Network Science & Telecommunications

Transcription

1 Network Science & Telecommunications Piet Van Mieghem 1 ITC30 Networking Science Vision Day 5 September 2018, Vienna Outline Networks Birth of Network Science Function and graph Outlook 1

2 Network: service(s) + topology A transport of items from A to B Service (function) B software, processes Topology (graph) hardware, structure Service and topology own specifications both are, generally, time-variant service is often designed independently of the topology often more than 1 service on a same topology 3 1 Three representations of a graph Topology domain N = 6 L = 9 4 Spectral domain! =! # = $Λ$ # Geometric domain 2 4 " $ $ $ A N N = $ $ $ # $ % ' ' ' ' ' ' &' $ & & : orthogonal eigenvector matrix Λ & & : diagonal eigenvalue matrix 1 Simplex in Euclidean (N-1)-dimensional space 3 Devriendt, K. and P. Van Mieghem, 2018, "The Simplex Geometry of Graphs", Delft University of Technology, report ( 2

3 Theory of processes on graphs Network Science What is new? Duality: both process & graph electrical & water flow processes proportional to the graph Laplacian relation between flow and potential conservation & physical laws services independently designed of the graph Internet actor networks 5 Outline Networks Birth of Network Science Function and graph Outlook 3

4 Technology Adoption (1997) Radio TV VCR Internet Cell Phone Microwave PC Car To reach 50 million users: Radio 38 years Telephone PC 16 TV 12 Electricity Internet 4 Airplane Forbes Magazine July 7th, World ICommerce Sales (1999) $Billions Low potential High potential Internet: a Hype or Fact? Internet Advertising $0 in 1993 $300 million in 1996 $3.7 billion in 1999 Internet Revolution? growth in GDP: 2.8% Internet economy: 175% from 1995 to

5 KPN stock over time 150 Euro March 2000 Internet bubble 2.3 Euro 29/8/2018 KPN: largest Dutch telecom provider (incumbent) 9 Google finance Focal technical question in the telecom world before 2000 Can a time-ignorant protocol as the Internet provide real-time services such as telephony and real-time video? Is the end-to-end delay below roughly 100ms between any pair of communicating nodes? 10 5

6 RIPE trace-route measurements ( ) Central Point MySQL Database Test Box A ISP A The results ISP C Test Box C Internal Internal networks networks Border Router Border Router Internal Internal networks networks Internal Internal networks networks Border Router Border Router Internal Internal networks networks Test Box B ISP B ISP D Test Box D Probe-packets RIPE NCC (the Network Coordination Centre of the Réseaux IP Européen) Amsterdam Hopcount of Internet paths (2004) Pr[H = k] = k] Asia Europe USA fit with log(n Asia ) = 13.5 fit with log(n Europe ) = 12.6 fit with log(n USA ) = 12.9 Hopcount: # links in path # intermediate routers End-to-end delay depends on the hopcount hop k Hop k P. Van Mieghem, Performance Analysis of Complex Networks and Systems, Cambridge University Press,

7 Trace-routes in the Internet J.-J. Pansiot and D. Grad, On Routes and Multicast Trees in the Internet, ACM Computer Communication Review, Vol. 28, pp , Internet: Power law degree distribution Pr[D > x] AS data Pr[D > x] ~ x-1.12 Degree D : number of direct neighbors of a node in a graph x 14 G. Siganos, M. Faloutsos, P. Faloutsos & C. Faloutsos, Power Laws and the AS-Level Internet Topology, IEEE Trans. On Networking, Vol. 11, No. 4, pp ,

8 15 Reprinted from Linked: The New Science of Networks by Albert-Laszlo Barabasi 15 Birth of Network Science (around 2000) Observation of power law degree in many more networks: o World-Wide Web o movie actor collaboration network, o the human respiratory system o the size and location of earthquakes, o stock-price fluctuations o biological cellular networks o scientific citation network o Pareto (1896), Zipf (1949), Bak (1996), Barabasi (1999) 16 A.-L. Barabasi, Network Science, Cambridge University Press,

9 Outline Networks Birth of Network Science Function and graph Process is linear in the graph Local rule global emergent process Outlook Linear dynamics on networks Linear dynamic process: proportional to (~) graph of network v i x i i link flow y ij Examples: water (or gas) flow ~ pressure displacement (in spring) ~ force heat flow ~ temperature electrical current ~ voltage # $ # & ~' $& Local rule v j j x j x = Q. v injected nodal current vector weighted Laplacian of the graph nodal potential vector 18 P. Van Mieghem, K. Devriendt and H. Cetinay, 2017, "Pseudoinverse of the Laplacian and best spreader node in a network", Physical Review E, vol. 96, No. 3, p

10 r ij i r ik k j Inverses: x = Qv v=! " x with voltage reference u T v = 0! " : pseudoinverse of the weighted Laplacian obeying!! " =! "! = $ & ' ( ( = )) * : all-one matrix u : all-one vector Unit current injected in node i x = e i 1/N u nodal potential of i " +, = -,, " The best spreader is the node k with minimum -,, Laplace-transformed x(s) = Q(s)v(s) v R L i = *+ *, 7 : = 0. 7 ; = - *. *, Laplace transform V(s) I(s) Z(s) C 7 8 = + 2? > = = AB5 C, *, 4 v = v R + v L + v C - *. *, * (0) = 7 4 D E = F E G E + H I E with < = = 0 + =- + 1 =2 20 J 4 = = 7 8(0) -. 0 = 10

11 Geometry of a graph (dual representation) Spectral decomposition:! " = $% " $ & = $ % " $ % " & The matrix ' " = $ % " & has rank N-1 and! " = ' " & ' " The i-th column vector ( " ) obeys ( " " ) ( + 1, = -)+, simplex (, ( 5 (, 2 ( 5 2 inverse simplex ( 6 2 ( From ' & ' " =. //0 1 : ( + & ( ) 2 = 1 4 ( ) & ( ) 2 = )+ is the effective resistance between node i and j Outline Networks Birth of Network Science Function and graph Process is linear in the graph Local rule global emergent process Outlook 11

12 Continuous-time SIS model on networks Constant infection rate b on all links Constant curing rate d for all nodes t = b /d : effective spreading rate X j X j ( t) =1 node j is infected at time t ( t) = 0 node j is healthy at time t d 1 Healthy 3 b 0 b Infected 2 Infected Infection and curing are independent Poisson processes 24 P. Van Mieghem, J. Omic, R. E. Kooij, Virus Spread in Networks, IEEE/ACM Transaction on Networking, Vol. 17, No. 1, pp. 1-14, (2009). Governing SIS equation for node j de[x j ] N = E δx j + (1 X j )β a kj X k dt k=1 time-change of E[X j ] = Pr[X j = 1], probability that node j is infected if infected: probability of curing per unit time if not infected (healthy): probability of infection per unit time N N de[x j ] = δe X j + β a kj E [ X k ] β a kj E X j X k dt k=1 R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Review of Modern Physics, 2015 k=

13 SIS Prevalence Fraction of infected nodes in the graph G S( t) = 1 N N j=1 X j ( t) (random variable!) Prevalence: Expected fraction of infected nodes in G ( ) = E S( t) y t!" # $ = 1 N N j=1 ( t) =1 Pr!" X j # $ also called order parameter in statistical physics P. Van Mieghem, F. Darabi Sahneh and C. Scoglio, 2014, "Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold", Proceedings of the 53rd IEEE Conference on Decision and Control (CDC 14), December 15-17, Los Angeles, CA, USA (also on 26 SIS Prevalence versus viral infectiousness No disease endemic τ (1) = 1 c A λ 1 ( ) < τ c Epidemic Threshold strongly depends on the network! 27 13

14 Local rule - global emergent properties class de[x j ( t)] # = E% δx j t dt $ ( ) + (1 X j t N ( ))β a kj X k t k=1 Local SIS rule ( ) & ( ' dy( t * ) = y( t * ) + τ dt * N E " # wt t * Global emergent SIS spread ( )Qw t * ( ) P. Van Mieghem, 2016, "Approximate formula and bounds for the time-varying SIS prevalence in networks", Physical Review E, Vol. 93 No. 5, p $ % The Laplacian Q = D A The normalized time t* = d t Bernoulli state vector w( t * ) = X 1 ( t * ), X 2 ( t * ),, X N t * ( ( )) 28 SIS prevalence dynamics ( ) = y( t * ) + τ dt * N E " # wt t * dy t * ( )Qw t * ( ) $ % Set of infected nodes at time t* NS( t * ) = 7 w T ( t * )Qw( t * ) =12 6 Cut-Set: set of links with 1 infected node at time t* Set of susceptible nodes at time t* P. Van Mieghem, 2016, "Universality of the SIS prevalence in networks", Delft University of Technology, report (arxiv ). Van Mieghem, P. and K. Devriendt, 2018, "An Epidemic Perspective on the Cut Size in Networks", Delft University of Technology, report

15 Outline Networks Birth of Network Science Function and graph Outlook Types of Networks Infrastructural networks (public, high value) o telecommunication, electricity, transport (train, air, car, ship), water, gas, waste Economic networks: o banks; flow of money, products,... Biological networks: o organisms, metabolic, brain,... Relational networks: private, graph clear; function? o Online social (Twitter, Facebook,...), family trees, company/nation organization, collaboration, software structure,... Network isomorphisms: what is common in all those networks? 31 15

16 Robust design of networks Robust/resilient networks against failures major difficult: what is robustness? o We need standardization to agree on a computable R-value Optimization problem? o network science: local-rule, global emergent properties o control/system theory: operational points around instabilities ( phase transition ) o game theory: discover the rules & strategy of the game autonomous networking (i.e. with minimum human interference) self-adaptive networking (brain) 32 Books Articles:

17 Open Postdoc position Optimal scheduling and routing with stringent end-to-end delay requirements (i.e. combining service and topology ) 34 More info? Please, me Thank You Piet Van Mieghem NAS, TUDelft P.F.A.VanMieghem@tudelft.nl 35 17