IBIS2006 p.1/32

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1 - #$ %'& -! "! IBIS2006 p.1/32

2 1. Introduction....!! IBIS2006 p.2/32 #$ %'&

3 Six Degrees of Separation 6 S.Milgram Physiology Today WWW 19 d = log 10 N R.Albert et al. Nature ! : #$ %'& IBIS2006 p.3/32

4 Examples: Internet and WWW Internet connectivity with selected backbone ISPs (CAIDA) Nature networks/ppt/sandiego.ppt/ #$ %'& IBIS2006 p.4/32

5 Scale-free networks SF : : : WWW P (k) k γ 2 < γ < 3 #$ %'& IBIS2006 p.5/32

6 q = 1 f Robustness IBIS2006 p.6/32 S s f #$ %'&

7 2. Generating function k. G 0 (x) def = k P (k)x k P (k) x λ λk P (k) = e k! G 0 (x) = k=0 P (k)x k = k = G 0(1) = k=0 [ ] d dx eλ(x 1) e λλk k! xk = e λ(x 1) x=1 = λ. #$ %'& IBIS2006 p.7/32

8 Generating function (cont.) k v v k kp (k)xk j jp (j) = x G 0(x) def G 0 (1) = xg 1 (x). 2 G 0 (G 1 (x)) = k P (k) [G 1 (x)] k. x x G1(x) G1(x) v v" x v (a) 1 step v (b) 2 step #$ %'& IBIS2006 p.8/32

9 3. Cluster size H 1 (x) = P 1 (s)x s s H 1 (x) = xg 1 (H 1 (x)) H 0 (x) = xg 0 (H 1 (x)). s = H 0(1) = 1 + G 0(1) 1 G 1 (1). N s G 1 (1) = 1 k2 / k = 2. #$ %'& IBIS2006 p.9/32

10 Cite Percolation q P (k) P ( k) = k= k P (k) k C kq k(1 q) k k k = k k P ( k) = n np (n)q = k q. k = 1: P (1) 1 C 1 q(1 q) 0 +P (2) 2 C 1 q(1 q) 2 1 +P (3) 3 C 1 q(1 q) k = 2: 2P (2) 2 C 2 q 2 (1 q) 0 +2P (3) 3 C 2 q 2 (1 q) k = 3: 3P (3) 3 C 3 q 3 (1 q) P (1)q +2P (2)q +3P (3)q +... IBIS2006 p.10/32

11 Cite Percolation (cont.) k 2 = k k 2 P ( k) = k 2 q 2 + k q(1 q). SF q c 2 = k 2 k = q c( k 2 k ) + k k q c = 1 k 2 / k 1. 2 < γ < 3 k 2 = k 2 P (k) k 2 γ q c 0: R.Cohen et al. PRL IBIS2006 p.11/32

12 4. Geographical constraints M.T.Gastner and M.E.J.Newman Euro.Phys. J. B 49(2) IBIS2006 p.12/32

13 4. Geographical constraints L M.T.Gastner and M.E.J.Newman Euro.Phys. J. B 49(2) IBIS2006 p.12/32

14 4. Geographical constraints L M.T.Gastner and M.E.J.Newman Euro.Phys. J. B 49(2) P (k) IBIS2006 p.12/32

15 Terminated Nodes q 1 (a) (b) H (1) def 1 (x) = 1 F 1 (1) + xf (1) 1 (H 1 (x)) F (1) 1 (x) def = q k kp (k)x k 2 = x 1 F 1 (x) : : q : k-2 q : k-2 (a) return (b) terminate IBIS2006 p.13/32

16 Effect of Cycles (a) H (m 1) 1 (x) = 1 F 1 (1) + xf (1) 1 (H 1 (x)) H (m 1 1) 1 (x) (b) n L (k) L 2 x nl... m1 m2 : : k nl (a) a cycle L.Huang et al. Europhys.Lett. 72(1) (b) nl-cycles IBIS2006 p.14/32

17 Effect of Cycles (cont.) H 1 (x) = 1 F 1 (1) + q k [H(m 1) 1 (x)h (m 2) 1 (x)] n L(k) [H 1 (x)] k 1 2n L(k) s = H 1 (1) s = q + 2 qm1 qm2 1 q q c = q 2 k kn L(k) k(k 1) 2 kn L (k) k ( 1 2 (q c ) m 1 (qc ) m 2 2(1 qc ) q c k(k 2) k ) IBIS2006 p.15/32

18 Effect of Triangles 3 m 1 = m 2 = 1 C(k) q c = k(k 1) ( 1 q c k k(k 2) k ) C(k) (k 1)2 2 : : : : q c > q IBIS2006 p.16/32

19 Geographical SF networks Random Apollonian: Delaunay-like SF: Delaunay Triangulation: = + 1st diagonal flip 2nd diagonal flip IBIS2006 p.17/32

20 Topological Structure RA DT RA+NN(one) 4 Y.Hayashi and J.Matsukubo PRE IBIS2006 p.18/32

21 Degree Distribution RA DT RA + NN (one) RA + NN (all) P(k) k RA: power law DT: lognormal RA+NN: power law with exponential cutoff (the deg. of hubs are reduced) IBIS2006 p.19/32

22 Randomly Rewired Nets : : : : : : : : : : : : : : : : Rewiring a pair of links with the same degree at each node Maslov et al. Physica A IBIS2006 p.20/32

23 Tolerance to Failures GC S/N s <s> RA RA+NN(one) RA+NN(all) DT RA RA+NN(one) RA+NN(all) DT <s> RA RA+NN(one) RA+NN(all) DT RA RA+NN(one) RA+NN(all) DT Relative size S/N f Relative size S/N f Fraction f of failures (a) geographical nets (RA: DT: RA+NN(one): +) Fraction f of failures (b) rewired nets IBIS2006 p.21/32

24 Damages by Failures Initial N=200 randomly removed 32 nodes RA DT IBIS2006 p.22/32

25 Tolerance to Attacks Relative size S/N <s> RA RA+NN(one) RA+NN(all) DT RA RA+NN(one) RA+NN(all) DT f Fraction f of attacks (a) geographical nets Relative size S/N RA RA+NN(one) RA+NN(all) DT f Fraction f of attacks <s> (b) rewired nets RA RA+NN(one) RA+NN(all) DT IBIS2006 p.23/32

26 Damages by Attacks Initial N=200 targeted attacks on 16 hubs RA DT RA IBIS2006 p.24/32

27 Damages by Attacks (cont.) Initial N=200 targeted attacks on 16 hubs RA+NN(one) RA+NN(all) 4 RA IBIS2006 p.25/32

28 8. Summary P (k) 1 qc = k 2 / k 1 qc 0 SF IBIS2006 p.26/32

29 A1. Random Apollonian Net Configuration: iterative subdivision of a randomly chosen triangle from an initial triangulation Initial triangulation Add a new node inside a chosen triangle Connect to its closest 3 nodes Some long-range links naturally appear in narrow collapsed triangles near the boundary edges Zhou Yan Wang Phys. Rev. E IBIS2006 p.27/32

30 A2. Voronoi and Delaunay Optimal triangulation in some geometric criteria: maximim angle minimax circumcircle short path length in the same order as the direct Euclidean dist. Dual Graph Consider the combination of RA (by triangulation on a plane) and DT to avoid the long-range links IBIS2006 p.28/32

31 A3. Efficient Routing Planar triangulation: reasonable math. abstraction of ad hoc net. (each triangle forms a service region) Moreover a memoryless no defeat and competitive online routing algorithm has been developed for such networks taking into account the face. Source Destination Bose and Morin SIAM J. of Comp. 33(4) 2004 IBIS2006 p.29/32

32 A4. Dense-get-denser Migration-in/out versus population density of Japanese prefectures in each year 1e+07 1e+06 "cum_in_1960" "cum_in_1965" "cum_in_1970" "cum_in_1975" "cum_in_1980" "cum_in_1985" "cum_in_1990" "cum_in_1995" "cum_in_2000" "cum_in_2003" 1e+07 1e+06 "cum_out_1960" "cum_out_1965" "cum_out_1970" "cum_out_1975" "cum_out_1980" "cum_out_1985" "cum_out_1990" "cum_out_1995" "cum_out_2000" "cum_out_2003" cum. migration-in cum. migration-out population density population density IBIS2006 p.30/32

33 A5. Classes of Geo. SF Nets Modulated BA: Π i k i l α rand. position of node SF on lattices: connect within r = A k 1/d i Space-filling: subdivision of a region (heterogeneous dist. of nodes) (a) BA (b) SFL (c) RAN Step 0 Step 1 Step 2 initial N0 nodes with m links assign a degree k initial trianglulation add new node saturated node select a node connect to the neighborhoods 1/2 in the radius r = A k add new node into a chosen triangle new m links : pref. attach. connect to its 3 nodes IBIS2006 p.31/32

34 A6. Planarity and Shortness class planarity of net shortness of links Modulated BA Manna 02 crossing links with disadvantaged Xulvi-Brunet 02 (not prohibited) long-range links SF on lattices ben-avraham 03 cross of regular long shortcuts Warren 02 links and shortcuts from hubs Space-filling Apollonian nets. by subdivision long-range links Doye 05 Zhou 04 of a selected region in narrow triangles IBIS2006 p.32/32