Information Network Analysis and Discovery

Cuiping LiGuoming He

Information School, Renmin University of China

Related Work1. Whole graph Level -Macro properties (Laws, generators) -Summary/Visualization -Index

2. Sub-graph Level

-Frequent Pattern Mining

-Clustering (Community/group detection) -Connected Sub-graph, Central Piece -Pattern Match

3. Node or Link Level -Ranking

-Proximity/Similarity-Node Classification-Outlier Detection (Abnormal nodes/links)

Node Proximity/Similarity: Why?

• Link prediction [Liben-Nowell+], [Tong+]• Ranking [Haveliwala], [Chakrabarti+]• Email Management [Minkov+]• Image caption [Pan+]• Neighborhooh Formulation [Sun+]• Conn. subgraph [Faloutsos+], [Tong+], [Koren+]• Pattern match [Tong+]• Collaborative Filtering [Fouss+]• Many more…

Node Similarity: Related Work(1)• Computer Network’99: Finding related pages in the World

Wide Web, Jeffrey Dean , Monika R. Henzinger (adapting from HITS)

• KDD’02: SimRank: A Measure of Structural-Context Similarity, Glen Jeh, Jennifer Widom (Adapting from PageRank)– Exploiting Hierarchical Domain Structure to Compute Simil

arity. P. Ganesan, H. Garcia-Molina, and J. Widom. Transactions on Information Systems, 21(1): 64-93, January 2003.

• Vertex similarity in networks: Phys. Rev. E 73, 026120 (2006)

• Optimization on simrank– WWW’05: Scaling link-base similarity search,

D.Fogaras, B. Racz (Approximate)– VLDB’08: Accuracy Estimate and Optimization

Techniques for SimRank Computation Dmitry Lizorkin, Pavel Velikhov , Maxim Grinev, Denis Turdakov .

Node Similarity: Related Work(2)

• Domain-Integrated of simrank:– VLDB’08: Simrank++: Query Rewriting through Link

Analysis of the Click Graph, Loannis Antonellis (Stanford University), Hector Garcia-Molina (Stanford University), Chi-Chao Chang (Yahoo!). (keywords, ads)

• Clustering using simrank:– SIGIR’03: ReCom: Reinforcement Clustering of multi-

type interrelated data objects, J. Wang, H.J. Zeng, Z. Chen, H.J. LU,L. Tao

– VLDB’06: LinkCLus:Efficient Clustering via Heterogeneous Semantic Links, Xiaoxin Yin, Jiawei Han, Philip Yu

Existing Research: Limitation 1

• Not Dynamic– Static Algorithm

• Iterative

– Challenges of Dynamic Network• Re-computation even one node or edge changes

– Our Solution• Non-iterative• Incremental Computation

Cuiping Li, Jiawei Han, Guoming He, Xin Jin, Yizhou Sun, Yintao Yu, Tianyi Wu, "Fast Computation of SimRank for Static and Dynamic Information Networks", Int. Conf. on Extending Data Base Technology (EDBT'10), Lausanne, Switzerland, March 2010

Existing Research: Limitation 2

• Not Efficient– Our Solution: employ the modern hardware

resource• GPU (Graphic Process Unit)• Multi-Processor

Compute Node Similarity for Dynamic Network

• SimRank formula

• Or

• Intuition– Two objects are similar if they are referenced by

similar objects.

How to Compute SimRank Incremetally

• Fist glance at SimRank formula– It is Iterative. Has no chance to be computed incrementally

• Key Observation– SimRank iteration formula has the same form as the well-known

Sylvester Equations, based on this, we can compute SimRank without iteration.

Vec-Operator and Kronecker Products

• Vec-Operator

– Vec flattens an n x n matrix A into an n2 x 1 vector

– It stacks the columns of the matrix on top of each other, from left to right

• Kronecker Product

– Product of two matrices A and B

– Each element of A is multiplied with the full matrix B:

Sylvester Equations

• Sylvester Equations: X=SXT + X0

– Given three n x n matrixes S, T, and X0

– We want to determine X– Solvable in O(n3)

Sylvester Equations• Rewrite the Sylvester Equations as

vec(X)=vec(SXT) + vec(X0)

• Exploit the well-known fact

vec(SXT) = (TTS)vec(X)

• We can get

vec(X)= (TTS)vec(X) + vec(X0)

• We can get (I - TTS)vec(X) = vex(X0)

• Now we have to solve

vec(X)=(I - TTS)-1 vec(X0)

SimRank

• SimRank has the same form as the Sylvester equationsX=cATXA +(1-c)e,

(A is the normalized adjacent matrix, e is an identity matirx)

• Similarly, for SimRank, we have to solve

vec(X)=(I -cAT AT)-1 vec((1-c)e)

vec(X)= (1-c) (I -cAT AT)-1 vec(e)– AT AT can be solved in O(n3)– More importantly, when A is sparse/skew, we can

improve the efficiently further.

15

Advantages of non-iterative method

vec(X)= (1-c) (I -cAT AT)-1 vec(e)

• It can be solved approximately

• It can be computed incrementally

• It can be computed pair-wisely

vec(X)= (1-c) (I -cWW)-1 vec(e)

• 利用奇异值分解 SVD 和 Sherman-Morrison方程求 L的逆

Approximation

W =

IvecVVUUcIcSvec

VVUUcIL

UUVVc

VUW

1

1

1

• W 的 low rank SVD分解

• k的大小– k越大，计算时间越长，精确度越高

• Error Bound

Approximation

kknkkkknnn VUW

• 预计算

• 计算某对结点 (i,j) 的 SimRank

Approximation

rlji

jil

VcVIcjiS

UUV

,

:,:,

1,

IvecVVV

UUVVc

r 1

Incremental Computation

只需要对 U,,V进行维护即可

Applications

• Similarity Tracking: return the N most similar nodes of i at each time step t.

• Centrality Tracking: return the N most central nodes at each time step t.

Experimental Result on DBLP

Top-10 Most Similar Terms for ‘Prof. Jennifer Widom’ up to Each Time Step

Experimental Result

Top-10 Most Similar Authors for ‘Prof. Jennifer Widom’ up to Each Time Step

• 预计算时间

• 计算不同个数结点对的时间

Wikepedia Data

• We set the threshold T to be 1.0e-6. For k=15– the pre-compute time of the Wikipedia dataset

is approx. 5.68 hours– the query time for every 1000 node pairs is

3.718 seconds