NETPLEXITY Networks and Complexity for the Real World

Transcription

1 SCCS 2012 University of Gloucestershire 9th - 12th August Tim Evans NETPLEXITY Networks and Complexity for the Real World Page 1

2 What is a Network? Mathematically, a network is a graph a set of vertices (or nodes) and a set of edges (or links) representing bilateral relationships Page 2

3 Explosion of interest since 1998 [Watts & Strogatz, Nature 1998] Fraction of papers with word starting NETWORK in title compared to number of papers with word MODEL Page 3 WHY?

4 Why so much recent interest? Social Network Analysis dates back 50 or more years e.g. Citation networks [Price 1965] Graph Theory in mathematics is centuries old e.g. Euler - Konigsburg bridge problem Arrival of BIG data sets and the computer power to analyse them is powering recent interest. Page 4

5 EXAMPLES

6 Types of Network by application Physical links/hardware based telephone links, internet hardware, power lines, transport Biological Networks neural, biochemical, protein, ecological Social Networks Questionaires, observation, electronic social networks Information Networks academic papers, patents, keywords, web pages, artefact networks Dimensions 2 Death of Distance? [Cairncross 1997] High Dimension Page 6

7 Physical Networks: Airline Transport Map Vertices = airports, geographical location Edges= flights from/to, thickness passengers Weighted Graph [Holten & van Wijk 2009] Page 7

8 Physical Networks: Minoan Aegean Stochastic network model representing links in Minoan Aegean (c1500bc) [TSE, Knappett and Rivers, 2008,2009] Page 8

9 Biological Networks:- Neuroscience Vertices from anatomy Vertices from recording sites Weighted Graphs Page 9 Network Analysis [Bullmore & Sporns, 2009]

10 Social Networks: Zachary Karate Club Vertices = Club Members Club split in two Simple Graph Edges = Observed Relationships ON or OFF Page 10 [Zachary, 1977; TSE & Lambiotte 2009]

11 Citation Networks Vertices = papers [TSE & Lambiotte, 2009] Directed Acyclic Graph Edges = citations from bibliography to paper [Zachary, 1977] [Blondel et al, 2008] Page 11

12 So many networks Networks are a useful way to describe many different data sets Physical links/hardware based Biological Networks Social Networks Information Networks Page 12

13 REPRESENTATIONS

14 Representations Data often has a `natural network There is no one way to view this natural network visualisation There are always many different networks representing the data Page 14

15 Visualisation In a network the location of a vertex is defined only relative to its neighbours i.e. defined by the edges This looks like a random network? Page 15

16 Visualisation In a network the location of a vertex is defined only relative to its neighbours i.e. defined by the edges [Evans, 2004] Identical Networks Periodic Lattice N=20, E=40 Same network with vertices arranged in regular order. Same network with vertices arranged in random order Page 16

17 Visualisation [Evans, 2004] Even if a network has a specific embedding in 2D space, this is not encoded in the network Periodic Lattice 2D spatial location everything Airlines 2D important but social & political spaces important Page 17

18 Visualisation Is this a random network? Visualisation can show non trivial structure:- Positioning of vertices using algorithms that reflect connections Application of Community Detection (clustering) Page 18

19 Visualisation Is this a random network? Visualisation can show non trivial structure:- Positioning of vertices using algorithms that reflect connections Application of Community Detection (clustering) Page 19

20 Visualisation Positioning of vertices using algorithms that reflect connections Application of Community Detection (clustering) Edge colour also Page 20 Ford- Fulkerson partition of Zachary Actual Split [Zachary, 1977; TSE & Lambiotte 2009]

21 Visualisation Choosing the right visualisation is a powerful practical tool, and its not just the vertices... [Holten & van Wijk 2009] e.g. Edge bundling for air network [Holten & van Wijk, 2009]

22 Different Networks, same data Vertices = brain regions e.g. From Correlation Matrices in Neuroscience Networks Matrix = Correlated Activity Edges = Minimum Correlation Page 22 [Lo et al, 2011]

23 Different networks, same data - Network Projections papers authorship authors p q r s bipartite network e.g. Coauthorship networks Company Board Membership Directors coauthorship q Codirectorship Page 23 authors p,q q,r s unipartite network Directors

24 Line Graphs Vertices = Intersections Edges = Streets Page 24

25 Line Graphs Vertices = Intersections Edges = Streets Vertices = Streets Edges = Intersections Space Syntax Space is the Machine, Hillier, 1996 Page 25

26 Line Graph problem One vertex degree k [Evans & Lambiotte 2009] Clique with k(k-1)/2 edges Graph G Line Graph L(G) Solution: weight line graph edges with 1/k Page 26

27 Line Graphs Produce line graph Cluster vertices of line graph = Edge colours Vertices in several communities Actual Split Ford-Fulkerson partition of Zachary Page 27 [TSE & Lambiotte, 2009]

28 Karate Club Analysis Vertices in One Edge Community # k Fraction k In Green C Mr Hi (the Instructor) bridges several groups % % % % % 0 (Mr_Hi) 16 25% Page 28 [TSE & Lambiotte, 2009]

29 Clique Graphs A Clique Graphs record the number of vertices common to two cliques in in the original graph A. A b a g a C (3) 2 2 [TSE, 2010] b 1 g

30 Karate Club Clique Graph Communities Red = multiple communities Colours = Unique Community Zachary partition [TSE, 2010]

31 Hypergraphs A collection of vertices A collection of hyperedges subsets of vertices of any size Subsets of different sizes n These indicate n-times relationships between people No longer just bilateral relationships A graph is just a 2-regular hypergraph i.e. n=2 for all hyperedges.

32 COMPLEXITY

33 Complexity is Difficult Interactions occur defined at small, local scales Emergence of large scale phenomena Just statistical mechanics applied to new problems? Page 33

34 Complexity and Networks Networks are a natural part of Complexity Real networks are difficult Mathematical proofs only for random graphs Computational algorithms often NP-complete Edges represent local interactions Emergent Phenomena Communities/Clusters Small Worlds Six degrees of separation Power Laws and Fat-Tailed Distributions Netplexity = Complex Networks Page 34

35 Critical Phenomena and Networks Large fluctuations Detailed microscopic rules irrelevant Universality classes Models of self-organised criticality often mix ideas from complexity with critical phenomena concepts e.g. Sand pile model Are critical phenomena relevant to networks? Page 35

36 Large Fluctuations in Networks Imperial Library Loans (Hook, Sooman, Warren, TSE) Equiv. Binomial Imperial Medical School Web Site (Hook, TSE) Logged user in degree (Sooman, Warren, TSE) Page 36 gallery future-i com [Barabási, Albert 1999] All log(k) vs. log(p(k)) except text log(rank) vs. log(freq.) Text frequency in my Contemporary Physics review

37 Power Law Degree Distributions Plots like these often described in terms of power laws. WARNING - Not all networks have power laws in their degree distributions p(k) despite impression in literature and folk tradition. Page 37

38 Power Law Degree Distributions What you can say is:- Many networks have fat tailed distributions A power law may be a useful summary of tail Many other functional forms may fit as well Usually too little data to be more precise [Stumpf and Porter, 2012; TSE Page 38

39 Growth with Preferential Attachment P(k) [Yule 1925, 1944; Simon 1955; Price 1965,1976; Barabasi,Albert 1999 ] P(k) = k / (2E) Preferential Attachment Rich get Richer 2/(2E) 5/(2E) 4/(2E) Result: Scale-Free Network n(k) ~ k -g g=3 2/(2E) Page 39

40 Preferential Attachment for Real Networks [Saramäki, Kaski 2004; TSE, Saramäki 2004] 1. Add a new vertex with ½<k> new edges 2. Attach to existing vertices, found by executing a random walk on the network of L steps Start Walk Here Page 40 (Can also mix in random attachment with probability p r )

41 Preferential Attachment for Real Networks Probability of arriving at a vertex number of ways of arriving at vertex = k, the degree Preferential Attachment g=3 Can also mix in random attachment with probability p r Start Walk Here Page 41

42 Naturalness of the Random Walk algorithm Gives preferential attachment from any network and hence a scale-free network Uses only LOCAL information at each vertex Simon/Barabasi-Albert models use global information in their normalisation Uses structure of Network to produce the networks a self-organising mechanism e.g. informal requests for work on the film actor s social network e.g. finding links to other web pages when writing a new one Page 42

43 Is the Walk Algorithm Robust? I varied: Length of walks <k> Starting point of walks Length distribution of walks.. L=0 Pure Random Attachment exponential <d>~5 graph L=7 L=1 YES - Good Power Laws but NOT Universal values - 10% or 20% variation Page 43

44 Finite Size Effects Networks are mesoscopic systems In practice a network of N ~ 1 million is still not large since many quatitities scale with the logarithm of system size e.g. Diameter scales as log(n) ~ 6. Page 44

45 Networks are not Critical Systems No universal behaviour Power law or other long tails? Finite size effects cause uncertainty Networks are mesoscopic systems Value of power in power laws not universal even if power law behaviour is. Page 45

46 Criticality, Complexity and Networks Networks are usually Complex Networks are not usually critical systems Netplexity = Complex Networks Page 46

47 NETWORK SCIENCE

48 What is Network Science? Based on analysis through networks Graphs, hypergraphs Part of wider studies in complexity Local interactions produce emergent phenomena Not new Social Network Analysis since 50 s Mathematical graph theory since Euler in 1735 New aspect is Information Age Large data sets and their analysis now possible Multidisciplinary Communication difficult between fields Page 48

49 Does Network Science really exist? No coherent definition Too broad to be a single area New name for old work = Hype Too early to say No need to define a new field [ Network Science, nap.edu, 2005] Page 49

50 Are networks providing new insights? Just another approach to statistical analysis and data mining Sometimes this is a better way to analyse Gives new questions e.g. Small world definition Gives new answers e.g. Small world models Brings the tools of Complexity Scaling Page 50

51 THANKS or search for Tim Evans Networks

52 Bibliography: Free General Network Reviews Evans, T.S. Complex Networks, Contemporary Physics, 2004, 45, [arxiv.org/cond-mat/ ] Easley, D. & Kleinberg, J. Networks, Crowds, and Markets: Reasoning About a Highly Connected World Cambridge University Press, 2010 [ ] van Steen, M. Graph Theory and Complex Networks Maarten van Steen, 2010 [ ] Hanneman, R. A. & Riddle, M. Introduction to social network methods 2005 Riverside, CA: University of California, Riverside [ ] Brandes, U. & Erlebach, T. (ed.) Network Analysis: Methodological Foundations 2005 [ Page 52

53 Bibliography for this talk Barabási, A. & Albert, R. Emergence of scaling in random networks, Science, 1999, 286, 173 Blondel, V. D.; Guillaume, J.-L.; Lambiotte, R. & Lefebvre, E., Fast unfolding of community hierarchies in large networks, J STAT MECH-THEORY E, 2008, P10008 [arxiv.org: ] Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems Nat Rev Neurosci, 2009, 10, F. Cairncross (1997) The Death of Distance, Harvard University Press, Cambridge MA. Evans, T.S. Complex Networks, Contemporary Physics, 2004, 45, [arxiv.org/cond-mat/ ] Watts, D. J. & Strogatz, S. H., Collective dynamics of 'small-world' networks, Nature, 1998, 393, Evans TS Power Laws Evans, T.; Knappett, C. & Rivers, R., Using Statistical Physics To Understand Relational Space: A Case Study From Mediterranean Prehistory in Complexity Perspectives on Innovation and Social Change, Lane, D.; Pumain, D.; van der Leeuw, S. & West, G. (Eds.), Springer, 2009, 7, Evans, T.S. & Lambiotte, R., Line Graphs, Link Partitions and Overlapping Communities, Phys.Rev.E, 2009, 80, [arxiv: ] Evans, T.S. Clique Graphs and Overlapping Communities, J.Stat.Mech, 2010, P12037 [arxiv.org: ] Hillier, Space is the Machine, 1996 Holten, D. & van Wijk, J. J. Force-Directed Edge Bundling for Graph Visualization, Eurographics/ IEEE-VGTC Symposium on Visualization 2009, H.-C. Hege, I. Hotz, and T. Munzner (Guest Editors) 28 (2009), Number 3 Knappett, C.; Evans, TS. & Rivers, R., Modelling Maritime Interaction In The Aegean Bronze Age Antiquity, 2008, 82, Chun-Yi Zac Lo, Yong He, Ching-Po Lin, Graph theoretical analysis of human brain structural networks, Reviews in the Neurosciences 2011 (on line) Price, D. S., Networks of Scientific Papers, Science, 1965, 149, Stumpf, MPH & Porter, MA, Critical Truths About Power Laws, Science 335, 2012, [DOI: /science ]. Watts, D. J. & Strogatz, S. H. Collective dynamics of 'small-world' networks. Nature, 1998, 393, Zachary, W. Information-Flow Model For Conflict And Fission In Small-Groups Journal Of Anthropological Research, Page , 33,

54 Background Information See or Netplexity - Networks and Complexity for the Real World. General talk on Networks and Complexity given by Dr Tim Evans, Theoretical Physics, Imperial College London given on Friday 10th August 2012 as part of the second annual Student Conference on Complexity Science, Oxstalls Campus, University of Gloucestershire, 9th - 12th August Abstract: I will look at some of the different ways the science of Complex Networks gives insights into the world around us. Networks are an excellent way to look at the many large data sets which have appeared over the last decade or so such as web pages, Facebook, digital document repositories. The explosion of interest in networks has produced many new tools and insights which can be applied to many different types of data, from biological systems to humanities. Bio: Dr Tim Evans is part of the Theoretical Physics group and the Complexity and Networks programme at Imperial College London. He finished his PhD at Imperial in 1987, applying statistical physics to quantum field theory for many body problems. He was then a researcher at the University of Alberta in Edmonton Canada, after which he held research positions at Imperial, including a final one as a Royal Society University Research Fellow. He was appointed to the staff at Imperial in He has always been interested in many body systems both in and out of equilibrium and his current focus is on complex systems in general and Complex Networks in particular. This is both from a theoretical perspective (such as line graph representations of networks) and in terms of applications to practical problems such as bibliometrics and cultural transmission, part of an interest in `sociophysics' in general. This includes an ongoing project in Archaeology. Page 54