Subsampling Graphs 1 RECAP of PageRank- - PowerPoint Presentation

Slide1

Subsampling Graphs

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RECAP of PageRank-NIbble

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Why I’m talking about graphs

Lots of large data

is

graphs

Facebook, Twitter, citation data, and other

social

networks

The web, the blogosphere, the semantic web, Freebase, Wikipedia, Twitter, and other information networksText corpora (like RCV1), large datasets with discrete feature values, and other bipartite networksnodes = documents or wordslinks connect document  word or word  documentComputer networks, biological networks (proteins, ecosystems, brains, …), …Heterogeneous networks with multiple types of nodespeople, groups, documents

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Our first operation on graphs

Local graph partitioning

Why?

it’s about the best we can do in terms of subsampling/exploratory analysis of a graph

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What is Local Graph Partitioning?

Global

Local

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What is Local Graph Partitioning?

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Main results of the paper (Reid et al)

An

approximate

personalized

PageRank computation that only touches nodes “near” the seed

but has small error relative to the true PageRank vector

A proof that a sweep over the approximate personalized PageRank vector finds a cut with conductance sqrt(α ln m)unless no good cut existsno subset S contains significantly more mass in the approximate PageRank than in a uniform distribution7Slide8

Approximate PageRank: Key Idea

p

is current approximation (start at

0

)

r

is set of “recursive calls to make”

residual errorstart with all mass on su is the node picked for the next call

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ANalYsis Of

PageRank-NIbble

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Analysis

linearity

re-group & linearity

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Approximate PageRank: Key Idea

By definition PageRank is fixed point of:

Claim:

Proof:

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Approximate PageRank: Algorithm

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Analysis

So, at every point in the

apr

algorithm:

Also, at each point, |

r|

1 decreases by α*

ε

*degree(

u

), so: after T push operations where degree(

i-th

u)=d

i

, we know

which bounds the size of

r

and

p

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Analysis

With the invariant:

This bounds the error of

p

relative to the PageRank vector.

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Comments – API

p,r

are hash tables – they are small (1/

ε

α)

push just needs

p, r,

and neighbors of

u

Could implement with API:

List<Node> neighbor(Node

u)

int

degree(Node

u

)

d(v)

=

api.degree

(

v

)

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Comments - Ordering

might pick the

largest

r(u)/d(u) …

or…

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Comments – Ordering for Scanning

Scan

repeatedly

through an adjacency-list encoding of

the graph

For every line you read

u, v1,…,v

d

(u)

such that

r(u)/d(u) >

ε

:

benefit: storage is O(

1

/

ε

α

) for the hash tables, avoids any

seeking

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Possible optimizations?

Much faster than doing random access the first few scans,

but then

slower the last few

…there

will be only a few ‘pushes’ per

scan

Optimizations you might imagine:Parallelize?Hybrid seek/scan:Index the nodes in the graph on the first scanStart seeking when you expect too few pushes to justify a scanSay, less than one push/megabyte of scanningHotspots:Save adjacency-list representation for nodes with a large r(u)/d(u) in a separate file of “hot spots” as you scanThen rescan that smaller list of “hot spots” until their score drops below threshold.

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After computing apr

…

Given a graph

that’s too big for memory, and/or

that’s only accessible via API

…we can extract a

sample

in an interesting areaRun the apr/rwr from a seed nodeSweep to find a low-conductance subsetThencompute statisticstest out some ideasvisualize it…19Slide20

Key idea: a “sweep”

Order vertices v

1

, v

2

, …. by highest

apr

(s)Pick a prefix S={ v1, v2, …. vk }:S={}; volS =0; B={

}

For k=1,…n:

S += {

v

k

}

volS

+= degree(

v

k

)

B += {u:

v

k

u

and u not in S}

Φ

[k] = |B|/

volS

Pick k:

Φ

[k]

is smallest

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Scoring a graph prefix

the edges leaving S

vol

(S) is sum of

deg

(x) for x in S

for small S:

Prob(random

edge

leaves S)

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Putting this together

Given a graph

that’s too big for memory, and/or

that’s only accessible via API

…we can extract a

sample

in an interesting area

Run the apr/rwr from a seed nodeSweep to find a low-conductance subsetThencompute statisticstest out some ideasvisualize it…

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Visualizing a Graph with Gephi

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Screen shots/demo

Gelphi

– java tool

Reads several inputs

.

csv

file

[demo]24Slide25

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