1. Alessandro Vespignani (CNRS, LPT-Paris)

2. Alain Barrat (CNRS, LPT-Paris) Yamir Moreno (University of Saragoza) Alexei Vazquez (University of Notre Dame) Romualdo Pastor-Satorras (UPC -Barcelona) Roberto Percacci (INFN) Marc Barthelemy (CEA-Paris) Luca Dall’Asta (CNRS, LPT-Paris) Ignacio Alvarez Hamelin (CNRS, LPT-Paris)

3. The Physical Internet Satellites Computers (routers) Modems (??) Phone cables Optic fibers EM waves Technological Heterogeneity

4. A network is a system that allows its abstract/mathematical representation as a graph Vertices (nodes) = elements of the system Edges (links) = interactions/relations among the elements of the system

5. Internet tomography Multi-probe reconstruction (router-level) Use of BGP tables for the Autonomuos System level (domains) CAIDA NLANR RIPE IPM PingER Claffy et al (1999). Topology and performance measurements Graph representation different granularities

7. CAIDA AS cross section map

8. Shortest paths = minimum (# hops) between two nodes Regular lattice with N= 10 4 d ~ 10 2 Small world with N= 10 4 d ~ ln N

9. Average fraction of nodes within a shortest path of lenght d Distribution of Shortest paths (# hops) between two nodes

10. Haphazard set of points and lines Randomness This does not imply complexity!!

11. Erdös-Rényi model (1960) Poisson distribution With probability p an edge is established among couple of vertices = p N

12. Where “the complications” are ?? (Late 90s large networks graphs and data become available)

13. Clustering coefficient = connected peers will likely know each other C = # of links between 1,2,…n neighbors n(n-1)/2 1 2 3 n Higher probability to be connected

14. Connectivity distribution Scale-free properties P(k) = probability that a node has k links P(k) ~ k -  (    3) = const   = const   Diverging fluctuations Faloutsos et al. 1999 Router level& AS level Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)

15. Scale-free topology generators INET (Jin, Chen, Jamin) BRITE (Medina & Matta) Classical Internet topology generators Waxman generator Structural generators Transit-stub Tiers Exponentially Bounded Degree distributions Modeling of the Internet structure with ad-hoc algorithms tailored on the properties we consider more relevant

16. The Internet growth 19972000 AS31129107 19972000 AS31129107 In 1999: 3410 new AS 1713 lost AS Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001) Qian, Govindan et al. (2002)

17. Main Features of complex networks Many interacting units Dynamical evolution Self-organization Many interacting units Dynamical evolution Self-organization Non-trivial architecture Unexpected emergent properties Cooperative phenomena Complexity Supervising entityProject/blueprint

18. Statistical physics approach to network modeling Microscopic processes of the many component units Macroscopic statistical and dynamical properties of the system Cooperative phenomena Complex topology Natural outcome of the dynamical evolution

19. Shift in focus : Dynamical processes Modeling starts from the understanding of the basic mechanisms underlying the networks’ growth Complex topology is spontaneously generated in the models (opposite to ad-hoc constructions) Richer understanding of the interplay among dynamics, traffic and economical requirements.

20. Preferential attachment mechanism Networks expand by the addition of new nodes Examples: WWW : addition of new documents Internet : connection of new routers Nodes are wired with higher probabibility to highly connected nodes Examples: WWW : links to well known web-pages Internet : links to well connected ISP

21. How to generate scale-free graph Growth : at each time step a new node is added with m links to be connected with previous nodes Preferential attachment : The probability that a new link is connected to a given node is proportional to the number of node’s links. by Barabasi & Albert (1999) The BA model The preferential attachment is following the probability distribution : The generated connectivity distribution is P(k) ~ k - 

22. BA network Degree distribution

23. Preferential Attachment in Internet Probability that a link connects to a node with connectivity k  (k) ~ k  p(k)  k    - 1.2  1.0 Linear preferential attachment Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001) Qian, govindan et al. (2002) Jeong, Neda and Barabasi (2003)

24. Shift of focus: Static construction Dynamical evolution Direct problem Evolution rulesEmerging topology Inverse problem Given topologyEvolution rules

25. More models Generalized BA model (Redner et al. 2000) (Mendes & Dorogovstev 2000) (Albert et al.2000) Non-linear preferential attachment :  (k) ~ k  Initial attractiveness :  (k) ~ A+k  Rewiring Highly clustered (Eguiluz & Klemm 2002) Fitness Model (Bianconi et al. 2001) Multiplicative noise (Huberman & Adamic 1999)

26. Heuristically Optimized Trade-offs (HOT) Papadimitriou et al. (2002) New vertex i connects to vertex j by minimizing the function Y(i,j) =  d(i,j) + V(j) d= euclidean distance V(j)= measure of centrality Optimization of conflicting objectives

27. What else…… Hierarchies and correlations (architecture) Robustness and resilience Spreading phenomena Routing and database updating

28. The Hierarchy of the Internet Stub AS : has only one connection to another AS Multi-homed AS: two or more connections to other ASs but does not carry transit traffic Transit AS: Two or more connections to other ASs and carries transit traffic HIERARCHICAL DECOMPOSITION Four level hierarchy (linear scale) (Govindan and Reddy 1994) Three-tier hierarchy (log scale) (Chang et al. 98) Jellyfish hierarchy (connectivity shells) (Tauro et al. 2001)

29. Connectivity correlations Degree correlation function =  k’ k’ p(k’|k) Average nearest neighbors degree Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)

30. Connectivity Hierarchy Degree correlation function =  k’ k’ p(k’|k) Average nearest neighbors degree Highly degree ASs connect to low degree ASs Low degree ASs connect to high degree ASs No hierarchy for the router map

31. Clustering Hierarchy Clustering coefficient as a function of the vertex degree Highly degree ASs bridge not connected regions of the Internet Low degree ASs have links with highly interconnected regions of the Internet No hierarchy at the router level

32. Scale-free hierarchy Continuum of levels Modular construction Small groups of networks organized in larger groups which act as the modules at the next level and so on “ad libitum”

33. Models validation (part II) Clustering hierarchyDegree hierarchy

34. Absence of any epidemic threshold (critical point) Active state for any value of The infection pervades the system whatever spreading rate In infinite systems the infection is infinitely persistent (indefinite stationary state) Density of infected individuals Scale-free connectivity Pastor-Satorras &Vespignani, PRL 86, 3200(2001)

35. Rationalization of computer virus data Lack of healthy phase = standard immunization cannot drive the system below thershold!!!

36. Distributed database updating Broadcast = each elements passes the update to neighbors Epidemics = the update is spread as an infective agent E= efficiency = # updated databases # of messages sent Warning => not deterministic Not all elements are contacted!! Trade-offs between efficiency and reliability Moreno, Nekovee,Vespignani(03)

37. Internet is ever changing at all levels Is it too ambitious the attempt to have a dynamical theory of the Internet at the large scale ?? The lesson of statistical physics and cooperative phenomena: Basic symmetry and principles win over the microscopic details when we look at emergent properties

38. One step back….

39. Deployement of measurement tools Active Passive Netscan (traceroute based tool) maps the paths to selected IP address from a testing host (single probe). Testing host = directed graph spanning tree One path to each node NO cross-paths Burch & Cheswick (1999)

40. Interconnected level maps Heuristic methods (Govindan et al.) Router level maps Very effective for intranetwork

41. Measurements infrastructures Merging partial spanning tress from multiple sources

42. Sampling is incomplete Lateral connectivity is missed (edges are underestimated) Finite size sample Govindan et al 2002

43. Introduction of Biases Vertices and edges best sampled in the proximity of sources Number of sources and target (total traceroute probes) Statistical properties of the sampled graph sharply different from the original one Crovella et al. 2002 Clauset & Moore 2004 De Los Rios & Petermann 2004

44. Mean-Field Theory of traceroute-like exploration  = N s N t N N t = # targets (  t -> density of targets) N s = # sources p ij = 1 –exp ( -  b ij ) p i = 1 – (1-  t ) exp ( -  b i ) =  t N s k* i = 2  +2  b i ) Edge detection probability Vertex detection probability Effective degree observed b i, b ij Betweenness

45. Betweenness centrality = # of shortest paths traversing a vertex or edge (flow of information ) if each individuals send a message to all other individuals

46. Scale-free graph are better discriminated Tail is sampled very effectively

47. Homogeneous graphs give rise to spurious effects Average connectivity always dominate

48. Heavy tails properties are a genuine feature of the Internet Quantitative analysis might be strongly biased however What else…. Router level very limited maps Optimized strategies Massive deployement traceroute@home

49. The dark side of the moon……Traffic and weights The internet is a weighted networks bandwidth, traffic, efficiency, routers capacity Data are scarse and on limited scale Interaction among topology and traffic Traffic and routing