Optimization of Network Topology Elias Boutros Khalil, Bistra - - PowerPoint PPT Presentation

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Optimization of Network Topology Elias Boutros Khalil, Bistra - - PowerPoint PPT Presentation

Oct 12, 2022 401 likes •984 views

Scalable Diffusion-Aware Optimization of Network Topology Elias Boutros Khalil, Bistra Dilkina, Le Song Georgia Institute of Technology Problem Given G(V,E), a set of source nodes X (infected nodes) Linear Threshold Model

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Scalable Diffusion-Aware Optimization of Network Topology

Elias Boutros Khalil, Bistra Dilkina, Le Song Georgia Institute of Technology

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Problem

  • Given
  • G(V,E),
  • a set of source nodes X (infected nodes)
  • Linear Threshold Model
  • Find a set of k edges to
  • remove, s.t., the spread of a certain

substance is minimized

  • add, s.t., the spread of a certain substance

is maximized

2

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Review: Diffusion Models

  • Linear Threshold Model
  • Each edge has a weight Wuv
  • each node u chooses a threshold uniformly

at random in [0,1]

  • Node v will be infected if
  • Independent Cascade Model
  • Each edge has a propagation probability

Puv

  • Each infected node u has only one chance

to infect its neighbor v with prob. Puv

3

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Review: Influence Maximization

  • Given
  • G(V,E)
  • LT model or IC model
  • To find k nodes to activate to maximize

the spread of a certain substance

  • Greedy algorithm
  • Objective function is submodular
  • (1-1/e)-appriximation

4

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Edge Deletion Problem

  • Given G, source set A,
  • Find k edges
  • Supermodular
  • Greedy algorithm provides (1-1/e)-

approximation

  • Scaling up tricks

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Edge Addition Problem

  • Given G, source set A,
  • Find k edges
  • Still supermodular (Equivalent to

constrained submodular minimization)

  • Algorithm: max. the lowerbound

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Edge Addition Problem

  • Marginal Gain is bounded
  • Apply an approach for constrained submodular

minimization with approximation guarantees

  • R. Iyer, S. Jegelka, and J. Bilmes. Fast semidifferential based

submodular function optimization. In ICML, 2013.

7

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Experiments

  • Datasets
  • Syntetic dataset: generated by Kronecker

graph model

  • (1) CorePeriphery, (2) ErdosRenyi and (3)

Hierarchical

  • Real datasets:

8

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Experiments

  • Competing heuristics
  • Random
  • Weights: highest weights
  • Betweenness
  • Eigen: k edges to max the leading

eigendrop

  • Degree: k edges whose destination nodes

have the highest out-degrees [8]

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Experiments

Edge deletion Edge addition

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Core Decomposition of Uncertain Graphs

Francesco Bonchi, Francesco Gullo, Andreas Kaltenbrunner, Yana Volkovich Yahoo Labs, Spain

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Core decomposition

  • k-core of a graph
  • a maximal subgraph in which every vertex

is connected to at least k other vertices within that subgraph

  • Core decomposition
  • The set of all k-cores of a graph G forms

the core decomposition of G

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K-core under uncertain graphs

  • A maximal subgraph whose vertices have at

least k neigbours in that subgraph with probability no less than η

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Example

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Motivation

  • core decomposition can be computed

efficiently in deterministic graphs

  • computed in linear time
  • However, does not guarantee efficiency

in uncertain graphs

  • even the simplest graph operations may

become computationally intensive.

  • uncertain graph
  • edges are assigned a probability of existence
  • E.g.:, protein-interaction, the influence of one

person on another

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Applications

  • Influence maximization
  • Idea: just reduce the input graph G by keeping only

the inner-most η-shells

  • the higher the core index is, the more likely the

vertex is an influential spreader [17]

  • Task-driven team formation
  • Node: individuals; edge: a probabilistic topic model
  • Given a pair <T,Q> where T is the set of terms, Q is

a set of nodes

  • Goal: Find a node of nodes A where Q⊆A, which a

good team to perform the task in T

  • Solution: find a connected component of (k,η)-core

which contains A

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Algorithm framework

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Follow the deterministic case the maximum degree such that the probability for v to have that degree is no less than η Non-trivial to compute

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Experiments

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Influence Maximization Task-driven Team-formation

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Fast Influence-based Coarsening for Large Networks

KDD, New York City August 26, 2014

Manish Purohit^, B. Aditya Prakash*, Chanhyun Kang^, Yao Zhang*, V S Subrahmanian^ *Virginia Tech ^University of Maryland

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Networks are getting huge!

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Flickr (friendship network): 87 million users and 8 billion photos until 2013 Amazon (friendship network): 237 million accounts until 2013 Twitter (follower network): 271 million monthly active users Facebook (friendship network): 829 million daily active users on average in June 2014

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Need for fast analysis

  • Ever growing list of applications of

network effects

  • Viral Marketing
  • Immunization
  • Information Diffusion
  • …

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However, scaling up traditional algorithms up to millions of nodes is hard

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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How to handle large-scale networks

  • Approaches
  • Use faster / simpler algorithms
  • Perform analysis locally
  • i.e., divide the large network into

smaller subgraphs

  • Zoom-out the network to
  • btain a smaller

representation of the network

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this paper

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Bird’s eye view of a network

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Bird’s eye view of a network

  • “Zoom-out” of the graph to get a quick

picture

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Called “coarsen” in this paper

Big graph Zoom-out A F E D C B Small representation

  • f the network

A C B E F D

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Outline

  • Motivation
  • Challenges
  • Problem Definition
  • Our Proposed Method
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Challenges

  • C1: How do we maintain diffusive

characteristics when coarsening networks?

  • C2: How do we merge node to get the

coarse network?

  • C3: how do we find the best node to

merge fast?

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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C1: Information Diffusion

  • Cascading behavior in networks

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Diffusion is graph induced by a time ordered propagation of information (edges)

Blogs Posts Links Information cascade Source: [McGlohon et. al., SDM2007] B1 B2 B4 B3 1 1 2 3 1 Blog network

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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C1: Model information diffusion

  • Information spreads over networks
  • e.g.:, rumor/meme spreads over Twitter following

network

  • Independent cascade model (IC) [Kempe+, KDD03]
  • Weights pij: propagation prob. from i to j
  • Each node has only one chance to infect its

neighbors

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

Meme spreading

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C1: Diffusive characteristics

  • First eigenvalue λ1 (of adjacency matrix)

is enough for most diffusion models. (Prakash et al. [ICDM’12])

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λ1 is the epidemic threshold “Safe”

“Vulnerable” “Deadly”

Increasing λ1 , Increasing vulnerability

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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C1: maintain diffusive characteristics

  • Goal: maintain the diffusive characteristics of

the original network in the coarsened network?

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Original network coarsen A F E D C B Coarsened network A C B E F D

Make the coarsened network has the least change in the first eigenvalue

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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C2: How to merge nodes

  • Goal: Merge nodes of graph G to get the

coarsened graph that “approximates” G with respect to diffusion

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Merge b and a can get the least change

  • f λ1

Is this correct?

0.375!

Original network

Influence from d to b: 0.5 Influence from d to a: 0.25 Average: 0.375

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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  • In general:

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C2: How to merge nodes

Merging a,b

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

Details

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C3: which nodes to merge

  • Goal:
  • Find the best nodes to merge
  • Fast, scalable to large network

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Talk about it later

Original network coarsen A F E D C B Coarsened network A C B E F D

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Outline

  • Motivation
  • Challenges
  • Problem Definition
  • Our Proposed Method
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Problem Definition

Graph Coarsening Problem (GCP) Given: large graph G(V, E), and reduction factor α Find: the best set of edges to merge Such that: |λG - λH| is minimized

  • (i.e. H is the coarsened graph with the

least change in the first eigenvalue)

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Naive Greedy Heuristic

Step:

  • Score every edge by the change in eigenvalue
  • Greedily choose the edge (a,b) with the least score,

and merge (a,b)

  • Re-evaluate the scores of every edge and repeat

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  • Too slow! O(m2) time to score all edges
  • Lose time benefits of analyzing the smaller

graph

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Outline

  • Motivation
  • Problem Definition
  • Challenges
  • Our Proposed Method
  • CoarseNet
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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CoarseNet: idea

  • Can we approximate the edge scores faster?
  • Yes!
  • Use matrix perturbation arguments to

estimate (up to first order terms) the score of an edge in constant time!

  • Score all edges in O(m) time
  • Naive Heuristic: O(m2) time

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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CoarseNet: details

  • Corollary 5.1: Given the first eigenvalue λ,

and corresponding eigenvectors u, v, the score of a node pair score(a, b) can be approximated in constant time.

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(a,b) is a node- pair

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

We want to characterize the change of λ after coarsening

a b f g e Coarsen merge (a,b) c f g e

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the out-adjacency vector of merged node c

CoarseNet

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See paper for details

A u = λ . u

u(i)

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

left eigenvector right eigenvector weight of (b,a) weight of (a,b)

Details

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CoarseNet: Complete algorithm

  • Step

1: compute scores for all edge pairs 2: Merge nodes with smallest score

  • 3. Goto step 1 until αn nodes left

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Original Network (weight=0.5) Assigning scores Merging edges Coarsened Network

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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CoarseNet: running time

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  • Running time: O(mln(m)+αnnθ)
  • m: number of edges
  • n: number of nodes
  • nθ : the maximum degree of any vertex during the

merging process

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Outline

  • Motivation
  • Challenges
  • Problem Definition
  • Our Proposed Method
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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How do we perform?

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The first eigenvalue gets preserved well up to large coarsening factors!

Amazon (See more results in the paper) DBLP

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Scalability w.r.t Reduction Factor (α)

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Scales linearly with the desired reduction factor

Amazon (334,863 vertices) DBLP (511,163 vertices) (See more results in the paper)

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Scalability w.r.t Graph Size (𝑜)

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Flickr

Scales linearly with the number of nodes

We extracted 6 connected components (with 500K to 1M vertices in steps of 100K) of the Flickr network

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Outline

  • Motivation
  • Challenges
  • Problem Definition
  • Our Proposed Method
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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  • How to market well?
  • Convince a subset of individuals to adopt a new

product

  • Then, trigger a large cascade of further adoptions
  • Influence maximization problem
  • [Kempe et. al, KDD03]
  • Find the best set of seeds in a network to achieve

highest diffusion

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Application 1: Influence Maximization

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

Who is the most influential person? Influence

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Application 1: Influence Maximization

  • Our fast algorithm CSPIN:

Step 1: Coarsen the large social network using CoarsenNet Step 2: Solve influence maximization on the coarsened network Step 3: Randomly select one node from each selected “supernode”

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Step 1: Coarsen A C B E F D Step 2: Solve influence maximization A C B E F D Step 3: Randomly select one node from C

We call it CSPIN

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Quality of CSPIN

  • We use and compare against the fast and

popular PMIA algorithm (Chen et al. [KDD’07])

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We obtain influence spread as good as by PMIA

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Quality of CSPIN w.r.t 𝛽

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We can merge up to 95% of the vertices are merged without significantly affecting the influence spread!

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Scalability w.r.t number of seeds

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Log scale

Finds good solutions in minutes instead of hours!

Portland (1.5 million vertices) (See more results in the paper)

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Application 2: Diffusion Characterization

  • Goal: use Graph Coarsening to understand

information cascades

  • Dataset: Flixster
  • a fridendship network with movie ratings
  • Cascade: the same movie rating from friends
  • Methodology
  • coarsen the network using CoarseNet with the

reduction factor α=0.5

  • study the formed groups (supernodes)

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Diffusion observation

Observation 1: a very large fraction of movies propagate in a small number of groups Observation 2: a multi-modal distribution

Stats:

  • 1891 groups
  • mean group size: 16.6
  • the largest group: 22061

nodes (roughly 40% of nodes)

(See more results in the paper)

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Can get non-network surrogates for super-nodes

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Outline

  • Motivation
  • Challenges
  • Problem Definition
  • Our Proposed Method
  • Experiments
  • Applications
  • Conclusion

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Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Conclusion

Graph Coarsening Problem

  • Given: a large graph and

the reduction factor

  • Find: "best" nodes to

coarsen CoarseNet

  • estimate edge score in

constant time

  • Sub-quadratic

Applications

  • Influence Maximization
  • Diffusion Characterization

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Original network

coarsen

A F E D C B Coarsened network A C B E F D

Purohit, Prakash, Kang, Zhang, Subrahmanian 2014

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Any Questions?

  • Code at:

http://www.cs.vt.edu/~badityap/

Funding:

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Original network

coarsen

A F E D C B Coarsened network A C B E F D