*Greedy Forwarding inDynamic Scale-Free NetworksEmbedded in Hyperbolic Metric SpacesDmitri Krioukov CAIDA/UCSDJoint work withF. Papadopoulos, M. Bogu, A. Vahdat

*Outline

Model of scale-free networks embedded in hyperbolic metric spacesGreedy forwarding in the modelConclusion

Motivation

MotivationRouting overhead is a serious scaling limitation in many networks (Internet, wireless, overlay/P2P networks, etc.)Search for infinitely scalable routing without any overheadDo not propagate any information about changing topologyRoute without any global topology knowledge, using only local informationHow is it possible?*

Greedy geometric forwarding as routing using only local informationNetwork topology is embedded in a geometric spaceTo reach a destination, each node forwards the packet to the neighbor that is closest to the destination in the space*

*Hidden space visualized

Desired properties of greedy forwarding, and related metricsProperty 1: Greedy routes should never get stuck at local minima, nodes that do not have any neighbor closer to the destination than themselvesSuccess ratio, percentage of successful greedy paths reaching their destinations, should be close to 1Property 2: Greedy paths should be close to shortest pathsStretch, ratio of the lengths of greedy to shortest paths, should be also close to 1Property 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)*

Problem formulation (high-level)Find a combination of network topology and underlying geometric space which would satisfy these desired propertiesAny 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)Lets focus on these topologies (which, luckily, also characterize the Internet and P2P networks)But what about the underlying space?*

Conjecture: space is hyperbolicNodes in real complex networks can often be classified hierarchicallyHierarchies are tree-like structuresHyperbolic geometry is the geometry of tree-like structuresFormally: trees embed almost isometrically in hyperbolic spaces, not in Euclidean ones

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*Main hyperbolic property: the exponential expansion of spaceCircle length and disc area grow with their radius R as ~ eRThey are exactly 2 sinh R 2 (cosh R 1)The numbers of nodes in a tree at or within R hops from the root grow as ~ bR where b is the tree branching factorThe metric structures of hyperbolic spaces and trees are essentially the same

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 topologiesCheck if greedy forwarding satisfies the desired properties in the resulting embedding*

*Outline

Motivation

Greedy forwarding in the modelConclusion

Model of scale-free networks embedded in hyperbolic metric spaces

*The model

*The model (cont.)

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*Average node degree at distance r from the disc center

*Node degree distribution

*Model vs. AS Internet

Growing networksThe model can be adjusted for networks growing in hyperbolic spacesAll results stay the same*

*Outline

MotivationModel of scale-free networks embedded in hyperbolic metric spaces Conclusion

Greedy forwarding in the model

*Two greedy forwarding algorithmsOriginal Greedy Forwarding (OGF): select closest neighbor to destination, drop the packet if no one closer than current hopModified Greedy Forwarding (MGF): select closest neighbor to destination, drop the packet if a node sees it twice

*Property 1:success ratio

*Property 2:average and maximum stretch

*Property 3: Robustness of greedy forwarding w.r.t. network dynamicsScenario 1: Randomly remove a percentage of links and compute the new success ratioScenario 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)

*Percentage of successful paths (dynamic networks, scenario 1)

*Percentage of successful paths (dynamic networks, scenario 2)

Shortest paths in scale-free graphs and hyperbolic spaces*

*Outline

MotivationModel of scale-free networks embedded in hyperbolic metric spacesGreedy forwarding in the model Conclusion

Conclusion (low-level)Hyperbolic geometry naturally explains the two main topological characteristics of complex networksscale-free degree distributionsstrong clusteringGreedy forwarding in complex networks embedded in hyperbolic spaces is exceptionally efficient*

Conclusion (mid-level)Complex network topologies are naturally congruent with hyperbolic geometriesGreedy paths follow shortest paths that approximately follow geodesics in the hyperbolic spaceBoth topology and geometry are tree-likeThis congruency is robust w.r.t. topology dynamicsThere are many link/node-disjoint shortest paths between the same source and destination that satisfy the above propertyStrong clustering (many by-passes) boosts up the path diversityIf some of shortest paths are damaged by link failures, many others remain available, and greedy routing still finds them *

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 worldsHidden hyperbolic geometry naturally explains how this balance is achieved*

ImplicationsGreedy forwarding mechanisms in these settings may offer virtually infinitely scalable information dissemination (routing) strategies for communication networksZero communication costs (no routing updates!)Constant routing table sizes (coordinates in the space)No stretch (all paths are shortest, stretch=1)*

ApplicationsInternet routing (hard): need to reverse the problem and find an embedding for a given Internet topology firstOverlay 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 satisfiedwithout 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*

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