3 forms of convexity in graphs & networks joint work with Lovro ˇ Subelj Tilen Marc University of Ljubljana University of Ljubljana Faculty of Computer and Institute of Mathematics, Information Science Physics and Mechanics COSTNET ’17
definitions of convexity convex / non-convex real functions, sets in R 2 & subgraphs ℝ 2 x 1 x 1 f(x) f(x) x 2 x 2 x x disconnected ⊇ connected ⊇ induced ⊇ isometric ⊇ convex subgraphs ( sna ) k -clubs & k -clans are convex k -cliques ( def ) subset S is convex if it induces convex subgraph ( def ) convex hull H ( S ) is smallest convex subset including S 1/14
expansion of convex subsets grow subset S by one node & expand S to convex hull H ( S ) — S = { random node i } — until S contains n nodes: 1. select i / ∈ S by random edge 2. expand S = H ( S ∪ { i } ) tree-like t = 2 clique-like t = 0 t = 1 S quantifies (locally) tree-like / clique-like structure of graphs 2/14
convex expansion in graphs s ( t ) = average fraction of nodes in S after t expansion steps s ( t ) = ( t + 1) / n in convex & s ( t ) ≫ ( t + 1) / n in non-convex graphs Random tree Triangular lattice Random graph 1 1 1 0.8 0.8 0.8 % nodes s(t) % nodes s(t) % nodes s(t) 0.6 0.6 0.6 random node random node random node central node central node central node 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 5 10 15 5 10 15 5 10 15 # steps t # steps t # steps t 6 3 3 3 3 3 3 3 4 3 3 11 3 3 3 3 13 3 3 14 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 15 3 3 0 8 5 3 3 3 1 3 3 3 3 6 2 3 3 3 3 3 5 12 11 11 3 3 2 11 15 3 3 3 3 11 4 3 7 8 4 3 3 3 3 10 5 11 3 3 3 3 2 3 8 15 3 3 9 3 3 3 3 1 3 7 3 3 3 3 3 3 8 12 3 3 3 3 1 0 15 3 3 3 3 13 7 3 3 3 3 3 6 12 3 3 3 2 14 3 3 3 3 7 3 3 3 3 13 9 12 3 3 3 2 9 14 3 3 3 0 3 12 3 3 3 13 14 3 3 3 3 10 3 3 3 3 12 3 3 3 3 14 3 3 3 3 10 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 7 3 3 3 3 3 3 3 8 3 9 s ( t ) quantifies (locally) tree-like / clique-like structure of graphs 3/14
convex expansion in networks convex infrastructure and collaborations & non-convex food web Western US power grid European highways Networks coauthorships 1 1 1 0.8 0.8 0.8 % nodes s(t) % nodes s(t) % nodes s(t) network 0.6 0.6 0.6 edge rewiring 0.4 0.4 0.4 Erdos−Renyi 0.2 0.2 0.2 0 0 0 5 10 15 5 10 15 5 10 15 # steps t # steps t # steps t Oregon Internet map US airports connections Little Rock food web 1 1 1 0.8 0.8 0.8 % nodes s(t) % nodes s(t) % nodes s(t) 0.6 0.6 0.6 0.4 0.4 0.4 network edge rewiring 0.2 0.2 0.2 Erdos−Renyi 0 0 0 5 10 15 5 10 15 5 10 15 # steps t # steps t # steps t random graphs fail to reproduce convexity in empirical networks random graphs convex for < O (ln n ) & non-convex for > O (ln 2 n ) core-periphery networks have convex periphery & non-convex c-core 4/14
global measure c -convexity n − 1 � � X c ≥ X RW ≥ X ER X c = 1 − c max(∆ s ( t ) − 1 / n , 0) c c t =1 X c highlights tree-like / clique-like networks (cliques connected tree-like) X RW X ER X RW X ER X 1 X 1 . 1 1 1 1 . 1 1 . 1 Western US power grid ∗ 0 . 95 0 . 32 0 . 24 0 . 91 0 . 10 0 . 01 European highways ∗ 0 . 66 0 . 23 0 . 27 0 . 44 − 0 . 02 0 . 06 Networks coauthorships 0 . 91 0 . 09 0 . 06 0 . 83 − 0 . 05 − 0 . 09 Oregon Internet map 0 . 68 0 . 36 0 . 06 0 . 53 0 . 20 − 0 . 09 Caenorhabditis elegans 0 . 57 0 . 54 0 . 07 0 . 43 0 . 40 − 0 . 13 US airports connections 0 . 43 0 . 24 0 . 00 0 . 30 0 . 16 − 0 . 07 Scientometrics citations 0 . 24 0 . 16 0 . 02 0 . 04 0 . 00 − 0 . 13 US election weblogs 0 . 17 0 . 12 0 . 00 0 . 06 0 . 04 − 0 . 08 Little Rock food web 0 . 03 0 . 03 0 . 02 − 0 . 06 − 0 . 02 − 0 . 02 X c measures global & regional (periphery) convexity in networks 5/14
local measure of convexity L 1 ≤ L ER L c = 1 + max { t | s ( t ) < ( t + c + 1) / n } ≈ ln n / ln � k � 1 L c highlights locally tree-like / clique-like networks & random graphs L ER L ER L t L 1 ln n / ln � k � t 1 Western US power grid 14 9 6 9 8 . 66 European highways 16 7 7 7 7 . 54 Networks coauthorships 17 4 7 4 3 . 77 Oregon Internet map 3 4 3 4 4 . 40 Caenorhabditis elegans 2 5 2 5 5 . 79 US airports connections 2 3 2 3 2 . 38 Scientometrics citations 3 4 3 4 4 . 30 US election weblogs 2 2 2 2 2 . 15 Little Rock food web 2 2 2 2 1 . 59 L c measures local & absolute (tree/clique) convexity in networks 6/14
convexity in graphs & networks 67 17 10 6 16 6 6 6 6 8 23 6 7 8 6 6 6 6 6 25 5 68 24 59 6 6 6 6 6 3 5 10 6 6 6 6 6 5 23 95 6 6 6 6 6 6 6 3 6 53 51 11 32 22 13 6 6 6 6 6 6 5 6 6 6 6 6 5 6 6 6 5 5 6 6 6 6 6 6 6 14 5 5 5 5 5 83 11 6 6 6 27 5 5 6 6 6 6 6 6 6 6 5 5 5 5 6 6 6 6 6 6 6 6 6 75 30 58 58 8 5 5 31 55 60 6 6 6 6 75 5 5 64 6 6 6 6 6 6 6 6 6 6 6 5 58 5 5 5 73 5 6 6 6 6 6 5 5 5 5 5 6 6 6 6 6 6 2 6 91 5 5 8 5 5 6 6 6 6 6 6 6 6 6 6 6 30 8 5 5 5 8 5 5 5 5 5 6 6 6 6 6 6 89 13 5 78 5 5 5 60 55 6 6 6 6 6 6 6 6 6 56 89 87 5 5 5 5 5 5 5 6 6 6 6 6 89 47 70 74 30 5 5 35 5 5 73 5 5 5 6 6 6 6 6 6 6 6 6 6 6 99 64 5 66 65 5 19 5 5 5 5 6 6 6 6 6 6 6 6 6 6 95 5 17 5 5 5 5 5 5 5 5 5 6 6 6 6 0 6 6 6 6 6 6 6 6 36 70 61 88 5 5 96 5 33 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 45 61 30 5 5 35 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 93 84 95 67 76 5 5 38 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 95 95 96 52 61 50 87 5 5 5 5 5 5 99 5 81 6 6 6 1 6 6 6 6 6 6 6 6 79 23 89 98 28 22 8 4 5 5 5 61 6 6 6 6 6 6 81 31 32 36 36 2 5 5 5 0 5 5 5 5 6 6 6 6 6 5 6 6 6 6 6 6 6 6 65 89 89 72 89 8 28 43 18 5 5 5 5 6 6 6 6 6 6 6 6 83 23 25 42 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 25 31 23 23 19 5 85 9 5 5 5 7 5 5 6 6 6 6 6 6 6 6 6 6 34 25 89 36 5 5 5 5 5 5 9 6 6 6 6 6 6 6 6 6 26 42 8 62 77 5 5 46 8 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 12 26 60 5 39 5 5 5 5 5 6 6 6 6 6 6 86 86 86 63 5 5 5 5 5 93 6 6 6 6 6 6 6 6 6 6 23 12 11 25 13 5 58 80 5 5 5 5 5 5 5 15 6 6 6 6 6 6 6 6 6 6 6 6 25 25 85 89 89 58 29 5 5 5 5 6 6 6 6 6 6 6 6 6 27 5 16 86 44 18 5 5 6 6 6 6 6 6 6 6 78 37 26 51 69 5 5 5 56 6 6 6 6 6 6 6 6 6 6 6 97 26 12 68 71 69 5 16 8 5 8 46 6 6 6 6 6 6 6 6 6 46 43 25 5 1 5 5 5 8 6 6 6 6 6 6 6 6 6 6 18 23 43 5 5 5 20 5 84 5 89 6 6 6 6 6 6 29 26 87 22 51 39 6 5 5 89 24 6 6 6 6 6 6 6 6 6 62 59 5 40 98 5 45 25 6 6 6 6 6 6 6 6 6 6 33 28 5 48 86 5 5 5 15 6 6 6 6 6 6 39 94 77 8 5 6 6 6 6 6 6 6 6 6 66 7 14 43 40 5 49 21 34 6 6 6 6 6 6 6 9 20 25 51 88 5 74 89 6 6 4 6 6 92 6 5 63 5 52 77 5 6 6 6 6 6 6 6 6 21 44 3 40 82 42 5 5 54 6 6 6 6 6 6 6 6 6 15 38 49 0 5 5 5 8 70 6 6 6 6 94 5 26 6 6 6 6 6 6 24 17 4 5 6 6 6 6 6 48 54 80 41 6 6 6 6 6 6 18 1 5 50 5 6 6 6 6 6 57 6 6 6 6 6 6 6 54 54 82 6 6 2 5 6 6 71 30 30 90 53 97 57 72 6 6 6 6 55 8 94 47 6 6 6 14 6 6 6 73 35 76 6 6 6 13 6 77 79 6 10 12 6 6 6 20 92 37 15 21 91 6 6 8 41 90 19 23 local convexity global convexity regional convexity random graphs tree/clique-like core-periphery networks networks etc. < ln n / ln � k � c -convexity � = standard measures & c-core � = k -cores robustness, navigation, optimization, abstraction, comparison etc. 7/14
to be continued. . . arXiv: 1608.03402v3 Marc & ˇ Subelj (2017) Convexity in complex networks, Network Science , pp. 27 Lovro ˇ Tilen Marc Subelj University of Ljubljana University of Ljubljana tilen.marc@imfm.si lovro.subelj@fri.uni-lj.si http://www.imfm.si http://lovro.lpt.fri.uni-lj.si
convex skeletons of networks Lovro ˇ Subelj University of Ljubljana Faculty of Computer and Information Science COSTNET ’17
convexity under randomization sn − 1 � � Xs = s − c max( s ∆ s ( t ) − 1 / n , 0) s = fraction of nodes in LCC t =1 Xs under degree-preserving / full randomization by edge rewiring Xs very sensitive to random perturbations of network structure 8/14
convex skeletons of networks convex skeleton = largest high- Xs subnetwork (every S is convex) spanning tree & convex skeleton of network scientists coauthorships convex skeleton is tree of cliques extracted by targeted edge removal 9/14
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