This document summarizes a research paper that models how recommender systems can influence product popularity in markets. It presents a model that simulates user purchases based on personal preferences and recommendations from social connections. Experiments on this model using real social network data found that the recommender system did not significantly distort the market shares of different products. However, adding a "super-node" that strongly recommends one product to all users did substantially distort the market in favor of that product.

1. 1 KYOTO UNIVERSITY
KYOTO UNIVERSITY
The Limits of Popularity-Based
Recommendations, and the Role of Social Ties
Daiki Tanaka
Kashima lab., Kyoto University
Research Seminar, 2017/5/12(Fri)

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Today’s paper:
n Title : The Limits of Popularity-Based Recommendations,
and the Role of Social Ties
n Venue: KDD2016
n Authors:
Marco Bressan
Sapienza University of
Rome Rome, Italy
Stefano Leucci
Sapienza University of
Rome Rome, Italy
Alessandro Panconesi
Sapienza University of
Rome Rome, Italy
Prabhakar Raghavan
Google Mountain
View, CA
Erisa Terolli
Sapienza University
of Rome Rome, Italy

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Background

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Background:
n To what extent can a market be altered by a Recommender
System(RS)?
l Ex) if an online bookstore starts adopting a RS, will unknown
books become hits? How will readers change?
l In this paper we try an approach by introducing a natural
model for markets that are governed by a RS.
n Focusing on popularity of products and connection among
users

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Related Work

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Related work:
n A study shows that recommendations of popular users are much
more likely to be accepted than recommendations of “average”
users.
n Another study finds that a large part of the items adopted by a user
are also adopted by his/her friends.
n our model allows users to influence each other
n ours is stochastic and allows for complex dependences on
past events
n We analyzed the interactions between RS’s and markets

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A MODEL FOR Recommender System

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Model:
Definition
n Products.
n Users (or Buyers).
n The graph.
n The purchasing process.
n The weight of history.

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Model : definition
Products
l set 𝒫 of m products
l each product can be bought multiple times during the
purchasing process.
l Products are bought one at a time in an infinite sequence
of time steps t = 1,2,3...

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Model : definition
Users (or Buyers)
We have a set U := {1,...,n} of n users.
l A fixed purchased rate 𝑓# ∈ 0 , 1 ∶ User u is chosen with 𝑓#
• ∑ 𝑓# = 1#∈-
l 𝐵# 𝑝 : 𝑢2s personal preference for p ∈ Ρ
l a fixed probability 𝛼# ∈ [0 , 1) : when u buys a product it follows a
recommendation with probability 𝛼#
l a list of products purchased in the past (before time 0)
𝑓#, Β# 𝑝 , 𝛼# are fixed parameters.

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Model : definition
The graph
Users are connected via a directed, weighted graph 𝐺 = 𝒰, 𝐸
l an edge 𝑣𝑢 denotes the fact that 𝑣 can influence 𝑢 when 𝑢 decides
to follow a recommendation.
l The arc has weight 𝑤=# which gives the strength of this trust
relationship.
u v
𝒘 𝒗𝒖

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Model : definition
The purchasing process
At each time step t = 1,2,..., a user u is chosen with probability 𝑓# and
buys a product in one of two ways:
1. With probability 1 − 𝛼#
u chooses a product according to its own distribution 𝐵# of
personal preferences.
2. With probability 𝛼#
u follows the recommendation.

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Model : definition
The purchasing process
n When user u does not follow recommendation
l Which product?
• at probability 𝑏#(𝑝), u buys p.
……..
p p’ p”
𝑏# 𝑝 𝑏# 𝑝2 𝑏#(𝑝”)
u

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Model : definition
The purchasing process
n When user u follows recommendation
l from whom?
• At probability 𝑤=#, u follows recommendation of v
l Which product?
• At probability 𝑥=
F
(𝑝) , u buys p because of v’s recommendation.
• 𝑢, 𝑣 ∈ 𝒰, 𝑝 ∈ 𝒫
v v’ v’’
…….
𝑤=#
𝑤=GG#
u

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Model : definition
past purchases
User u have made 𝑘# purchases before starting the system.
Now we are at time t > 0
n ℎ#
F,b
∶ the weight of a purchase made by u at time i < t
products
time
m
1
2
…………….
-𝑘# … … … … … … … . −1 0 1 𝑡-1

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Model : definition
The weight of history
n ti
−𝑘#, … , −1 𝐼#
F ⊆ 0, … 𝑡 − 1
𝐼#
F ∶ 𝑠𝑒𝑡 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑤ℎ𝑒𝑛 𝑢 𝑏𝑜𝑢𝑔ℎ𝑡 𝑠𝑜𝑚𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡
(t > 0)

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Model : definition
The weight of history
n ti
−𝑘#, … , −1 𝐼#
F ⊆ 0, … 𝑡 − 1
p
• 𝐽#,r
F ⊆ −𝑘#, … , −1 ⋃ 𝐼#
F ∶ 𝑠𝑒𝑡 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑤ℎ𝑒𝑛 𝑢 𝑏𝑜𝑢𝑔ℎ𝑡 𝑝.
• 𝑥#
F 𝑝 ≔ ∑ ℎ#
F,b
b∈uv,w
x

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Analysis of the model

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Analysis: preparation
We focus on a particular product p*.
n u is the recommender node at time t.
l 𝑥#
F : denoting 𝑥#
F (𝑝∗)
l 𝒙F ≔ (𝑥{
F
, … , 𝑥|
F )
l 𝑏# : u’s personal preference for p*.
l b := (𝑏{, … , 𝑏|)
l A := diag 𝛼{, … 𝛼| =
𝛼{ ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 𝛼|
l M : 𝑀#= = 𝑤=#
l f := (𝑓{, 𝑓•, … , 𝑓|)

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Analysis:
Does 𝑥F
converge?
The past fades away : l𝑖𝑚F→„ 𝑬[ℎ#
F,b
] = 0.
[Theorem 1.]
If the past fades away, then
limF→„ 𝑬 𝑥F
= 𝒙„
where 𝒙„
= 𝑳𝒃, L= 𝑰 − 𝑨𝑴 Œ{
𝑰 − 𝑨
Otherwise 𝑬 𝑥F might not converge, or converge but not to 𝒙„.

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Analysis:
user’s influence on the market
n 𝑳 ∶= 𝑰 − 𝑨𝑴 Œ{
𝑰 − 𝑨 (2)
l 𝐿=# is how much u influences v.
l So, ∑ 𝐿Ž#Ž quantifies u’s influence on the entire market.
n We collect these individual market influences :
𝜸• ≔ 𝒇• 𝑳 = (𝛾{, ⋯ , 𝛾| ) (3)
where f = (𝑓{, 𝑓•, … , 𝑓|)
n The larger value of γ”, the greater influence of u on the final market
share of p*.

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Analysis:
market share of p*
l with recommender
n 𝒇 • 𝒙„
= 𝜸 • 𝒃
l without recommender
n 𝒇 • 𝒃
Define market distortion as:
△≔
𝒇•𝒙—
𝒇•𝒃
=
𝜸•𝒃
𝒇•𝒃
(5)
no RS with RS
Market share of p*
f・b 𝛾・b

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Analysis:
computing 𝛾
n 𝑳 ∶= 𝑰 − 𝑨𝑴 Œ{
𝑰 − 𝑨 (2)
n 𝜸•
≔ 𝒇•
𝑳 (3)
l To compute 𝛾 directly through (2) and (3) is expensive
l So rewrite 𝑰 − 𝑨𝑴 Œ{
as ∑ 𝑨𝑴 b˜„
b™š
n 𝜸•
= 𝒇•
𝑳 = 𝑓• ∑ 𝑨𝑴 b
(𝑰 − 𝑨)˜„
b™š (9)
l Using this throughout experiments

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Experiments

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Experiments:
data sets
l Google+ : 107,614 nodes、 13,673,453 arcs
l Twitter SNAP : 81,306 nodes、 2,420,766 arcs
l Twitter LAW : 41,652,230 nodes、 1,468,365,182 arcs
l Slashdot : 82,144 nodes、 425,072 arcs
l Yelp : 365,759 nodes、 1,288,031 edges
l Facebook : 63,731 nodes、 817,090 edges

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Experiment:
parameters
n H𝑜𝑤 𝑡𝑜 𝑠𝑒𝑡 f, w, h, 𝛼? : follow Basic Scenario(Definition1.)
l f: who buys? → 𝑓# =
{
|
l w: whose recommendation? → 𝑤=# =
{
b|œ•ž(#)
l h: ℎ#
F,Œb
= ℎ#
F,b
=
{
Ÿv˜ v
x
l 𝛼: 0.2

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Experiment1:
n This experiment checks that the purchasing process unrolls
as predicted
l Remaining products are coalesced into one]
n Simulating the purchasing process for 10,000n steps

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Result:
𝒙F
converges to 𝒙„
.

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Experiment2:
b
n b was instantiated with different types of distributions
l uniform in [0,1]
l exponential of mean ½
l a power-law of exponent 0.01
l normal of mean 1/2 and standard deviation 1/6

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Result:
b
n Market distortion doesn’t change so much
n In all cases the value of △ was always in the range 1±0.002
n social graphs prevent the recommender from distorting the
market.

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Experiment3:
insert a super-node
n Insert a super-node to G.
l It points to every user and always recommends p*.
…...
supernode
Users

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Experiment3: result
when there is a supernode
Δ ∶ 105.3 % 〜 149.7 %

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Conclusion

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n They modeled the recommender system by introducing natural
model for markets
n According to experiments,
l real-world social graphs prevent the recommender from distorting
the market.
l When we insert a super-node to the graph, the market is distorted.
Conclusion: