Influential People

Analysis of Boolean FunctionsandComplexity TheoryEconomicsCombinatoricsEtc.Slides prepared with help of Ricky Rozen

Influential People • The theory of the Influence of Variables on Boolean Functions[KKL,BL,R,M], has been introduced to tackle Social Choice problems and distributed computing. • It has motivated a magnificent body of work, related to • Sharp Threshold [F, FG] • Percolation[BKS] • Economics: Arrow’s Theorem[K] • Hardness of Approximation[DS] Utilizing Harmonic Analysis of Boolean functions… • And the real important question:

Where to go for Dinner? The alternatives Diners would cast their vote in an (electronic) envelope The system would decide –not necessarily according to majority… And what ifsomeone(in Florida?)can flipsome votes influence Power

Boolean Functions • Def: ABoolean function Power set of [n] Choose the location of -1 Choose a sequence of -1 and 1

Noise Sensitivity • The values of the variables may each, independently, flip with probability  • It turns out: one cannot design an f that would be robust to such noise --that is, would, on average, change value w.p. < O(1)-- unless determining the outcome according to very few of the voters

Voting and influence • Def: theinfluence ofi on f is the probability, over a random input x, that f changes its value when i is flipped -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 1 -1

-1 1 -1 -1 ? 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 • Majority:{1,-1}n {1,-1} • Theinfluence of i on Majority is the probability, over a random input x, Majority changes with i • this happens when half of the n-1 coordinate (people) vote -1 and half vote 1. • i.e.

-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 • Parity: {1,-1}n {1,-1} Always changes the value of parity

-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 • Dictatorshipi:{1,-1}20 {1,-1} • Dictatorshipi(x)=xi • influence of i on Dictatorshipi= 1. • influence of ji on Dictatorshipi=0.

Average Sensitivity • Def: theAverage Sensitivity off(as) is the sum of influences of all coordinates i  [n] : • as(Majority) = n½ • as(Parity) = n • as(dictatorship) =1

When as(f)=1 Def: f is abalancedfunction if it equals -1 exactly half of the times:Ex[f(x)]=0 Can a balanced f have as(f) < 1? What about as(f)=1? Beside dictatorships? Prop: f isbalancedandas(f)=1f is adictatorship.

Representing f as a Polynomial • What would be the monomials over x  P[n] ? • All powers except 0 and 1 cancel out! • Hence, one for each characterS[n] • These are all the multiplicative functions

Fourier-Walsh Transform • Consider all characters • Given any functionlet the Fourier-Walsh coefficients of f be • thus f can be described as

Norms Def:Expectation norm on the function Def:Summation norm on the transform Thm [Parseval]: Hence, for a Boolean f

Distribution over Characters • We may think of the Transform as defining a distribution over the characters.

SimpleObservations • Claim: • For any function f whose range is {-1,0,1}:

Variables` Influence • Recall: influence of an index i [n] on a Boolean function f:{1,-1}n {1,-1} is • Which can be expressed in terms of the Fourier coefficients of fClaim: • And the as:

Fourier Representation of influence Proof: consider the influence function which in Fourier representation is and

If s s.t |s|>1 and then as(f)>1 Balanced f s.t. as(f)=1 is Dict. • Since f is balanced and • So f is linear • For any i s.t. Only i has changed

Expectation and Variance • Claim: • Hence, for any f

First Passage Percolation [BKS] Each edge costs a w/probability ½ and bw/probability½

First Passage Percolation • Consider the Grid • For each edge e of chooseindependentlywe = 1 or we = 2, each with probability ½ • This induces a shortest-path metric on • Thm: The variance of the shortest path from the origin to vertex v is bounded from above by O( |v|/ log |v|) [BKS] • Proof idea: The average sensitivity of shortest-path is bounded by that term

Proof outline • LetGdenote the grid • SPG– the shortest path in G from the origin to v. • Let denote the Grid which differ from G only on we i.e. flip the value of e in G. • Set

Observation If e participates in a shortest path then flipping its value will increase or decrease the SP in 1 ,if e is not in SP - the SP will not change.

Proof cont. • And by [KKL] there is at least one variable whose influence is at least (logn/n)

Graph properties Def: A graph property is a subset of graphs invariant under isomorphism. Def: a monotone graph property is a graph property P s.t. • If P(G) then for every super-graph H of G (namely, a graph on the same set of vertices, which contains all edges of G) P(H) as well. In other words:P: {-1, 1}V2{-1, 1}

Examples of graph properties • G is connected • G is Hamiltonian • G contains a clique of size t • G is not planar • The clique number of G is larger than that of its complement • The diameter of G is at most s • ... etc . • What is the influence of different e on P?

Erdös–Rényi G(n,p)Graph TheErdös-Rényidistribution of random graphs Put an edge between any two vertices w.p.p

Definitions • P – a graph property • (P) - the probability that a random graph on n vertices with edge probability p satisfies P. • GG(n,p) - G is a random graph of n vertices and edge probability p.

Example-Max Clique Probability for choosing an edge • Consider GG(n,p) • The size of the interval of probabilities pfor which the clique number of Gis almost surely k(where k  log n) is of order log-1n. • The threshold interval: The transition between clique numbers k-1 and k. Number of vertices

The probability of having a clique of size k is 1- The probability of having a clique of size k is  • The probability of having a (k + 1)-clique is still small (log-1n). • The value of pmust increase byclog-1n before the probability for having a (k + 1)-clique reaches and another transition interval begins.

Def: Sharp threshold • Sharp threshold in monotone graph property: • The transition from a property being very unlikely to it being very likely is very swift. G satisfies property P GDoes not satisfies property P

Thm: every monotone graph property has a Sharp Threshold[FK] • Let P be any monotone property of graphs on n vertices . If p(P) >  then q(P) > 1- for q=p + c1log(½)/logn Proof idea: show asp’(P), for p’>p, is high

Thm [Margulis-Russo]: For monotonef HenceLemma:For monotonef > 0,  q[p, p+] s.t. asq(f)  1/ Proof: Otherwise p+(f) > 1

Proof [Margulis-Russo]:

Mechanism Design Problem • Nagents, each agent i hasprivateinput tiT. All other information ispublicknowledge. • Each agent i has avaluationfor all items: Each agent wishes to optimize her own utility. • Objective: minimize the objective function, the total payment. • Means: protocol between agents and auctioneer.

Vickery-Clarke-Groves (VCG) • Sealed bid auction • A Truth Revealing protocol, namely, one in which each agent might as well reveal her valuation to the auctioneer • Whereby each agent gets the best (for her) price she could have bid and still win the auction

Shortest Path using VGC • Problem definition: • Communication networkmodeled by a directed graph G and two vertices source s and target t. • Agents= edges in G • Each agent has a cost for sending a single message on her edge denote by te. • Objective: find the shortest (cheapest) path from s to t. • Means: protocol between agents and auctioneer.

10$ 10$ 50$ 50$ VCG for Shortest-Path Always in the shortest path

How much will we pay? • SP • Every agent will get 1$ more. 1$ 2$ 2$ 1$ 1$ 1$ 1$ 1$ 2$ 1$ 2$ 1$ 1$ 2$

-1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 Juntas • A function is a J-junta if its value depends on only J variables. 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 • A Dictatorship is 1-junta -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

[n] [n] I I z x Noise-Sensitivity How often does the value of f changes when the input is perturbed? [n] [n] I I z x

[n] [n] I I z x Noise-Sensitivity • Def(,p,x[n] ): Let 0<<1, and xP([n])Then y~,p,x, if y = (x\I) z where • I~[n] is a noise subset, and • z~ pI is a replacement. Def(-noise-sensitivity): let 0<<1, then [ When p=½ equivalent to flipping each coordinate in x independently w.p. /2.]

Noise-Sensitivity – Cont. • Advantage: very efficiently testable (using only two queries) by a perturbation-test. • Def(perturbation-test): choose x~p, and y~,p,x, check whether f(x)=f(y)The success is proportional to the noise-sensitivity of f. • Prop: the -noise-sensitivity is given by

Relation between Parameters Prop: small ns small high-freq weight Proof: therefore: if ns is small, then Hence the high frequenciesmust have small weights (as ). Prop: small as small high-freq weight Proof:

High vs. Low Frequencies Def: The section of a function f above k is and the low-frequency portion is

Low-degree B.f are Juntas [KS] Theorem:  constant >0 s.t. any Boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k) Corollary: fix a p-biased distribution p overP([n])Let >0 be any parameter. Set k=log1-(½)Then  constant >0 s.t. any Boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k)

Freidgut Theorem Thm: any Boolean f is an [, j]-junta for Proof: • Specify the junta J • Show the complement ofJ has little influence

Long-Code In the long-code the set of legal-words consists of all monotone dictatorships This is the most extensive binary code, as its bits represent all possible binary values over n elements

Long-Code • Encoding an element e[n]: • Eelegally-encodes an element e if Ee = fe T F F T T

Influential People