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Survey on Evolving Graphs Research

Speaker: Chenghui RenSupervisors: Prof. Ben Kao,

Prof. David Cheung

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MotivationEvolving graphs are everywhere

Social networks Users join social networks Friendships are established

The Web New Web pages are created Hyperlinks are established

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MotivationEvolving graphs are everywhere

P2P networks New routers appear Routing table size (vertex degree)

changes Spatio networks

Transportation cost (edge weight) changes

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Research branches

Evolution of graphs How do graphs evolve over time? Example

The networks are becoming denser over time with the average degree increasing [J. Leskovec 2007]

Querying evolving graphs Apply queries on evolving graphs to extract

information Example

How to update the PageRank efficiently as graphs evolve?

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Roadmap

Motivation Why we are interested in evolving graphs

Evolution of graphs How graphs evolve over time

Macroscopic evolution Microscopic evolution

Querying evolving graphs How to process queries on evolving graphs

Incremental computation Key moment detection Find-verify-fix framework

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Evolution of graphs

Macroscopic evolution of graphs How do global properties (e.g., degree

distribution, diameter) evolve? Microscopic evolution of graphs

Example How do a user link to other users?

Microscopic node behavior results in macroscopic behavior

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Macroscopic evolution Stable degree distributions[R. Albert 1999]

Power law distribution: P(degree = k) is proportional to 1/k^a The major hubs are closely followed by smaller ones The nodes tend to form communities

Examples Social networks, including collaboration networks. An example that

has been studied extensively is the collaboration of movie actors in films.

Protein-Protein interaction networks. Sexual partners in humans, which affects the dispersal of sexually

transmitted diseases. Many kinds of computer networks, including the internet and

the World Wide Web. Semantic networks Airline networks.

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Macroscopic evolution Densification and shrinking diameters [J.

Leskovec 2007] Densification formula

E(t) is proportional to N(t) ^ a (1 < a < 2) Shrinking diameters

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Microscopic evolution

Preferential attachment model [R. Albert 1999] New vertices attach preferentially to

sites that are already well connected Obey the power law distribution Global model: new vertices can

connect to any vertex in the whole network

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Microscopic evolution

Forest fire model [J. Leskovec 2007] Intuition: how do authors identify

references? Find first paper and cite it Copy a few citations from first Continue recursively From time to time use bibliographic tools

(e.g. CiteSeer) and chase back-links

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Microscopic evolution

Forest fire model [J. Leskovec 2007] A node arrives Randomly chooses an “ambassador” Starts burning nodes (with probability p)

and adds links to burned nodes “Fire” spreads recursively, with

exponential decay

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Microscopic evolution

Forest fire model [J. Leskovec 2007] Obey the densification, shrinking

diameter and power law distribution Local model: A newcomer will have a

lot of links near the community of his/her ambassador, a few links beyond this, and significantly fewer farther away

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Roadmap

Motivation Why we are interested in evolving graphs

Evolution of graphs How graphs evolve over time

Macroscopic evolution Microscopic evolution

Querying evolving graphs How to process queries on evolving graphs

Incremental computation Key moment detection Find-verify-fix framework

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Querying evolving graphs

A number of queries in literature PageRank queries Diameter queries Minimum spanning tree (MST) queries Shortest path queries Centrality queries …

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Querying evolving graphs Methodologies

Incremental computation PageRank queries Diameter queries

Key moment detection Minimum spanning tree queries

Our work: find-verify-fix framework Shortest path queries Centrality queries

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Incremental computation

Typically, the difference between two consecutive snapshots G1 and G2 is small

Compute the solution for G2 based on the solution for G1

The incremental algorithms are expected to be fast

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PageRank queries

Rank of a web page depends on the rank of the web pages pointing to it

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PageRank queries

Computing PageRank for large graphs at each time instance is expensive

Incremental algorithms are proposed [P. Desikan 2005]

Principle idea: PageRank depends only on the pages that point to it and is independent of the pages pointed by the page

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PageRank queriesDetect a changed portion of graphPartition the graph into scalable P and non-scalable Q such that there are no incoming links from Q to PCompute PageRank for QMerge the rankings of the two independent partitionsPageRank values of partition P

are obtained by simple scaling with scaling factor n(G1)/n(G2)

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Diameter queries

In a P2P network, an important and fundamental question is how many neighbors should a computer have, i.e., what size the routing table should be

Network diameter corresponds to the number of hops a query needs to travel in the worst case

If the diameter is large, the routing table size should be increased

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Diameter queries

G-Scale [Y. Fujiwara 2011] First study to address diameter

detection problem that guarantees exactness and efficiency on both single big graph and evolving graphs

Weak point: It assumes that one node and its connected edges are added to a time-evolving graph at each time tick. General edge insertions and edge deletions are not considered

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Key moment detection

Given an evolving graph and a query, a key moment detection algorithm tries to detect those moments at which the solution to the query changes

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MST queries

MSTs can be used to solve energy-efficient problems in spatio networks

A time aggregated graph is a graph in which each edge is associated with an edge weight function

A time-sub-interval is defined as a maximal sub interval of time horizon which has a unique MST

An efficient solution to determine time-sub-intervals is available [V. Gunturi 2010]

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MST queries

Methodology [V. Gunturi et al 2010] Edge order interval: a sub interval of

time horizon during which there is clear ordering of edge weight functions, i.e., none of them intersect with each other

Principle idea: An edge-order-interval has a unique MST

Inspired by Prim’s algorithm

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MST queries

An edge-order-interval

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MST queries

V. Gunturi et al proposed methods to efficiently determine at which moments to partition the edge-order-intervals

They also provided methods to incrementally compute MST based on the MST for the preceding edge-order-interval

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Our find-verify-fix framework

Given an evolving graph (G1, G2, G3, …, Gn), FVF Find representative solutions (RS’s) for

G1~Gn Verify whether these RS’s are indeed the

solution for each individual snapshot If the verification fails, try to fix the RS’s

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Our find-verify-fix framework

FVF can now handle: Exact shortest path (SP) queries on un-

weighted evolving graph Approximate SP queries on weighted

evolving graphs Approximate centrality queries

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Future work

Find more interesting queries Incorporate the ideas of incremental

algorithms and key moment detection algorithms to the FVF framework

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Thanks!