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Searching and Sampling Take a Walk Through a Network! Antonio - - PowerPoint PPT Presentation

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Searching and Sampling Take a Walk Through a Network! Antonio Carzaniga Faculty of Informatics Universit della Svizzera italiana Mach 4, 2020 Outline Applications The network as a linear transformation Other applications of linear algebra

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Searching and Sampling

Take a Walk Through a Network! Antonio Carzaniga

Faculty of Informatics Università della Svizzera italiana

Mach 4, 2020

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Outline

Applications The network as a linear transformation Other applications of linear algebra

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v

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v

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v a very limited local view of the network

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v local view Networks

◮ peer-to-peer ◮ . . .

Services

◮ address-based ◮ content-based ◮ multicast ◮ search ◮ sampling ◮ . . .

Algorithms

◮ random walks ◮ . . .

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 9 remaining hops: 9

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 8 remaining hops: 8

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 7 remaining hops: 7

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 6 remaining hops: 6

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 5 remaining hops: 5

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 4 remaining hops: 4

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 3 remaining hops: 3

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 2 remaining hops: 2

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 1 remaining hops: 1

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

remaining hops: 0! remaining hops: 0!

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GE VS TI VD FR BE OW NW UR GR NE LU ZG SZ GL SG JU SO BS BL AG SH ZH TG AR AL

node SG selected node SG selected

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Other Applications

Relevance score for hyper-linked documents (PageRank)

◮ Input: a large collection of linked documents such as Web

pages

◮ Output: a ranking of the pages by reputation ◮ a page that is linked by reputable pages acquires more

reputation

◮ equivalent to a random walk over the Web

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Problem: given a directed graph G = (V, A), compute the probability pu that a sufficiently long random walk would end at node u ∈ V for all nodes u.

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Problem: given a directed graph G = (V, A), compute the probability pu that a sufficiently long random walk would end at node u ∈ V for all nodes u. Approaches:

  • 1. Simulation
  • 2. Math! (linear algebra)
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v local view Random Walks

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v

0.2 0.2 0.5 0.1

local view Random Walks Execution

◮ trivial, local process

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v

0.2 0.2 0.5 0.1

local view Random Walks Execution

◮ trivial, local process

Configuration and bias

◮ how do we choose

transition probabilities?

◮ is the sample biased?

How?

◮ how long do we walk?

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AL AG AR BE BL BS FR GE GL GR JU LU NE NW OW SG SH SO SZ TG TI UR VD VS ZG ZH

stationary distribution (hops → ∞)

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v1 v2 v3

1 1 0.5 0.5

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t)

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0)

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0)

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

A is stochastic: 1 = |λ1| > |λ2| ≥ |λ3| ≥ . . .

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

λ1 = 1 λ2,3 = −0.5 ± 0.5i

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

λ1 = 1 λ2,3 = −0.5 ± 0.5i π

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

λ1 = 1 λ2,3 = −0.5 ± 0.5i π ǫt ≈ |λ2|t → 0

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

π ǫt ≈ |λ2|t → 0

Stationary distribution

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v1 v2 v3

1 1 0.5 0.5

let pi(t) = Pr[walk is at node vi at time t] p(0) = [0 1 0]T means the walk starts at v2 p1(t + 1) = 0.5 · p3(t) p2(t + 1) = p1(t) + 0.5 · p3(t) p3(t + 1) = p2(t) p(t + 1) = Ap(t) p(t) = Atp(0) p(0) = c1x1 + c2x2 + · · · + cnxn p(t) = Atp(0) = λt

1c1x1 + λt 2c2x2 + · · · + λt ncnxn

π ǫt ≈ |λ2|t → 0

Stationary distribution Mixing Time: τ ≈ log|λ2 | ǫ s.t.

ǫt < ǫ for t > τ

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Notice that, if the network is ergodic, then we know for sure that there is a stationary distribution π that satisfies the equation

π = Aπ.

So, we can compute the stationary distribution directly by solving this system of equations: Aπ = π

n

  • i=1

πi = 1