Scalable overlay Networks

Transcription

1 overlay Networks Dr. Samu Varjonen 1

2 Lectures MO C122 Introduction. Exercises. Motivation. TH DK117 Unstructured networks I MO C122 Unstructured networks II TH DK117 Bittorrent and evaluation MO C122 Privacy (Freenet etc.) and intro to power-law networks. TH DK117 Consistent hashing. Distributed Hash Tables (DHTs) MO C122 DHTs continued TH DK117 Power-law networks MO C122 Power-law networks and applications. TH DK117 Applications I MO C122 Applications II TH DK117 Advanced topics MO C122 Conclusions and summary TH DK117 Reserved 2

3 Contents Internet Indirection Architecture (i) Error and Attack Tolerance of Complex Networks Navigability of Complex Networks Mathematics and the Internet: A Source of Enormous Confusion and Great Potential 3

4 Internet Indirection Infrastructure (I3) Ion Stoica et al Internet indirection infrastructure. In SIGCOMM '02. 4

5 Recap of Chord Distributed lookup protocol Assign a unique m-bit key (identifier) to a node Every node has predecessor and successor A key k is assigned to a node whose identifier is equal to or follows k the identifier space. Consistent hashing (of IP address) generates keys Identifiers (nodes) ordered in a circle module as 2m Given a key, map the key to a node k stored on successor(k) Routing table (at most m entries on each node) ith entry first node succeeds node by at least 2 i-1 5

6 Chord Properties Each node responsible for K/N keys K total #keys, N total #nodes When a node joins or leaves the network only O(K/N) keys will be relocated Relocation is local to the node Lookups take O(log N) messages O(log2 N) messages required to re-establish routing invariants after join/leave Each node's successor is correctly maintained For every key (k), the node responsible for k is successor(k) 6

7 Chord finger table Finger Maps to Real node 1,2,3 x+1,x+2,x+4 N14 4 x+8 N21 5 x+16 N32 6 x+32 N42 m=6 Current node at p for j=1,...,m the fingers of p+2j-1 2m-1 0 N56 N1 Predecessor node N8 N N14 N N21 N38 N

8 Chord lookup m=6 2m-1 0 N56 N1 N8 N51 N14 N42 N21 N38 N32 8

9 Recap of Chord Distributed lookup protocol Assign a unique m-bit key (identifier) to a node Every node has predecessor and successor A key k is assigned to a node whose identifier is equal to or follows k the identifier space. Consistent hashing (of IP address) generates keys Identifiers (nodes) ordered in a circle module as 2m Given a key, map the key to a node k stored on successor(k) Routing table (at most m entries on each node) ith entry first node succeeds node by at least 2 i-1 9

10 Packet's Perspective of Internet Services Unicast: One fixed source to one fixed destination Broadcast: One source to all destinations Multicast: One fixed source to multiple destinations who are part of a group Anycast: One source to exactly one destination who is a member of a group Internet Services using Unicast, Broadcast, Multicast, and Anycast are built over the point-to-point abstraction What would be another abstraction? 10

11 Rendezvous-based Communication Source sends packets to a logical identifier. Receivers express interest in packets sent to an identifier Packet is a pair (id, data) id data Receivers use triggers (id, addr) to express interest id host/object/session/ m bits Forward packet with identifier (id) to receiver with IP address (addr) Packet sent to receivers if the the interest (idt) from receiver is a longest prefix match the match is longer than matching threshold k (k < m) Abstraction decouples the act of sending from the act of receiving 11

12 API to Implement Indirection SendPacket (p) InsertTrigger(t) RemoveTrigger(t) API Implemented in an i3 Overlay Network Overlay Consists of i3 Servers Store Triggers Forward packets using IP between i3 nodes and endhosts Packets are not stored at the Servers Implemented using Chord (could be any other DHT) 12

13 R inserts a trigger (id, R) and receives all packets with identifier id. Mobility is transparent for the sender the host changes its address from R1 to R2, it updates its trigger from (id, R1) to (id, R2). Source: 13

14 A multicast tree using a hierarchy of triggers Source: 14

15 Anycast using the longest matching prefix rule. Source: 15

16 Benefits of i3 Support for mobility On moving to new address (addr'), receiver sends new trigger (id, addr') Receivers periodically refresh triggers Multicast Source is agnostic to the set of receivers Receiver agnostic to the set of sources Trigger chains can be used to minimize triggers Anycast Id contains a common prefix component and a suffix Anonymity Service Composition Stacked identifiers 16

17 Modeling Overlay Networks (contd) 17

18 Recap Milgram's Experiment Duncan Watts Random Rewiring Model Scale Free Networks (Power-Law Networks) Preferential attachment Evolving Copying Model (Copying Generative Model) Navigation in Small World Complex Networks Overlay Networks P2P 18

19 Error and Attack Tolerance of Complex Networks Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no (2000):

20 Scale-Free Model for ASGraph High Degree Nodes (Hubs) AS Topology of skitter dataset parsed by SNAP team skitter.html 20

21 Importance of Hubs (Random Graph) Albert, Réka, et al. "Error and attack tolerance of complex networks." (2000): nature 406, no

22 Error vs Attack Error (Node Failure) random node fails (malfunction) Attack Selected node with a given property is made to fail Which nodes would you target if you knew the network is a scale-free network? Nodes with the highest degree 22

23 Graph Diameter Impact of Errors and Attacks (Graph Diameter) Why? Fraction of nodes removed Changes in diameter when a small fraction f of the nodes is removed The malfunctioning (absence) of any node in general increases the distance between the remaining nodes, as it can eliminate some paths that contribute to the system's interconnectedness Albert, Réka, et al. "Error and attack tolerance of complex networks." (2000): nature 406, no

24 Graph Diameter Impact of Errors (exponential) Fraction of nodes removed The exponential network s diameter increases monotonically with f despite its redundant wiring it is increasingly difficult for the remaining nodes to communicate with each other. This behaviour is rooted in the homogeneity of the network Since all nodes have approximately the same number of links They all contribute equally to the network's diameter The removal of each node causes the same amount of damage. 24

25 Graph Diameter Impact of Errors (scale-free) Fraction of nodes removed Scale-free network the diameter remains unchanged under an increasing level of errors Even when as many as 5% of the nodes fail, the communication between the remaining nodes in the network is unaffected. This robustness of scale-free networks is rooted in their extremely inhomogeneous connectivity distribution because the power-law distribution implies that the majority of nodes have only a few links, nodes with small connectivity will be selected with much higher probability. The removal of these small nodes does not alter the path structure of the remaining nodes, and thus has no impact on the overall network topology. But the attacker would try to take out the hubs... 25

26 Graph Diameter Impact of attacks (exponential) Fraction of nodes removed Informed agent that attempts to deliberately damage a network will not eliminate the nodes randomly Attacker will preferentially target the most connected nodes. To simulate an attack the most connected nodes (degree k) are removed first and then in decreasing order Measuring the diameter of an exponential network under attack there is no substantial difference whether the nodes are selected randomly or in decreasing order of connectivity Owing to the homogeneity of the network 26

27 Graph Diameter Impact of attacks (scale-free) Fraction of nodes removed Drastically different behaviour is observed for scale-free networks When the most connected nodes are eliminated, the diameter of the scale-free network increases rapidly Doubling its original value if 5% of the nodes are removed This vulnerability to attacks is rooted in the inhomogeneity of the connectivity distribution The connectivity is maintained by a few highly connected nodes whose removal drastically alters the network's topology, and decreases the ability of the remaining nodes to communicate with each other. 27

28 Relative <s> and S Impact of Errors and Attacks (Size of Largest Cluster) Fraction of nodes removed When nodes are removed from a network Clusters of nodes whose links to the system disappear may be cut off (fragmented) from the main cluster Size of the largest cluster S Fraction f of the nodes are removed either randomly or in an attack mode average size of isolated clusters s 28

29 Relative <s> and S Impact of Errors and Attacks (Size of Largest Cluster) Fraction of nodes removed When nodes are removed from a network It was found that for the exponential network, as f increases S displays a threshold-like behaviour such that for f > fec 0.28 we have S 0 That is, all the clusters except the largest one finding that s increases rapidly until s 2 at fec Similar behaviour is observed when we monitor the average size s of the isolated clusters After which it decreases to s = 1. 29

30 Relative <s> and S Impact of Errors and Attacks (Size of Largest Cluster) Fraction of nodes removed Response of a scale-free network to attacks and failures is rather different For random failures no threshold for fragmentation is observed, instead, the size of the largest cluster slowly decreases. s 1 for most f values indicates that the nodes are breaking off one by one In contrast with the catastrophic fragmentation of the exponential network at fec, the scale-free network stays together as a large cluster for very high values of f Robust to failures Swifter degradation on attacks Scale-free networks are topological stabel under random failures and fall apart only after the main cluster has been completely deflated (error tolerance). The response to attack of the scale-free network is similar (but swifter) to the response to attack and failure of the exponential network Critical threshold fsfc 0.18, smaller than the value fec 0.28 observed for the exponential network, the system breaks apart, forming many isolated clusters. 30

31 Network Response to Attacks and Failures f=0 f 0.05 f 0.18 f 0.45 f=1 Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no (2000):

32 Critical Threshold (random node failures) m=1 Cohen's technique can be extended to errors (No closed form for fc for errors ) Cohen, Reuven et al. "Resilience of the Internet to random breakdowns." Physical review letters 85, no. 21 (2000):

33 Summary on Attack and Error Tolerance of Complex Networks Scale-free networks resilient to random failures but vulnerable to targetted attacks 33

34 Navigability of Complex Networks Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009):

35 Clustering of Nodes Serrano, M. Angeles, Dmitri Krioukov, and Marián Boguná. "Self similarity of complex networks and hidden metric spaces." Physical review letters 100, no. 7 (2008):

36 Clustering of Nodes Degree-thresholding renormalization procedure: Produces a hierarchy of subgraphs within a given graph G as above For each degree threshold kt = 0, 1, 2..., First extract from G the subgraph G(k T ) induced by nodes with degrees k > kt. Second, for each node in G(kT ), Compute its internal degree ki, i.e., the number of links that connect a given node to other nodes in G(kT ) Finally, rescale ki s by the average internal degree to obtain the rescaled quantity k i/ ki(kt). ki(kt) in G(kT ) Serrano, M. Angeles, Dmitri Krioukov, and Marián Boguná. "Self similarity of complex networks and hidden metric spaces." Physical review letters 100, no. 7 (2008):

37 Clustering of Nodes Average clustering coefficient as a function of the threshold degree kt for renormalized real networks and their randomized counterparts. Internal average degree as a function of kt for the same networks. Constant clustering coeff. Average Degree Increases More similar nodes are more likely to be connected How to Generate Scale-Free Graphs with Strong Clustering? Serrano, M. Angeles, Dmitri Krioukov, and Marián Boguná. "Self similarity of complex networks and hidden metric spaces." Physical review letters 100, no. 7 (2008):

38 Generating Scale-Free Graphs with Strong Clustering Node similarity and hidden metric spaces Milgram s experiment Social networks are navigable without global information. The only information that people used to make their routing decisions in Milgram s experiment was a set of descriptive attributes of the destined recipient, such as place of living and occupation. In many networks observed in nature, including those in society and biology (signalling pathways, neural networks, etc.), nodes efficiently find intended communication targets even though they do not possess any global view of the system. 40

39 Generating Scale-Free Graphs with Strong Clustering Take all nodes and uniformly distribute them on underlying ring (hidden metric space) Assign each node its expected degree k from power-law degree distribution Connect each pair of nodes with probability r(d;k,k ) d is the distance between the nodes in the circle dc ~ kk is the characteristic distance scale Probability of link connection between nodes decreases with hidden distance between them and increases with their degree 41

40 Generating Scale-Free Graphs with Strong Clustering Hubs will be connected with a high probability because of large dc Low degree nodes connected only if d is small Hubs connected to low degree nodes at moderate distance parameter determines the hidden distance Larger the more preferred are connections between close nodes in hiddent space Pairs of high-degree nodes are connected with high propability regardles of distance Pairs of low-degree nodes are connected only if their d is close to dc 42

41 Path Length Avg. Length of Greedy Routing Paths (Greedy Routing) Network Size Exponent Path length grows polylogarithmically with the network size Paths shorter for smaller exponents and stronger clustering Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009):

42 Greedy Routing Hidden Space as the coordinate space Hidden space is circle in this example Greedy Routing: Send to neighbor who is closer to the destination (in hidden space) Unsuccessful Paths: None of your neighbors are closer to the destination in the hidden space 44

43 Success Probability (Greedy Routing) P(success) increases with N when exp < 2.6 decays if exp >2.6 P(success) decays with N Weak clustering Strong clustering P(success) intertwined Maximises path success if exponent and clustering above 1.5 N = 105 Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009):

44 Navigation in Scale Free Networks PGP BGP 46

45 Implications of Result Internet Routing Routers currently exchange signals to keep coherent view of network Network size increasing with time (leads to increased signal traffic) Hidden metric space eliminates the need for control signals exchanged to notify changes in network How to proceed to discover the hidden metric space Does Shortest Path imply Shortest Time to destination? What happens in case of congestion at hubs? 47

46 Mathematics and the Internet: A Source of Enormous Confusion and Great Potential W Willinger et al. Mathematics and the internet: A source of enormous confusion and great of the AMS potential. In Notices 48

47 Scale-Free Model for ASGraph AS Topology of skitter dataset parsed by SNAP team skitter.html 49

48 Is the Scale-Free Internet A Myth? What we have seen till now wrt to Preferential Attachment Preferential attachment results in Hubs Hubs vulnerable to coordinated attacks Why is the Internet still up and running Is the Scale-Free modeling paradigm consistent with the engineered nature of the Internet and the design constraints imposed by existing technology? Is the simplistic toy model too generic? Do the available measurements, their analysis, and their modeling efforts support the claims made by Error and Attack Tolerance paper? 50

49 Importance of Measurements Tool for measurement study for AS-measurements Traceroute Biases of traceroute Uses IPv4 Protocol What about non-ipv4 protocols like MPLS? What about IPv6 Entry points to non-ipv4 regions can aggregate to Hubs Only reports the interfaces traversed by the packet Routers can have multiple interfaces and appear on different routes with different IP addresses 51

50 Leverage Domain Knowledge Device Constraints What about Overlay Networks? Finite capacity of routers Placement of High Degree Nodes Finite number of interfaces on routers Edge vs Core How would you deploy the network if you are a network engineer? Leverage domain knowledge to identify driving forces behind the design of high engineered systems such as the Internet 52

51 Summary (Modeling Overlay Networks) 54

52 Recap of Modeling Overlay Networks Milgram's Experiment Duncan Watts Random Rewiring Model Scale-Free Networks Preferential attachment Evolving Copying Model (Copying Generative Model) Scale-Free with Strong Clustering Error and Fault Tolerance of Complex Networks Navigation (Greedy Routing) In Small World (Kleinberg's Small World) In Complex Networks (Scale-Free with Strong Clustering) Mathematics and the Internet: A Source of Enormous Confusion and Great Potential 55

53 Commonly used metrics Clustering Coefficient Diameter Degree Distribution 56

54 Methodology 1)Make observations (conduct measurement studies) 2)Build model to explain observations Choose the right level of granularity (zoom level) Strip the problem to a simple form Attempt to formulate the problem and model the system 3)Validate model Reproduce observations/measurements Explain observations 4)Revisit step 2 (and 1) to improve understanding 57

55 Important Articles Milgram, Stanley. "The small world problem." Psychology today 2.1 (1967): Watts, Duncan and Strogatz, Steven. "Collective dynamics of small world networks." Nature (1998): Barabási, Albert László, and Albert, Réka. "Emergence of scaling in random networks." Science 286, no (1999): Kleinberg, Jon. "The small world phenomenon: An algorithmic perspective." In ACM Symposium on Theory of computing, pp Ravi Kumar et al. "Stochastic models for the web graph." In Annual Symposium on Foundations of Computer Science, Albert, Réka, and Barabási, Albert László. "Statistical mechanics of complex networks." Reviews of modern physics 74.1 (2002): 47. Newman, Mark. "The structure and function of complex networks." SIAM review 45, no. 2 (2003): Mitzenmacher, M. (2004). "A brief history of generative models for power law and lognormal distributions." Internet mathematics, 1(2), Mark Newman. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46, no. 5 (2005): Jure Leskovec et al. "Graphs over time: densification laws, shrinking diameters and possible explanations." In ACM SIGKDD, pp Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009): W Willinger et al. Mathematics and the internet: A source of enormous confusion and great potential. In Notices of the AMS