Local Sparsification for Scalable Module Identification in Networks

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Local Sparsification for Scalable Module Identification in

NetworksSrinivasan Parthasarathy

Joint work with V. Satuluri, Y. Ruan, D. Fuhry, Y. Zhang

Data Mining Research LaboratoryDept. of Computer Science and

EngineeringThe Ohio State University

“Every 2 days we create as much information as we did up

to 2003”- Eric Schmidt, Google ex-CEO

The Data Deluge

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600$ to buy a disk drive that can store

all of the world’s music[McKinsey Global Institute Special Report, June ’11]

Data Storage Costs are Low

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Data does not exist in isolation.

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Data almost always exists in connection with other data.

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Social networksProtein Interactions Internet

VLSI networks Data dependencies Neighborhood graphs

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All this data is only useful if we can scalably extract useful knowledge

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Challenges

1. Large Scale

Billion-edge graphs commonplace

Scalable solutions are needed

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Challenges

2. Noise

Links on the web, protein interactions

Need to alleviate

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Challenges

3. Novel structure

Hub nodes, small world phenomena, clusters of varying densities and

sizes, directionality

Novel algorithms or techniques are needed

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Challenges

4. Domain Specific Needs

E.g. Balance, Constraints etc.

Need mechanisms to specify

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Challenges

5. Network Dynamics

How do communities evolve? Which actors have influence? How do clusters change as a function of external factors?

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Challenges

6. Cognitive Overload

Need to support guided interaction for human in the

loop

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Our Vision and Approach

Graph Pre-processing

SparsificationSIGMOD ’11, WebSci’12

Near Neighbor SearchFor non-graph data

PVLDB ’12

SymmetrizationFor directed graphs

EDBT ’10

Core Clustering

Consensus Clustering KDD’06, ISMB’07

Viewpoint Neighborhood Analysis

KDD ’09

Graph Clustering via Stochastic FlowsKDD ’09, BCB ’10

Dynamic Analysis and Visualization

Event Based Analysis KDD’07,TKDD’09

Network VisualizationKDD’08

Density PlotsSIGMOD’08, ICDE’12

Scalable Implementations and Systems Support on Modern Architectures

Multicore Systems (VLDB’07, VLDB’09), GPUs (VLDB’11), STCI Cell (ICS’08),

Clusters (ICDM’06, SC’09, PPoPP’07, ICDE’10)

Application Domains

Bioinformatics (ISMB’07, ISMB’09, ISMB’12, ACM BCB’11, BMC’12)Social Network and Social Media Analysis (TKDD’09, WWW’11, WebSci’12,

WebSci’12)

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Graph Sparsification for Community Discovery

SIGMOD ’11, WebSci’12

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Is there a simple pre-processing of the graph to reduce the edge set

that can “clarify” or “simplify” its cluster structure?

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Given a graph, discover groups of nodes that are strongly connected to one another but weakly connected to the rest of the graph.

Graph Clustering and Community Discovery

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Social Network and Graph Compression

Direct Analytics on compressed representation

Graph Clustering : Applications

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Optimize VLSI layout


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Protein function prediction


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Data distribution to minimize communication and balance load


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Is there a simple pre-processing of the graph to reduce the edge set

that can “clarify” or “simplify” its cluster structure?

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Preview

Original Sparsified

[Automatically visualized using Prefuse]

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The promise

Clustering algorithms can run much faster and be more accurate on a

sparsified graph.

Ditto for Network Visualization

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Utopian Objective

Retain edges which are likely to be intra-cluster edges, while

discarding likely inter-cluster edges.

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A way to rank edges on “strength” or similarity.

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Algorithm: Global Sparsification (G-Spar)

Parameter: Sparsification ratio, s

1. For each edge <i,j>:(i) Calculate Sim ( <i,j> )

2. Retain top s% of edges in order of Sim, discard others

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Dense clusters are over-represented, sparse clusters under-represented

Works great when the goal is to just find the top communities

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Algorithm: Local Sparsification (L-Spar)

Parameter: Sparsification exponent, e (0 < e < 1)

1. For each node i of degree di:(i) For each neighbor j:

(a) Calculate Sim ( <i,j> )(ii) Retain top (d i)e

neighbors in order of Sim, for node i

Underscoring the importance of Local Ranking

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Ensures representation of clusters of varying densities

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But...

Similarity computation is expensive!

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A randomized, approximate solution based on

Minwise Hashing[Broder et. al., 1998]

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Minwise Hashing

{ dog, cat, lion, tiger, mouse}

[ cat, mouse, lion, dog, tiger]

[ lion, cat, mouse, dog, tiger]

Universe

A = { mouse, lion }

mh1(A) = min ( { mouse, lion } ) = mouse

mh2(A) = min ( { mouse, lion } ) = lion

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Key Fact

For two sets A, B, and a min-hash function mhi():

Unbiased estimator for Sim using k hashes:

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Time complexity using Minwise Hashing

EdgesHashes

Only 2 sequential passes over input. Great for disk-resident data

Note: exact similarity is less important – we really just care about relative ranking lower k

Theoretical Analysis of L-Spar: Main Results

• Q: Why choose top de edges for a node of degree d?

• A: Conservatively sparsify low-degree nodes, aggressively sparsify hub nodes. Easy to control degree of sparsification.

• Proposition: If input graph has power-law degree distn. with exponent , then sparsified graph also has power-law degree distn. with exponent

• Corollary: The sparsification ratio corresponding to exponent e is no more than

• For = 2.1 and e = 0.5, ~17% edges will be retained.

• Higher (steeper power-laws) and/or lower e leads to more sparsification.

e

e 1

1

2

e

Experiments• Datasets

• 3 PPI networks (BioGrid, DIP, Human)

• 2 Information (Wiki, Flickr) & 2 Social (Orkut , Twitter) networks

• Largest network (Orkut), roughly a Billion edges

• Ground truth available for PPI networks and Wiki

• Clustering algorithms

• Metis [Karypis & Kumar ‘98], MLR-MCL [Satuluri & Parthasarathy, ‘09], Metis+MQI [Lang & Rao ‘04], Graclus [Dhillon et. al. ’07], Spectral methods [Shi ’00], Edge-based agglomerative/divisive methods [Newman ’04]

• Compared sparsifications

• L-Spar, G-Spar, RandomEdge and ForestFire

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Dataset Dataset ((n, mn, m))

SparsSpars. .

RatioRatio

RandomRandom G-SparG-Spar L-SparL-Spar

SpeedSpeed QualitQualityy


SpeedSpeed QualityQuality

Yeast_NoiYeast_Noisysy(6k, (6k, 200k)200k)

17% 11x -10% 30x -15% 25x +11%

WikiWiki(1.1M, (1.1M, 53M)53M)

15% 8x -26% 104x -24% 52x +50%

OrkutOrkut(3M, (3M, 117M)117M)

17% 13x +20% 30x +60% 36x +60%

Results Using Metis

[Hardware: Quad-core Intel i5 CPU, 3.2 GHz, with 16GB RAM ]

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SparsSpars. .

RatioRatio






17% 11x -10% 30x -15% 25x +11%


15% 8x -26% 104x -24% 52x +50%


17% 13x +20% 30x +60% 36x +60%

Results Using Metis

Same sparsification ratio for all 3 methods.


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SparsSpars. .

RatioRatio






17% 11x -10% 30x -15% 25x +11%


15% 8x -26% 104x -24% 52x +50%


17% 13x +20% 30x +60% 36x +60%

Results Using Metis

Good speedups, but typically loss in quality.


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SparsSpars. .

RatioRatio






17% 11x -10% 30x -15% 25x +11%


15% 8x -26% 104x -24% 52x +50%


17% 13x +20% 30x +60% 36x +60%

Results Using Metis

Great speedups and quality.


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Spars. Spars. RatioRatio

L-SparL-Spar


Yeast_NoisyYeast_Noisy(6k, 200k)(6k, 200k)

17% 17x +4%

WikiWiki(1.1M, 53M)(1.1M, 53M)

15% 23x -4.5%

OrkutOrkut(3M, 117M)(3M, 117M)

17% 22x 0%

L-Spar: Results Using MLR-MCL


L-Spar: Qualitative Examples

Node Retained neighbors Discarded neighbors

Graph (Wiki article)

Graph Theory, Adjacency list, Adjacency matrix,Model theory

Tessellation,Roman letters used in Mathematics, Morphism

Jack Dorsey (Twitter user, and co-founder)

Biz Stone, Evan Williams, Jason Goldman, Sarah Lacy

Alyssa Milano, JetBlue Airways, WholeFoods, Parul Sharma

Gladiator (Flickr tag)

colosseum, world-heritage, site, italy

europe, travel, canon, sky, summer

Twitter executives, Silicon Valley figures

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Impact of Sparsification on Noisy Data

As the graphs get noisier, L-Spar is increasingly beneficial.

Impact of Sparsification on Spectrum: Yeast PPI

Global Sparsification results in multiple components

Local sparsification seems to match trends of original graph

Impact of Sparsification on Spectrum: Epinion

Impact of Sparsification on Spectrum: Human PPI

Impact of Sparsification on Spectrum: Flickr

Anatomy of density plot

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Some measure of density

Specific ordering of the vertices in the graph

Density Overlay Plots

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Visual Comparison between Global vs Local Sparsification

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Summary

Sparsification: Simple pre-processing that makes a big difference

• Only tens of seconds to execute on multi-million-node graphs.

• Reduces clustering time from hours down to minutes.

• Improves accuracy by removing noisy edges for several algorithms.

• Helps visualization

• Ongoing and future work

• Spectral results suggests one might be able to provide theoretical rationale – Can we tease it out?

• Investigate other kinds of graphs, incorporating content, novel applications (e.g. wireless sensor networks, VLSI design)

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Prior Work

•Random edge Sampling [Karger ‘94]

•Sampling in proportion to effective resistances: good guarantees but very slow [Spielman and Srivastava ‘08]

•Matrix sparsification [Arora et. al. ’06]: Fast, but same as random sampling in the absence of weights.

Topological Measures

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Modularity (from Wikipedia )

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•Modularity is the fraction of the edges that fall within the given groups minus the expected such fraction if edges were distributed at random. The value of the modularity lies in the range [−1/2,1). It is positive if the number of edges within groups exceeds the number expected on the basis of chance.

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The MCL algorithm

Expand: M := M*M

Inflate: M := M.^r (r usually 2), renormalize columns

Converged?

Input: A, Adjacency matrixInitialize M to MG, the canonical transition matrix M:= MG:= (A+I) D-1

Yes

Output clusters

No

Prune

Enhances flow to well-connected nodes (i.e. nodes within a community).

Increases inequality in each column. “Rich get richer, poor get poorer.”(reduces flow across communities)

Saves memory by removing entries close to zero. Enables faster convergence

Clustering Interpretation: Nodes flowing into the same sink node are assigned same cluster labels

Documents

Local Sparsification for Scalable Module Identification in Networks