Recommendation systems in the scope of opinion formation: a model Marcel Blattner Matus Medo Laboratory for Web Science Physics Department University of Applied Sciences FFHS University of Fribourg Regensdorf, Switzerland Fribourg, Switzerland marcel.blattner@ffhs.ch matus.medo@unifr.ch ABSTRACT on its own in the 90s [42, 20, 28, 21, 11]. The interest in rec- Aggregated data in real world recommender applications of- ommendation systems increased steadily in recent years, and ten feature fat-tailed distributions of the number of times attracted researchers from different fields [43]. The success individual items have been rated or favored. We propose a of highly rated Internet sites as Amazon, Netflix, YouTube, model to simulate such data. The model is mainly based on Yahoo, Last.fm and others is to a large extent based on social interactions and opinion formation taking place on a their recommender engines. Corresponding applications rec- complex network with a given topology. A threshold mech- ommend everything from CD/DVD’s, movies, jokes, books, anism is used to govern the decision making process that web sites to more complex items such as financial services. determines whether a user is or is not interested in an item. The most popular techniques related to recommendation We demonstrate the validity of the model by fitting atten- systems are collaborative filtering [8, 26, 11, 24, 28, 21, 41, dance distributions from different real data sets. The model 45] and content-based filtering [14, 40, 35, 5, 30]. In ad- is mathematically analyzed by investigating its master equa- dition, researchers developed alternative methods inspired tion. Our approach provides an attempt to understand rec- by fields as diverse as machine learning, graph theory, and ommender system’s data as a social process. The model can physics [16, 17, 37, 52, 51, 10, 48, 50]. Furthermore, rec- serve as a starting point to generate artificial data sets useful ommendation systems have been investigated in connection for testing and evaluating recommender systems. with trust [2, 39, 47, 32, 33] and personalized web search [9, 12, 46], which constitutes the new research frontier in search engines. Categories and Subject Descriptors However, there are still many open challenges in the re- H.1.m [Information Systems]: Miscellaneous search field of recommendation systems [1, 22, 25, 18, 24, 43, 15]. One key question is connected to the understanding of the user rating mechanism. We build on a well documented General Terms influence of social interactions with peers on the decision to Experimentation, Theory vote, favor, or even purchase an item [44, 27]. We propose a model inspired by opinion formation taking place on a complex network with a predefined topology. Our model is Keywords able to generate data observed in real world recommender recommender systems, opinion formation, complex networks systems. Despite its simplicity, the model is flexible enough to generate a wide range of different patterns. We mathe- matically analyze the model using a mean field approach to 1. INTRODUCTION the full Master Equation. Our approach provides an under- This is the information age. We are witnessing informa- standing of the data in recommender systems as a product tion production and consumption in a speed never seen be- of social processes. The model can serve as a data genera- fore. The WEB2.0 paradigm enables consumers and produc- tor which is valuable for testing and evaluation purposes for ers to exchange data in a collaborative way benefiting both recommender systems. parties. However, one of the key challenges in our digitally- The rest of the paper is organized as follows. The model driven society is information overload [7]. We have the ’pain is outlined in Sec. (2). Methods, data set descriptions, and of choice’. Recommendation systems represent a possible so- validation procedures are in Sec. (3). Results are presented lution to this problem. They have emerged as a research area in Sec. (4). Discussion and an outlook for future research directions are in Sec. (5). 2. MODEL 2.1 Motivation Paper presented at the 2012 Decisions@RecSys workshop in conjunction Our daily decisions are heavily influenced by various infor- with the 6th ACM conference on Recommender Systems. Copyright 2012 mation channels: advertisement, broadcastings, social inter- for the individual papers by the papers’ authors. Copying permitted for private and academic purposes. This volume is published and copyrighted actions, and many others. Social ties (word-of-mouth) play by its editors. a pivotal role in consumers buying decisions [44, 27]. It was 32 demonstrated by many researchers that personal communi- the connections of the corresponding Influence-Network. From cation and informal information exchange not only influence our model’s point of view this means the following: an in- purchase decisions and opinions, but shape our expectations dividual’s IIA for a particular item i may be shifted due of a product or service [49, 4, 3]. On the other hand, it was to social interactions with directly connected peers (these shown [23], that social benefits are a major motivation to interactions thus take place on the corresponding IN), who participate on opinion platforms. If somebody is influenced already experienced the product or service in question. This by recommendations on an opinion platform like MovieLens process can shift the Intrinsic-Item-Anticipation of an indi- or Amazon, social interactions and word-of-mouth in general vidual who did not yet experience product/object i closer to are additional forces governing the decision making process or beyond the critical-anticipation-threshold. to purchase or even to rate an object in a particular way We now summarize the basic ingredients of our model. [31]. An individual user’s opinions on objects are assembled in Our model is formulated within an opinion formation frame- two consecutive stages: i) opinion making based on different work where social ties play a major role. We shall discuss external sources, including suggestions by recommendation the following main ingredients of our model: systems and ii) opinion making based on social interactions in the Influence-Network. The second process may shift the • Influence-Network (IN) opinions generated by the first process. • Intrinsic-Item-Anticipation (IIA) 2.2 Mathematical formulation of the model • Influence-Dynamics (ID) In this section we firstly describe how individuals’ Intrinsic- Item-Anticipations may change due to social interactions Influence Network. taking place on a particular Influence-Network. Secondly, we We call the network where context-relevant information introduce dynamical processes governing the opinion prop- exchange takes place an Influence-Network (IN). Nodes of agation. the IN are people and connections between nodes indicate the influence among them. Note that we put no constraints IIA shift. on the nature of how these connections are realized. They We model a possible shift in the IIA as: may be purely virtual (over the Internet) or based on physi-  (1−γ) cal meetings. We emphasize that INs are domain dependent, Θj fˆij = fij + . (1) i.e., for a given community of users, the Influence Network kj concerning books may differ greatly (in topology, number of ties, tie strength, etc.) from that concerning another subject where fˆij is the shifted Intrinsic-Item-Anticipation of indi- such as food or movies. Indeed, one person’s opinion lead- vidual j for object i, fij is the unbiased IIA, Θj is the num- ers (relevant peers) concerning books may be very different ber of j’s neighbors, who already experienced and liked item from those for food or other subjects. In this scope, we see i, kj denotes the total number of j’s neighbors in the cor- the INs as domain-restricted views on social networks. It responding IN, and γ ∈ (0, 1) quantifies trust of individuals is thus reasonable to assume that Influence Networks are to their peers. An individual j will consume, purchase, or similar to social interaction networks which often exhibit a positively rate an item i only if scale-free topology [6]. However, our model is not restricted fˆij ≥ ∆. (2) to a particular network structure. We identify ∆ as the Critical-Anticipation-Threshold. Val- Intrinsic-Item-Anticipation. ues of fij are drawn from a probability distribution fi . Since Suppose a new product is launched on the market. Ad- the IIA for each individual is an aggregate of many different vertisement, marketing campaigns, and other efforts to at- and largely independent contributions, we assume that fi tract customers predate the launching process and continue is normally distributed, fi ∈ N (µi , σ). (Unless stated oth- after the product started to spread on the market. These erwise.) To mimic different item anticipations for different efforts influence product-dependent customer anticipation. objects i, we draw the mean µi from a uniform distribution It is clear that the resulting anticipation is a complex com- U (−, ). We maintain µi , , and σ, so that fi is roughly bination of many different components including intrinsic bounded by (−1, 1), i.e., −1 ≤ µ − 3σ < µ + 3σ ≤ 1. Note product quality and possibly also suggestions from recom- that fˆij can exceed these boundaries after a shift of the corre- mendation systems. sponding IIA occurs. The second term on the right hand side In our model we call the above-described anticipation Intrinsic- of Eq. (1) is the influence of j’s neighborhood weighted by Item-Anticipation (IIA) and measure it by a single number. trust γ. To better understand the interplay between γ and It is based on many external sources, except for the influence the density of attending users in the neighborhood of user generated by social interactions. It is the opinion on some- i, ρ := Θj /kj , we refer to Fig. 1. Trust γ ≈ 1 causes a big thing taken by individuals, before they start to discuss the shift on the IIA’s even for ρ ≈ 0. On the other hand, γ ≈ 0 subject with their peers. Furthermore, we assume that an needs high ρ to yield a significant IAA shift. These prop- individual will invest resources (time/money) into an object erties are understood as follows: people trusting strongly in only, if the Intrinsic-Item-Anticipation is above a particular their peers need only few positive opinions to be convinced, threshold, which we call Critical-Anticipation-Threshold. whereas people trusting less in their social environment need considerable more signals to be influenced. Influence-Dynamics. The Influence-Dynamics describes how individuals’ Intrinsic- Influence-Dynamics. Item-Anticipations are altered by information exchange via The Influence-Dynamics proceeds as follows. Firstly, we 33 Algorithm 1 RecSysMod algorithm. P contains the con- figuration parameter for the network. ∆ is the Anticipation Threshold and γ denotes the trust. O ∈ N is the number of objects to simulate. G(N, E) is the network. N is the set of nodes and E is the set of edges. 1: procedure RecSysMod I(P, ∆, γ, O) 2: G(N, E) ← GenNetwork(P) 3: for all Objects in O do 4: generate distribution fi from N (µi , σ) 5: for each node j ∈ N in G do 6: draw fij from fi 7: if fij < ∆ then 8: jstate ← S 9: else 10: jstate ← A 11: end if Figure 1: Contour plot for γ and ρ = Θj /kj . Num- 12: end for bers inside the plot quantify the shift in the IAA as 13: repeat a function of γ and ρ. 14: for all j with jstate = S AND Θj > 0 do h i(1−γ) Θ 15: fˆij ← fij + j kj draw an Influence-Network IN(P) with a fixed network topol- 16: if fˆij < ∆ then ogy (power-law, Erdős-Rényi, or another). P refers to a 17: jstate ← D set of appropriate parameters for the Influence-Network in 18: else question (like network type, number of nodes, etc.). The 19: jstate ← A network’s topology is not affected by the dynamical pro- 20: end if cesses (opinion propagation) taking place on it. We justify 21: end for this static scenario by assuming that the time scale of the 22: until |{j|jstate = S AND Θj > 0}| = 0 topology change is much longer then the time scale 1 of opin- 23: end for ion spreading in the network. Each node in the Influence- 24: end procedure Network corresponds to an individual. For each individual j we draw an unbiased Intrinsic-Item-Anticipation fij from the predefined probability distribution fi . At each time step, every individual is in one of the following states: {S, A, D}. tion for the dynamics. As already said before, two things can S refers to a susceptible state and corresponds to the initial happen when a non-attender is connected to an attender: state for all nodes at t = 0. A refers to an attender state a) he/she becomes an attender too, or b) he/she becomes and corresponds to an individual with the property fˆij ≥ ∆. a denier who will not attend/favor the item. For these two interaction types we formally write: D refers to a denier state with the property fˆij < ∆ after an information exchange with his/her peers in the Influence- λ S + A −→ 2A Network happened. An individual in state D or A can not α change his/her state anymore. It is clear that an individual S + A −→ D + A (3) in state A cannot back transform to the susceptible state Here λ denotes the probability that a susceptible node con- S, since he/she did consume or favor item i and we do not nected to an attender becomes an attender too, and α is account for multiple attendances in our model. An indi- the probability that a susceptible node attached to an at- vidual in state D was influenced but not convinced by his tender becomes a denier. To take into account the underly- opinion leaders (directed connected peers). We make the ing network topology of the Influence Network it is com- following assumption here: if individual j’s opinion leaders mon to introduce compartments k [19]. Let NkA be the are not able to convince individual j, meaning that individ- number of nodes in state A with k connections, NkS the ual’s j Intrinsic Item Anticipation fˆij stays below the criti- number of nodes in state S with k connections, and NkD cal threshold ∆ after the influence process, then we assume the number of nodes in state D with k connections, respec- that j’s opinion not to attend object i remains unchanged tively. Furthermore we define the corresponding densities: in the future. Therefore we have the following possible tran- ak (t) = NkA /Nk , sk (t) = NkS /Nk and dk (t) = NkD /Nk . Nk is sitions for each node in the influence network: jS → jA or the total number of nodes with k connections in the network. jS → jD . Node states are updated asynchronously which Since every node from Nk must be in one of the three states, is more realistic than synchronous updating, especially in ∀t : ak (t) + sk (t) + dk (t) = 1. A weighted sum over all k social interaction models [13]. The Influence-Dynamics is compartments gives the total fraction of attenders at time t, summarized in Algorithm 1. P a(t) = k P (k)ak (t) where P (k) is the degree distribution of the network (it also holds that a(t) = N A (t)/N ). The Master Equation. time dependence of our state variables ak (t), dk (t), sk (t) is We are now in the position to formulate the Master Equa-  ȧk (t) = λksk (t)Ω  1 The term time scale denotes a dimensionless quantity and  ˙ dk (t) = αksk (t)Ω (4) specifies the devisions of time. A shorter time scale means  a faster spreading of opinions in the network. ṡk (t) = −(α + λ)ksk (t)Ω  34 where Ω is the density of attenders in the neighborhood of 17632 artists, and 92834 user-listended artists relations in susceptible node with k connections averaged over k total. In addition, the data set contains 12717 bi-directional X user friendship relations. These data sets are chosen because Ω= P (k)(k − 1)ak / hki (5) they exhibit very different attendance distributions and thus k provide an excellent playground to validate our model in dif- where hki denotes the mean degree of the network. As ferent settings. outlined above, λ is the probability that a node in state S transforms to state A if it is connected to a node in Experiments. state A. This happens when fˆij > ∆. Therefore, we have Data topologies. We firstly investigate the simulated ∆− < fij < ∆ where ∆− = ∆ − (1/k)1−γ . From this we attendance distributions as a function of trust γ, the an- R∆ have λ = ∆ f (x)dx, where f (x) is the expectation dis- ticipation threshold ∆, and the network topology. For this − R∆ purpose we simulate the dynamics on a toy network with 500 tribution. Similarly we write for α = l − f (x)dx, where nodes and record the final attendance number of 300 objects. l denotes the lower bound of the expectation distribution The simulation is conducted for ER and PL networks and f (x). A crude mean field approximation can be obtained by performed as outlined in the simulations paragraph above. multiplying the right hand sides of Eq. (4) with P (k) and In Fig.(2) and Fig.(3) we investigate the skewness [53] of the summing over k, which yields a set of differential equations attendance distributions and the maximal attendance ob- tained for the corresponding parameter settings. The skew-  ȧ(t) = λ hki s(t)a(t),  ˙  ness of a distribution is a measure for the asymmetry around d(t) = α hki s(t)a(t), (6)  its mean value. A positive skewness value means that there ṡ(t) = −(α + λ) hki s(t)a(t).  is more weight to the left from the mean, whereas a negative value indicates more weight in the right from the mean. which is later used to obtain analytical results for the atten- Fitting real data. We explore the model’s ability to fit dance fraction a(t). real world recommendation attendance distributions found in the described data sets. For this purpose we fix for the 3. METHODS Netflix data set a network with 480189 nodes and perform a We describe here our simulation procedures, datasets, ex- simulation for 17770 objects. In the MovieLens case we do periments, and analytical methods. the same for 943 nodes and 1682 objects and for the Lastfm data set we simulate on a network with 1892 nodes and 17632 Simulations. objects. In the case of Lastfm we have the social network Our simulations employ Alg. (1). As outlined in the model data as well. We validate our model on that data set by two section, we do not change the network topology during the experiments: a) we use the provided user friendship network dynamical processes. We experiment with two different net- as simulation input and fit the attendance distribution and work types, Erdős-Rényi (ER), and power law (PL) which b) we fit the attendance distribution like in the MovieLens are both generated by a so-called configuration model [34]. and Netflix case with an artificially generated network. ER and PL represent two fundamentally different classes Mathematical analysis. We investigate the Master Equa- of networks. The former is characterized by a typical de- tions Eq. (4) and Eq. (6). We provide a full analytical solu- gree scale (mean degree of the network), whereas the latter tion for Eq. (6) and an analytical approximation for Eq. (4) exhibits a fat-tailed degree distribution which is scale free. in the early spreading stage. The networks are random and have no degree correlations and no particular community structure. To obtain repre- 4. RESULTS sentative results we stick to the following approach: we fix Data topologies. The landscape of attendance distribu- the network type, number of nodes, number of objects, and tions of our model is demonstrated in Fig. (2) and Fig. (3). network type relevant parameters to draw an ER or PL net- To obtain these results, simulations were performed as de- work. We call this a configuration P. In addition, we fix the scribed in Sec. (3). The item anticipation fi was drawn from variance σ of the anticipation distributions fi . We perform a normal distribution with mean values µi ∈ U (−0.1, 0.1) each simulation on 50 different networks belonging to the and variance σ = 0.25 fixed for all items. Both networks same configuration P and on each network we simulate the have 500 nodes. In the Erdős-Rényi case, we used a wiring dynamics 50 times. Then we average the obtained atten- probability p = 0.03 between nodes. The Power Law net- dance distributions over all 2500 simulations. work was drawn with an exponent δ = 2.25. The simulated attendance distributions in Fig.(2) and Fig.(3) show a wide Datasets. range of different patterns for both ER and PL Influence- To show the validity of our model we use real world rec- Networks. In particular, both network types can serve as ommender datasets. MovieLens (movielens.umn.edu), a a basis for attendance distributions with both positive and web service from GroupLens (grouplens.org) where ratings negative skewness. Therefore, the observed fat-tailed distri- are recorded on a five stars scale. The data set contains butions are not a result of the heterogeneity of a scale free 1682 movies and 943 users. Only 6, 5% of possible votes are network but they are emergent properties of the dynamics expressed. Netflix data set (netflix.com). We use the Net- produced by our model. The parameter region for highly flix grand prize data set which contains 480189 users and positively-skewed distributions is the same for both network 17770 movies and also uses a five stars scale. Lastfm data types. The parameters γ and ∆ can be tuned so that all set (Lastfm.com). This data set contains social networking, items are attended by everybody or all items are attended tagging, and music artist listening information from users by nobody. While not relevant for simulating realistic atten- of the Last.fm online music system. There are 1892 users, dance distributions, these extreme cases help to understand 35 the model’s flexibility. Figure 4: Fit of the MovieLens attendance dis- tribution with trust γ = 0.50, critical anticipation Figure 2: Skewness of the attendance distributions threshold ∆ = 0.6, anticipation distribution variance as a function of trust γ and the critical anticipation σ = 0.25, and power law network with exponent threshold ∆ for Erdős-Rény networks with 500 nodes δ = 2.25, 943 nodes, and 1682 simulated objects. and 300 simulated items. Figure 5: Fit of the Netflix attendance distribution with trust γ = 0.52, critical anticipation threshold Figure 3: Skewness of the attendance distributions ∆ = 0.72, anticipation distribution variance σ = 0.27, as a function of trust γ and the critical anticipation and power law network with exponent δ = 2.2, 480189 threshold ∆ for power-law networks with 500 nodes nodes, and 17770 simulated objects. and 300 simulated items. Ru Fitting real data We fit real world recommender data ditions for the first movers a0 = ∆ f (x)dx, s(0) = 1 − a(0), from MovieLens, Netflix and Lastfm with results reported and d(0) = 0. In the following we use the bra-ket nota- in Fig. (4), Fig. (5), Fig. (6), Fig. (7), and Tab. (1), re- tion hxi to represent the average of a quantity x. Standard spectively. The real and simulated distributions are com- methods can now be used to arrive at2 pared using Kullback-Leibler (KL) divergence [29]. We re- port the mean, median, maximum, and minimum of the (τ hki)−1 exp(t/τ ) a(t) = . (7) simulated and real attendance distributions. Trust γ, antic- (α + λ) [exp(t/τ ) − 1] + (τ hki a0 )−1 ipation threshold ∆, and anticipation distribution variance Here τ is the time scale of the propagation which is defined σ are reported in figure captions. We also compare the aver- as aged mean degree, maximum degree, minimum degree, and clustering coefficient of the real Lastfm social network and τ = (a0 α hki + λ hki)−1 . (8) networks obtained to fit the data. Results are reported in −1 This is similar to the time scale τ = (λ hki) in the well Tab. (2) and Fig. (8). Note that thus obtained parameter known SI Model [38, 6]. Eq.(7) can be very useful in predict- values can be useful also in real applications where, assum- ing the average behavior of users in a recommender system. ing that our social opinion formation model is valid, one Since Eq. (4) is not accessible to a full analytical solution, could detect decline of the overall trust value in an online we investigate it for the early stage of the dynamics. As- community, for example. Mathematical analysis. Eq. (6) can be solved analyti- 2 We give here only the solution for a(t) because we are cally. We have ∀t : a(t)+s(t)+d(t) = 1 with the initial con- mainly interested in the attendance dynamics. 36 D KL Med Mean Max Min ML 0.046 27/26 59/60 583/485 1/1 NF 0.030 561/561 5654/5837 232944/193424 3/16 LFM1 0.05 1/1 5.3/5.2 611/503 1/1 LFM2 0.028 1/1 5.3/5.8 611/547 1/1 Table 1: Simulation results. ML: Movielens, NF: Netflix, LFM1: Lastfm with real network, LFM2: Lastfm with simulated network, KL: Kullback- Leibler divergence, Med: Median, Mean, Max: maximal attendance (data/simulated), Min: mini- mal attendance (data/simulated). D hki kmin kmax δ C LFM1 13.4 1 119 2.3 0.186 LFM2 12.0 1 118 2.25 0.06 Figure 6: Fit of the Lastfm attendance distribution with trust γ = 0.4, critical anticipation threshold ∆ = Table 2: Mean, minimum, maximum degree, clus- 0.8, anticipation distribution variance σ = 0.24, and tering coefficient C, and estimated exponent δ of the real Lastfm user friendship network with 1892 nodes real (LFM1) and simulated (LFM2) social network and 17632 simulated objects. for the Lastfm data set. Figure 7: Fit of the Lastfm attendance distribution Figure 8: Log-log plot of real (red) and simulated with trust γ = 0.6, critical anticipation threshold (blue) social network degree distribution P (k) for ∆ = 0.8, anticipation distribution variance σ = 0.24, the Lastfm data set. Inset: plot of the cumulative and power law network with exponent δ = 2.25, 1892 degree distribution. nodes and 17632 simulated objects. We emphasize that Eq.(10) is valuable in predicting users’ suming a(0) = a0  0, we can neglect the dynamics of d(t) behavior of a recommender system in an early stage. to obtain ! k2 Ω̇(t) = hki − 1 Ω(t). 5. DISCUSSION Social influence and our peers are known to form and in- In addition, Eq. (4) yields fluence many of our opinions and, ultimately, decisions. We ) propose here a simple model which is based on heteroge- ȧk (t) = λk(1 − ak (t))Ω(t) (9) neous agent expectations, a social network, and a formalized ṡk (t) = −(α + λ)k(1 − ak (t))Ω(t) social influence mechanism. We analyze the model by nu- merical simulations and by master equation approach which Neglecting terms of order a2k (t) and summing the solution is particularly suitable to describe the initial phase of the of ak (t) over P (k), we get a result for the early spreading social “contagion”. The proposed model is able to generate stage a wide range of different attendance distributions, includ-   a(t) = a(0) 1 + τ λ exp(t/τ ) − 1 , (10) ing those observed in popular real systems (Netflix, Lastfm, and Movielens). In addition, we showed that these patterns with the timescale τ = k2 / λ( k2 − hki) . The obtained   are emergent properties of the dynamics and not imposed time scale τ valid in the early stage of the opinion spreading by topology of the underlying social network. Of particular is clearly dominated by the network heterogeneity. This re- interest is the case of Lastfm where the underlying social sult is in line with known disease models, e.g., SI,SIR [38, 6]. network is known. Calibrating the observed attendance dis- 37 tribution against the model then leads not only to social [9] A. Birukov, E. Blanzieri, and P. Giorgini. 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