Dynamic Networks & Evolving Graphs Afonso Ferreira CNRS I3S & INRIA Sophia Antipolis Afonso.Ferreira@sophia.inria.fr With Aubin Jarry.

Dynamic Networks

&

Evolving Graphs

Afonso FerreiraCNRS

I3S & INRIA Sophia Antipolis

Afonso.Ferreira@sophia.inria.fr

With Aubin Jarry

Dynamic Networks

• Mobile Wireless Networks (eg, Ad-hoc)• Fixed Packet Networks (eg, Internet)• Fixed Connected Networks (eg, WDM)• Fixed Schedule Fixed Networks

– (eg, Sensors, Transport)

• Fixed Schedule Mobile Networks(eg, LEO Satellites, Robots)

T1

T2

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T1

T2

T3

T4

Distance = 3

= 4

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T2

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T4

Distance= 3 hops / 1 TU

= 1 hop / 4 TU

Outline

• Motivation: grasp dynamic networks• The Evolving Graph• Distances, Paths, Journeys, Connectivity,

...• Old questions - New insights• A direct application• Conclusions

2003 NSF report onFundamental Research in

Networking• Understanding about networks

– Needs: Substantial innovation and paradigm shifts

• Scalable design and control of networks– Needs: Fundamental understanding

• Reproducibility of experiments – Needs: Reference models and benchmarks

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The Evolving Graph

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The Evolving Graph

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Evolving Graphs

• Given a graph G(V,E) and an ordered sequence of

its subgraphs, SSG=Gt0, Gt1, ..., GtT.

The system EG = (G, SSG) is called an evolving

graph.

• Input coding: list of presence intervals for each edge and for each vertex (this can be evidently relaxed in case of a valid mobility model, eg)

• Dynamics: – Size of edge and node lists.

[Algotel’02]

Journeys in EGs

• Sequence of edges {e1, e2, …, ek} 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).

• Observations:– Journeys cannot go to the past– A round journey is J(u,u, s). Like a usual cycle,

but not quite.

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The Evolving Graph

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Routing Issues

• Minimum hop count = Distance– shortest journey

• Timed evolving graphs (TEGs): – traversal time on the edges.

• Minimum arrival date = Earliest arrival date– foremost journey

• Minimum journey time = Delay – fastest journey

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. Insert it in the ‘shortest paths’ tree.

(TEGs are complex: Prefix journeys of foremost journeys are not necessarily foremost.)

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Algorithm

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Analysis

• For each closed vertex, the algorithm performs O(log + log N) operations per neighbor.

• Total number of operations is

O(vV [| +(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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Algorithm for fading memory

Source:

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Analysis

• O(M (log + log N)) operations.• Again bounded by the actual dynamics

of the evolving graph.

Journey Issues

• Minimum arrival date = Earliest arrival date– O(M (log + log N))

• Minimum hop count = Usual distance– O(NM log )

• Minimum journey time = Delay – O(NM 2 )

• Many others to explore

[WiOpt’03, IJFCS 03]

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.

Example I: CC

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Example I: CC

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CC:

Example II: CC

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CC:

Example II: o-CC

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O-CC:

Complexity result

• Computing (o-)CCs is NP-Complete.– It is in NP: computing journeys is

polynomial.– Reduction from Clique

[Ad-Hoc Now 03]

The GadgetGiven G =(V,E) and integer k, create an EG:For each ui in V create a vi and a hii.• Time step 1:

– Create a CC connecting all h-nodes.

• Time step 2:– Create edges {vi,hii}, – For each edge {ui, uj} in E, create edges {vi,hij}.

• Time step 3:– For each edge {ui, uj} in E, create edges {hij,vj}.

• Time step 4:– Create a CC connecting all h-nodes.

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A Key Issue?

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An application

• Minimum Energy Broadcast & Range Assignment Problem (STACS 97, Infocom 00)

• E~d2

The MEB&RAP

• NP- Complete• 12-Approximation

– Direct computation of the MST of the underlying weighted complete graph

– Analysis using geometric arguments

The MEB&RAP

• Static x (Low) Dynamic

• What is a MST over time??• Take an Ad-Hoc network where nodes do

not move, but can alternate sleep/awake modes according to a predefined schedule:– A MST over time is a rooted MST allowing for

journeys from the root to the leafs in the corresponding weighted evolving graph

Computing a MST over time

• NP-Complete (Reduction from Steiner)• Precludes the use of the MST-based

heuristic to solve the MEB&RAP in dynamic networks

[WiOpt 04]

Current & Future Work

• Rooted MST is NP-Complete– But Local Minima RST is Polynomial!

• Flows in EGs

• Algorithms for EGs (eg Connected Components)

• Distributed algorithms for EGs– Competitive analysis of protocols

• Harness Dynamic Networks

Related Combinatorial Models

used in Dynamic Networks• Graphs• Random Graphs & Adversaries [Scheideler’02]

• Dynamic Graph Algorithms [Frigioni et al’00]

• Time-Expanded Graphs [FoFu’58]

• MERIT [FaSy’01]

– A sequence of historic network snapshots– Competitive analysis of protocols

NSF report onFundamental Research in

Networking• Understanding about networks

– Needs: Substantial innovation and paradigm shifts

• Scalable design and control of networks– Needs: Fundamental understanding

• Reproducibility of experiments – Needs: Reference models and benchmarks

Conclusion

• Evolving Graphs– Graphs + Time Domain– A model for complexity, combinatorics, algorithms

• Old questions - New insights– Hardness induced by time

• Many applications in Dynamic Networks– Wireless nets, evolving request matrices,

transports...

• Many new ways to explore!

The End

Afonso.Ferreira@sophia.inria.fr

The idea

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hki hkk

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Networks

• Valued graphs• Weights = costs, distance, traversal time, etc.

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Networks

• Road networks, railway systems– Traversal times are arbitrary but finite– Arcs are closed at certain periods– Parking is allowed at vertices whenever

possible• Computing shortest paths in loaded

networks– Earliest Arrival Times [HP’74,D’66]– Dijkstra-like, no complexity analysis

Networks

• Weights = traversal time: Time-expanded graphs [FF’58]

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Networks

• Weights = traversal time: Time-expanded graphs [FF’58]

• Complexity increases– Pseudo-polynomial (weights must be

integers)• Time-dependent networks [CH’66]

– Weights depend on the number of flow units entering the link

– Computation of quickest flows (ESA’02, SODA’02)

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Fading memory

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Example of Journeys

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07h0010h00

10h0011h0013h0015h00

Fixed-Schedule Dynamic Networks

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Models for Dynamic Networks

• Graphs• Random Graphs • Dynamic Graphs

– Discrete step is one link/node change– Focus on data-structures & amortized

analyses– Time is not an issue

Motivation: Formalize the notion of time in graphs

EGs & Dynamic Networks

• Fixed-Schedule– Satellite constellations– Transportation networks– Robot networks

• History– Competitive analysis– MERIT

• Stochastic?– Mobility model

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

Combinatorial Models for Dynamic Networks

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