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1 Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces Dmitri Krioukov CAIDA/UCSD Joint work with F. Papadopoulos, M. Boguñá, A. Vahdat

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

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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

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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

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5 Hidden space visualized

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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

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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

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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

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9 Main hyperbolic property: the exponential expansion of space Circle length and disc area grow with their radius R as ~ e R They 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 ~ b R where b is the tree branching factor The metric structures of hyperbolic spaces and trees are essentially the same

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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

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

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12 The model

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13 The model (cont.)

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15 Average node degree at distance r from the disc center

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16 Node degree distribution

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17 Model vs. AS Internet

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

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

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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

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21 Property 1: success ratio

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22 Property 2: average and maximum stretch

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23 Property 3: Robustness of greedy forwarding w.r.t. network dynamics Scenario 1: Randomly remove a percentage of 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)

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

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

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Shortest paths in scale-free graphs and hyperbolic spaces 26

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

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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

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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

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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

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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

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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

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