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Algorithmic and Economic Aspects of Networks
Nicole Immorlica
Beliefs in Social Networks
Given that we influence each other’s beliefs,- will we agree or remain divided?- who has the most influence over our beliefs?- how quickly do we learn?- do we learn the truth?
Observational Learning
Key Idea: If your neighbor is doing better than you are, copy him.
Bayesian Updating Model
n agentsconnected in a social networkat eachtime t= 1, 2, …, each agent selects an action from a finite setpayoffsto actions are random and depend on the state of nature
Agent Goal
maximize sum of discounted payoffs∑t > 0δt∙πitwhereδ< 1is discount factor andπitis payoff to i at time t.
Example
Two actionsaction Ahas payoff 1action Bhas payoff 2 with probability p and 0 with probability (1-p)If p > ½, agents prefer B, else agents prefer A.
Example
Agents havebeliefsμi(pj)representing probability agent i assigns to event that p =pj.Multi-armed bandit…with observations.
Example
B: 0
B: 0
A: 1
B: 2
A: 1
B: 2
B: 0
B: 0
Center agent, Day 0:Pr[p=1/3] = 0, Pr[p=2/3] = 1Play action B, payoff 0
Center agent, Day 1:Pr[p=1/3] > 0, Pr[p=2/3] < 1Play action A, payoff 1
Center agent, Day 2:Now must take into account “echoes” for optimal update
B: 2
A: 1
A: 1
B: 0
A: 1
A: 1
B: 0
A: 1
Example
Ignoring echoes,Theorem [Bala and Goyal]: With prob. 1, all agents eventually play the same action.Proof: By strong law of large numbers, if B is played infinitely often, beliefs converge to correct probability.
Example
Note, all agents play same action, but- don’t necessarily have same beliefs- don’t necessarily pick “right” action ** unless someone is optimistic about B
Imitation and Social Influence
At time t, agent i has anopinionpi(t)in [0,1].Letp(t) = (p1(t), …,pn(t))be vector of opinions.Matrix Trepresents interactions:
T11T12T13
T21T22T23
T31T32T33
How much agent 2 believes agent 1
Rows sum to 1
Updating Beliefs
Update rule: p(t) = T ∙ p(t-1)
T11T12T13
T21T22T23
T31T32T33
p1(t-1)p2(t-1)p3(t-1)
T11p1(t-1)T12p1(t-1)T13p1(t-1)
T21p2(t-1)T22p2(t-1)T23p2(t-1)
T31p3(t-1)T32p3(t-1)T33p3(t-1)
Example
1/31/31/3
1/21/20
01/43/4
Example
2
1
3
1/3
1/3
1/3
1/2
1/2
1/4
3/4
Example
Suppose p(0) = (1, 0, 0). Thenp(1) = T p(0) = = (1/3, 1/2, 0)p(2) = T p(1) = (5/18, 5/12, 1/8)p(3) = T p(2) = (0.273, 0.347, 0.198)p(4) = T p(3) = (0.273, 0.310, 0.235)… p(∞) (0.2727, 0.2727, 0.2727)
1/31/31/3
1/21/20
01/43/4
100
Incorporating Media
Media is listened to by (some) agents, but not influenced by anyone.Represent media by agent i withTii= 1,Tij= 0 for j not equal to i. Media influences agents k for whichTki> 0.
Converging Beliefs
When does process have a limit?Note p(t) = T p(t-1) =T2p(t-2) = … =Ttp(0).Process converges whenTtconverges.Final influence weights are rows ofTt.
Example
01/21/2
100
010
t
2/52/51/5
2/52/51/5
2/52/51/5
Example
01/21/2
100
100
Does not converge!
Example
01/21/2
100
100
1/2
1/2
1
1
Aperiodic
Definition. T is aperiodic if the gcd of allcycle lengths is one (e.g., if T has a selfloop).
Convergence
T is aperiodic and strongly connectedT converges
(standard results inMarkov chain theory)
Everyone should trust themselves a little bit.
Can be relaxed, see book.
Consensus
For any aperiodic matrix T, any “closed” and strongly connected group reaches consensus.
Social Influence
We look for a unit vector s = (s1, …,sn) such thatp(∞) = s ∙ p(0)Then s would be the relative influences of agents in society as a whole.
Social Influence
Note p(0) & T p(0) have same limiting beliefs, sos ∙ p(0) = s ∙ (T p(0))And since this holds for every p, it must be thats T = s
Social Influence
The vector s is an eigenvector of T with eigenvalue one.If T is strongly connected, aperiodic, and has rows that sum to one, then s is unique.Another interpretation: s is the stationary distribution of the random walk.
Computing Social Influence
Sinces ∙ p(0) = p(∞) = T∞∙ p(0)it must be that each row of T converges to s.
Who’s Influential?
Note, since s is an eigenvector,si=Tjisj, so an agent has high influence if they are listened to by influential people.
PageRank
Compute influence vector on web graph and return pages in decreasing order of influence.- each page seeks advice from all outgoing links (equally)- add restart probabilities to make strongly connected- add initial distribution to bias walk
Time to Convergence
If it takes forever for beliefs to converge, then we may never observe the final state.
Time to Convergence
Two agents1. similar weightings (T11~T21) implies fast convergence2. different weightings (T11>>T21) implies slow convergence
Diagonal Decomposition
Want to explore how farTtis from T∞Rewrite T in its diagonal decomposition soT =u-1Λufor a matrix u and adiagonal matrixΛ.1. Compute eigenvectors of T2. Let u be matrix of eigenvectors3. LetΛbe matrix of eigenvalues
Exponentiation
NowTtbecomes:(u-1Λu) (u-1Λu) … (u-1Λu)=u-1ΛtuandΛtis diagonal matrix, so easy exponentiate.
Speed of Convergence
1 0
0T11–T12
1 0
0 (T11–T12)t
t
Since (T11- T12) < 1, (T11- T12)tconverges to zero.Speed of convergence is related to magnitute of 2ndeigenvalue,… and to how different weights are.
More Agents
Speed of convergence now relates to how much groups trust each other.
Finding the Truth
When do we converge to the correct belief?
Assume Truth Exists
There is aground truthμ.There aren agents(to make formal, study sequence of societies with n∞).Each agent has asignalpi(0)distributed with meanμand varianceσi2.
Wisdom
Definition. Networks arewiseif p(∞) converges toμwhen n is large enough.
Truth Can Be Found
By law of large numbers, averaging all beliefs with equal weights converges to truth.Sufficient: agents have equal influence.
Necessary Conditions
Necessary that- no agent has too much influence- no agent has too much relative influence- no agent has too much indirect influence
1-δ
1-δ
1-δ
1-δ
1-δ
1-δ
δ
δ
δ
δ
δ
δ
Sufficient Conditions
Sufficient that the society exhibits-balance: a smaller group of agents does not get infinitely more weight in from a larger group than it gives back-dispersion: each small group must give some minimum amount of weight to larger groups
Assignment:
Readings:Social and Economic Networks, Chapter 8PageRankpapersReaction to paperPresentation volunteer?