Aggregated data in real world recommender applications often feature fat-tailed distributions of the number of times individual items have been rated or favored. We propose a model to simulate such data. The model is mainly based on social interactions and opinion formation taking place on a complex network with a given topology. A threshold mechanism is used to govern the decision making process that determines whether a user is or is not interested in an item. We demonstrate the validity of the model by tting attendance distributions from di erent real data sets. The model is mathematically analyzed by investigating its master equation. Our approach provides an attempt to understand recommender system's data as a social process. The model can serve as a starting point to generate arti cial data sets useful for testing and evaluating recommender systems.
This is the information age. We are witnessing
information production and consumption in a speed never seen
before. The WEB2.0 paradigm enables consumers and
producers to exchange data in a collaborative way bene ting both
parties. However, one of the key challenges in our
digitallydriven society is information overload [
The most popular techniques related to recommendation
systems are collaborative ltering [
However, there are still many open challenges in the
research eld of recommendation systems [
The rest of the paper is organized as follows. The model is outlined in Sec. (2). Methods, data set descriptions, and validation procedures are in Sec. (3). Results are presented in Sec. (4). Discussion and an outlook for future research directions are in Sec. (5). 2. 2.1
Our daily decisions are heavily in uenced by various
information channels: advertisement, broadcastings, social
interactions, and many others. Social ties (word-of-mouth) play
a pivotal role in consumers buying decisions [
Our model is formulated within an opinion formation framework where social ties play a major role. We shall discuss the following main ingredients of our model:
We call the network where context-relevant information
exchange takes place an In uence-Network (IN). Nodes of
the IN are people and connections between nodes indicate
the in uence among them. Note that we put no constraints
on the nature of how these connections are realized. They
may be purely virtual (over the Internet) or based on
physical meetings. We emphasize that INs are domain dependent,
i.e., for a given community of users, the In uence Network
concerning books may di er greatly (in topology, number of
ties, tie strength, etc.) from that concerning another subject
such as food or movies. Indeed, one person's opinion
leaders (relevant peers) concerning books may be very di erent
from those for food or other subjects. In this scope, we see
the INs as domain-restricted views on social networks. It
is thus reasonable to assume that In uence Networks are
similar to social interaction networks which often exhibit a
scale-free topology [
We now summarize the basic ingredients of our model. An individual user's opinions on objects are assembled in two consecutive stages: i) opinion making based on di erent external sources, including suggestions by recommendation systems and ii) opinion making based on social interactions in the In uence-Network. The second process may shift the opinions generated by the rst process. 2.2
In this section we rstly describe how individuals' IntrinsicItem-Anticipations may change due to social interactions taking place on a particular In uence-Network. Secondly, we introduce dynamical processes governing the opinion propagation.
We model a possible shift in the IIA as: f^ij = fij +
j kj (1 )
: where f^ij is the shifted Intrinsic-Item-Anticipation of individual j for object i, fij is the unbiased IIA, j is the number of j's neighbors, who already experienced and liked item i, kj denotes the total number of j's neighbors in the corresponding IN, and 2 (0; 1) quanti es trust of individuals to their peers. An individual j will consume, purchase, or positively rate an item i only if ^ fij : (1) (2) We identify as the Critical-Anticipation-Threshold. ValIntrinsic-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 di erent vertisement, marketing campaigns, and other e orts to at- and largely independent contributions, we assume that fi tract customers predate the launching process and continue is normally distributed, fi 2 N ( i; ). (Unless stated othafter the product started to spread on the market. These erwise.) To mimic di erent item anticipations for di erent e orts in uence product-dependent customer anticipation. objects i, we draw the mean i from a uniform distribution It is clear that the resulting anticipation is a complex com- U ( ; ). We maintain i, , and , so that fi is roughly bination of many di erent components including intrinsic bounded by ( 1; 1), i.e., 1 3 < + 3 1. Note product quality and possibly also suggestions from recom- that f^ij can exceed these boundaries after a shift of the corremendation 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 in uence of j's neighborhood weighted by Item-Anticipation (IIA) and measure it by a single number. trust . To better understand the interplay between and It is based on many external sources, except for the in uence the density of attending users in the neighborhood of user generated by social interactions. It is the opinion on some- i, := j=kj , we refer to Fig. 1. Trust 1 causes a big thing taken by individuals, before they start to discuss the shift on the IIA's even for 0. On the other hand, 0 subject with their peers. Furthermore, we assume that an needs high to yield a signi cant IAA shift. These propindividual 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 in uenced.
The In uence-Dynamics describes how individuals' IntrinsicItem-Anticipations are altered by information exchange via
The In uence-Dynamics proceeds as follows. Firstly, we
draw an In uence-Network IN(P) with a xed network
topology (power-law, Erdo}s-Renyi, or another). P refers to a
set of appropriate parameters for the In uence-Network in
question (like network type, number of nodes, etc.). The
network's topology is not a ected by the dynamical
processes (opinion propagation) taking place on it. We justify
this static scenario by assuming that the time scale of the
topology change is much longer then the time scale 1 of
opinion spreading in the network. Each node in the In
uenceNetwork corresponds to an individual. For each individual
j we draw an unbiased Intrinsic-Item-Anticipation fij from
the prede ned probability distribution fi. At each time step,
every individual is in one of the following states: fS; A; Dg.
S refers to a susceptible state and corresponds to the initial
state for all nodes at t = 0. A refers to an attender state
and corresponds to an individual with the property f^ij .
D refers to a denier state with the property f^ij < after
an information exchange with his/her peers in the In
uenceNetwork happened. An individual in state D or A can not
change his/her state anymore. It is clear that an individual
in state A cannot back transform to the susceptible state
S, since he/she did consume or favor item i and we do not
account for multiple attendances in our model. An
individual in state D was in uenced but not convinced by his
opinion leaders (directed connected peers). We make the
following assumption here: if individual j's opinion leaders
are not able to convince individual j, meaning that
individual's j Intrinsic Item Anticipation f^ij stays below the
critical threshold after the in uence process, then we assume
that j's opinion not to attend object i remains unchanged
in the future. Therefore we have the following possible
transitions for each node in the in uence network: jS ! jA or
jS ! jD. Node states are updated asynchronously which
is more realistic than synchronous updating, especially in
social interaction models [
We are now in the position to formulate the Master Equa1The term time scale denotes a dimensionless quantity and speci es the devisions of time. A shorter time scale means a faster spreading of opinions in the network. (3) (4) 15: tion for the dynamics. As already said before, two things can happen when a non-attender is connected to an attender: a) he/she becomes an attender too, or b) he/she becomes a denier who will not attend/favor the item. For these two interaction types we formally write:
S + A S + A ! 2A !
D + A
Here denotes the probability that a susceptible node
connected to an attender becomes an attender too, and is
the probability that a susceptible node attached to an
attender becomes a denier. To take into account the
underlying network topology of the In uence Network it is
common to introduce compartments k [
We describe here our simulation procedures, datasets, experiments, and analytical methods.
Our simulations employ Alg. (1). As outlined in the model
section, we do not change the network topology during the
dynamical processes. We experiment with two di erent
network types, Erdo}s-Renyi (ER), and power law (PL) which
are both generated by a so-called con guration model [
To show the validity of our model we use real world recommender datasets. MovieLens (movielens.umn.edu), a web service from GroupLens (grouplens.org) where ratings are recorded on a ve stars scale. The data set contains 1682 movies and 943 users. Only 6; 5% of possible votes are expressed. Net ix data set (net ix.com). We use the Netix grand prize data set which contains 480189 users and 17770 movies and also uses a ve stars scale. Lastfm data set (Lastfm.com). This data set contains social networking, tagging, and music artist listening information from users of the Last.fm online music system. There are 1892 users, (5) (6) 17632 artists, and 92834 user-listended artists relations in total. In addition, the data set contains 12717 bi-directional user friendship relations. These data sets are chosen because they exhibit very di erent attendance distributions and thus provide an excellent playground to validate our model in different settings.
Data topologies. We rstly investigate the simulated
attendance distributions as a function of trust , the
anticipation threshold , and the network topology. For this
purpose we simulate the dynamics on a toy network with 500
nodes and record the nal attendance number of 300 objects.
The simulation is conducted for ER and PL networks and
performed as outlined in the simulations paragraph above.
In Fig.(2) and Fig.(3) we investigate the skewness [
Fitting real data. We explore the model's ability to t real world recommendation attendance distributions found in the described data sets. For this purpose we x for the Net ix data set a network with 480189 nodes and perform a simulation for 17770 objects. In the MovieLens case we do the same for 943 nodes and 1682 objects and for the Lastfm data set we simulate on a network with 1892 nodes and 17632 objects. In the case of Lastfm we have the social network data as well. We validate our model on that data set by two experiments: a) we use the provided user friendship network as simulation input and t the attendance distribution and b) we t the attendance distribution like in the MovieLens and Net ix case with an arti cially generated network.
Mathematical analysis. We investigate the Master Equations Eq. (4) and Eq. (6). We provide a full analytical solution for Eq. (6) and an analytical approximation for Eq. (4) in the early spreading stage. 4.
Data topologies. The landscape of attendance distributions of our model is demonstrated in Fig. (2) and Fig. (3). To obtain these results, simulations were performed as described in Sec. (3). The item anticipation fi was drawn from a normal distribution with mean values i 2 U ( 0:1; 0:1) and variance = 0:25 xed for all items. Both networks have 500 nodes. In the Erdo}s-Renyi case, we used a wiring probability p = 0:03 between nodes. The Power Law network was drawn with an exponent = 2:25. The simulated attendance distributions in Fig.(2) and Fig.(3) show a wide range of di erent patterns for both ER and PL In uenceNetworks. In particular, both network types can serve as a basis for attendance distributions with both positive and negative skewness. Therefore, the observed fat-tailed distributions are not a result of the heterogeneity of a scale free network but they are emergent properties of the dynamics produced by our model. The parameter region for highly positively-skewed distributions is the same for both network types. The parameters and can be tuned so that all items are attended by everybody or all items are attended by nobody. While not relevant for simulating realistic attendance distributions, these extreme cases help to understand the model's exibility.
Fitting real data We t real world recommender data
from MovieLens, Net ix and Lastfm with results reported
in Fig. (4), Fig. (5), Fig. (6), Fig. (7), and Tab. (1),
respectively. The real and simulated distributions are
compared using Kullback-Leibler (KL) divergence [
Mathematical analysis. Eq. (6) can be solved analytically. We have 8t : a(t)+s(t)+d(t) = 1 with the initial conditions for the rst movers a0 = R u f (x)dx, s(0) = 1 a(0), and d(0) = 0. In the following we use the bra-ket notation hxi to represent the average of a quantity x. Standard methods can now be used to arrive at2 a(t) =
( hki) 1 exp(t= )
( + ) [exp(t= )
1] + ( hki a0) 1
:
(7)
Here is the time scale of the propagation which is de ned
as
= (a0 hki +
hki) 1 :
(8)
This is similar to the time scale = ( hki) 1 in the well
known SI Model [
Since Eq. (4) is not accessible to a full analytical solution, we investigate it for the early stage of the dynamics. As2We give here only the solution for a(t) because we are mainly interested in the attendance dynamics.
59=60
5654=5837
5:3=5:2
5:3=5:8
Neglecting terms of order a2k(t) and summing the solution
of ak(t) over P (k), we get a result for the early spreading
stage
a(t) = a(0) 1 +
exp(t= )
1 ;
(10)
with the timescale = k2 = ( k2 hki) . The obtained
time scale valid in the early stage of the opinion spreading
is clearly dominated by the network heterogeneity. This
result is in line with known disease models, e.g., SI,SIR [
We emphasize that Eq.(10) is valuable in predicting users' behavior of a recommender system in an early stage. 5.
Social in uence and our peers are known to form and inuence many of our opinions and, ultimately, decisions. We propose here a simple model which is based on heterogeneous agent expectations, a social network, and a formalized social in uence mechanism. We analyze the model by numerical simulations and by master equation approach which is particularly suitable to describe the initial phase of the social \contagion". The proposed model is able to generate a wide range of di erent attendance distributions, including those observed in popular real systems (Net ix, Lastfm, and Movielens). In addition, we showed that these patterns are emergent properties of the dynamics and not imposed by topology of the underlying social network. Of particular interest is the case of Lastfm where the underlying social network is known. Calibrating the observed attendance distribution against the model then leads not only to social in uence parameters but also to the degree distribution of the social network which agrees with that of the true social network.
The Kullback-Leibler distances (KL) for the simulated and real attendance distributions are below 0:05 in all cases, thus demonstrating a good t. However, the maximum attendances could not be reproduced exactly by the model. One reason may be missing degree correlations in the simulated networks in contrast to real networks where positive degree correlations (so-called degree assortativity) are common. For the Lastfm user friendship network we observe a higher clustering coe cient C 0:18 compared to the clustering coe cient C 0:06 in the simulated network. To compensate for this, a higher trust parameter is needed to t the real Lastfm attendance distribution with simulated networks.
We are aware that our statistics to validate the model is not complete. But we are con dent, that our approach points to a fruitful research direction to understand recommender systems' data as a social driven process.
The proposed model can be a rst step towards a data
generator to simulate bipartite user-object data with
realworld data properties. This could be used to test and
validate new recommender algorithms and methods. Future
research directions may expand the proposed model to
generate ratings within a prede ned scale. Moreover, it could
be very interesting to investigate the model in the scope of
social imitation [