Recommendation systems in the scope of opinion formation: a model

Laboratory for Web Science (LWS)University of Applied Sciences Switzerland http://lws.ffhs.ch Follow @blattnerma , Dr. Marcel Blattner

Recommendation systems in the scope of opinion formation: a model Dr. Marcel Blattner, Laboratory for Web Science University of Applied Sciences Switzerland Dr. Matus Medo, Physics Department, University of Fribourg University of Fribourg Switzerland , Dr. Marcel Blattner

Motivation , Dr. Marcel Blattner

MotivationReal world recommender system data are theresult of complex processes. Social interactionsplay a major role. , Dr. Marcel Blattner

MotivationReal world recommender system data are theresult of complex processes. Social interactionsplay a major role.How can we model those mechanisms to reproduceobserved data (opinions) in recommendationsystems? , Dr. Marcel Blattner

MotivationReal world recommender system data are theresult of complex processes. Social interactionsplay a major role.How can we model those mechanisms to reproduceobserved data (opinions) in recommendationsystems?How can we beneﬁt from such a model? , Dr. Marcel Blattner

ts are always ’bound’ on used To highlight various aspects of B-Rank, a to ty to obtain similar results for The Aim Fig.(1) is introduced. For simplicity all links be jects and users are equally weighted wi = 1 8i. mpared to ZLZ-II, is achieved result highlights the fact, that The model should personality. From real world objects users igher diversity is positive cor-n general [22]. However, bipartite generate user re in off-line user-object data experiments and raw robust conclusions.Z-II algorithm was proposed reached a comparable perfor- Figure 1: Toy net to illustrate B-Rank. CirclesRank in the movielens dataset. hyperedges (users), squares are hyperverticesning parameter l . B-Rank in jects. The votes are illustrated as links betwe nd therefore easier to imple- and users.may increase improvements First, some general aspects are discussed, see presented basic B-Rank al- shown, how all aspects are well captured by th eight matrix W . This will be algorithm. per. Another extension is to Case A: huge audience in common. Intuipagation (indirect connections objects a and b are similar to each other, when , Dr. Marcel Blattner

ts are always ’bound’ on used To highlight various aspects of B-Rank, a to ty to obtain similar results for The Aim Fig.(1) is introduced. For simplicity all links be jects and users are equally weighted wi = 1 8i. mpared to ZLZ-II, is achieved result highlights the fact, that The model should personality. From real world objects users igher diversity is positive cor-n general [22]. However, bipartite generate user re in off-line user-object data experiments and raw robust conclusions.Z-II algorithm was proposed reached a comparable perfor- Figure 1: Toy net to illustrate B-Rank. CirclesRank in the movielens dataset. hyperedgesWe trysquares are hypervertices (users), to understandning parameter l . B-Rank in jects. The these data as a as links betwe votes are illustrated result nd therefore easier to imple- and users. of social processes.may increase improvements First, some general aspects are discussed, see presented basic B-Rank al- shown, how all aspects are well captured by th eight matrix W . This will be algorithm. per. Another extension is to Case A: huge audience in common. Intuipagation (indirect connections objects a and b are similar to each other, when , Dr. Marcel Blattner

The ModelModel Assumptions (I): Objects generate anticipation distributions. (IIA - intrinsic item anticipation)

The ModelModel Assumptions (I): Objects generate anticipation distributions. (IIA - intrinsic item anticipation) Individuals will invest resources only, if their anticipation exceed some threshold. anticipation threshold

The ModelModel Assumptions (I): Objects generate anticipation distributions. (IIA - intrinsic item anticipation) Individuals will invest resources only, if their anticipation exceed some Potential threshold. Adopters Adopters anticipation threshold

The ModelModel Assumptions (I): Objects generate anticipation distributions. (IIA - intrinsic item anticipation) Individuals will invest resources only, if their anticipation exceed some Potential threshold. Adopters Adopters Potential adopters are able to become adopters or anticipation deniers threshold

The ModelModel Assumptions (II): The shift from a potential adopter to an adopter is caused by information exchange on a network with a specific topology. Potential Adopters (Influence Network) Adopters anticipation threshold , Dr. Marcel Blattner

The ModelModel Assumptions (II): The shift from a potential adopter to an adopter is caused by information exchange on a network with a specific topology. Potential Adopters (Influence Network) Adopters anticipation threshold adopter potential adopter denier Inﬂuence Network , Dr. Marcel Blattner

The ModelModel Assumptions (II): The shift from a potential adopter to an adopter is caused by information exchange on a network with a specific topology. Potential Adopters (Influence Network) Adopters anticipation threshold adopter Possible transitions: potential adopter denier Inﬂuence Network , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation threshold  (1 ) ˆ ⇥j fij = fij + kj , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation threshold  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) Intrinsic ItemAnticipation , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation Neighbors, who threshold already adopted  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) Intrinsic ItemAnticipation , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation Neighbors, who threshold already adopted  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) Intrinsic Item Total # ofAnticipation neighbors , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation Neighbors, who threshold already adopted Trust  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) Intrinsic Item Total # ofAnticipation neighbors , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation Shifted Neighbors, who threshold Item already adopted TrustAnticipation  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) Intrinsic Item Total # ofAnticipation neighbors , Dr. Marcel Blattner

The ModelMathematicalformulation IIA-Shift Potential Adopters Adopters anticipation Shifted Neighbors, who threshold Item already adopted TrustAnticipation  (1 ) ˆ ⇥j fij = fij + kj , fij 2 fi 2 N (µi , ) ˆ fij adopter Intrinsic Item Total # of ˆ fij < denierAnticipation neighbors , Dr. Marcel Blattner

The Model Simulation of Algorithm 1 RecSysMod algorithm. P contains the con- dynamics for a set figuration parameter for the network. is the Anticipation Threshold and denotes the trust. O 2 N is the number of objects to simulate. G(N, E) is the network. N is the set of of objects nodes and E is the set of edges. 1: procedure RecSysMod I(P, , , O) 2: G(N, E) GenNetwork(P) 3: for all Objects in O do 4: generate distribution fi from N (µi , ) 5: for each node j 2 N in G do 6: draw fij from fi 7: if fij < then 8: jstate S 9: else 10: jstate A 11: end ifContour plot for and ⇢ = ⇥j /kj . Num- 12: end for the plot quantify the shift in the IAA as 13: repeatof and ⇢. 14: for all j with jstate = S AND ⇥j > 0 do h i(1 ) ˆ ⇥ 15: fij fij + jkjuence-Network IN(P) with a fixed network topol- 16: ˆ if fij < then law, Erd˝s-Rńyi, or another). P refers to a o e 17: jstate D priate parameters for the Influence-Network in 18: elseke network type, number of nodes, etc.). The 19: jstate Aopology is not a↵ected by the dynamical pro- 20: end if on propagation) taking place on it. We justify 21: end forcenario by assuming that the time scale of the 22: until |{j|jstate = S AND ⇥j > 0}| = 0ange is much longer then the time scale 1 of opin- 23: end for g in the network. Each node in the Influence- 24: end procedure responds to an individual. For each individual n unbiased Intrinsic-Item-Anticipation fij from , Dr. Marcel Blattnered probability distribution fi . At each time step,

Distribution Landscape Simulation ofdynamics for a set of objects , Dr. Marcel Blattner

Figure 4: Fit of the MovieLens attendance dis-Figure 2: Skewness of the attendance distributions tribution with trust threshold Results = 0.50, critical anticipation = 0.6, anticipation distribution varianceas a function of trust and the critical anticipation = 0.25, and power law network with exponentthreshold for Erd˝s-Rńy networks with 500 nodes o e = 2.25, 943 nodes, and 1682 simulated objects.and 300 simulated items. Figure 4: Fit of the MovieLens attendance dis- Figure 5: Fit of the Netflix attendance distribution tribution with trust = 0.50, critical anticipation with trust = 0.52, critical anticipation thresholdFigure 3: Skewness of the attendance distributions threshold = 0.6, anticipation distribution variance = 0.72, anticipation distribution variance = 0.27,as a function of trust law network with exponent = 0.25, and power and the critical anticipation and power law network with exponent = 2.2, 480189threshold943 for power-law networks with 500 nodes = 2.25, nodes, and 1682 simulated objects. nodes, and 17770 simulated objects.and 300 simulated items. MovieLens Netflix 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. Standardspectively. The real and simulated distributions are com- methods can now be used to arrive at2pared using Kullback-Leibler (KL) divergence [29]. We re-port the mean, median, maximum, and minimum of the (⌧ hki) 1 exp(t/⌧ ) a(t) = 1 . (7)simulated and real attendance distributions. Trust , antic- (↵ + ) [exp(t/⌧ ) 1] + (⌧ hki a0 )ipation threshold , and anticipation distribution variance Here ⌧ is the time scale of the propagation which is defined are reported in figure captions. We also compare the aver- as , Dr. Marcel Blattneraged mean degree, maximum degree, minimum degree, and

Figure 6: Fit of the Lastfm attendance distribution with trust = 0.4, critical anticipation threshold = Table 2: Mean, minimum, maximum degree, clus- Results 0.8, anticipation distribution variance = 0.24, and tering coe cient C, and estimated exponent 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. D KL ML 0.0 D KL Med Mean Max Min NF 0.0 ML 0.046 27/26 59/60 583/485 1/1 LFM1 0.0 NF 0.030 561/561 5654/5837 232944/193424 3/16 LFM2 0.0 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: Si Netflix, LF Table 1: Simulation results. ML: Movielens, NF: Netflix, LFM1: Lastfm with real network, LFM2: Lastfm wit Lastfm with simulated network, KL: Kullback- Leibler div Leibler divergence, Med: Median, Mean, Max: maximal at maximal attendance (data/simulated), Min: mini- mal attenda mal attendance (data/simulated). D hki D hki kmin kmax C LFM1 13. LFM1 13.4 1 119 2.3 0.186 LFM2 12.0 1 118 2.25 0.06 LFM2 12. Figure 7: Fit of the Lastfm attendance distributiongure 6: Fit of the Lastfm attendance distribution Figure 8: 6: Fit of the Lastfm attendance distribution Figure Log-log plot of real (red) and simulated with anticipation0.6, critical= anticipation Mean, minimum,(blue) social network degree distribution P (k) for =th trust = 0.4, criticaltrust = threshold Table 2: threshold with trust = 0.4, critical anticipation threshold maximum degree, clus- Table 2: M, anticipation distributionanticipation distribution tering coe cient C, and estimatedanticipation distribution variance cumulative = 0.8, variance = 0.24, and variance = 0.24, the Lastfm data of the Inset: plot of the = 0.24, and 0.8, exponent set. tering coe l Lastfm user friendship network network with exponent = 2.25, 1892 and power law with 1892 nodes real (LFM1) and simulated (LFM2) social user friendship network with 1892 nodes degreeLastfm network real distribution. real (LFM1d 17632 simulated objects. 17632 simulated objects. the Lastfm data set. nodes and for and 17632 simulated objects. for the Last 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 ⌦ ↵ ! 2 k ˙ ⌦(t) = hki 1 ⌦(t). 5. DISCUSSION LastFM 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- ak (t) = k(1 ak (t))⌦(t) ˙ (9) neous agent expectations, a social network, and a formalized sk (t) = (↵ + )k(1 ak (t))⌦(t) ˙ social influence mechanism. We analyze the model by nu- merical simulations and by master equation approach which Neglecting terms of order a2 (t) and summing the solution k 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 ⇣gure 7: Fit of the Lastfm attendance distribution Figure ⌘ Log-log plot of a wide range of di↵erent attendance distributions, includ- 8: real (red) and simulatedth trust a(t) = a(0) 1 + ⌧ exp(t/⌧ ) 1 social network degree distribution P in for = 0.6, critical anticipation threshold (blue) , (10) ing those observed (k) popular real systems (Netflix, Lastfm, ⌦ 2 0.24, ⇤ and Movielens). In addition, we showed that these patterns= 0.8, anticipation distribution variance ↵= ⇥ ⌦ ↵ the Lastfm data set. Inset: plot of the cumulatived power law networkthe timescale ⌧ = =k / 1892 k with with exponent 2.25, ( 2 hki) . The obtained degree distribution. are emergent Fit of theof the dynamics and not imposed Figure 7: properties Lastfm attendance distribution Figure 8: Ldes and 17632 simulated objects. in the early stage of the opinion spreading time scale ⌧ valid bywith trust the = 0.6, critical anticipationparticular topology of underlying social network. Of threshold (blue) socia , Dr. Marcel Blattner is clearly dominated by the network heterogeneity. This re- interest 0.8, anticipation distribution variance social = is the case of Lastfm where the underlying = 0.24, the Lastfm We emphasize that Eq.(10) is valuable in predicting users’

Results D KL Med Mean Max Min ML D 0.046 KL 27/26 Med 59/60 Mean 583/485Max 1/1 Min NF ML 0.030 561/561 5654/5837 232944/193424 3/16 1/1 0.046 27/26 59/60 583/485 LFM1 NF 0.05 0.030 1/1 561/561 5.3/5.2 5654/5837 611/503 1/1 232944/193424 3/16 LFM2 0.028 1/1 5.3/5.8 611/547 1/1 LFM1 0.05 1/1 5.3/5.2 611/503 1/1 Table 1: Simulation results. 5.3/5.8Movielens, NF: LFM2 0.028 1/1 ML: 611/547 1/1 Netﬂix, LFM1: Lastfm with real network, LFM2: Lastfm 1: Simulation results. ML: Movielens, NF: Table with simulated network, KL: Kullback- Leibler divergence, Med: with real Mean, Max: Netﬂix, LFM1: Lastfm Median, network, LFM2: Lastfm with simulated network, KL: Kullback- maximal attendance (data/simulated), Min: mini- Leibler divergence, Med: Median, Mean, Max: mal attendance (data/simulated). maximal attendance (data/simulated), Min: mini- D hki kmin kmax C mal attendance (data/simulated). LFM1 13.4 1 119 2.3 0.186 LFM2 12.0 1 118 2.25 0.06on D hki kmin kmax C = Table 2: Mean, minimum, maximum0.186 LFM1 13.4 1 119 2.3 degree, clus-nd tering coe 12.0 LFM2 1 118 2.25 0.06 cient C, and estimated exponent of the nes real (LFM1) and simulated (LFM2) social network= Table 2: Mean, minimum, maximum degree, clus- for the Lastfm data set. d tering coe cient C, and estimated exponent of thees real (LFM1) and simulated (LFM2) social network for the Lastfm data set. , Dr. Marcel Blattner

Mathematical analysisCoupled differential equations Coupled differential equations for k-compartments (mean ﬁeld approximation) 9 9 ak (t) = ksk (t)⌦ ˙ > = a(t) = < k > s(t)a(t), ˙ > = ˙ dk (t) = ↵ksk (t)⌦ ˙ d(t) = ↵ < k > s(t)a(t), > ; > ; sk (t) = ˙ (↵ + )ksk (t)⌦ s(t) = ˙ (↵ + ) < k > s(t)a(t) P k P (k)(k 1)ak ⌦= <k> Z Z = f (x)dx, ↵= f (x)dx, = (1/k)1 l , Dr. Marcel Blattner

Mathematical analysis - ResultsCoupled differential equations Coupled differential equations for k-compartments (mean ﬁeld approximation) ⇣ ⌘ (⌧ hki) 1 exp(t/⌧ )a(t) = a(0) 1 + ⌧ exp(t/⌧ ) 1 a(t) = 1 (↵ + ) [exp(t/⌧ ) 1] + (⌧ hki a0 ) ⌦ 2 ↵ k 1⌧= ⌧ = (a0 ↵ hki + hki) [ (hk 2 i hki)] , Dr. Marcel Blattner

Use cases? , Dr. Marcel Blattner

Use cases?1. First step for a data generator, useful to testnew methods and algorithms , Dr. Marcel Blattner

Use cases?1. First step for a data generator, useful to testnew methods and algorithms2. Especially useful when you have a friendship-network(like in the LastFM case) to extrapolatefuture data topologies. , Dr. Marcel Blattner

Future work , Dr. Marcel Blattner

Future workExpand the proposed model to generate ratingswithin a predeﬁned scale (like 5-star) , Dr. Marcel Blattner

Future workExpand the proposed model to generate ratingswithin a predeﬁned scale (like 5-star)Work out use cases and show the beneﬁt ofthe proposed a model , Dr. Marcel Blattner

...this is the end my friend...Questions? , Dr. Marcel Blattner

Recommendation systems in the scope of opinion formation: a model

by Marcel Blattner on Sep 09, 2012

Accessibility

Categories

Upload Details

Usage Rights

2 Embeds 4

Statistics

Recommendation systems in the scope of opinion formation: a model Presentation Transcript

https://twitter.com	3
https://www.xing.com	1