on Evolving Graphs Research Speaker Chenghui Ren Supervisors Prof Ben Kao Prof David Cheung 1 Motivation Evolving graphs are everywhere Social networks Users join social networks ID: 277182
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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 networksUsers join social networksFriendships are establishedThe WebNew Web pages are createdHyperlinks are established
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MotivationEvolving graphs are everywhere
P2P networks New routers appearRouting table size (vertex degree) changesSpatio networksTransportation cost (edge weight) changes
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Research branchesEvolution of graphs
How do graphs evolve over time?ExampleThe networks are becoming denser over time with the average degree increasing [J. Leskovec 2007]Querying evolving graphsApply queries on evolving graphs to extract informationExampleHow to update the
PageRank
efficiently as graphs evolve?
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RoadmapMotivation
Why we are interested in evolving graphsEvolution of graphsHow graphs evolve over timeMacroscopic evolutionMicroscopic evolutionQuerying evolving graphsHow to process queries on evolving graphs
Incremental computation
Key moment detection
Find-verify-fix framework
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Evolution of graphsMacroscopic evolution of graphsHow do global properties (e.g., degree distribution, diameter) evolve?
Microscopic evolution of graphsExampleHow do a user link to other users?Microscopic node behavior results in macroscopic behavior6Slide7
Macroscopic evolution
Stable degree distributions[R. Albert 1999]Power law distribution: P(degree = k) is proportional to 1/k^aThe 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 evolutionDensification and shrinking diameters [J.
Leskovec 2007]Densification formulaE(t) is proportional to N(t) ^ a (1 < a < 2)Shrinking diameters
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Microscopic evolutionPreferential attachment model [R. Albert 1999]
New vertices attach preferentially to sites that are already well connectedObey the power law distributionGlobal model: new vertices can connect to any vertex in the whole network
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Microscopic evolutionForest fire model [J. Leskovec
2007]Intuition: how do authors identify references?Find first paper and cite itCopy a few citations from firstContinue recursivelyFrom time to time use bibliographic tools (e.g.
CiteSeer
) and chase back-links
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Microscopic evolutionForest fire model [J. Leskovec
2007]A node arrivesRandomly 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 evolutionForest fire model [J. Leskovec
2007]Obey the densification, shrinking diameter and power law distributionLocal 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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RoadmapMotivation
Why we are interested in evolving graphsEvolution of graphsHow graphs evolve over timeMacroscopic evolutionMicroscopic evolutionQuerying evolving graphsHow to process queries on evolving graphs
Incremental computation
Key moment detection
Find-verify-fix framework
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Querying evolving graphsA number of queries in literature
PageRank queriesDiameter queriesMinimum spanning tree (MST) queriesShortest path queriesCentrality queries…
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Querying evolving graphs Methodologies
Incremental computationPageRank queriesDiameter queriesKey moment detectionMinimum spanning tree queries Our work: find-verify-fix framework
Shortest path queries
Centrality queries
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Incremental computationTypically, the difference between two consecutive snapshots
G1 and G2 is smallCompute the solution for G2 based on the solution for G1The 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 expensiveIncremental 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 queries
Detect a changed portion of graphPartition the graph into scalable P and non-scalable Q such that there are no incoming links from Q to P
Compute
PageRank
for Q
Merge the rankings of the two independent partitions
PageRank
values of
partition
P are obtained by simple scaling with scaling factor n(G1)/n(G2)
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Diameter queriesIn 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 caseIf the diameter is large, the routing table size should be increased20Slide21
Diameter queriesG-Scale [Y. Fujiwara 2011]First study to address diameter detection problem that guarantees
exactness and efficiency on both single big graph and evolving graphsWeak 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 detectionGiven 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 queriesMSTs can be used to solve energy-efficient problems in
spatio networksA time aggregated graph is a graph in which each edge is associated with an edge weight functionA 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 queriesMethodology [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 otherPrinciple 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 queriesV. Gunturi
et al proposed methods to efficiently determine at which moments to partition the edge-order-intervalsThey also provided methods to incrementally compute MST based on the MST for the preceding edge-order-interval26Slide27
Our find-verify-fix frameworkGiven an evolving graph (G1, G2, G3, …,
Gn), FVF Find representative solutions (RS’s) for G1~GnVerify whether these RS’s are indeed the solution for each individual snapshotIf the verification fails, try to fix the RS’s27Slide28
Our find-verify-fix frameworkFVF can now handle:
Exact shortest path (SP) queries on un-weighted evolving graphApproximate SP queries on weighted evolving graphsApproximate centrality queries28Slide29
Future workFind more interesting queriesIncorporate the ideas of incremental algorithms and key moment detection
algorithms to the FVF framework29Slide30
Thanks!
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