Introduction to Dynamic Networks Models, Algorithms, and Analysis Rajmohan Rajaraman, Northeastern U. www.ccs.neu.edu/home/rraj/Talks/DynamicNetworks/DYNAMO/ June 2006 Dynamo Training School, Lisbon Introduction to Dynamic Networks 1 Many Thanks to Filipe Araujo, Pierre Fraigniaud, Luis Rodrigues, Roger Wattenhofer, and organizers of the summer school All the researchers whose contributions will be discussed in this tutorial Dynamo Training School, Lisbon
Introduction to Dynamic Networks 2 What is a Network? General undirected or directed graph Dynamo Training School, Lisbon Introduction to Dynamic Networks 3 Classification of Networks Synchronous: Messages delivered within one time unit Nodes have access to a common clock Asynchronous:
Message delays are arbitrary No common clock Dynamo Training School, Lisbon Static: Nodes never crash Edges maintain operational status forever Dynamic: Nodes may come and go Edges may crash and recover Introduction to Dynamic Networks 4 Dynamic Networks: What? Network dynamics:
The network topology changes over times Nodes and/or edges may come and go Captures faults and reliability issues Input dynamics: Load on network changes over time Packets to be routed come and go Objects in an application are added and deleted Dynamo Training School, Lisbon Introduction to Dynamic Networks 5 Dynamic Networks: How? Duration: Transient: The dynamics occur for a short period, after which the system is static for an extended time period Continuous: Changes are constantly occurring and the system has to constantly adapt to them
Control: Adversarial Stochastic Game-theoretic Dynamo Training School, Lisbon Introduction to Dynamic Networks 6 Dynamic Networks are Everywhere Internet The network, traffic, applications are all dynamically changing Local-area networks Users, and hence traffic, are dynamic Mobile ad hoc wireless networks Moving nodes
Changing environmental conditions Communication networks, social networks, Web, transportation networks, other infrastructure Dynamo Training School, Lisbon Introduction to Dynamic Networks 7 Adversarial Models Dynamics are controlled by an adversary Adversary decides when and where changes occur Edge crashes and recoveries, node arrivals and departures Packet arrival rates, sources, and destinations For meaningful analysis, need to constrain adversary Maintain some level of connectivity
Keep packet arrivals below a certain rate Dynamo Training School, Lisbon Introduction to Dynamic Networks 8 Stochastic Models Dynamics are described by a probabilistic process Neighbors of new nodes randomly selected Edge failure/recovery events drawn from some probability distribution Packet arrivals and lengths drawn from some probability distribution Process parameters are constrained Mean rate of packet arrivals and service time distribution moments Maintain some level of connectivity in network Dynamo Training School, Lisbon
Introduction to Dynamic Networks 9 Game-Theoretic Models Implicit assumptions in previous two models: All network nodes are under one administration Dynamics through external influence Here, each node is a potentially independent agent Own utility function, and rationally behaved Responds to actions of other agents Dynamics through their interactions Notion of stability: Nash equilibrium Dynamo Training School, Lisbon Introduction to Dynamic Networks
10 Design & Analysis Considerations Distributed computing: For static networks, can do pre-processing For dynamic networks (even with transient dynamics), need distributed algorithms Stability: Transient dynamics: Self-stabilization Continuous dynamics: Resources bounded at all times Game-theoretic: Nash equilibrium Convergence time Properties of stable states: How much resource is consumed? How well is the network connected? How far is equilibrium from socially optimal? Dynamo Training School, Lisbon Introduction to Dynamic Networks
11 Five Illustrative Problem Domains Spanning trees Transient dynamics, self-stabilization Load balancing Continuous dynamics, adversarial input Packet routing Transient & continuous dynamics, adversarial Queuing systems Adversarial input Network evolution Stochastic & game-theoretic Dynamo Training School, Lisbon Introduction to Dynamic Networks 12
Spanning Trees Dynamo Training School, Lisbon Introduction to Dynamic Networks 13 Spanning Trees One of the most fundamental network structures Often the basis for several distributed system operations including leader election, clustering, routing, and multicast Variants: any tree, BFS, DFS, minimum spanning trees Dynamo Training School, Lisbon Introduction to Dynamic Networks 14
Spanning Tree in a Static Network 8 7 3 2 1 6 5 4 Assumption: Every node has a unique identifier The largest id node will become the root Each node v maintains distance d(v) and next-hop h(v) to largest id node r(v) it is aware of: Node v propagates (d(v),r(v)) to neighbors If message (d,r) from u with r > r(v), then store (d+1,r,u) If message (d,r) from p(v), then store (d+1,r,p(v))
Dynamo Training School, Lisbon Introduction to Dynamic Networks 15 Spanning Tree in a Dynamic Network 8 7 3 2 1 5 6 4 Suppose node 8 crashes Nodes 2, 4, and 5 detect the crash
Each separately discards its own triple, but believes it can reach 8 through one of the other two nodes Can result in an infinite loop How do we design a self-stabilizing algorithm? Dynamo Training School, Lisbon Introduction to Dynamic Networks 16 Exercise Consider the following spanning tree algorithm in a synchronous network Each node v maintains distance d(v) and next-hop h(v) to largest id node r(v) it is aware of In each step, node v propagates (d(v),r(v)) to neighbors On receipt of a message: If message (d,r) from u with r > r(v), then store (d+1,r,u) If message (d,r) from p(v), then store (d+1,r,p(v))
Show that there exists a scenario in which a node fails, after which the algorithm never stabilizes Dynamo Training School, Lisbon Introduction to Dynamic Networks 17 Self-Stabilization Introduced by Dijkstra [Dij74] Motivated by fault-tolerance issues [Sch93] Hundreds of studies since early 90s A system S is self-stabilizing with respect to predicate P Once P is established, P remains true under no dynamics From an arbitrary state, S reaches a state satisfying P within finite number of steps Applies to transient dynamics Super-stabilization notion introduced for
continuous dynamics [DH97] Dynamo Training School, Lisbon Introduction to Dynamic Networks 18 Self-Stabilizing ST Algorithms Dozens of self-stabilizing algorithms for finding spanning trees under various models [Gr03] Uniform vs non-uniform networks Fixed root vs non-fixed root Known bound on the number of nodes Network remains connected Basic idea: Some variant of distance vector approach to build a BFS
Symmetry-breaking Use distinguished root or distinct ids Cycle-breaking Use known upper bound on number of nodes Local detection paradigm Dynamo Training School, Lisbon Introduction to Dynamic Networks 19 Self-Stabilizing Spanning Tree 8 7 3 2 1 6
5 4 Suppose upper bound N known on number of nodes [AG90] Each node v maintains distance d(v) and parent h(v) to largest id node r(v) it is aware of: Node v propagates (d(v),r(v)) to neighbors If message (d,r) from u with r > r(v), then store (d+1,r,u) If message (d,r) from p(v), then store (d+1,r,p(v)) If d(v) exceeds N, then store (0,v,v): breaks cycles Dynamo Training School, Lisbon Introduction to Dynamic Networks 20 Self-Stabilizing Spanning Tree Suppose upper bound N not known [AKY90] Maintain triple (d(v),r(v),p(v)) as before If v > r(u) of all of its neighbors, then store
(0,v,v) If message (d,r) received from u with r > r(v), then v joins this tree Sends a join request to the root r On receiving a grant, v stores (d+1,r,u) Other local consistency checks to ensure that cycles and fake root identifiers are eventually detected and removed Dynamo Training School, Lisbon Introduction to Dynamic Networks 21 Spanning Trees: Summary Model: Transient adversarial network dynamics Algorithmic techniques: Symmetry-breaking through ids and/or a distinguished root
Cycle-breaking through sequence numbers or local detection Analysis techniques: Self-stabilization paradigm Other network structures: Hierarchical clustering Spanners (related to metric embeddings) Dynamo Training School, Lisbon Introduction to Dynamic Networks 22 Load Balancing Dynamo Training School, Lisbon Introduction to Dynamic Networks 23
Load Balancing Each node v has w(v) tokens Goal: To balance the tokens among the nodes Imbalance: maxu,v |w(u) - wavg| In each step, each node can send at most one token to each of its neighbors Dynamo Training School, Lisbon Introduction to Dynamic Networks 24 Load Balancing
In a truly balanced configuration, we have |w(u) - w(v)| 1 Our goal is to achieve fast approximate balancing Preprocessing step in a parallel computation Related to routing and counting networks [PU89, AHS91] Dynamo Training School, Lisbon Introduction to Dynamic Networks 25 Local Balancing Each node compares its number of tokens with its neighbors In each step, for each edge (u,v): If w(u) > w(v) + 2d, then u
sends a token to v Here, d is maximum degree of the network Purely local operation Dynamo Training School, Lisbon Introduction to Dynamic Networks 26 Convergence to Stable State How long does it take local balancing to converge? What does it mean to converge? Imbalance is constant and remains so What do we mean by how long? The number of time steps it takes to achieve the above imbalance Clearly depends on the topology of the network
and the imbalance of the original token distribution Dynamo Training School, Lisbon Introduction to Dynamic Networks 27 Expansion of a Network Edge expansion : Minimum, over all sets S of size n/2, of the term |E(S)|/|S| Lower bound on convergence time: (w(S) - |S|wavg)/E(S) = (w(S)/|S| - wavg)/ Expansion = 12/6 = 2 wavg = 3 Lower bound = (29 - 18)/12
Dynamo Training School, Lisbon Introduction to Dynamic Networks 28 Properties of Local Balancing For any network G with expansion , any token distribution with imbalance converges to a distribution with imbalance O(dlog(n)/ ) in O(/ ) steps [AAMR93, GLM+99] Analysis technique: Associate a potential with every node v, which is a function of the w(v) Example: (w(v) - avg)2, cw(v)-avg Potential of balanced configuration is small Argue that in every step, the potential decreases by a desired amount (or fraction) Potential decrease rate yields the convergence time There exist distributions with imbalance that
would take (/ ) steps Dynamo Training School, Lisbon Introduction to Dynamic Networks 29 Exercise For any graph G with edge expansion , show that there is an initial distribution with imbalance such that the time taken to reduce the imbalance by even half is (/ ) steps Dynamo Training School, Lisbon Introduction to Dynamic Networks 30 Local Balancing in Dynamic Networks The purely local nature of the algorithm useful
for dynamic networks Challenge: May not know the correct load on neighbors since links are going up and down Key ideas: Maintain an estimate of the neighbors load, and update it whenever the link is live Be more conservative in sending tokens Result: Essentially same as for static networks, with a slightly higher final imbalance, under the assumption that the the set of live edges form a network with edge expansion at each step Dynamo Training School, Lisbon Introduction to Dynamic Networks 31 Adversarial Load Balancing
Dynamic load [MR02] Adversary inserts and/or deletes tokens In each step: Balancing Token insertion/deletion For any set S, let dt(S) be the change in number of tokens at step t Adversary is constrained in how much imbalance can be increased in a step Local balancing is stable against rate 1 adversaries [AKK02] Dynamo Training School, Lisbon dt(S) (avgt+1 avgt)|S| r e(S) Introduction to Dynamic Networks
32 Stochastic Adversarial Input Studied under a different model [AKU05] Any number of tokens can be exchanged per step, with one neighbor Local balancing in this model [GM96] Select a random matching Perform balancing across the edges in matching Load consumed by nodes One token per step Load placed by adversary under statistical constraints Expected injected load within window of w steps is at most rnw The pth moment of total injected load is bounded, p > 2 Local balancing is stable if r < 1 Dynamo Training School, Lisbon
Introduction to Dynamic Networks 33 Load Balancing: Summary Algorithmic technique: Local balancing Design technique: Obtain a purely distributed solution for static network, emphasizing local operations Extend it to dynamic networks by maintaining estimates Analysis technique: Potential function method Martingales Dynamo Training School, Lisbon Introduction to Dynamic Networks
34 Packet Routing Dynamo Training School, Lisbon Introduction to Dynamic Networks 35 The Packet Routing Problem Given a network and a set of packets with sourcedestination pairs Path selection: Select paths between sources and respective destinations Packet forwarding: Forward the packets to the destinations along selected paths Dynamics: Network: edges and their capacities Input: Packet arrival rates and locations Interconnection networks [Lei91], Internet [Hui95],
local-area networks, ad hoc networks [Per00] Dynamo Training School, Lisbon Introduction to Dynamic Networks 36 Packet Routing: Performance Static packet set: Congestion of selected paths: Number of paths that intersect at an edge/node Dilation: Length of longest path Dynamic packet set: Throughput: Rate at which packets can be delivered to their destination Delay: Average time difference between packet release at source and its arrival at destination Dynamic network: Communication overhead due to a topology change In highly dynamic networks, eventual delivery?
Compact routing: Sizes of routing tables Dynamo Training School, Lisbon Introduction to Dynamic Networks 37 Routing Algorithms Classification Global: All nodes have complete topology information Decentralized: Nodes know information about neighboring nodes and links Static: Routes change rarely over time
Dynamic: Topology changes frequently requiring dynamic route updates Proactive: Nodes constantly react to topology changes always maintaining routes of desired quality Reactive: Nodes select routes on demand Dynamo Training School, Lisbon Introduction to Dynamic Networks 38 Link State Routing Each node periodically broadcasts state of its links to the network
Each node has current state of the network Computes shortest paths to every node Dijkstras algorithm Stores next hop for each destination Dynamo Training School, Lisbon E F A G B C H D
Introduction to Dynamic Networks 39 Link State Routing, contd When link state changes, the broadcasts propagate change to entire network Each node recomputes shortest paths High communication complexity Not effective for highly dynamic networks Used in intra-domain routing E F A G
B C H D OSPF Dynamo Training School, Lisbon Introduction to Dynamic Networks 40 Distance Vector Routing Distributed version of Bellman-Fords algorithm Each node maintains a distance vector G
Exchanges with neighbors Maintains shortest path distance and next hop Basic version not selfstabilizing Use bound on number of nodes or path length Poisoned reverse Dynamo Training School, Lisbon A 4E B 5E H D A 45 D G B 6G A 3C B 6G
Introduction to Dynamic Networks 41 Distance Vector Routing Basis for two routing protocols for mobile ad hoc wireless networks DSDV: proactive, attempts to maintain routes AODV: reactive, computes routes on-demand using distance vectors [PBR99] G H D Dynamo Training School, Lisbon
A 4E B 5E A 4D B 6G A 3C B 6G Introduction to Dynamic Networks 42 Link Reversal Routing Aimed at dynamic networks in which finding a single path is a challenge [GB81] Focus on a destination D Idea: Impose direction on links so that all paths lead to D Each node has a height Height of D = 0
Links are directed from high to low D is a sink By definition, we have a directed cyclic graph Dynamo Training School, Lisbon 4 E 5 5 A 13 B F 4 G 31 C 2 H
0 D Introduction to Dynamic Networks 43 Setting Node Heights If destination D is the only sink, then all directed paths lead to D If another node is a sink, then it reverses all links: Set its height to 1 more than the max neighbor height Repeat until D is only sink A potential function argument shows that this procedure is selfstabilizing Dynamo Training School, Lisbon 74 E
5 5 A F 4 G 61 B 3 C 2 H 0 D Introduction to Dynamic Networks 44 Exercise For tree networks, show that the link reversal algorithm self-stabilizes from an
arbitrary state Dynamo Training School, Lisbon Introduction to Dynamic Networks 45 Issues with Link Reversal A local disruption could cause global change in the network The scheme we studied is referred to as full link reversal Partial link reversal When the network is partitioned, the component without sink has continual reversals Proposed protocol for ad hoc networks (TORA) attempts to avoid these [PC97] Need to maintain orientations of each edge for each destination
Proactive: May incur significant overhead for highly dynamic networks Dynamo Training School, Lisbon Introduction to Dynamic Networks 46 Routing in Highly Dynamic Networks Highly dynamic network: The network may not even be connected at any point of time Problem: Want to route a message from source to sink with small overhead Challenges: Cannot maintain any paths May not even be able to find paths on demand May still be possible to
route! Dynamo Training School, Lisbon E F A G B C H D Introduction to Dynamic Networks 47 End-to-End Communication Consider basic case of one source-destination pair
Need redundancy since packet sent in wrong direction may get stuck in disconnected portion! Slide protocol (local balancing) [AMS89, AGR92] Each node has an ordered queue of at most n slots for each incoming link (same for source) Packet moved from slot i at node v to slot j at the (v,u)queue of node u only if j < i All packets absorbed at destination Total number of packets in system at most C = O(nm) Dynamo Training School, Lisbon Introduction to Dynamic Networks 48 End-to-End Communication End-to-end communication using slide For each data item: Sender sends 2C+1 copies of item (new token added only if queue is not full) Receiver waits for 2C+1 copies and outputs majority Safety: The receiver output is always prefix of sender input
Liveness: If the sender and the receiver are eventually connected: The sender will eventially input a new data item The receiver eventually outputs the data item Strong guarantees considering weak connectivity Overhead can be reduced using coding e.g. [Rab89] Dynamo Training School, Lisbon Introduction to Dynamic Networks 49 Routing Through Local Balancing Multi-commodity flow [AL94] Queue for each flows packets at head and tail of each edge In each step: New packets arrive at sources Packet(s) transmitted along each edge using local balancing
Packets absorbed at destinations Queues balanced at each node Local balancing through potentials Packets sent along edge to maximize potential drop, subject to capacity Queues balanced at each node by simply distributing packets evenly Dynamo Training School, Lisbon k(q) = exp(q/(8Ldk) L = longest path length dk= demand for flow k Introduction to Dynamic Networks 50 Routing Through Local Balancing Edge capacities can be dynamically and adversarially changing
If there exists a feasible flow that can route dk flow for all k: This routing algorithm will route (1eps) dk for all k Crux of the argument: Destination is a sink and the source is constantly injecting new flow Gradient in the direction of the sink As long as feasible flow paths exist, there are paths with potential drop k(q) = exp(q/(8Ldk) L = longest path length Follow-up work has looked at packet dk= demand for flow k delays and multicast problems [ABBS01, JRS03] Dynamo Training School, Lisbon Introduction to Dynamic Networks 51
Packet Routing: Summary Models: Transient and continuous dynamics Adversarial Algorithmic techniques: Distance vector Link reversal Local balancing Analysis techniques: Potential function Dynamo Training School, Lisbon Introduction to Dynamic Networks 52 Queuing Systems Dynamo Training School, Lisbon
Introduction to Dynamic Networks 53 Packet Routing: Queuing We now consider the second aspect of routing: queuing Edges have finite capacity When multiple packets need to use an edge, they get queued in a buffer Packets forwarded or dropped according to some order Dynamo Training School, Lisbon D D
E F A D B D G D C D H D D D Introduction to Dynamic Networks 54 Packet Queuing Problems In what order should the packets be forwarded? First in first out (FIFO or FCFS) Farthest to go (FTG), nearest to go (NTG)
Longest in system (LIS), shortest in system (SIS) Which packets to drop? Tail drop Random early detection (RED) Major considerations: Buffer sizes Packet delays Throughput Our focus: forwarding Dynamo Training School, Lisbon Introduction to Dynamic Networks 55 Dynamic Packet Arrival Dynamic packet arrivals in static networks Packet arrivals: when, where, and how? Service times: how long to process?
Stochastic model: Packet arrival is a stochastic process Probability distribution on service time Sources, destinations, and paths implicitly constrained by certain load conditions Adversarial model: Deterministic: Adversary decides packet arrivals, sources, destinations, paths, subject to deterministic load constraints Stochastic: Load constraints are stochastic Dynamo Training School, Lisbon Introduction to Dynamic Networks 56 (Stochastic) Queuing Theory Rich history [Wal88, Ber92] Single queue, multiple parallel queues very wellunderstood
Networks of queues Hard to analyze owing to dependencies that arise downstream, even for independent packet arrivals Kleinrock independence assumption Fluid model abstractions Multiclass queuing networks: Multiple classes of packets Packet arrivals by time-invariant independent processes Service times within a class are indistinguishable Possible priorities among classes Dynamo Training School, Lisbon Introduction to Dynamic Networks 57
Load Conditions & Stability Stability: Finite upper bound on queues & delays Load constraint: The rate at which packets need to traverse an edge should not exceed its capacity Load conditions are not sufficient to guarantee stability of a greedy queuing policy [LK91, RS92] FIFO can be unstable for arbitrarily small load [Bra94] Different service distributions for different classes For independent and time-invariant packet arrival distributions, with class-independent service times [DM95, RS92, Bra96] FIFO is stable as long as basic load constraint holds Dynamo Training School, Lisbon Introduction to Dynamic Networks
58 Adversarial Queuing Theory Directed network Packets, released at source, travel along specified paths, absorbed at destination In each step, at most one packet sent along each edge Adversary injects requests: A request is a packet and a specified path Queuing policy decides which packet sent at each step along each edge [BKR+96, BKR+01] Dynamo Training School, Lisbon E F
A G B C H D Introduction to Dynamic Networks 59 Load Constraints Let N(T,e) be number of paths injected during interval T that traverse e (w,r)-adversary: For any interval T of w consecutive time steps, for every edge e: N(T,e) w r Rate of adversary is r
e # paths using e injected at t (w,r) stochastic adversary: For any interval [t+1t+w], for every edge e: E[N(T,e)|Ht] w r w t Area w r Dynamo Training School, Lisbon Introduction to Dynamic Networks 60
Stability in DAGs Theorem: For any dag, any greedy policy is stable against any rate-1 adversary At(e) = # packets in network at time t that will eventually use e Qt(e) = queue size for e at time t Proof: time-invariant upper bound on At(e) e1 e2 e 3 e Large queue: Qt-w(e) w At(e) At-w(e) Small queue: Qt-w(e) < w At-w(e) w + j At-w(ej) At(e) 2w + j At-w(ej) Dynamo Training School, Lisbon
Introduction to Dynamic Networks 61 Extension to Stochastic Adversaries Theorem: In DAGs, any greedy policy is stable against any stochastic 1- rate adversary, for any >0 Cannot claim a hard upper bound on At(e) Define a potential t, that is an upper bound on the number of packets in system Show that if the potential is larger than a specified constant, then there is an expected decrease in the next step Invoke results from martingale theory to argue that E[t] is bounded by a constant Dynamo Training School, Lisbon Introduction to Dynamic Networks 62
FIFO is Unstable [A+ 96] Initially: s packets waiting at A to go to C Next s steps: rs packets for loop rs packets for B-C A B D C Next rs steps: r2s packets from B to A r2s packets for B-C Next r2s steps: r3s packets for C-A
Now: s+1 packets waiting at C going to A FIFO does not use edges most effectively Dynamo Training School, Lisbon Introduction to Dynamic Networks 63 Stability in General Networks LIS and SIS are universally stable against rate <1 adversaries [AAF+96] Furthest-To-Go and Nearest-To-Origin are stable even against rate 1 adversaries [Gam99] Bounds on queue size: Mostly exponential in the length of the shortest path For DAGs, Longest-In-System (LIS) has poly-sized queues Bounds on packet delays: A variant of LIS has poly-sized packet delays
Dynamo Training School, Lisbon Introduction to Dynamic Networks 64 Exercise Are the following two equivalent? Is one stronger than the other? A finite bound on queue sizes A finite bound on delay of each packet Dynamo Training School, Lisbon Introduction to Dynamic Networks 65 Queuing Theory: Summary Focus on input dynamics in static networks Both stochastic and adversarial models Primary concern: stability
Finite bound on queue sizes Finite bound on packet delays Algorithmic techniques: simple greedy policies Analysis techniques: Potential functions Markov chains and Markov decision processes Martingales Dynamo Training School, Lisbon Introduction to Dynamic Networks 66 Network Evolution Dynamo Training School, Lisbon Introduction to Dynamic Networks 67
How do Networks Evolve? Internet New random graph models Developed to support observed properties Peer-to-peer networks Specific structures for connectivity properties Chord [SMK+01], CAN [RFH+01], Oceanstore [KBC+00], D2B [FG03], [PRU01], [LNBK02], Ad hoc networks Connectivity & capacity [GK00] Mobility models [BMJ+98, YLN03, LNR04] Dynamo Training School, Lisbon Introduction to Dynamic Networks 68 Internet Graph Models Internet measurements [FFF99, TGJ+02, ]:
Degrees follow heavy-tailed distribution at the AS and router levels Frequency of nodes with degree d is proportional to 1/d, 2 < < 3 Models motivated by these observations Preferential attachment model [BA99] Power law graph model [ACL00] Bicriteria optimization model [FKP02] Dynamo Training School, Lisbon Introduction to Dynamic Networks 69 Preferential Attachment Evolutionary model [BA99] Initial graph is a clique of size d+1 d is degree-related parameter
In step t, a new node arrives New node selects d neighbors Probability that node j is neighbor is proportional to its current degree Achieves power law degree distribution Dynamo Training School, Lisbon Introduction to Dynamic Networks 70 Power Law Random Graphs Structural model [ACL00] Generate a graph with a specified degree sequence (d1,,dn) Sampled from a power law degree distribution Construct dj mini-vertices for each j Construct a random perfect matching Graph obtained by adding an edge for every edge
between mini-vertices Adapting for Internet: Prune 1- and 2-degree vertices repeatedly Reattach them using random matchings Dynamo Training School, Lisbon Introduction to Dynamic Networks 71 Bicriteria Optimization Evolutionary model Tree generation with power law degrees [FKP02] All nodes in unit square When node j arrives, it attaches to node k that minimizes: djk + hk If 4 o(n): Degrees distributed as power
law for some , dependent on Can be generalized, but no provable results known Dynamo Training School, Lisbon hk: measure of centrality of k in tree Introduction to Dynamic Networks 72 Connectivity & Capacity Properties Congestion in certain uniform multicommodity flow problems: Suppose each pair of nodes is a source-destination pair for a unit flow What will be the congestion on the most congested edge of the graph, assuming uniform capacities Comparison with expander graphs, which would tend to have the least congestion
For power law graphs with constant average degree, congestion is O(n log2n) with high probability [GMS03] (n) is a lower bound For preferential attachment model, congestion is O(n log n) with high probability [MPS03] Analysis by proving a lower bound on conductance, and hence expansion of the network Dynamo Training School, Lisbon Introduction to Dynamic Networks 73 Network Creation Game View Internet as the product of the interaction of many economic agents Agents are nodes and their strategy choices create the network Strategy sj of node j:
Edges to a subset of the nodes Cost cj for node j: |sj| + k dG(s)(j,k) Hardware cost plus quality of service costs Dynamo Training School, Lisbon 3 + sum of distances to all nodes Introduction to Dynamic Networks 74 Network Creation Game In the game, each node selects the best response to other nodes strategies Nash equilibrium s: For all j, cj(s) cj(s) for all s that differ from s only in the jth
component Price of anarchy [KP99]: Maximum, over all Nash equilibria, of the ratio of total cost in equilibrium to smallest total cost Bound, as a function of [AEED06]: O(1) for = O(n) or (n log n) Worst-case ratio O(n1/3) Dynamo Training School, Lisbon Introduction to Dynamic Networks 75 Other Network Games Variants of network creation games Weighted version [AEED06] Cost and benefit tradeoff [BG00]
Cost sharing in network design [JV01, ADK04, GST04] Congestion games [RT00, Rou02] Each source-destination pair selects a path Delay on edge is a function of the number of flows that use the edge Dynamo Training School, Lisbon Introduction to Dynamic Networks 76 Network Evolution: Summary Models: Stochastic Game-theoretic Analysis techniques: Graph properties, e.g., expansion, conductance Probabilistic techniques Techniques borrowed from random graphs
Dynamo Training School, Lisbon Introduction to Dynamic Networks 77
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