Complex networks: an introduction

Transcription

1 Alain Barrat Complex networks: an introduction CPT, Marseille, France ISI, Turin, Italy

2 Plan of the lecture I. INTRODUCTION II. I. Networks: definitions, statistical characterization II. Real world networks DYNAMICAL PROCESSES I. Resilience, vulnerability II. III. IV. Random walks Epidemic processes (Social phenomena) V. Some perspectives

3 What is a network Network=set of nodes joined by links very abstract representation very general convenient to describe many different systems

4 Some examples Social networks Nodes Individuals Links Social relations Internet WWW Routers AS Webpages Cables Commercial agreements Hyperlinks Protein interaction networks Proteins Chemical reactions and many more ( , P2P, foodwebs, transport.)

5 Interdisciplinary science Science of complex networks: -graph theory -sociology -communication science -biology -physics -computer science

6 Interdisciplinary science Science of complex networks: Empirics Characterization Modeling Dynamical processes

7 G=(V,E) Paths Path of length n = ordered collection of n+1 vertices i 0,i 1,,i n V n edges (i 0,i 1 ), (i 1,i 2 ),(i n-1,i n ) E i 3 i 4 i 5 i 0 i 1 i 2 Cycle/loop = closed path (i 0 =i n )

8 Paths and connectedness G=(V,E) is connected if and only if there exists a path connecting any two nodes in G is connected is not connected is formed by two components

9 Paths and connectedness G=(V,E)=> distribution of components sizes Giant component= component whose size scales with the number of vertices N Existence of a giant component Macroscopic fraction of the graph is connected

10 Paths and connectedness: Paths are directed directed graphs Giant IN Component Giant SCC: Strongly Connected Component Giant OUT Component Disconnected components Tendrils Tube Tendril

11 Shortest paths Shortest path between i and j: minimum number of traversed edges i j distance l(i,j)=minimum number of edges traversed on a path between i and j Diameter of the graph= max(l(i,j)) Average shortest path= ij l(i,j)/(n(n-1)/2) Complete graph: l(i,j)=1 for all i,j Small-world : small diameter

12 Centrality measures How to quantify the importance of a node? Degree=number of neighbours= j a ij k i =5 i (directed graphs: k in, k out ) Closeness centrality g i = 1 / j l(i,j)

13 Betweenness centrality for each pair of nodes (l,m) in the graph, there are σ lm shortest paths between l and m σ lm i shortest paths going through i b i is the sum of σ lm i / σ lm over all pairs (l,m) path-based quantity j i b i is large b j is small NB: similar quantity= load l i = σ i lm NB: generalization to edge betweenness centrality

14 Structure of neighborhoods k 3 Clustering coefficient of a node i 2 C(i) = # of links between 1,2, n neighbors k(k-1)/2 1 Clustering: My friends will know each other with high probability! (typical example: social networks)

15 Structure of neighborhoods Average clustering coefficient of a graph C= i C(i)/N NB: slightly different definition from the fraction of transitive triples: C = 3 x number of fully connected triples number of triples

16 Statistical characterization Degree distribution List of degrees k 1,k 2,,k N Not very useful! Histogram: N k = number of nodes with degree k Distribution: P(k)=N k /N=probability that a randomly chosen node has degree k Cumulative distribution: P > (k)=probability that a randomly chosen node has degree at least k

17 Statistical characterization Degree distribution P(k)=N k /N=probability that a randomly chosen node has degree k Average= k = i k i /N = k k P(k)=2 E /N Sparse graphs: k N Fluctuations: k 2 - k 2 k 2 = i k 2 i /N = k k2 P(k) k n = k k n P(k)

18 Statistical characterization Multipoint degree correlations P(k): not enough to characterize a network Large degree nodes tend to connect to large degree nodes Ex: social networks Large degree nodes tend to connect to small degree nodes Ex: technological networks

19 Statistical characterization Multipoint degree correlations Measure of correlations: P(k,k, k (n) k): conditional probability that a node of degree k is connected to nodes of degree k, k, Simplest case: P(k k): conditional probability that a node of degree k is connected to a node of degree k often inconvenient (statistical fluctuations)

20 Statistical characterization Multipoint degree correlations Practical measure of correlations: average degree of nearest neighbors k=4 i k=4 k=7 k=3 k i =4 k nn,i =( )/4=4.5

21 Statistical characterization average degree of nearest neighbors Correlation spectrum: putting together nodes which have the same degree class of degree k

22 P(k k) Statistical characterization independent of k case of random uncorrelated networks proba that an edge points to a node of degree k number of edges from nodes of degree k number of edges from nodes of any degree P unc (k k)=k P(k )/ k proportional to k itself

23 Typical correlations Assortative behaviour: growing k nn (k) Example: social networks Large sites are connected with large sites Disassortative behaviour: decreasing k nn (k) Example: internet Large sites connected with small sites, hierarchical structure

24 Correlations: Clustering spectrum P(k,k k): cumbersome, difficult to estimate from data Average clustering coefficient C=average over nodes with very different characteristics Clustering spectrum: putting together nodes which have the same degree (link with hierarchical structures) class of degree k

25 Weighted networks Real world networks: links carry trafic (transport networks, Internet ) have different intensities (social networks ) General description: weights i w ij j a ij : 0 or 1 w ij : continuous variable

26 Weights: examples Scientific collaborations: number of common papaers Internet, s: traffic, number of exchanged s Airports: number of passengers Metabolic networks: fluxes Financial networks: shares usually w ii =0 symetric: w ij =w ji

27 Weighted networks Weights: on the links Strength of a node: s i = j V(i) w ij =>Naturally generalizes the degree to weighted networks =>Quantifies for example the total trafic at a node

28 Weighted clustering coefficient i w ij =1 w ij =5 i s i =16 c iw =0.625 > c i k i =4 c i =0.5 s i =8 c iw =0.25 < c i

29 Weighted clustering coefficient k w ik i w ij (w jk ) j Average clustering coefficient C= i C(i)/N C w = i C w (i)/n Random(ized) weights: C = C w C < C w : more weights on cliques C > C w : less weights on cliques Clustering spectra

30 Weighted assortativity i k i =5; k nn,i =1.8

31 Weighted assortativity i k i =5; k nn,i =1.8

32 Weighted assortativity i k i =5; s i =21; k nn,i =1.8 ; k nn,iw =1.2: k nn,i > k nn,i w

33 Weighted assortativity i k i =5; s i =9; k nn,i =1.8 ; k nn,iw =3.2: k nn,i < k nn,i w

34 Participation ratio 1/k i if all weights equal close to 1 if few weights dominate

35 Plan of the lecture I. INTRODUCTION II. I. Networks: definitions, statistical characterization II. Real world networks DYNAMICAL PROCESSES I. Resilience, vulnerability II. III. IV. Random walks Epidemic processes (Social phenomena) V. Some perspectives

36 Two main classes Natural systems: Biological networks: genes, proteins Foodwebs Social networks Infrastructure networks: Virtual: web, , P2P Physical: Internet, power grids, transport

37 Metabolic Network Protein Interactions Nodes: metabolites Links:chemical reactions Nodes: proteins Links: interactions

38 Scientific collaboration network Nodes: scientists Links: co-authored papers Weights: depending on number of co-authored papers number of authors of each paper number of citations

39 Transportation network: Urban level TRANSIMS project Nodes=locations (homes, shops, offices ) Weighted links=flow of individuals

40 World airport network complete IATA database l l V = 3100 airports E = weighted edges l w ij #seats / (time scale) > 99% of total traffic

41 Meta-population networks Each node: internal structure Links: transport/traffic City a City i City j

42 Internet Computers (routers) Satellites Modems Phone cables Optic fibers EM waves different granularities

43 Internet mapping continuously evolving and growing intrinsic heterogeneity self-organizing Largely unknown topology/properties Mapping projects: Multi-probe reconstruction (router-level): traceroute Use of BGP tables for the Autonomous System level (domains) CAIDA, NLANR, RIPE, IPM, PingER, DIMES Topology and performance measurements

44 The World-Wide-Web Virtual network to find and share informations web pages hyperlinks CRAWLS

45 Sampling issues social networks: various samplings/networks transportation network: reliable data biological networks: incomplete samplings Internet: various (incomplete) mapping processes WWW: regular crawls possibility of introducing biases in the measured network characteristics

46 Networks characteristics Networks: of very different origins Do they have anything in common? Possibility to find common properties? the abstract character of the graph representation and graph theory allow to answer.

47 Social networks: Milgram s experiment Milgram, Psych Today 2, 60 (1967) Dodds et al., Science 301, 827 (2003) Six degrees of separation SMALL-WORLD CHARACTER

48 Small-world properties Average number of nodes within a chemical distance l Scientific collaborations Internet

49 Small-world properties N points, links with proba p: static random graphs short distances (log N)

50 Clustering coefficient n 3 Empirically: large clustering coefficients 2 Higher probability to be connected 1 Clustering: My friends will know each other with high probability (typical example: social networks)

51 Small-world networks N nodes forms a regular lattice. With probability p, each edge is rewired randomly =>Shortcuts N = 1000 Large clustering coeff. Short typical path Watts & Strogatz, Nature 393, 440 (1998)

52 Topological heterogeneity Statistical analysis of centrality measures: P(k)=N k /N=probability that a randomly chosen also: P(b), P(w). node has degree k Two broad classes homogeneous networks: light tails heterogeneous networks: skewed, heavy tails

53 Topological heterogeneity Statistical analysis of centrality measures Broad degree distributions (often: power-law tails P(k) k -γ, typically 2< γ <3) Internet No particular characteristic scale

54 Topological heterogeneity Statistical analysis of centrality measures: linear scale Poisson vs. Power-law log-scale

55 Exp. vs. Scale-Free Poisson distribution Power-law distribution Exponential Scale-free

56 Power-law tails P(k) k -γ Consequences Average= k = k P(k)dk Fluctuations k 2 = k 2 P(k) dk k c 3-γ k c =cut-off due to finite-size N => diverging degree fluctuations for γ < 3 Level of heterogeneity:

57 Other heterogeneity levels Weights Strengths

58 Other heterogeneity levels Betweenness centrality

59 Clustering and correlations non-trivial structures

60 Complex networks Complex is not just complicated Cars, airplanes => complicated, not complex Complex (no unique definition): many interacting units no centralized authority, self-organized complicated at all scales evolving structures emerging properties (heavy-tails, hierarchies ) Examples: Internet, WWW, Social nets, etc

61 Example: Internet growth

62 Main features of complex networks Many interacting units Self-organization Small-world Scale-free heterogeneity Dynamical evolution Standard graph theory Random graphs Static Ad-hoc topology Example: Internet topology generators Modeling of the Internet structure with ad-hoc algorithms tailored on the properties we consider more relevant

63 Statistical physics approach 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 Development of new modeling frameworks

64 New modeling frameworks Example: preferential attachment (1)GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system). (2)PREFERENTIAL ATTACHMENT : The probability that a new node will be connected to node i depends on the connectivity k i of that node Π ( k ) i = ki Σ k j j and many other mechanisms and models P(k) ~k -3 A.-L.Barabási, R. Albert, Science 286, 509 (1999)