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Further Investigations on Heat Diffusion Models

Haixuan Yang

Supervisors: Prof Irwin King and Prof Michael R. Lyu

Term Presentation 2006

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Outline Introduction Input Improvement – Three candidate graphs Outside Improvement – DiffusionRank Inside Improvement – Volume-based heat difusion model Summary

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Introduction

PHDC Volume-based HDM

DiffusionRank

HDM on Graphs

Inside Improvement

Input Improvement

Outside Improvement

PHDC: the model proposed last year

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PHDC PHDC is a classifier motivated by

Tenenbaum et al (Science 2000) Approximate the manifold by a KNN graph Reduce dimension by shortest paths

Kondor & Lafferty (NIPS 2002) Construct a diffusion kernel on an undirected graph Apply to a large margin classifier

Belkin & Niyogi (Neural Computation 2003) Approximate the manifold by a KNN graph Reduce dimension by heat kernels

Lafferty & Kondor (JMLR 2005) Construct a diffusion kernel on a special manifold Apply to SVM

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PHDC Ideas we inherit

Local information relatively accurate in a nonlinear manifold.

Heat diffusion on a manifold a generalization of the Gaussian density from Euclidean space

to manifold. heat diffuses in the same way as Gaussian density in the ideal

case when the manifold is the Euclidean space. Ideas we think differently

Establish the heat diffusion equation on a weighted directed graph. The broader settings enable its application on ranking on the

Web pages. Construct a classifier by the solution directly.

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Heat Diffusion Model in PDHC Notations

Solution

Classifier G is the KNN Graph: Connect a directed edge (j,i) if j is one of the

K nearest neighbors of i. For each class k, f(i,0) is set as 1 if data is labeled as k and 0

otherwise. Assign data j to a label q if j receives most heat from data in class

q.

. at time nodeat heat the:),(

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)0()0()( fefetf HtH

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Input Improvement Three candidate graphs

KNN Graph Connect points j and i from j to i if j is one of

the K nearest neighbors of i, measured by the Euclidean distance.

SKNN-Graph Choose the smallest K*n/2 undirected edges,

which amounts to K*n directed edges. Minimum Spanning Tree

Choose the subgraph such that It is a tree connecting all vertices; the sum of

weights is minimum among all such trees.

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Input Improvement Illustration

Manifold KNN Graph SKNN-Graph Minimum Spanning Tree

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Input Improvement Advantages and disadvantages

KNN Graph Democratic to each node Resulting classifier is a generalization of KNN May not be connected Long edges may exit while short edges are removed

SKNN-Graph Not democratic May not be connected Short edges are more important than long edges

Minimum Spanning Tree Not democratic Long edges may exit while short edges are removed Connection is guaranteed Less parameter Faster in training and testing

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Experiments Experimental Setup

Experimental Environments Hardware: Nix Dual Intel Xeon

2.2GHz OS: Linux Kernel 2.4.18-27smp

(RedHat 7.3) Developing tool: C

Data Description 3 artificial Data sets and 6

datasets from UCI Comparison

Algorithms: Parzen window

KNNSVM KNN-HSKNN-HMST-H

Results: average of the ten-fold cross validation

Dataset Cases Classes Variable

Syn-1 100 2 2

Syn-2 100 2 3

Syn-3 200 2 3

Breast-w 683 2 9

Glass 214 6 9

Iono 351 2 34

Iris 150 3 4

Sonar 208 2 60

Vehicle 846 4 18

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Experiments Results Dataset SVM KNN PWA KNN-H MST-H SKNN-H

Syn-1 66.0 67.0 80.0 93.0 95.0 95.0

Syn-2 34.0 67.0 83.0 94.0 94.0 89.0

Syn-3 54.0 79.5 92.0 91.0 90.0 92.0

Breast-w 96.8 94.1 96.6 96.9 95.9 99.4

Glass 68.1 61.2 63.5 68.1 68.7 70.5

Iono 93.7 83.2 89.2 96.3 96.3 96.3

Iris 96 97.3 95.3 98.0 92.0 94.7

Sonar 88.5 80.3 53.9 90.9 91.8 94.7

Vehicle 84.8 63.0 66.0 65.5 83.5 66.6

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Conclusions KNN-H, SKNN-H and MST-H

Candidates for the Heat Diffusion Classifier on a Graph.

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Application Improvement PageRank

Tries to find the importance of a Web page based on the link structure.

The importance of a page i is defined recursively in terms of pages which point to it:

Two problems: The incomplete information about the Web structure. The web pages manipulated by people for commercial

interests. About 70% of all pages in the .biz domain are spam About 35% of the pages in the .us domain belong to spam

category.

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Why PageRank is susceptible to web spam? Two reasons

Over-democratic All pages are born equal--equal voting ability of one

page: the sum of each column is equal to one. Input-independent

For any given non-zero initial input, the iteration will converge to the same stable distribution.

Heat Diffusion Model -- a natural way to avoid these two reasons of PageRank

Points are not equal as some points are born with high temperatures while others are born with low temperatures.

Different initial temperature distributions will give rise to different temperature distributions after a fixed time period.

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DiffusionRank On an undirected graph

Assumption: the amount of the heat flow from j to i is proportional to the heat difference between i and j.

Solution:

On a directed graph Assumption: there is extra energy imposed on

the link (j, i) such that the heat flow only from j to i if there is no link (i,j).

Solution:

On a random directed graph Assumption: the heat flow is proportional to the

probability of the link (j,i). Solution:

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DiffusionRank On a random directed graph

Solution:

The initial value f(i,0) in f(0) is set to be 1 if i is trusted and 0 otherwise according to the inverse PageRank.

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Computation consideration Approximation of heat kernel

N=? When N>=30, the real eigenvalues of are

less than 0.01; when N>=100, they are less than 0.005. We use N=100 in the paper.

When N tends to infinity

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Discuss γ γcan be understood as the thermal conductivity. When γ=0, the ranking value is most robust to

manipulation since no heat is diffused, but the Web structure is completely ignored;

When γ= ∞, DiffusionRank becomes PageRank, it can be manipulated easily.

Whenγ=1, DiffusionRank works well in practice

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DiffusionRank Advantages

Can detect Group-group relations Can cut Graphs Anti-manipulation

+1

-1

γ= 0.5 or 1

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DiffusionRank Experiments

Data: a toy graph (6 nodes) a middle-size real-world graph (18542 nodes) a large-size real-world graph crawled from CUHK

(607170 nodes) Compare with TrustRank and PageRank

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Results The tendency of

DiffusionRank when γ becomes larger

On the toy graph

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Anti-manipulation On the toy graph

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Anti-manipulation on the middle graph and the large graph

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Stability--the order difference between ranking results for an algorithm before it is manipulated and those after that

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Conclusions This anti-manipulation feature enables

DiffusionRank to be a candidate as a penicillin for Web spamming.

DiffusionRank is a generalization of PageRank (when γ=∞).

DiffusionRank can be employed to detect group-group relation.

DiffusionRank can be used to cut graph.

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Inside Improvement Motivations

Finite Difference Method is a possible way to solve the heat diffusion equation. the discretization of time the discretization of space and time

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Motivation Problems where we cannot employ FD

directly in the real data analysis: The graph constructed is irregular; The density of data varies; this also results in an

irregular graph; The manifold is unknown; The differential equation expression is unknown

even if the manifold is known.

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Intuition

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Volume-based Heat Diffusion Model Assumption

There is a small patch SP[j] of space containing node j; The volume of the small patch SP[j] is V (j), and the heat

diffusion ability of the small patch is proportional to its volume.

The temperature in the small patch SP[j] at time t is almost equal to f(j,t) because every unseen node in the small patch is near node j.

Solution

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Volume Computation Define V(i) to be the volume of the

hypercube whose side length is the average distance between node i and its neighbors.

a maximum likelihood estimation

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Experiments

K: KNN P: Parzen window U: UniverSvm L: LightSVMC: consistency method

VHD-v: by the best vVHD: v is found by the estimation HD: without volume considerationC1: 1st variation of CC2: 2nd variation of C

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Conclusions The proposed VHDM has the following

advantages: It can model the effect of unseen points by

introducing the volume of a node; It avoids the difficulty of finding the explicit

expression for the unknown geometry by approximating the manifold by a finite neighborhood graph;

It has a closed form solution that describes the heat diffusion on a manifold;

VHDC is a generalization of both the Parzen Window Approach (when the window function is a multivariate normal kernel) and KNN.

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Summary The input improvement of PHDC provide

us more choices for the input graphs. The outside improvement provides us a

possible penicillin for Web spamming, and a potentially useful tool for group-group discovery and graph cut.

The inside improvement shows us a promising classifier.