ネットワーク科学 空間システムデザイン

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1. 1 2019 ( ) 1 / 34
2. 1. or S.H.Yook, H.Jeong, and A.-L.Barab´ asi, PNAS 99(21), 13382, 2002 ( ) 2 / 34
3. M.T.Gastner and M.E.J.Newman, EPJB 49, 2004 ( ) 3 / 34
4. Typical Spatial Net Constructions Step 0 Step 1 Step 2 : in the radius r = A k (a) BA (b) SFL (c) RAN 1/2 Add new node New m-links Preferential attachment Connect to neighbors Add a new node inside a chosen triangle Connect to its closest 3 nodes Initial triangulation Assign a degree k Initial N0 nodes with m-links Select a node Saturated node Y.Hayashi, IPSJ Journal 47(3), 2006 ( ) 4 / 34
5. 2. BA BA j Πj ∝ d−α ij popβ j kγ j 011 101 110 111 S.S.Manna et al., PRE 66, 026118/066114, 2002, PRE 68, 026104, 2003, New J. of Phy. 9(30), 2007. ( ) 5 / 34
6. 3. A B C E H F D (c) From a random Sierpinski gasket to the correspondig Sierpinski Net Mapping from links a-i (b) Random Pseurdofractal SF Net (a) Random Apollonian Net A B C A B C D D E E to nodes A-I G I h g a b c d e i f S.N.Dorogovtsev et al., PRE 65, 066122, 2002, Z.Zhang et al., EPJB 65, 2008 ( ) 6 / 34
7. Random Appolonian Network , , . Add a new node inside a chosen triangle Connect to its closest 3 nodes Initial triangulation ⇒ ( ) 7 / 34
8. N + 1 k + 1 n(k + 1, N + 1) = k N△ n(k, N) + 1 − k + 1 N△ n(k + 1, N) P(k) ≈ n(k, N)/N , k(P(k + 1) − P(k)) + N + N△ N P(k + 1) = 0 , k k dP dk = −γRA P , P(k) ∼ k−γRA . , γRA = (N△ + N)/N ≈ 3, N△ = N△0 + 2N. T.Zhou, G.Yan, and B.-H.Wang, PRE 71, 046141, 2005 ( ) 8 / 34
9. Descartes ( ) 9 / 34
10. 4. (Optimal Traffic Tree) f def = e bewe = e be le te , e te = T ⇔ g def = T − e te = 0. te e , le , be ( e ) , we = le /te . , F = f − λg . ∂F ∂te = − bele t2 e + λ = 0, , λ e , te ∝ √ bele . , ∂F/∂λ = 0 te = T √ be le e √ be le f , F = 1 T e √ bele 2 . ( ) 10 / 34
11. ∼ µ, ν > 0 f = e bµ e lν e . µ = 0, ν = 1 , µ = 1/2, ν = 1/2 OTT, µ = 1, ν = 0 M.Barth´ elemy, and A.Flammini, J. of Stat. Mech. L0702, 2006 arXiv:physics/0601203v2 ( ) 11 / 34
12. : (1 − δ) × + δ × δ = 0.0 δ = 1.0 ο < δ < 1 M.T.Gastner, and M.E.J.Newman, PRE 74, 2006 ( ) 12 / 34
13. D.J.Ashton et al., PRL 94, 2005 ( ) 13 / 34
14. 5. dwij dt = f (Qij ) − wij Qij = wij (pi − pj ) lij , A.Tero et al., Science 327, 439, 2010 Constructal Law ( ) 14 / 34
15. Initial Configuration: UDG 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 Connectivity Ratio Transmission Range A N0 =102 N0 =103 N0 =104 Unit disk graph : Unit Disk Graph dij < A/ √ N0 , i j ⇒ A Y.Hayashi, and Y.Meguro, Physica A 391, 872-879, 2011 ( ) 15 / 34
16. Network Genetration Methods R = 0.1NT , Link Survival GreedyRouting eij wij → wij + 1 pd = 0.1 wij → wij − 1 T = 3 × 104 v w s t u s t u v w s t u Modified Rule 1 Modified Rule 2 (Self-avoiding) v X w dead-end Greedy Routing vt ut wt vt wt d < d < d d < d < d ut ( ) 16 / 34
17. Modification: Adding Shortcuts Path Reinforcement T , , LS 10 30% , , , Random Shortcut , LS 10 30% LS wij = 0 , eij , ( ) 17 / 34
18. From Organizational Theory 1997 , , 2006-07 Silicon Valley ( ) 18 / 34
19. Quantitative Analysises Shortcut Effect: Internet AS, Delaunay Tri., Apollonian SF : Y.Hayashi, and J.Matsukubo, Improvement of the robustness on geographical networks by adding shortcuts, Physica A, 380, 2007. Y.Hayashi, Necessary Backbone of Super-highways for Transport on Geographical Complex Networks, Advances in Complex Systems 12(1), 2009. Multi-Scale Quatered Net : Y.Hayashi, Evolutionary Construction of Geographical Networks with Nearly Optimal Robustness and Efficient Routing Properties, Physica A 388, 2009. ( ) 19 / 34
20. Simulation Results UDG LS PR10% RS10% ( ) 20 / 34
21. Robustness against Failures 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 S/N Fraction f of failures LS PR 30% RS 30% 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 <s> Fraction f of failures LS PR 30% RS 30% ⇒ fc , LS 0.6 PR RS 0.8 ( ) 21 / 34
22. Robustness against Attacks 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 S/N Fraction f of attacks LS PR 30% RS 30% 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 <s> Fraction f of attacks LS PR 30% RS 30% ⇒ fc , LS 0.4 PR RS 0.6 ( ) 22 / 34
23. Average Path-Length: SW Property 100 101 102 102 103 104 <Lij > NT LS PR 10% PR 30% RS 10% RS 30% 100 101 102 102 103 104 LS PR 10% PR 30% RS 10% RS 30% ⇒ LS O( √ NT ) PR RS O(log NT ) ! ( ) 23 / 34
24. 6. (a) triangle (b) square (c) triomino chair tiling (d) sphinx tiling ( ) 24 / 34
25. Multi-Scale Quartered Net 1 , : k1 = 2, k2 = 4(Tri) or 3(Squ), k3 = 6(Tri) or 4(Squ). 80km 160 × 160 500m ( ) 25 / 34
26. Good Properties of MSQ Nets (t-spanner, t = 2) Online Routing v1 v2 v3 v4 v5 v0: source v6: terminal v1’ v3’ v5’ v5’’ Y.Hayashi, Physica A 388, 991, 2009, PRE 82, 016108, 2010. , . ( ) 26 / 34
27. Generalized MSQ Nets , L × L ( ) (a) Initial square (b) chosen face (c) subdivision Y.Hayashi, T.Komaki, Y.Ide, T.Machida, and N.Konno, Physica A 392, 2013 ( ) 27 / 34
28. Random Walks with Splitting x′ × y′ x × y ⇔ (x′, y′) (x, y) Division of Rectangle Directional random Walk (x′ − x) × y, x × (y′ − y), (x′ − x) × (y′ − y) , ( ) 28 / 34
29. Universality Generalized-MSQ : ( ) ( ) , ⇒ G-MSQ Science 318(11), 742-743, 2007 Y.Hayashi, Chapter 4, In ”Networks -Emerging Topics in Computer Science,” A.Rezazadeh, L.Momeni, and I.Bilogrevic(Eds), iConcept Press, 2012 ( ) 29 / 34
30. Biological Foraging L´ evy lij P(lij ) ∼ l−µ ij µ → 1 , µ ≥ 3 Brown , µ ≈ 2 G.M.Viswanathan et al. Nature 401, 911, 1999, M.C.Santos et al. PRE 72, 046143, 2005. ⇒ ( ) 30 / 34
31. Inhomogeneous Adaptive Search G-MSQ α = 0- L´ evy Nt = 200 Ns = 50 ( ) 31 / 34
32. DTN Delay/Disruption-Tolerant Network (L) http://www.conceptdraw.com/examples/vehicular-network, (R) https://www.rt.com/usa/185128-drone-nasa-robot-highway ( ) 32 / 34
33. DTN , (MFs) Routes Stochastic Deterministic for moving decision (one- or multi-) Interactions Synchronous Asynchronous for data trans. by encounters by node-relaying of agents with temporal store J.J.P.C.Rodrigues(Eds), Advences in DTNs, Woodhead Pub., 2015, E.Shah, Int. J. of Com. Theo. and Eng. 3(4), 2011, W.Zhao et al., INFOCOM, 2005. ( ) 33 / 34
34. MF MF x disappeared cycle reset until the cycle at the damaged layer unification of cycles udpate for the smaller cycles by the division in re-assignments of ferries shallow layer deep layer notification to recover MF - - Y.Hayashi, IEEE Xplore Digital Library FoCAS 2015 ( ) 34 / 34