video. Moreover, the table 3, provide a high regression coefficient which is the additional evidence that the recommendation system has an important impact on the view count and a popular video can affect only the videos located in the same region of the popularity and the age. 4. STABILITY AND VIDEO VIEW DIVER- SITY Consider X n, n 1 a sequence of random process that satisfies the Markov property and takes values from a set S = {1,2,..,n} where n is the view count of the visited video. We construct this random process by using a random walker moving in the graph as defined in the previous section, but we take into account only the view count of the video. In this section, we investigate the stability of Markov chain defined above in order to identify the impact of Youtube recommendation system on view diversity and how suggest best long tail videos that are undiscovered because of its less view count. We are interested to the stability in order to show if the random walker can remain in a small region starting from an initial video and move to other video using YouTube recommendation system. In order to study the stability we will define the first hitting time as follows Definition 1. We define the hitting time of a set of states (b) N=10 fig5 Figure 3: The age of one video on the X-axis, and the average age of the top N = 3, 15 of its recommendation list Table 3 regression coefficients R2 1 videos days videos days videos days videos days videos days videos days videos days videos days (a) table2 Figure 4: Regression coefficients and coefficient of determination R-square for different values of N

5 B by τ B = inf{n 1 : X n B} This set is called a positive recurrent if supe xτ B < x B The Markov chain is stable if it is a positive class recurrent. From different experiments (see for example Table 3), we observe that for N = {3,4,5} and for all i N we have E [ log(x n+1) X n = x ] <, and E [ log(x n+1) log(x n) X n = x ] 0 for log(x) K > 0 It follows from Theorem 3 in [11], that he Markov chain cannot be positive recurrent. The instability of the Markov chain for N = {3,4,5} (which is thecase ifthecomputer sscreenis small) hasseveral interpretations and consequences. Indeed, by the rapid adoption of smartphone, tablets and e-reader which are characterized by a small screen, our result may explain some results in [1] which showed how Youtube recommendation can improve the view diversity and help users to discover videos in long tail. 5. CONCLUSION This paper has several contributions. Firstly we showed through the measurements the relation between function of views of a video and the function averaged over the number of views of videos in its recommendation list. Based on this relationship we explored the evolution on number of views that a random walker sees when traveling through recommendation graph. We showed in particular if the number of videos in the list is small, the number of views tends to increase along the trajectory of random walker. We conclude that the random trajectory defined as Markov chain, is not sable which means that the trajectory does t not contain cycles. 6. REFERENCES [1] R. Zhou, S. Khemmarat, and L. Gao, The impact of youtube recommendation system on video views, in Proc. of IMC 2010, Melbourne, November [2] M. Cha, H. Kwak, P. Rodriguez, Y.-Y. Ahn, and S. Moon, I tube, you tube, everybody tubes: analyzing the world s largest user generated content video system, in Proc. of ACM IMC, San Diego, California, USA, October , pp [3] R. Crane and D. Sornette, Viral, quality, and junk videos on YouTube: Separating content from noise in an information-rich environment, in Proc. of AAAI symposium on Social Information Processing, Menlo Park, California, CA, March [4] P. Gill, M. Arlitt, Z. Li, and A. Mahanti, YouTube traffic characterization: A view from the edge, in Proc. of ACM IMC, [5] J. Ratkiewicz, F. Menczer, S. Fortunato, A. Flammini, and A. Vespignani, Traffic in Social Media II: Modeling Bursty popularity, in Proc. of IEEE SocialCom, Minneapolis, August [6] G. Chatzopoulou, C. Sheng, and M. Faloutsos, A First Step Towards Understanding Popularity in YouTube, in Proc. of IEEE INFOCOM, San Diego, March , pp [7] M. Cha, H. Kwak, P. Rodriguez, Y.-Y. Ahn, and S. Moon, Analyzing the video popularity characteristics of large-scale user generated content systems, IEEE/ACM Transactions on Networking, vol. 17, no. 5, pp , [8] G. Szabo and B. A. Huberman, Predicting the popularity of online content, Comm. of the ACM, vol. 53, no. 8, pp , Aug [9] X. Cheng, C. Dale, and J. Liu, Statistics and social network of youtube videos, in International Workshop on Quality of Service, June [10] Google adwords click through rates per position, October [11] S. Foss, Coupling again: the renovation theory, Lectures on Stochastic Stability, Lecture 6, Heriot-Watt University. [Online]. Available: Lecture6.pd

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