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IEEE ICDCS, Toronto, Canada, June 2007 (LA-UR-06-8032) 1 Scale-Free Overlay Topologies with Hard Cutoffs for Unstructured Peer-to-Peer Networks Hasan Guclu Los Alamos National Laboratory guclu@lanl.gov Murat Yuksel University of Nevada – Reno yuksem@cse.unr.edu

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IEEE ICDCS, Toronto, Canada, June 2007 2 Outline Motivation and Problem Statement Topology Generation Mechanisms Barabási-Albert (Preferential Attachment) Model Configuration Model Hop-and-Attempt Preferential Attachment Discover-and-Attempt Preferential Attachment Search Methods Flooding Normalized Flooding Random Walk Summary and Conclusions

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IEEE ICDCS, Toronto, Canada, June 2007 3 Motivation Diameter d Exponent Number of stubs m O(lnln N)(2,3)≥1 O(ln N/lnln N)3≥2 O(ln N)31 >3≥1 Search Efficiency vs. Exponent and Connectedness Ultra-small Small-world Characteristics of the p2p overlay topology has significant effects on the search performance.

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IEEE ICDCS, Toronto, Canada, June 2007 4 Motivation Key Question: How to construct the overlay topology by using local information in p2p nets such that the search efficiency is good? Scale-freeness (i.e. power-law exponent) is related to search efficiency Key Constraints: No global knowledge No peer wants to take on the load – hard cutoff on the degree When a new peer joins, how should it construct its list of neighbors? A local decision affecting global behavior (emergence).

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IEEE ICDCS, Toronto, Canada, June 2007 Fat-tailed power-law degree distribution: No typical scale Two well-known topology generation algorithms: Preferential Attachment (PA) by Barabasi and Albert. Dynamic model (fixed exponent) Configuration Model (CM) Static model Pre-defined degree distribution with a parameterized exponent 5 Scale-Free Topologies

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IEEE ICDCS, Toronto, Canada, June 2007 Definition of natural cutoff: For scale-free networks with power-law degree distribution (m: minimum degree) Natural cutoff Natural cutoff for PA model ( ) Hard cutoff is the value of the maximum degree imposed on nodes. 6 Natural and Hard Cutoff

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IEEE ICDCS, Toronto, Canada, June 2007 Preferential attachment (Barabási- Albert, PA) model (PA) Configuration model (CM) Hop-and-attempt PA model (HAPA) Discover-and-attempt PA model (DAPA) 7 Network Generation Mechanisms

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IEEE ICDCS, Toronto, Canada, June 2007 8 Preferential Attachment (PA) Connect to an existing peer with probability proportional to its current degree. prefer the peers with larger degree simply skip the existing peers already saturated their hard cutoffs Requires global info

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IEEE ICDCS, Toronto, Canada, June 2007 PA with Hard Cutoff At steady state: Total rate: Probability to connect to the nodes with degree k

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IEEE ICDCS, Toronto, Canada, June 2007 PA with Hard Cutoff

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IEEE ICDCS, Toronto, Canada, June 2007 PA with Hard Cutoff

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IEEE ICDCS, Toronto, Canada, June 2007 12 Configuration Model (CM) Given a target hard cutoff and a power-law exponent, generate the perfect scale-free degree distribution… allows multiple links and self loops may have disconnected components not practical, but does generate the best possible scale-freeness within the hard cutoff constraint – i.e., good for studying

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IEEE ICDCS, Toronto, Canada, June 2007 13 Hop-and-attempt PA Model (HAPA) At every time step a new node is added to the network This new node attempts to connect to a randomly chosen existing node A by using the preferential attachment rule Then it attempts to connect to a randomly chosen node B which is a neighbor of A The node repeats this procedure until it fills all its stubs (or the number of links it has reaches m)

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IEEE ICDCS, Toronto, Canada, June 2007 14 Discover-and-attempt PA Model (DAPA) First, a substrate network with a specific topology and a large number of nodes (we use geometric random network) is generated A finite number of nodes are selected randomly and put into p2p network which is empty at the beginning A node is randomly selected from the substrate network and let it send a broadcasting message to its neighbors reachable in sub steps The selected node finds all the nodes in its horizon belonging to the peers network and attempts to connect by using the preferential attachment rule until having m links if possible If it is connected to at least one peer it is added to the peers network This process is repeated until the number of peers reaches to the number desired

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IEEE ICDCS, Toronto, Canada, June 2007 15 Discover-and-attempt PA Model (DAPA)

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IEEE ICDCS, Toronto, Canada, June 2007 16 ProcedureGlobal Information PAYes CMYes HAPAPartial DAPANo Global versus Local Information

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IEEE ICDCS, Toronto, Canada, June 2007 17 Search Methods Flooding Source node sends a message to all its neighbors and every node which receives the message forwards it to all its neighbors except the node the message is received from until the target node receives the message Normalized flooding Similar to flooding but the nodes send the messages to at most m (minimum number of links in the network) neighbors Random walk Similar to flooding but the nodes send the messages only to one of their neighbors except the source node

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IEEE ICDCS, Toronto, Canada, June 2007 18 Flooding PA is better due to nodes at the edge

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IEEE ICDCS, Toronto, Canada, June 2007 19 Flooding HAPA rocks, DAPA not bad

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IEEE ICDCS, Toronto, Canada, June 2007 20 Normalized Flooding PA likes cutoff, CM does not.

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IEEE ICDCS, Toronto, Canada, June 2007 21 Normalized Flooding The lower the cutoff the better the performance

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IEEE ICDCS, Toronto, Canada, June 2007 22 Normalized Flooding Cutoff is goooood. Not so short-sighted network gives good results.

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IEEE ICDCS, Toronto, Canada, June 2007 23 Random Walk (The same number of messages in NF and RW) PA likes cutoff, CM does not.

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IEEE ICDCS, Toronto, Canada, June 2007 24 Random Walk The lower the cutoff the better the performance

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IEEE ICDCS, Toronto, Canada, June 2007 25 Random Walk The lower the cutoff the better the performance.

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IEEE ICDCS, Toronto, Canada, June 2007 26 Conclusions In flooding the lower the hard cutoff the lower the number of hits. HAPA without cutoff does especially good in flooding due to the star-like topology. Increasing the minimum degree eliminates the negative effect of the hard cutoff. There exists an interplay between connectedness (m) and the degree distribution exponent if there is a hard cutoff, except CM. Harder cutoffs may improve search efficiency in normalized flooding and random walk except CM. Extended version of the paper in http://arxiv.org/abs/cs/0611128 Acknowledgments DOE (DE-AC52-06NA25396), NSF (0627039) and Sid Redner (Boston University).

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IEEE ICDCS, Toronto, Canada, June 2007 27 Thank you! THE END

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