Statistical Physics of Routing David Saad * Bill C. H. Yeung* K.Y - PowerPoint PPT Presentation

Light, Polymers and Automobiles - Statistical Physics of Routing David Saad * Bill C. H. Yeung* K.Y Michael Wong # Caterina De Bacco $ Silvio Franz $ *Nonlinearity and Complexity Research Group – Aston # Hong Kong University of Science and Technology $ LPTMS, Université Paris-Sud, Orsay

2 Outline Motivation – why routing? The models – two scenarios  One universal source  Ordinary routing Two approaches: cavity, replica and polymer methods Results: microscopic solutions, macroscopic phenomena Applications: e.g. subway, air traffic networks Conclusions

3 Why routing? Are existing algorithms any good? - Routing tables computed by shortest-path, or minimal weight on path (e.g. Internet) - Geographic routing (e.g. wireless networks) Des 1: k Des 2: j D j … i Des 1 source destination k - Insensitive to other path choices  congestion, or low occupancy routers/stations for sparse traffic - Heuristics- monitoring queue length  sub-optimal

4 Global optimization 1. A difficult problem with non-local variables Unlike most combinatorial problems such as Graph coloring, Vertex cover, source destination K-sat, etc. 2. Non-linear interaction between communications: avoid congestion  repulsion consolidate traffic  attraction paths interact with each other Interaction is absent in similar problems: spanning trees and Stenier trees M.Bayati et al , PRL 101, 037208 (2008)

5 Communication Model N nodes ( i , j , k …) M communications ( ν ,.. ) each with a fixed source and destination Denote, σ j ν = 1 (communication ν passes through node j ) σ j ν = 0 (otherwise) cost γ >1 γ =1 Traffic on j  I j = Σ ν σ j ν γ <1 I j Find path configuration which globally minimizes γ γ H = Σ j ( I j ) H = Σ ( ij ) ( I ij ) or - γ >1 repulsion (between com.)  avoid congestion - γ <1 attraction  aggregate traffic (to  idle nodes) - γ =1 no interaction, H = Σ ν j σ j ν  shortest path routing

6 Analytical approach  Map the routing problem onto a model of resource allocation: Each node i has initial resource Ʌ i Ʌ i = +∞ - Receiver (base station, router) Ʌ i = -1 - Senders (e.g. com. nodes) Ʌ i = 0 - others γ Minimize H = Σ ( ij ) ( I ij ) Constraints: (i) final resource R i = Ʌ i + Σ j  ∂ i I ji = 0 , all i (ii) currents are integers resource Central router com. nodes (integer current)  each sender establishes a single path to the receiver

7 The cavity method E i ( I il ) = optimized energy of the tree terminated at node i without l At zero-temperature, we use the following recursion to obtain a stable P [ E i ( I il )]         E ( I ) min | I | E ( I ) i il il j ji    {{ I } | R 0 }  ji i j L \ { l } i Algorithm: However, constrained minimization E  ( ij I ) over integer domain  difficult i j E  ( ij I ) j i γ >1 , we can show that E i ( I il ) is convex  computation greatly simplified Yeung and Saad, PRL 108 , 208701 (2012); Yeung, IEEE Proc NETSTAT (2013)

8 Results - Non-monotonic  L   2 i.e. γ =2  avoid congestion H = Σ ( ij ) ( I ij ) M – number of senders ??? Small deviations between simulation - finite size effect, N  , deviation  Average path length per communication Random regular graph k=3 Initial  in  L  - as short Final  in  L  - when routes are being occupied traffic is dense, longer routes are chosen everywhere is congested

9 Results - balanced receiver ??? Algorithmic convergence time Random regular graphs Example: M=6,  k  =3 M / N M / N 2 H = Σ ( ij ) ( I ij ) Small peaks in  L  are multiples of  k  , 1 1 balance traffic around receiver 1 1 Consequence  peaks occur in convergence time  T c  1 1 1 1 1 Studied - random network, scale-free 1 2 networks, qualitatively similar behavior 1 2 2 2

10 RS/RSB multiple router types One receiver “type” γ , γ >1 - H = Σ ( ij ) ( I ij ) - E j ( I ji ) is convex - RS for any M / N Cost RS • Two receiver “types”: A & B Solution - Senders with Ʌ A = - 1 or Ʌ B = -1 space γ , γ >1 - H = Σ ( ij ) (| I ij A |+| I ij B |) A , I ij B ) not always convex - E j ( I ij - Experiments exhibit RSB-like Cost behavior RSB Solution space

11 Node disjoint routing  Random communicating pairs  = 1…M Routes do not cross Node i has initial resource Ʌ  i : - Receiver Ʌ  i = -1 - Senders Ʌ  i = +1 Ʌ  - others i = 0 Currents: - Route  passes through ( i,j ) i → j I  ij = +1 - Route  passes through ( i,j ) j → i I  ij = -1 I  - otherwise ij = 0 Minimize H = Σ ( ij ) f ( Σ  |I  ij |) - but no crossing Constraints:(i) final resource Ʌ  i + Σ j  ∂ i I  ji = 0 , all i,  (ii) currents are integers and I  ij = - I  ji

12 The cavity method At zero-temperature, we use recursion relation to obtain a stable P [ E ij ({ I ij })] where { I ij }= I 1 ij , I 2 ij…, I M ij f ( I ) = I for I=0,1 and ∞ otherwise        E ({ I }) min E ({ I }) f (|| { I } ||) ij ij ki ki il    {{ I } | R 0 }  ki i k L \{ j } i Messages reduced from 3 M to 2M+1

13 Results De Bacco, Franz, Saad, Yeung JSTAT P07009 (2014)

14 Do you see the light? Node disjoint routing is important for optical networks Task: accommodate more communications per wavelength Same wavelength communications cannot share an edge/vertex Approaches used: greedy algorithms, integer-linear programming… Greedy algorithms (such as breadth first-search) usually calculate shortest path and remove nodes from the network

15 General routing - analytical approach More complicated, cannot map  to resource allocation Use model of interacting polymers - communication  polymer with fixed ends ν = 1 (if polymer ν passes through j ) - σ j polymers σ j ν = 0 (otherwise) - I j = Σ ν σ j ν (no. of polymers passing through j ) γ - minimize H= Σ j ( I j ) , of any γ We use polymer method+ replica

16 Analytical approach Replica approach – averaging topology, start/end 𝑎 𝑜 − 1 log 𝑎 = lim 𝑜 𝑜→0 2 =1 Polymer method – p -component spin such that 𝑇 𝑏 and 𝑇 𝑏 2 =p , when p  0 , 𝑏 The expansion of ( Π i 𝑒𝜈(𝑻 𝑗 ) ) Π ( kl ) ( 1+A kl S k ∙S l ) results in S ka S la S la S ja S ja S ra S ra ….. describing a self-avoiding loop/path between 2 ends M. Daoud et al (and P. G. de Gennes) Macromolecules 8, 804 (1976)

17 Related works Polymer method+ replica approach was used to study travelling salesman problem (Difference: one path, no polymer interaction) Cavity approach was used to study interacting polymers (Diff: only neighboring interactions considered, here we consider overlapping interaction) Here: polymer + replica approach to solve a system of polymers with overlapping interaction  recursion + message passing algorithms (for any γ ) M. Mezard, G. Parisi, J. Physique 47, 1284 (1986) A. Montanari, M. Muller, M. Mezard, PRL 92, 185509 (2004) E. Marinari, R. Monasson, JSTAT P09004 (2004); EM, RM, G. Semerjian. EPL 73, 8 (2006)

18 The algorithm

19 Extensions Edge cost Weighted edge costs Combination of edge/node costs Directed edges

20 Results – Microscopic solution cost γ >1 γ =1 convex vs. concave cost γ <1 I j - source/destination of a communication - shared by more than 1 com. Size of node  traffic N =50, M =10 γ =0.5 γ =2 γ >1 repulsion (between com.)  avoid congestion - γ <1 attraction  aggregate traffic (to  idle nodes)  to save energy -

21 London subway network 275 stations Each polymer/communication – Oyster card recorded real passengers source/destination pair Oyster card London tube map

22 Results – London subway with real source destination pairs recorded by Oyster card cost γ >1 γ =2 γ =1 γ <1 M =220 I j

23 Results – London subway with real source destination pairs recorded by Oyster card cost γ >1 γ =0.5 γ =1 γ <1 M =220 I j

24 Results – Airport network γ =2, M =300

25 Results – Airport network γ =0.5, M =300

26 Results – comparison of traffic γ >1 cost γ =1 γ <1 I j γ =2 Orly γ =0.5 γ =2 vs γ =0.5 - Overloaded station/airport has lower traffic - Underloaded station /airport has higher traffic

27 Comparison with Dijkstra algorithm Comparison of energy E and path length L obtained by polymers-inspired (P) and Dijkstra (D) algorithms γ =2 2 γ =0. 0.5 E P −E D L P −L D E P −E D L P −L D E D L D E D L D − 20.5 ± 0 . 5% +5.8 ± 0 . 1% − 4 . 0 ± 0 . 1% +5.8 ± 0 . 3% London subway Global airport − 56 . 0 ± 2 . 0% +6 . 2 ± 0 . 2% − 9 . 5 ± 0 . 2% +8 . 6 ± 1 . 2%

28 and with a Multi-Commodity flow algorithm Comparison of energy E and Based on node-weighted path length L obtained by shortest paths d i using total polymers-inspired (P) and Multi- current I i ; rerouting longest Commodity flow (MC) algorithms paths below edge capacity (Awerbuch, Khandekar (2007) 𝑓 𝛽𝐽 𝑗 with optimal α ) 𝑒 𝑗 = 𝑓 𝛽𝐽 𝑘 𝑘 γ =2 2 γ =0. 0.5 E P −E MC ( α) L P −L MC ( α) No algorithm identified for E MC ( α) L MC ( α) comparison − 0.7 ± 0 . 04% London +0.72 ± 0 . 10% subway − 3.9 ± 0.59% Global +0.90 ± 0 . 64% airport