The Mathematical Description of Networks

Transcription

1 Modelling Complex Systems University of Manchester, 21 st 23 rd June 2010 Tim Evans Theoretical Physics The Mathematical Description of Networs Page 1

2 Notation I will focus on Simple Graphs with multiple edges allowed (no values or directions on edges, no values for vertices) N = number of vertices in graph E = number of edges in graph = degree of a vertex <> = average degree = 2E/N Degree =2 Degree Distribution n() = number of vertices with degree p() = n()/n = normalised distribution Page 2

3 1. Random Graphs 2. Scale-Free Models 3. Different Networ Views 4. Summary How to excite a Mathematician give them the simplest networ model

4 Classical Random Graphs [Solomonoff-Rapoport 51, Erdős-Réyni 59] For every pair of distinct vertices add a single edge with probability (1-p) p = <>/(N-1), otherwise with probability (1-p) no edge is added p Page 4

5 Classical Random Graph Gives Binomial Degree Distribution exp p( )! with <>=(N-1)p Exponential cutoff so no hubs e.g. N=10 6, <>=4.0, typically has <17 Page 5

6 Example of Classical Random Graph N=200 <>~4.0 < 11 In figure vertex size Diffuse, no tight cores Page 6

7 Generalised Random Graphs The Molloy-Reed Construction [1995,1998] i. Fix N vertices ii. Attach stubs to each vertex, where is drawn from given distribution p() iii. Connect pairs of stubs chosen at random Page 7

8 No Vertex-Vertex Correlations Generalised Random Graphs have given p() but otherwise completely random in particular - Properties of all vertices are the same For any given source vertex, the properties of neighbouring vertices independent of properties of the source vertex Page 8

9 Random Wals on Random Graphs The degree distribution of a neighbour is not simply p() You are more liely to arrive at a high degree vertex than a low degree one p( ) n p( n i n ) n Degree of neighbour n Page 9 independent of degree of starting point i i

10 A random friend is more popular than you n n p( ) n i 2 (Number of friends neighbour has) - (Number of your friends) n 2 0 Give a random friend that life saving vaccine (if social networs are random and uncorrelated) Page 10

11 Length of Random Wals on Random Graphs Suppose we follow a random wal where we never go bac along the edge we just arrived on, then for infinite graphs (N ) Wals always end if n < 2 No GCC 1 2 OUT ( n -1) Wals never end if n > 2 GCC n IN (GCC= Giant Connected Component) Page 11

12 Length of Random Wals on Random Graphs PROVIDED there are no loops. True for sparse random graphs in limit of infinite size (N ) Wals always end if n < 2 No GCC 1 2 OUT ( n -1) Wals never end if n > 2 GCC n IN (GCC= Giant Connected Component) Page 12

13 GCC (Giant Connected Component) transition GCC= Giant Connected Component, where a finite fraction of vertices in infinite graph are connected GCC exists if z>1 where z n OUT 1 ( n -1) = Fractional measure of how much more popular your friends are Page 13 n IN

14 Other properties of General Random Graphs All global properties depend on same z 1 e.g. GCC size, component distribution, average path lengths 2 Page 14

15 Ensembles of Graphs Mathematically we do not consider a single instance of a random graph but an ensemble of random graphs Page 15

16 Ensembles of Graphs e.g. The probability of creating a particular simple graph with E edges and Ē empty edges is P( G) p 1 E p E Classical Random Graphs Page 16

17 Ensemble Averages Averages of quantities are strictly over both a) different graphs and b) over some element of a graph e.g. vertices G P( G ) 1 N i i V ( G) Page 17

18 Exponential Random Graphs (p* models) General ensemble of graphs, those with highest probability obey any given constraints f G P( G) f ( G) P( G) 1 Z e H ( G) H(G) chosen so that graphs with preferred properties are most liely Page 18

19 Example Graph Hamiltonians H(G) H( G) be Classical random graph with p=2e/(n(n-1)) H ( G) vv b v v Random Graph with given degree distribution. In both cases Lagrange multipliers b, b v fixed by specifying desired values of < E > and < v > Page 19

20 MR Random Graph Calculations Calculations lie those above wor because lac of correlations between vertices few loops for large sparse graphs, graphs are basically trees These can be reasonable approximations for many models and perhaps for a few real graphs too. Otherwise use as a null model. Page 20

21 Summary of Random Graphs Calculations wor because lac of correlations between vertices few loops for large sparse graphs, graphs are basically trees Accessible analytically so can suggest typical behaviour even if very wea e.g. diameter vs N These can be reasonable approximations for many theoretical models Probably not for real world so then use these as a null model.

22 1. Random Graphs 2. Scale-Free Models 3. Different Networ Views 4. Summary How to excite a physicist give them a power law

23 log p() Distribution p() log(p()) Long Tails in Real Data Period 2 (excluding Haldane), degree distribution Imperial Library Loans (Hoo, Sooman, Warren, TSE) log() Imperial Medical School (URL, 4 wees, < 3rd Jan 2005) N=14805 <>=22.95 slope= E E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 Imperial Medical School Web Site (Hoo, TSE) p Po 1.E Degree distribution, ebay Crawl (max 1000) Logged user in degree 1.E-08 1.E-09 Equiv. Binomial Degree (Sooman, Warren, TSE) Page 23 log gallery future-i com All log() vs. log(p()) except text log(ran) vs. log(freq.) Text frequency in TSE Contemporary Physics review

24 Growth with Preferential Attachment [Yule 1925, 1944; Simon 1955; Price 1965,1976; Barabasi,Albert 1999 ] 1. Add new vertex attached to one end of m=½<> new edges 2. Attach other ends to existing vertices chosen with by picing random end of an existing edge chosen randomly, so probability is P() = / (2E) Preferential Attachment Rich get Richer Page 24 2/(2E) P() 2/(2E) 5/(2E) 4/(2E) Result: Scale-Free n() ~ -g g=3

25 N=200, <>~4.0, vertex size Classical Random Scale-Free = Power-Law p()~ 1/ 3 Diffuse, small degree vertices max =O(ln(N)) Page 25 Tight core of large hubs max =O(N 1/2 )

26 Master Equation Approach Let n(,t) represent the average number of vertices at time t. (I should really use <n(,t)>) Again average means we loo at an ensemble of such networs. The master equation the equation for evolution of the degree distribution averaged over different instances of networ in the ensemble n(,t) to n(,t+1) Page 26

27 Master Equation Processes n(,t) changes in one of three ways:- Increases as we add an edge to existing vertex of degree (-1). Decreases as we add an edge to existing vertex of degree. Number of vertices of degree =m= ½<> always increase by 1 as add new vertex. Page 27

28 Mean Field Degree Distribution Master Equation n(, t 1) n(, t) n( 1, t) mp( 1) n(, t) mp( ), m P() = Probability of attaching to a vertex of degree in simplest preferential attachment models Page 28

29 The Mean Field Approach is an Approximation Distribution n i () different in each instance i Page 29 Ensembles over many instances i at one time t ni ( ) n ( ) i b b n i ( ) b 1 n ( ) Normalisation of probabilities not usually same for different i If P() is a function of degree then normalisation of this probability is different in each instance of a networ in the ensemble at a single time t. i b

30 Ensemble Invariants n(, t 1) n(, t) n( 1, t) P( 1) n(, t) P( ), m Adding one vertex and m= ½<> edges at each time means that the number of edges Page 30 E(t) = mt + E(0) number of vertices N(t) = t + N(0) are the same for all instances of networ in the ensemble at any one time t.

31 The Mean Field Approach Can Be Exact Distribution n i () different in each instance i ni ( ) n ( ) i b b n i ( ) b 1 n ( ) i b Ensembles over many instances i at one time t Normalisation of probabilities the same for different i if b=0 or 1 YES only if Page 31 n i ( ) b n i ( ) b b =0 or b=1

32 Exact Solution of Master Equation Possible if P( ) p p 2E p r 1 N, Preferential Attachment Random Attachment Note probability so 0 P( ) 1 & p p p r 1 Lowest degree is 1 min m 2 Thus 0 p p min 21! Page 32

33 Page 33 Loo for asymptotic solutions Find for > m = ½<> Exact Solution of Master Equation m p p m p p N N p p r p r p P P 2) (1 1 1) ( 2) (1 ) (. 1 1) (. 1) ( ) ( ) ( ) ( ), ( p t N t n

34 Exact Solution of Master Equation Hence ) 1 ( ) ( ) ( b a a A p Large limit:- A p ) ( lim, 2, p p r p b p p a p p where &

35 Scale-Free Growing Model comments Illustrates use of master equations and their approximations statistical physics experience Exact solutions for ensemble average asymptotic value of degree distribution p() if P( ) (1 2E N Interpretation of parameters p p >1 allowed Finite Size effects? real networs are mesoscopic [TSE, Saramäi 2004] Fluctuations in ensemble? Networ not essential =frequency of previous choices Growth not essential networ rewiring g ~ 1.0 [Moran model, see TSE,Plato, 2008] p r ) p r 1, Page 35

36 Scale-Free in the Real World Attachment probability used was 1 P( ) p p 2E p r N, BUT if lim P() a for any a1 then a power law degree distribution is not produced! Page 36

37 Preferential Attachment for Real Networs [Saramäi, Kasi 2004; TSE, Saramäi 2004] 1. Add a new vertex with ½<> new edges 2. Attach to existing vertices, found by executing a random wal on the networ of L steps Probability of arriving at a vertex number of ways of arriving at vertex =, the degree Preferential Attachment g=3 Page 37 Start Wal Here (Can also mix in random attachment with probability p r )

38 Naturalness of the Random Wal algorithm Automatically gives preferential attachment for any shape networ and hence tends to a scale-free networ Uses only LOCAL information at each vertex Simon/Barabasi-Albert models use global information in their normalisation Uses structure of Networ to produce the networs a self-organising mechanism e.g. informal requests for wor on the film actor s social networ e.g. finding lins to other web pages when writing a new one Page 38

39 Is the Wal Algorithm Robust? d~5 I have varied: Length of wals <> Starting point of wals Length distribution of wals.. YES - Good Power Laws L=0 Pure Random Attachment exponential graph L=7 L=1 but NOT Universal values - 10% or 20% variation Page 39

40 1. Random Graphs 2. Scale-Free Models 3. Different Networ Views 4. Summary

41 What is a Networ? A Networ is BOTH a set of vertices AND a set of edges Page 41

42 Vertex Centric Viewpoint YET we tend to have a very VERTEX centred viewpoint:- Degree of a vertex Communities usually a partition of vertices Distances are vertex to vertex distances Word frequencies in Networ Review [TSE, Contemporary Physics 2004] Networ Words Vertex Words Edge Words Page 42

43 Word Frequencies in Networ Review Word Ran Count Word Ran Count networ distribut vertic scale edg 3 86 problem random 3 86 simpl graph 5 81 idea degre 6 78 physic power 7 68 size lattic 8 67 find law 9 65 real vertex type number case distanc hub model show connect area data neighbour lin studi world point larg term small figur averag form comput site Stop words removed and stemmed [TSE, Contemporary Physics 2004]

44 Edge Centric Viewpoint? Create a new graph L(G) whose vertices represent the edges of the old graph, G, and whose edges record the overlap of the old edges i.e. the common vertices in G. This is called the LINE GRAPH Name given by Harary who was using the word line for edges but this construction predates this wor. Page 44

45 Vertices of a Line Graph 1. For every edge a in original graph G create a vertex a in the line graph L(G) a a b b G L(G) Page 45

46 Edges of a Line Graph 2. Connect the vertices a and b in the line graph L(G) if the corresponding edges in original graph G were coincident a a i i i b i b Page 46 G L(G)

47 Properties of a Line Graph Not usually a duality transformation L(L(G)) G (Almost) always reversible [Whitney 1932] L(G) G G L(G) Page 47

48 The Problem with a Standard Line Graph High degree vertices in original graph G over represented by edges in Line Graph L(G). Page 48 G Degree vertex L(G) (-1)/2 edges

49 Solution Weighted Line Graphs [Evans & Lambiotte 2009] Original graph vertex of degree produces line graph edges of weight 1/(-1) Strength 1 G Degree vertex Page 49 Weight 1/(-1) WL(G) (-1)/2 edges, Total weight

50 Example Bow Tie Graph Graph G Line Graph L(G) 1 1/3 Page 50

51 Community Detection - Vertex Centric version Partition vertices into communities Perform random wal on vertices Compare number of random walers which stay within community after one step against number which remain within communities after infinite number of steps = Optimisation of Modularity [Girvan & Newman 2002; Lambiotte, Delvenne & Barahona 2008] Page 51

52 Community Detection - Edge Centric version Partition edges into communities Perform random wal on edges = Random wal on line graph vertices Compare number of random walers which stay within community after one step against number which remain within communities after infinite number of steps = Optimisation of Modularity of Weighted Line Graph [Evans & Lambiotte 2009] Page 52

53 Karate Club Edge partition means individuals vertices can be members of more than one community Page 53 Ford-Fulerson partition of Zachary

54 Karate Club Analysis Vertices in One Edge Community # Fraction In Green C Mr Hi (the Instructor) bridges several groups % % % % % 0 (Mr_Hi) 16 25% Page 54

55 Give Cliques a Chance a clique centric viewpoint Can we shift out view point from vertices to cliques? Use a Clique Graph to represent the way that n-cliques overlap. Example Graph A has three 3-cliques = triangles A b a Page 55

56 Edge Weights in Basic Clique Graph Weight of an edge in the basic Clique Graph C records the number of vertices common to two n-cliques in in the original graph A. A b a a C 2 2 b 1

57 Summary Line graphs, clique graphs, and graphs of the overlap of other structures, move focus from vertices to edges/cliques etc with minimal effort Weighted line graphs avoid problem of over representation of high degree vertices Example: Community detection on line graph produces overlapping vertex communities for original graph Page 57

58 1. Random Graphs 2. Scale-Free Models 3. Different Networ Views 4. Summary

59 Conclusions Considerable input possible from from a mathematical approach to networs Google Tim Evans Networs to find my web pages on networs Page 59