Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces

1. Greedy Forwarding inDynamic Scale-Free NetworksEmbedded in Hyperbolic Metric Spaces Dmitri KrioukovCAIDA/UCSD Joint work with F. Papadopoulos, M. Boguñá, A. Vahdat

2. Outline • Model of scale-free networks embeddedin hyperbolic metric spaces • Greedy forwarding in the model • Conclusion Motivation

3. Motivation • Routing overhead is a serious scaling limitation in many networks (Internet, wireless, overlay/P2P networks, etc.) • Search for infinitely scalable routing without any overhead • Do not propagate any information about changing topology • Route without any global topology knowledge, using only local information • How is it possible? 3

4. Greedy geometric forwarding as routing using only local information • Network topology is embedded in a geometric space • To reach a destination, each node forwards the packet to the neighbor that is closest to the destination in the space 4

5. Hidden space visualized

6. Desired properties of greedy forwarding, and related metrics • Property 1: Greedy routes should never get stuck at local minima, nodes that do not have any neighbor closer to the destination than themselves • Success ratio, percentage of successful greedy paths reaching their destinations, should be close to 1 • Property 2: Greedy paths should be close to shortest paths • Stretch, ratio of the lengths of greedy to shortest paths, should be also close to 1 • Property 3: Even if topology changes, success ratio and stretch should stay close to 1 without any recomputation (e.g., without nodes changing their positions in the space) 6

7. Problem formulation (high-level) • Find a combination of network topology and underlying geometric space which would satisfy these desired properties • Any suggestions? • Nature offers some: many dynamic networks in nature and society do route information without any topology knowledge (brain, regulatory, social networks, etc.) • All these complex networks have power-law degree distributions (scale-free) and strong clustering (many triangular subgraphs) • Let’s focus on these topologies (which, luckily, also characterize the Internet and P2P networks) • But what about the underlying space? 7

8. Conjecture: space is hyperbolic • Nodes in real complex networks can often be classified hierarchically • Hierarchies are tree-like structures • Hyperbolic geometry is the geometry of tree-like structures • Formally: trees embed almost isometrically in hyperbolic spaces, not in Euclidean ones 8

9. Main hyperbolic property: the exponential expansion of space Circle length and disc area grow with their radius R as~ eR They are exactly2 sinh R2 (cosh R  1) The numbers of nodes in a tree at or within R hops from the root grow as~bRwhere b is the tree branching factor The metric structures of hyperbolic spaces and trees are essentially the same 9

10. Problem formulation (low-level) • Verify the conjecture: check if hyperbolic geometry, in the simplest possible settings, can naturally give rise to scale-free, strongly clustered topologies • Check if greedy forwarding satisfies the desired properties in the resulting embedding 10

11. Outline • Motivation • Greedy forwarding in the model • Conclusion Model of scale-free networks embeddedin hyperbolic metric spaces

12. The model

13. The model (cont.)

14. 14

15. Average node degree at distance r from the disc center

16. Node degree distribution

17. Model vs. AS Internet

18. Growing networks • The model can be adjusted for networks growing in hyperbolic spaces • All results stay the same 18

19. Outline Motivation Model of scale-free networks embeddedin hyperbolic metric spaces Conclusion Greedy forwarding in the model 19

20. Two greedy forwarding algorithms • Original Greedy Forwarding (OGF): select closest neighbor to destination, drop the packet if no one closer than current hop • Modified Greedy Forwarding (MGF): select closest neighbor to destination, drop the packet if a node sees it twice

21. Property 1:success ratio

22. Property 2:average and maximum stretch

23. Property 3: Robustness of greedy forwarding w.r.t. network dynamics • Scenario 1: Randomly remove a percentageof links and compute the new success ratio • Scenario 2: Remove a link and compute the percentage of paths that were going through it and are still successful (that is, the percentage of paths that found a by-pass)

24. Percentage of successful paths (dynamic networks, scenario 1)

25. Percentage of successful paths (dynamic networks, scenario 2)

26. Shortest paths in scale-free graphs and hyperbolic spaces 26

27. Outline Motivation Model of scale-free networks embeddedin hyperbolic metric spaces Greedy forwarding in the model Conclusion 27

28. Conclusion (low-level) • Hyperbolic geometry naturally explains the two main topological characteristics of complex networks • scale-free degree distributions • strong clustering • Greedy forwarding in complex networks embedded in hyperbolic spaces is exceptionally efficient 28

29. Conclusion (mid-level) • Complex network topologies are naturally congruent with hyperbolic geometries • Greedy paths follow shortest paths that approximately follow geodesics in the hyperbolic space • Both topology and geometry are tree-like • This congruency is robust w.r.t. topology dynamics • There are many link/node-disjoint shortest paths between the same source and destination that satisfy the above property • Strong clustering (many by-passes) boosts up the path diversity • If some of shortest paths are damaged by link failures, many others remain available, and greedy routing still finds them 29

30. Conclusion (high-level) • To efficiently route without topology knowledge, the topology should be both hierarchical (tree-like) and have high path diversity (not like a tree) • Complex networks do borrow the best out of these two seemingly mutually-exclusive worlds • Hidden hyperbolic geometry naturally explains how this balance is achieved 30

31. Implications • Greedy forwarding mechanisms in these settings may offer virtually infinitely scalable information dissemination (routing) strategies for communication networks • Zero communication costs (no routing updates!) • Constant routing table sizes (coordinates in the space) • No stretch (all paths are shortest, stretch=1) 31

32. Applications • Internet routing (hard): need to reverse the problem and find an embedding for a given Internet topology first • Overlay networks: • with underlay (easier; examples: existing P2P): have freedom of constructing a name space and its embedding according to the model, so that all the desired properties are satisfied • without underlay (harder; examples: CCN, pocket switching): need to make sure that the underlay network topology and its dynamics are congruent with the overlay name space and its dynamics 32