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1 Complex Networks: Connectivity and Functionality Milena Mihail Georgia Tech.

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2 Search and routing networks, like the WWW, the internet, P2P networks, ad-hoc (mobile, wireless, sensor) networks are pervasive and scale at an unprecedented rate. Performance analysis/evaluation in networking: measure parameters hopefully predictive of performance. Important in network simulation and design. Which are critical network parameters/metrics that determine algorithmic performance? Predictive of routing and searching performance is conductance, expansion, spectral gap.

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3 How can network models capture the parameters/metrics that are critical in network performance? Can we design network algorithms/protocols that optimize these critical network parameters? This talk: The case of internet routing topology This talk: The case of P2P networks

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4 The case of modeling the internet routing topology Nodes are routers or Autonomous Systems Two nodes connected by a link if they are involved in direct exchange of traffic Sparse small-world graphs with large degree-variance But are degrees the right parameter to measure? Current Models for Internet Routing Topologies focus on large degree-variance Erdos-Renyi-like, Configurational : A random graph with given degrees Evolutionary, macroscopic and microscopic : The graph grows one vertex at a time and attaches preferentially to degrees or according to some optimization criterion Chung&Lu Barabasi&Albert Bollobas&Riordan Fabrikant,Koutsoupias,Papadimitriou

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5 An important metric: Conductance and the second eigenvalue of the stochastic normalization of the adjacency matrix characterize routing congestion under link capacities, mixing rate, cover time. Leighton-Rao Jerrum-Sinclair How does the second eigenvalue (more generally the principal eigenvalues) scale as the size of the network increases? Broder-Karlin computationally soft Matlab does 1-2M node sparse graphs

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6 Open problem: Erdos-Renyi like, configurational models which include spectral gap parameter? This is also another point of view of the small-world phenomenon random graph configurational model Gkantsidis,M,Zegura M,Papadimitriou,Saberi Gkantsidis,M,Saberi Second eigenvalue of internet is larger than that of random graphs but spectral gap remains constant as number of nodes increases. This also says that congestion under link capacities scales smoothly

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7 Some evolutionary random graph models may capture clustering One vertex at a time New vertex attaches to existing vertices Growth & Preferential Attachment

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8 Open Problem: characterize clustering as a function of model parameter ie, indicate which parameter ranges are important in simulations ? plots as number of nodes increases M,Saberi,Papadimitriou Flaxman,Frieze,Vera

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9 real network random graph, evolutionary model random graph, configurational model Other discrepancies of random graph models from real internet topologies: high degree nodes away from network core what do internet topologies optimize ? Li, Alderson, Willinger, Doyle high degrees mostly connected to low degrees core of low degrees

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10 Given total length l and n random points in a metric space construct a graph with total link length l that - maximizes spectral gap, conductance - minimizes congestion under node capacities Open Problem: Research direction: Algorithms improving congestion conductance and spectral gap Boyd&Saberi Rao&Vazirani

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11 Algorithms optimizing connectivity How do you maintain a P2P network with good search performance ?

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12 The case of Peer-to-Peer Networks n nodes, d-regular graph each node has resources O(polylogn) and knows a constant size neighborhood Distributed, decentralized Search for content, e.g. by flooding or random walk ? Must maintain well connected topology, e.g. a random graph, an expander Chawathe&al Gkantsidis&al Lv&al Jerrum-Sinclair Broder-Karlin

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13 Theorem [ Feder, Guetz, M, Saberi 06 ]: The Markov chain on d-regular graphs is rapidly mixing, even under local 2-link switches or flips. P2P Network Topology Maintenance by Constant Randomization Theorem [ Cooper, Frieze & Greenhill 04 ]: The Markov chain corresponding to a 2-link switch on d-regular graphs is rapidly mixing. Question: How does the network pick a random 2-link switch? In reality, the links involved in a switch are within constant distance. random graph, expander Gnutella: constantly drops existing connections and replaces them with new connections

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14 Space of d-regular graphs general 2-link switch Markov chain Space of connected d-regular graphs local Flip Markov chain Define a mapping from to such that (a) (b) each edge in maps to a path of constant length in The proof is a Markov chain comparison argument

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15 Question: How do we add new nodes with low network overhead? Question: How do we delete nodes with low network overhead? ? ? ? Gkantsidis,M,Saberi Padurangan,Raghavan,Upfal Law,Siu Ajtai,Komlos,Szemeredi Impagliazzo,Zuckerman

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16 Link Criticality Algorithms developing topology awareness Boyd,Diaconis,Xiao

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17 Generalized Search: 7$ 3$ 2S 1$ local information local information A node has a query and a budget Arbitrarily partition the remaining budget and forward the parts to the neighbors Subtract 1 from budget Link Criticality Gkantsidis,M,Saberi Boyd,Diaconis,Xiao

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18 Fastest Mixing Markov Chain Boyd,Diaconis,Xiao s.t. Let be a graph. Assign symmetric transition probabilities to links in (and self loops) so that the resulting matrix is stochastic and the second in absolute value largest eigenvalue is minimized. SDP formalization

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19 Fastest Mixing Markov Chain Subgradient Algorithm is some vector on of initial transition probabilities is the eigenvector corresponding to second in absolute value largest eigenvalue is a vector on with repeat subgradient step projection to feasible subspace Open Question: Is there a decentralized implementation or algorithm?

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20 How does Capacity/Throughput/Delay Scale? Mobility Increases Capacity, Grossgaluser & Tse, 2001 Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003 Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004 The Case of Ad-Hoc Wireless Networks Capacity of Wireless Networks, Gupta & Kumar, 2000Is there a connection with Lipton & Tarjans separators for planar graphs?

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