Mini course on Complex Networks

Transcription

1 Mini course on Complex Networks Massimo Ostilli 1 1 UFSC, Florianopolis, Brazil September 2017 Dep. de Fisica

2 Organization of The Mini Course Day 1: Basic Topology of Equilibrium Networks Day 2: Percolation and Magnetism Day 3: Growing Networks

3 First of all What is a graph? Preliminary Definitions A historical example

4 Formally, a graph G is a pair of sets G = (V, E), where V is a set of N = V nodes, and E a set of L = E edges (or links)

5 G is called Simple if there are no multiple links and no self-links

6 G is Connected if any node can be reached from any other node via a path of links

7 G is Disconnected if it is Not Connected

8 Node-degree k k 1 = 1, k 2 = 2, k 3 = 4,...

9 Mean-degree (or connectivity of G) k k = i k i/n = 2L/N N = 10, L = 17 2L/N = 3.4

10 G is a Tree if there are No Loops

11 Undirected and Directed Graphs k 4 = 4 k IN 4 = 3, k OUT 4 = 2

12 G is sparse if L = O(N) G is dense if L = O(N α ) with α > 1

13 To summarize G is a pair of sets G = (V, E), where V is a set of N = V nodes, and E a set of L = E edges (links) G is called simple if there are no multiple links and no self-links G is called sparse if L = O(N) G is called dense if L = O(N α ) with α > 1 The degree k i (or connectivity) of a node i, is the number of links emanating from the node k = 1 N N i=1 k i = 2L N, for the moment refers to the mean over a single G

14 Why studying graphs: An historical example

15 Why studying graphs: An historical example

16 Day 1: Basic Topology of Equilibrium Networks Cayley Trees and Bethe Lattices Adjacency Matrix Main Graph Metrics: P(k); C ; l The Random Graph Model The Configuration Model Main Graph Metrics: Simple Evaluations for locally Tree-like nets

17 Cayley Tree = Finite Regular Tree Here q = 3 and k =?

18 Cayley Tree = Finite Regular Tree Here q = 3 and k = 2 2/N (holds for any finite tree)

19 Bethe Lattice = Infinite Regular Tree Here q = 3 and k = q

20 Adjacency Matrix a Any graph G can be encoded via the Adjacency Matrix a. We label the nodes by an index i = 1,..., N { 1, if there a link between i and j, a i,j = 0, otherwise In other words G = (V, E) a

21 Adjacency Matrix a The degree of the vertex i k i = j a i,j, L(G) = 1 2 i,j a i,j = N k 2, The number of triangles passing through the vertex i N T (i) = j,k a i,j a j,k a k,i, N T (G) = 1 3 Tr(a3 ) The number of non self overlapping paths of length l passing between i and j N Paths (i, j; l) (a l ) i,j + O((a l 1 )) Random Matrix Theory approach...

22 Degree Distribution P(k) N(k) is the number of nodes with degree k P(k) = N(k) N, In a Regular d-dimensional Lattice P(k) = δ k,2d, In the Random Graph (classical) P(k) = k k k! e k, In Complex Networks (typically, for large k) P(k) k γ, γ > 2

23 Compare Random and Scale-Free Complex Network

24 Compare Random and Scale-Free Complex Network

25 Degree Distribution P(k) Note that, if P(k) k γ N P(k)k α k=1 N 1 dk P(k)k α, lim k < and lim k 2 k 2 < γ > 3, N N lim k < and lim k 2 k 2 = 2 < γ < 3, N N lim k = and lim k 2 k 2 = γ < 2. N N Most of the complex networks observed in nature and technology have γ 3 and very often γ 2.

26 Link-Degree Distribution (or Excess Distribution) P L (k) P L (k) = Prob. (that the end of a link points to a node of degree k) It is simple to see that We define also P L (k) = P(k)k k P L (k, k ) = Prob. (that the two ends of a link point to a node of degree k and to a node of degree k, resp. ) A graph G is called Uncorrelated if P L (k, k ) = P L (k)p L (k ).

27 Clustering Coefficient C C(i) = Prob. (that between two neighbors of node i there is a link) C = 1 C(i) Average Clustering Coefficient N i C = 0, Trees and Bethe Lattices ( C = O N 1), Random Graph C = O ( N α), α < 1 C = O (1), C = 1, Uncorrelated Complex Networks Lattices and Strongly Correlated Complex Networks Complete Graph Most of the complex networks observed in nature and technology have a small but not negligible C.

28 Average Path Length l Given G, let l i,j the length (i.e., the number of links), of the shortest path between i and j. Their mean is l = 2 N(N 1) In most cases of interest l scales very slowly with N (Small-World) and, furthermore the distribution of the l i,j is quite picked around l, and l Diameter(G) = max i,j l i,j. i<j l i,j

29 Regular d-dimensional Lattice

30 Regular d-dimensional Lattice In a cube of side R, N R d and l D R l N 1/d.

31 Bethe Lattice = Infinite Regular Tree R=distance between the central node, chosen as reference and the nodes on the boundary of this sub-graph having N nodes. We have N = 3 2 R 1 R 1 = log(n/3)/ log(2) the maximal distance between two randomly chosen nodes in the sub-graph will be D = 2R = log(n/3)/ log(2), from which we guess also l = O(log(N)/ log(2)).

32 The Random Graph (A Finite Random Bethe Lattice ) Given N and a parameter 0 p 1, for each pair of nodes put a link with probability p. In other words the a i,j s are i.i.d random variables with We have and from Prob. ( a i,j = 1 ) = p, Prob. ( a i,j = 0 ) = 1 p. k = 2L N, L = N(N 1) p k = (N 1)p 2

33 A Random Graph with N=20

34 Sparse Random Graph Since k = (N 1)p if we choose p = c N 1, c = O(1) k = c = O(1) (Sparse Graph) It is immediate to see that ( C = O c N 1) Locally Tree like... to be discussed later we can use the analogy with the Bethe Lattice and find-out the Small-World property: l log(n) log( k ) Six Degrees of Separation... to be discussed later

35 Sparse Random Graph We have in the sparse case, p = ( ) N 1 P(k) = p k (1 p) N 1 k k c N 1, we have ck lim P(k) = N k! e c This Degree Distribution is not representative of real-world networks!

36 How Real Nets look like: Flickr-User

37 How Real Nets look like: Air-Traffic

38 How Real Nets look like: Bio

39 How Real Nets look like: Internet

40 How Real Nets look like: WWW of Universities

41 How Real Nets look like: Neural Network

42 How Real Nets look like: Food Web

43 How Real Nets look like: Food Web

44 24 A. Clauset, C. R. Shalizi and M. E. J. Newman Power Law of Real Nets (A. Clauset et.al. SIAM 2009) P(x) P(x) P(x) P(x) (a) (b) 10 1 (c) words 10 3 proteins metabolic (d) (e) 10 2 (f) Internet 10 6 calls wars (g) (h) (i) terrorism HTTP species (j) 10 0 (k) 10 0 (l) birds blackouts book sales x x x

45 Power Law of Real Nets (A. Clauset et.al. SIAM 2009) quantity n x σ x max ˆx min ˆα count of word use ± (2) protein interaction degree ± 2 3.1(3) metabolic degree ± 1 2.8(1) Internet degree ± (9) telephone calls received ± (1) intensity of wars ± (2) terrorist attack severity ± 4 2.4(2) HTTP size (kilobytes) ± (5) species per genus ± 2 2.4(2) bird species sightings ± (2) blackouts ( 10 3 ) ± (3) sales of books ( 10 3 ) ± (3) population of cities ( 10 3 ) ± (8) address books size ± (6) forest fire size (acres) ± (3) solar flare intensity ± (2) quake intensity ( 10 3 ) ± (4) religious followers ( 10 6 ) ± (1) freq. of surnames ( 10 3 ) ± (2) net worth (mil. USD) ± (1) citations to papers ± (6) papers authored ± (1) hits to web sites ± (8) links to web sites ± (9) Table 6.1 Basic parameters of the data sets described in this section, along with their power-law fits and the corresponding p-value (

46 The Configuration Model Consider a sequence of non negative integers k 1,..., k N such that N k i = 2L i=1 Then connect in all the possible ways the 2L stubs

47 The Configuration Model

48 The Configuration Model In principle we can build graphs with any given P(k) = N(k)/N Due to the random way by which we join the stubs, these graphs are approximately uncorrelated: Prob. ( a i,j = 1 ) k i k j /2L Self-links and multiple-links exist but are negligible We can evaluate C We can evaluate l We can understand when correlations are important

49 The Configuration Model: C We use C = Prob. (that between two neighbors of a given node there is a link) C = 1 k(k 1) 2 N k 3

50 The Configuration Model: l We use N l = Number of Paths of length l ( k(k 1) N l = k k ) l 1 l = ln (N/ k ) ) ln ( k(k 1) k This makes us to understand also that the shortest loops are of length O(ln N)

51 The Configuration Model: About Correlations?? Prob. ( a i,j=1 ) = k i k j N k?? max i,j k i k j N k = k 2 max N k If P(k) k γ we use k max N 1 γ 1 k 2 max N k N 3 γ γ 1

52 To summarize We have static network models where P(k) is the only arbitrary input (maximally random) and where, in particular, if P(k) k γ with γ > 2: C 0 (Locally Tree-Like) l ln(n) for γ > 3 (Small-World), or l ln(ln(n)) for γ < 3 (Ultra-Small-World) Shortest Loops have lenght ln(n) (Locally Tree-Like) Correlations do exist for γ < 3 (Degree-Degree Corr.)

53 Day 2: Percolation and Magnetism Examples Node- and Link-Percolation Percolation in Uncorrelated Complex Networks Anomalous Mean-Field behavior Magnetism

54 Percolation

55 Link- and Node-Percolation Bond- (or Link-) Percolation: Links are kept with probability p Node- (or Site-) Percolation: Nodes are kept with probability p For the moment being we can avoid making use of p.

56 Percolation Without Removal in the Random Graph

57 Percolation Without Removal in Uncorrelated Nets F.C.C. = Finite Connected Component G.C.C. = Giant Connected Component x = Prob. (that the end of a randomly chosen link points to a F.C.C.) S =(Number of Nodes belonging to the G.C.C.)/N x = k=1 P L (k)x k 1 1 S = k=0 P(k)x k k(k 1) k k(k 1) k < 1 No Percolation > 1 Percolation

58 Anomalous Mean-Field Behavior (No Removal) Within a certain limit, we can choose P(k) such that ( ) k(k 1) k = 1 Percolation Threshold If in particular P(k) k γ, we find: if γ > 4, k 3 <, and S ( k k c ) β, with β = 1 (classical limit) if 3 < γ < 4, k 2 <, k 3 =, and S ( k k c ) β, with β = 1 γ 3 if 2 < γ < 3, k 2 =, to satisfy ( ) we need to introduce random node removal

59 Percolation via Node-Removal in Uncorrelated Nets Now we remove randomly each node with probability 1 p x = 1 p + p k=1 P L (k)x k 1 1 S = 1 p + p k=0 P(k)x k k(k 1) p k k(k 1) p k < 1 No Percolation > 1 Percolation

60 Anomalous Mean-Field Behavior (Node-Removal) p c = k k(k 1) and if P(k) k γ we find if γ > 4, k 3 <, and Percolation Threshold S (p p c ) β, with β = 1 (classical limit) if 3 < γ < 4, k 2 <, k 3 =, and S (p p c ) β, with β = 1 γ 3 if 2 < γ < 3, k 2 =, p c 0 and S p β with β = 1 3 γ

61 Anomalous Ferromagnetic Mean-Field Behavior If P(k) k γ we have: if γ > 5, k 4 <, and m (T T c ) β, with β = 1 2 (classical limit) if 3 < γ < 5, k 2 <, k 4 =, and m (T T c ) β, with β = 1 γ 3 if 2 < γ < 3, k 2 =, T c =, and m T β with β = 1 3 γ

62 In some networks T c = : how is it possible? If for k large P(k) e k/ k or P(k) k γ with γ 5, we have in both cases almost homogeneous networks, k k and m (T T c ) β, with β = 1 2 (cl. mean-field) If for k large P(k) k γ with γ 2, we have an extremely heterogeneous network, k fluctuates a lot and T c = with m T β An extreme example: The Star Graph

63 Day 3: Growing Networks Examples Random Growing Model Barabasi-Albert Model Linear Model Continuum Approximation

64 Growing Networks

65 Random Growing Model (Dorogovtsev) Prob. (that the new link goes to a node of degree k) = 1 t Master Equation for P(k; t): P(k; t + 1)(t + 1) P(k; t)t = P(k 1; t) P(k; t) + δ k,1 lim t P(k; t) = P(k) = 1 2 k

66 Barabasi-Albert Model Prob. (that the new link goes to a node of degree k) = lim t P(k; t) = P(k) k t i=1 k i 1 (k + 2)(k + 1)k k 3 C 1 N 3/4 l ln(n) ln (ln(n))

67 Barabasi-Albert Model Generalized Prob. (that one of the new m links goes to a node of degree k) = lim t P(k; t) = P(k) 1 (k + 2)(k + 1)k k 3 k t i=1 k i C 1 N 3/4 l ln(n) ln (ln(n))

68 Linear Model Here k 0 0 and k refers to IN-degree only Pr. (that one of the new m links goes to a node of degree k) = k+k 0 t i=1 (k i +k 0 ) lim t P(k; t) = P(k) k γ, γ = 2 + k 0 m

69 References: 1 R. Albert, A.L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002). 2 S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks (University Press: Oxford, 2003). 3 M.E.J. Newman, The structure and function of complex networks, SIAM Review 45, 167 (2003). 4 S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys. 80, 1275 (2008). 5 S.N. Dorogovtsev, Lectures on complex networks (Oxford Master Series in Statistical, Computational, and Theoretical Physics, 2010).