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Evolving Graphs & Dynamic Networks : Old questions New insights Afonso Ferreira S. Bhadra CNRS I3S & INRIA Sophia Antipolis Afonso.Ferreira@sophia.inria.fr

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Technology aware - Problem driven behavior Develop combinatorial models for networks –graphs, hypergraphs, etc. Identify underlying optimisation problems –coloring, flows, connectivity, etc. Design (combinatorial) algorithms –exact, distributed, on-line, approximation, randomized, (you may use linear programming), etc. Apply solutions to technology –improvements spread several technologies

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Combinatorial Models for Dynamic Networks....

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Graphs Random Graphs Dynamic Graphs Time-Expanded Graphs (MERIT) Our small dot: Formalize the notion of time evolution in graphs

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Outline Dynamic Networks & Evolving Graphs –Modeling time evolution in a formal way –Paths & Journeys –Old problems - New complexities Connected Components –Multicast trees in Mobile Networks –Current & Future work

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Mobile Dynamic Networks

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T1T1 T2T2 T3T3 T4T4 Distance = 3 = 4

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T1T1 T2T2 T3T3 T4T4 Distance = 3 hops / 1 TU = 1 hop / 4 TU

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1,2,3 1,3 1,2,4 1,3,4 2 2 3 1 4 The Evolving Graph 4

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1,2,3 1,3 1,2,4 1,3,4 2 2 3 1 4 The Evolving Graph 4

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Evolving Graphs SGiven a graph G(V,E) and an ordered sequence of its subgraphs, S G =G t 0, G t 1,..., G t T. S The system EG = (G, S G ) is called an evolving graph. Timed evolving graphs (TEGs): –Traversal time on the edges.

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Evolving Graphs Coding: Linked adjacency lists –Sorted edge schedule attached to each neighbor. –Sorted node schedule attached to head nodes. Dynamics: –Size of edge and node lists. A compact and tractable representation of Time-Expanded Graphs [FF’58]

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EGs & Dynamic Networks Fixed-Schedule –Satellite constellations –Transportation networks –Robot networks History –Competitive analysis –MERIT Stochastic –Mobility model

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Journeys in EGs Sequence of edges {e 1, e 2, …, e k } of G called a Route R(u,v) (= a path in G). A schedule s respecting EG and R, defines a journey J(u,v, s). Some facts: –Journeys cannot go to the past –A round journey is J(u,u, s). Like a usual circuit, but not quite.

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1,2,3 1,3 1,2,4 1,3,4 2 2 3 1 4 The Evolving Graph 4

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Some Distances Minimum hop count = Usual distance –shortest journey Minimum arrival date = Earliest arrival date – foremost journey Minimum journey time = Delay –fastest journey

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Algorithm for Foremost Journeys –Delete root of heap into x. –For each open neighbor v of x: Compute first valid edge schedule time greater or equal to current time step Insert v in the heap if it was not there already. –If needed, update distance to v and its key. –Update the heap. –Close x and insert it in the ‘shortest paths’ tree. (TEGs are complex: Prefix journeys of foremost journeys are not necessarily foremost.) [IJFCS’03]

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2/3/5/9 1/2/4/10 6/8 5/6/7 1/2 5/6/7 9 2/3/6 10 3/8 2 4 6 5 6 9 5 1/7 7 Algorithm Source: 0 Time:12345678

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Analysis For each closed vertex, the algorithm performs O(log + log N) operations. Total number of operations is O( v V [| + (v) | (log + log N)]) = O(M (log + log N)). Bounded by the actual size of the schedule lists, which measures the number of changes in the network topology.

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1/3/5/9 2/6 7/8 5/6 2/10 3/6 7 2/3/6 10 5/10 1 2 7 3 6 9 6 1/9 7 Algorithm for fading memory Source: 0 2/3/5/6/10 Time:12345678

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Connectivity Issues An EG is said to be connected if for every pair (u,v) there is a journey from u to v and a journey from v to u. A connected component of EG is defined as a maximal subset U of V, such that for every pair (u,v) there is a journey from u to v and a journey from v to u.

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Example I: CC 24 1 1

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24 1 1 CC:

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Example II: CC 24 1 1 CC:

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Example II: o-CC 24 1 1 O -CC:

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CCs and o-CCs A connected component of EG is defined as a maximal subset U of V, such that the EG induced by U is connected. An open connected component of EG is defined as a maximal subset U of V, such that for every pair (u,v) of U, there is a journey in EG from u to v and vice-versa.

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Complexity of (o-)CCs Computing (o-)CCs is NP-Complete. –It is in NP: computing journeys is polynomial. –Reduction from Clique

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The Gadget Given G =(V,E) and integer k, create an EG: For each u i in V create a v i and a h ii. Time step 1: –Create a CC connecting all h-nodes. Time step 2: –Create edges {v i,h ii }, –For each edge {u i, u j } in E, create edges {v i,h ij }. Time step 3: –For each edge {u i, u j } in E, create edges {h ij,v j }. Time step 4: –Create a CC connecting all h-nodes.

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The idea vivi h ii h ik h ki h kk vkvk 3 1,4 2 2 22 3

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A Foremost Multicast Tree 1 1 1 1 2 3 1 4

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Current & Future Work Rooted timed trees ( SSSPs) in TEGs –Foremost, shortest, fastest Rooted MST is NP-Complete –But Min Max RST is Polynomial! Applications: –Multicast trees, Energy aware routing, etc. Flows, coloring, scheduling,... in EGs Distributed algorithms for EGs –Competitive analysis of protocols (MERIT) Harness Dynamic Networks

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Thank you Afonso.Ferreira@sophia.inria.fr

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The End

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Foremost Journeys in EGs Setting: –Deterministic, Off-line, Centralized With LOG, packet transmission time is normalized so as to coincide with the duration of a time step. For TEGs is more complex: –Prefix journeys of foremost journeys are not necessarily foremost.

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Evolving Graphs Representations Imagine two presence matrices: P E and P V –Presence of edges and nodes at time step t i. Coding: Linked adjacency lists –Sorted edge schedule attached to each neighbor. –Sorted node schedule attached to head nodes. Dynamics: .

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1,2,3 1,3 1,2,4 1,3,4 2 2 3 1 4 Example of Journeys 4

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Dynamic Networks & Evolving Graphs Afonso Ferreira CNRS I3S & INRIA Sophia Antipolis Afonso.Ferreira@sophia.inria.fr With S. Bhadra, B. Bui Xuan, A. Jarry

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22h00 18h00 17h0016h00 13h00 12h00 07h00 10h00 11h00 13h00 15h00 Fixed-Schedule Dynamic Networks 07h00 10h00

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Some Issues in Dynamic Networks Property maintenance –E.g., Minimum Spanning Tree Fault tolerance –Link/node failure Congestion avoidance –Time dependency Topology prediction –The Web

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Dynamic Networks Mobile Wireless Networks (eg, Ad-hoc) Fixed Packet Networks (eg, Internet) Fixed Connected Networks (eg, WDM) Fixed-Schedule Networks (eg, LEO Satellites, Robots, Transport)

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Dynamic Networks Fixed-Schedule Networks (eg, LEO Satellites, Robots, Transport)

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Examples of Scenarios for Node Absence Scenario 1: There is information conservation. Scenario 2: There is no information conservation.

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Evolving Graphs S SGiven a graph G(V,E) and an ordered sequence of its subgraphs, S G =G t 0, G t 1,..., G t T. Then, the system EG = (G, S G ) is called an evolving graph.

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Analysis O(M (log + log N)) operations. Again bounded by the actual dynamics of the evolving graph.

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Journey Issues Distances –Hop count –Arrival date –Journey time

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Fixed Dynamic Networks

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2/3/5/9 1/2/4/10 6/8 5/6/7 1/2 5/6/7 9 2/3/6 10 3/8 2 4 6 5 6 9 5 1/7 7 Algorithm Source: 0 Time:12345678

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1/3/5/9 2/6 7/8 5/6 2/10 3/6 7 2/3/6 10 5/10 1 2 7 3 6 9 6 1/9 7 Algorithm for fading memory Source: 0 2/3/5/6/10 Time:12345678

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