Overview of Network Theory

Transcription

1 Overview of Network Theory MAE 298, Spring 2009, Lecture 1 Prof. Raissa D Souza University of California, Davis

2 Example social networks (Immunology; viral marketing; aliances/policy) M. E. J. Newman

3 The Internet (Robustness to failure; optimizing future growth; testing protocols on sample topologies) H. Burch and B. Cheswick

4 A typical web domain (Web search/organization and growth centralized vs. decentralized protocols) M. E. J. Newman

5 The airline network (Optimization; dynamic external demands) Continental Airlines

6 The power grid (Mitigating failure; Distributed sources) M. E. J. Newman

7 Biology: Networks at many levels Control mechanisms / drug design/ gene therapy / biomarkers of disease Cellular networks: GENOME Genome, Proteome: Dandekar Lab Metabolome: Fiehn Lab protein-gene interactions PROTEOME protein-protein interactions Data intergration BIOshare Lin, Genome Center Citrate Cycle METABOLISM Bio-chemical reactions Network structure / search for biomarkers: D Souza

8 Software systems (Highly evolveable, modular, robust to mutation, exhibit punctuated eqm) Open-source software as a systems paradigm. Networks: Function calls communication Socio-Technical congruence Bird, Devanbu, D Souza, Filkov, Saul, Wen

9 Networks: Physical, Biological, Social Geometric versus virtual (Internet versus WWW). Natural /spontaneously arising versus engineered /built. Each network optimizes something unique. Identifying similarities and fundamental differences can guide future design/understanding. Interplay of topology and function? Unifying features: Broad heterogeneity in node degree. Small Worlds (Diameter log(n)).

10 Explosion of work and tools R, Graphviz, Pajek, igraph, Network Workbench, NetworkX, Netdraw, UCInet, Bioconductor, Ubigraph...

11 Natn Acam Sciences/Natn Research Council Study (2005) all our modern critical infrastructure relies on networks... too much emphasis on specific applications/jargon/disciplinary stovepipes... need a cross-cutting science of networks... Research for the 21st century

12 How do we represent a network as a mathematical object?

13 NETWORK TOPOLOGY Connectivity matrix, M: M ij = { 1 if edge exists between i and j 0 otherwise = M Node degree is number of links.

14 Typical measures of network topology Degree distribution (fraction of nodes with degree k, for all k) Clustering coefficient (fraction of triangles in the graph/transitivity: Are my friends friends with each other?) Also a local measure, for each node c i is number of connections existing between neighbors/total number of possible connections.

15 Typical measures of network topology, cont Diameter (Greatest distance between any two connected nodes) Small world if d log N and strong clustering. (Watts Stogatz, Nature 393, 1998.) Betweenness centrality (Fraction of shortest paths passing through a node, i.e., is a node a bottleneck for flow?)

16 Typical measures of network topology, cont Assortative/dissortative mixing (Are nodes with similar attributes more or less likely to link to each other? Mixing by node degree common. Also, in social networks mixing by gender and race.) (Example of assortative mixing by race. Friendship network of HS students: White, African American and Other.)

17 Network Activity: FLOWS on NETWORKS (Spread of disease, routing data, materials transport/flow) Random walk on the network has state transition matrix, P : 1/4 1/3 1/2 1/4 0 1/4 1/3 0 1/4 0 1/4 0 1/ /4 1/3 0 1/4 1/ /4 1/2 = P The eigenvalues and eigenvectors convey much information. Markov Chains, Spectral Gap.

18 Random walk on the WWW is the Page Rank Page Rank of a node is the steady-state random walk occupancy probabilty.

19 Example Eigen-technique: Community structure (Political Books 2004) M. Girvan and M. E. J. Newman

20 The classic random graph, G(N, p) (The Null Model) P. Erdös and A. Rényi, On random graphs, Publ. Math. Debrecen P. Erdös and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci E. N. Gilbert, Random graphs, Annals of Mathematical Statistics, Start with N isolated vertices. Add random edges one-at-a-time. N(N 1)/2 total edges possible. After E edges, probability p of any edge is p = 2E/N(N 1) What does the resulting graph look like? (Typical member of the ensemble)

21 N=300 p = 1/400 = p = 1/200 = 0.005

22 Emergence of a giant component p c = 1/N. p < p c, C max log(n) p > p c, C max A N (Ave node degree t = pn so t c = 1.) Branching process (Galton-Watson); tree -like at t c = 1.

23 Is connectivity a good thing? Communication, transportation networks Spreading of a virus (human or computer)

24 Random graphs as real-world networks? What about degree distribution, clustering, assortativity...? Shown later, Erdos-Renyi yields a Poisson degree distribution, but configuration models work around this. Still need null models to match other properties. Network Analysis in the Social Sciences, S. P. Borgatti, A. Mehra, D. J. Brass, G. Labianca, Science 323, , Why would a real network look like a random one? Local properties of nodes and edges, not statistics of the network. Developing the correct null models?

25 Degree distribution of real-world networks Extremely broad range of node degree observed: from biological, to technological, to social.

26 Typical distribution in node degree The Internet Faloutsos 3, SIGCOMM 1999 p(k) k 2.16 Who-is-Who network Szendröi and Csányi p(k) = ck γ e αk " out" exp( ) * x ** ( ) Small data sets, power laws vs other similar distributions? What is the Internet / what level? (e.g., router vs AS)

27 Power law with exponential tail Ubiquitous empirical measurements: System with: p(x) x B exp( x/c) B C Full protein-interaction map of Drosophila High-confidence protein-interaction map of Drosophila Gene-flow/hydridization network of plants as function of spatial distance m Earthquake magnitude Nm Avalanche size of ferromagnetic materials L 1.4 ArXiv co-author network MEDLINE co-author network PNAS paper citation network

28 What is a power law? (Also called a Pareto Distribution in statistics). p k k γ ln p k γ ln k p(k) 1e 10 1e 07 1e 04 1e k

29 Power Laws versus Bell Curves: Heavy tails Power law distribution: p k k γ. Gaussian distribution: p k exp( k 2 /2σ 2 ). p(k) p(k) 1e 56 1e 44 1e 32 1e 20 1e k k If 1 < γ < 2, mean and variance. If 2 < γ < 3 mean is finite, but variance.

30 Degree distribution and Network Growth Models Heterogeneity in real networks. Concentrated, Poisson Distribution in Erdös-Rényi: Probability to connect to k nodes is p k. Probability to be disconnected from remaining (n k) is (1 p) (n k). Probability for a vertex to have degree k follows a binomial distribution: p k = ( n k) p k (1 p) n k. Seek alternate mechanisms...

31 Known Mechanisms for Power Laws Phase transitions (singularities) Random multiplicative processes (fragmentation) Combination of exponentials (e.g. word frequencies) Preferential attachment / Proportional attachment (Polya 1923, Yule 1925, Zipf 1949, Simon 1955, Price 1976, Barabási and Albert 1999) Attractiveness is proportional to size: ds dt s Add in saturation [Amaral 2000, Börner 2004], get power laws with exponential decay.

32 Origins of preferential attachment 1923 Polya, urn models Yule, explain genetic diversity Zipf, distribution of city sizes (1/f ) Simon, distribution of wealth in economies. ( The rich get richer ). [Interesting note, in sociology this is referred to as the Matthew effect after the biblical edict, For to every one that hath shall be given... (Matthew 25:29)]

33 Preferential attachment in networks D. J. de S. Price, Networks of scientific papers Science, First observation of power laws in a network context. Studied paper co-citation network. D. J. de S. Price, A general theory of bibliometric and other cumulative advantage processes J. Amer. Soc. Info. Sci., The rate at which a paper gains citations is proportional to the number it already has. (Probability to learn of a paper proportional to number of references it currently has). A.-L. Barabási and R. Albert, Emergence of Scaling in Random Networks Science, (Citations: 1000 in in in April, in Feb, 2009) (Together with Watts-Strogatz Collective dynamics of small-world networks Nature 1998, launched flurry of activity in network science.)

34 Preferential Attachment random graphs: A discrete time process. Start with single isolated node. At each time step, a new node arrives. Probability incoming node attaches to a particular node of degree j: P ij j Explicitly: P ij = j/ j d j = j/(2mt). We are interested in the limit of large graph size.

35 Rate equations (a typical analysis tool) (Let n k,t number of nodes of degree k at time t, and n t total number of nodes at time t: Note n t = t) For each arriving link: n k,t+1 = n k,t + (k 1) 2mt n k 1,t k 2mt n k,t Probability: p k,t = n k,t /n(t) Assume steady state: p k,t p k.

36 Recursion for p m p k = (k 1)(k 2) (m) (k+2)(k+1) (m+3) p m = m(m+1)(m+2) (k+2)(k+1)k 2 (m+1) p k = 2m(m+1) (k+2)(k+1)k For k 1 p k k 3 Get power law with γ = 3.

37 Concepts covered today Social, physical and biological networks Simple network metrics (recapped next page) Random walks on networks Random graphs Phase transitions in connectivity Preferential attachment and network growth Next time: Robustness, Internet structure, optimization, biological networks.

38 Outstanding challenges How do we connect network structure to function? Degree Clustering Coefficient Motifs Betweeness Centrality Assortativity Flow and transport Growth/evolution mechanisms. Interacting networks Strategic interactions / Game theory on networks