TRUST-BASED RECOMMENDATION SYSTEMS

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TRUST-BASED RECOMMENDATIONSYSTEMS

• an axiomatic approach

Microsoft Research, Redmond WA
Christian Borgs
Reid Anderson
Adam Kalai
Vahab Mirrokni
Uri Feige
Moshe Tennenholtz
Jennifer Chayes
Abie Flaxman
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TRUST, REC RANKING SYSTEMS

• What is the right model?

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OLD-FASHIONED MODEL

• I want a recommendation about an item, e.g.,
• Professor
• Product
• Service
• Restaurant
• I ask my trusted friends
• Some have a priori opinions (first-hand
  experience)
• Others ask their friends, and so on
• I form my own opinion based on feedback, which I
  may pass on to others as a recommendation

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OUR MODEL

• Trust graph
• Node set N, one node per agent
• Edge multiset E µ N2
• Edge from u to v means u trusts v
• Multiple parallel edges indicate more trust
• Votes disjoint V, V µ N
• V is set of agents that like the item
• V is set of agents that dislike the item
• Rec. system (software) assigns ,0,
  rec.Rs(N,E,V,V) to each nonvoter s 2
  Nn(VV)

0

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FAMOUS VOTING NETWORKS











ME voters
AL voters
WY voters





electoral college
AL (9)
ME (4)
WY (3)
congress

• U.S. presidential election majority-of-majorities
  system

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OUTLINE

• Trust-based recommendation systems
• Our voting network model
• Our approach the axiomatic approach
• Previously used separately for voting and
  ranking systems (e.g., AltmanTennenholtz05)
• We give three theorems
• An axiomatization ? random walk system
• Variation of above (transitivity) leads to
  impossibility
• An axiom generalizes majority-of-majorities to
  min-cut system on undirected graphs
• Future directions

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RANDOM WALK SYSTEM

• Input voting network, source (nonvoter) s.
• Consider hypothetical random walk
• Start at s
• Follow random edges
• Stop when you reach a voter
• Let ps Prwalk stops at voter
• Let qs Prwalk stops at voter (psqs1)
• Output rec. for s

0



0
if ps gt qs 0 if ps qs if ps lt qs


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AXIOMATIZATION 1

• 1. Symmetry
• Neutrality flipping vote signs flips rec signs
  8(N,E,V,V) 8s2Nn(VV) Rs(N,E,V,V)
  Rs(N,E,V,V)
• Anonymity Isomorphic graphs have isomorphic
  recs
• 2. Positive response
• If ss rec is 0 or and an edge is added to a
  brand new voter, then ss rec becomes

0

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AXIOMATIZATION 1

• 1. Symmetry
• 2. Positive response
• 3. Scale invariance (edge repl.)
• Replicating a node's outgoing edges k times
  doesnt change any recs.
• 4. Independence of Irrelevant Stuff
• A node's rec is independent of unreachable nodes
  and edges out of voters.
• 5. Consensus nodes
• If u's neighbors unanimously vote , and they
  have no other neighbors, then us may be taken to
  vote , too.

?

s

r

u
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AXIOMATIZATION 1

• 1. Symmetry
• 2. Positive response
• 3. Scale invariance (edge repl.)
• 4. Independence of Irrelevant Stuff
• 5. Consensus nodes
• 6. Trust Propogation
• If u trusts (nonvoter) v, then an equal number of
  edges from u to v can be replaced directly by
  edges from u to the nodes that v trusts (without
  changing any recs).

?

s
v
u
?
THM Axioms 1-6 are satisfied uniquely by random
walk system.
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AXIOMATIZATION 2

• 1. Symmetry
• 2. Positive response
• 3. Scale invariance (edge repl.)
• 4. Independence of Irrelevant Stuff
• 5. Consensus nodes
• 6. Trust Propogation
• Def s trusts A more than B in (N,E) if (VA
  and V B) ) ss rec is
• 7. Transitivity (Disjoint A,B,C µ N)
• If s trusts A more than B and
• s trusts B more than C then
• s trusts A more than C

s
A
B


THM 2 Axioms 1-2, 4-5, and 7 are a minimal
inconsistent set of axioms.
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AXIOMATIZATION 3

• Majority Axiom
• The rec. for a node is equal to the majority of
  the votes/recommendations of its trusted
  neighbors.

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GROUPTHINK

• No Groupthink Axiom
• If a set S of nonvoters are all recs, then a
  majority of the edges from S to N \ S are to
  voters or recs
• If a set S of nonvoters are all or 0 recs,
  then it cannot be that a majority of the edges
  from S to N \ S are to voters or recs

(and symmetric conditions)




THM 3 The No groupthink axiom uniquely
implies the min-cut system


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MIN-CUT SYSTEM

• (Undirected graphs only)
• Def A cut is a subset of edges that, when
  removed, leaves no path between / voters

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MIN-CUT SYSTEM




0

• (Undirected graphs only)
• Def A cut is a subset of edges that, when
  removed, leaves no path between / voters
• Def A mincut is a cut of minimal size
• The rec for node s is if in every mincut s
  is connected to a voter, if in every mincut
  s is connected to a voter,0 otherwise

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OPEN PROBLEM

• The no-groupthink axiom is impossible to satisfy
  on general undirected graphs. ?
• What is the right axiom that generalizes the
  majority-of-majorities?
• Starting idea
• Consistency axiom
• If a node has rec, then we can assign it vote
  without changing other recs.
• Open Problem Find a natural system obeying
  consistency ( symmetry, etc.) on directed
  graphs?

0


0
0
0

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BONUSINCENTIVE COMPATIBILITY

• To maximally influence a recommendation to , a
  group of voters might try to
• Misrepresent trust links amongst themselves.
• Create millions of new nodes with arbitrary votes
  and arbitrary trust links amongst this larger
  set.
• It turns out that This is no more effective than
  simply all voting
• This type of incentive compatibility holds for
  all of our systems.

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Conclusions

• Simple voting network model of trust-based rec
  systems
• Simplify matters by rating one item (at a time)
• Generalizes to real-valued weights, votes recs
• Two axiomatizations leading to unique sysetms
• Random walk system for directed graphs
• Min-cut system for undirected graphs (generalizes
  US presidential election system)
• One impossibility theorem
• Future work find other nice systems/axioms