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Dsouza Supernova 2008
 

Dsouza Supernova 2008

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    Dsouza Supernova 2008 Dsouza Supernova 2008 Presentation Transcript

    • Layers of Networks (Towards a Science of Networks) Raissa D’Souza UC Davis Dept of Mechanical and Aeronautical Eng. Complexity Sciences Center Santa Fe Institute
    • Transportation Networks/ Networks: Power grid (distribution/ collection networks) Computer networks Biological networks - protein interaction Social networks - genetic regulation - Immunology - drug design - Information 22 January 2007 CSE Advance - Commerce 2
    • 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. 1. How do we build a coherent distributed energy system integrating solar, wind, hydropower, bio-diesel, hydrogen, etc. 2. Is old infrastructure introducing vulnerabilities in telecom? • Definition of node can depend on level of representation.
    • Studying each network individually (Though we know they interact) • Topology (Statistical properties of node and edges) – degree and degree distribution (extremely varied) – diameter (“small-world”) – clustering coefficients – assortative mixing – betweenness, communities/partitioning, etc. • Activity (Information flows) – epidemiology (humans and computers) – Web search (ranking the web map) – consensus formation / tipping points / phase transitions Interactions between structure and function.
    • Software call graphs and OSS Developer networks • Highly evolveable, modular, robust to mutation, exhibit punctuated eqm • Open-source software as a “systems” / organization paradigm. D’Souza, Filkov, Devanbu, Swaminathan, Hsu
    • NETWORK TOPOLOGY Connectivity matrix, M : 1 if edge exists between i and j Mij = 0 otherwise.   1 1 1 1 0   1 1 0 1 0   1 0 1 0 0 =M    1 1 0 1 1     0 0 0 1 1 Node degree is number of links.
    • Broad Heterogeneity in node degree e.g., The “Who-is-Who” network in Budapest ´ ¨ ´ ´ (Balazs Szendroi and Gabor Csanyi) Bayesian curve fitting → p(k) = ck −γ e−αk
    • istribution peaked at k and decaying 20 10 Random Power Law Graphs: Attack (e.g., “Preferential Attachment”, Barabasi and Albert, Science 1999) 15 Hubs and leaves 5 b Failure 0 10 Albert, Jeong and Barabasi, Nature, 406 (27) 2000. 0.00 0.01 0.02 0.00 N=130, E=215 f Red five highest degree nodes; Figure 2 Changes in theGreen theirthe network as a func diameter d of neighbors. removed nodes. a, Comparison between the exponential (E) a models, each containing N ¼ 10;000 nodes and 20,000 links “Robust” to random failure, symbols correspond to the diameter of to targeted. (triang fragile the exponential (squares) networks when a fraction f of the nodes are removed Red symbols show the response of the exponential (diamonds) Is connectivity a good thing? networks to attacks, when the most connected nodes are rem Scale-free dependence of the diameter for different system sizes (N ¼ 1 rence between an exponential and a scale-free found that the obtained curves, apart from a logarithmic size s homogeneous: most nodes have approximately those shown in a, indicating that the results are independent o Engineered networks (e.g., the Internet) are not random! e-free network is inhomogeneous: the majority of note that the diameter of the unperturbed (f ¼ 0) scale-free n few nodes have a large number of links, of the exponential network, indicating that scale-free network connected. Red, the five nodes with the highest them more efficiently, generating a more interconnected web
    • Optimization in network growth (D’Souza, Borgs, Chayes, Berger, Kleinberg, PNAS 2007) o oo o o o o o o oo o o o o o o o o oo o o o o o o o o o oo oo o o o o o o o o o o o o o o o o o o o o oo o o o o o o o o o oo o o o o o o oo o o o o o o o o oo o oo o oo o o o o o o o o o o o o o o o o o o oo o o o o oo oo oo o o o o o o o o o o oo o o oo oo o o o o o o oo (Competing objectives)
    • Network Activity: FLOWS on NETWORKS (Spread of disease, routing data, materials transport/flow, gossip spread/marketing) 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 1/2 0 0 =P    1/4 1/3 0 1/4 1/2     0 0 0 1/4 1/2 The eigenvalues and eigenvectors convey much information. Markov Chains, Spectral Gap.
    • Feedback and network growth of Hierarchical organizations • Functional = efficient information flow throughout organization. • More functional → grow faster (but each new attachment less optimal) • Less functional → grow slower but more balanced (each new attachment more considered) (more balanced, efficient structures: respond to changing circumstances)
    • Building a “science of networks” • Last ten years, since 1999. • Understanding activity and topology of individual networks. • “Nodes”, “Robustness” (e.g., connectivity) context dependent. “all our modern critical infrastructure relies on networks”
    • Our modern infrastructure Layered, interacting networks • MATHEMATICS NEEDED: Multiple info streams; Layered interactions; PDEs (calculus)