Analyzing Probabilistic Graphs

Analyzing Probabilistic Graphs

Michalis Potamias

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Information propagation

The probabilistic view of graphs

Protein-protein interaction network

Mobile ad hoc network

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Michalis Potamias : Analyzing Probabilistic Graphs

Research Approach

Graph Analysis

Define tasks

Useful?

Practical?

Design algorithms

Nearest-neighbors, clustering, learning

Data

social, biological, mobile ad hoc …

Uncertainty

3Michalis Potamias : Analyzing Probabilistic Graphs

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Outline

• Distance and k-nearest neighbors– Distance definition– Sampling– kNN Pruning– Predicting known relationships from PPI networks

• Clustering– Edit distance and cluster graphs– Clustering probabilistic graphs

• Learning in information propagation– The problem– Anecdotes

• Ongoing and Future work– Information propagation– Random walks– Daily deals


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Outline






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Distance and Nearest Neighbors

How do we define a distance function in probabilistic graphs?

– Reliability(Valiant, SIAM J. Comput. 1979) (Asthana et al., Genome Research 2004)

– Probability of the most probable path (Sevon et al., DILS 2006)

– Based on the shortest path PDF(Potamias et al., VLDB 2010)

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PPI Nearest Neighbors: Filter candidate interactions.


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Possible worlds


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PDF

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Distance Functions based on the PDF

a world wA

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(Frank, OR 1969)

Compute the PDF of the shortest path distance between B and D

•Find the shortest path distance between B and D in each world

)),(1))(,(1))(,(1(),(),(

)Pr(

DApDCpCBpDBpBAp

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the graph


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Distance Functions based on the PDF

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the graph• A distance function is a scalar.• Use the Shortest Path PDF:

– Expected Infinity problem

– Median– Mode (majority)– ExpectedReliable

• Hard (Valiant, SIAM J. Comput. 1979)

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Computing the functions: Sampling

• Algorithm:1. Sample M worlds.2. In each world w perform a Dijkstra

traversal to compute the shortest path distance.

3. Compute the sample median distance.

• In practice, a small number of worlds yields a good approximation.


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K-Nearest Neighbor Query

• kNN query example:Given a probabilistic PPI, and a source protein find the set of k proteins closest to the source.

• Algorithm:– Sample M worlds.– In each world perform a Dijkstra traversal.– Approximate the median shortest path

distance of the source to all nodes in the graph.

– Process the kNN query.


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kNN Processing

1nn - medianSource node: ASample: 5 worlds

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kNN Processing


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Shortest Path Distance PDF from A to each other node


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kNN Processing


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kNN Processing


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kNN Processing


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kNN Processing


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kNN Processing


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kNN Pruning


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• The pruning algorithm:– Idea: Increase the horizon of each dijkstra one hop at

a time and maintain truncated PDFs.

– Node insertion to kNN set: Once its median distance is found.

– Termination condition: kNN set has size equal to k.


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kNN Pruning


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kNN Pruning

1nn - medianSource node: ASample: 5

worlds

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Shortest Path Distance PDF from A to discovered nodes

kNN set


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kNN Pruning


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kNN Pruning


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kNN Pruning


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kNN Pruning


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•1NN set is complete with B•2 nodes visited•Same answer as previously

•Overhead: dijkstra state needs to be maintained in memory for all worlds


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kNN Pruning


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kNN Pruning


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kNN Pruning


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kNN Pruning

BIOMINEDatabase of biological entities and uncertain interactions fromUHelsinki1M nodes, 10M edges

FLICKRUsers from flickr.com. Edges have been created assuming homophily based on jaccard of flickr groups77K nodes, 20M edges

DBLPAuthors from dblp. Probabilities have been assigned based on number of coauthored papers226K nodes, 1.4M edges

5NN query with a sample of 200 worlds.Speedups: 247x (BIOMINE), 111x (FLICKR), 269x (DBLP)


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Practical?

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Useful Distance Functions

• Dataset:– Probabilistic PPI network

[Krogan et al, Nature 06]– Known protein co-complex

relationships (ground truth)[Mewes et al, Nuc Acids Res 04]

• Experiment:– Choose a ground truth pair of

proteins (A,B)– Choose a protein C such that

there is no ground truth relationship for pair (A,C)

– Classification task: Distinguish between the two types of pairs using the PPI network


Useful?

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Outline






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Clustering Probabilistic Graphs


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Graph Edit Distance and Cluster Graphs

Cluster Graph: A set of disjoint cliques


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ClusterEdit

ClusterEdit:[Shamir et al., Disc. Applied Math.,

2004]:Given graph G, find a cluster-graph

C such that the edit distance between G and C is minimized

pClusterEdit: Given probabilistic graph G, find a

cluster-graph C such that the expected edit distance between a

world W and C is minimizedMichalis Potamias : Analyzing Probabilistic Graphs

Parameter free; the

number of clusters is part of the output.

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pClusterEdit

Correlation clustering. [Bansal et al., ML 2004]

Linear time randomized expected 5-approximation algorithm. [Ailon et al., JACM 2008]


Find cluster graph C such that minimizes

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Work on Probabilistic Graphs

•Most Probable Path [Sevon et al., DILS 2006]•Reliable Subgraphs

[Hintsanen and Toivonen, ECML/PKDD 2008]•Nearest Neighbors

[Potamias et al., VLDB 2010]•Frequent Subgraphs

[Zou et al., KDD 2010]•Top-k Maximal Cliques

[Zou et al., ICDE 2010]•Clustering

[Potamias et al., ongoing work]


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Probabilistic Graphs Roadmap






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Learning the probabilities of the edges in Information

Propagation


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The Problem

• Observe the time series of an information item’s spread in a given network. How endogenous and exogenous is the information item?

• Previous studies on inference of propagation probabilities:– Treat all information items as if they were

identical.– Assume that the network explains the

observed spread.[Rodriguez et al., KDD 2010][Goyal et al., WSDM 2010]

Michalis Potamias : Analyzing Probabilistic Graphs 66

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Endogenous Information


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Exogenous Information


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E2 model

• Model: each information item is characterized by its endogeneity and its exogeneity.

• Inference: Given the time-series of the spread and the network topology we find the maximum likelihood endogeneity and exogeneity for each item.


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Blogs and quotesMemetracker dataset

[Leskovec et al., KDD 2009]– Blogger.com and wordpress.com– US Presidential elections era, 8/2008 – 4/2009– Nodes: Bloggers (50K)– Information Items: Quotes (60K) – Activations: Bloggers use quotes in their posts

(600K)– Network: Bloggers link to other bloggers (300K)


Useful?

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Blogs and quotes


Blogs and quotes


Blogs and quotes


Blogs and quotes


Blogs and quotes


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Probabilistic Graphs Roadmap






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1. Disentangle the factors that explain it[Anagnostopoulos et al. KDD 2008 ] [Aral et al. PNAS 2009]

•Homophily•Users’ influence•Items’ endogeneity•Exogeneity

2. In the presence of incentives


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Random walksp1

p2

p3

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p5

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p5


Daily deals: Groupon


Research Approach

Graph Analysis

Define tasks

Useful?

Practical?

Design algorithms

Nearest-neighbors, clustering, learning

Data

social, biological, mobile ad hoc …

Uncertainty


Thank you

This talk consists of completed and ongoing work with the following collaborators:

Rakesh Agrawal (MSR)Aris Anagnostopoulos (U Rome)

Francesco Bonchi (YR)John Byers (BU)

Carlos Castillo (YR)Aristides Gionis (YR)George Kollios (BU)Evimaria Terzi (BU)

Konstantin Voevodski (BU)Georgios Zervas (BU)


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Analyzing Probabilistic Graphs

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