[PPT] - 3 forms of convexity in graphs & networks joint work with Lovro PowerPoint Presentation

SLIDE 1

3 forms of convexity in graphs & networks

Lovro ˇ Subelj

University of Ljubljana Faculty of Computer and Information Science joint work with

Tilen Marc

University of Ljubljana Institute of Mathematics, Physics and Mechanics

COSTNET ’17

SLIDE 2

definitions of convexity

convex/non-convex real functions, sets in R2 & subgraphs

f(x) f(x)

ℝ2

x1 x2 x1 x2

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

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SLIDE 3

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})

t = 0 t = 1 t = 2 tree-like clique-like

S quantifies (locally) tree-like/clique-like structure of graphs

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SLIDE 4

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

5 10 15 0.2 0.4 0.6 0.8 1

Random tree

% nodes s(t) # steps t

random node central node

5 10 15 0.2 0.4 0.6 0.8 1

Triangular lattice

% nodes s(t) # steps t

random node central node

5 10 15 0.2 0.4 0.6 0.8 1

Random graph

% nodes s(t) # steps t

random node central node

7 2 6 9 13 10 11 15 8 12 1 14 3 5 4 11 8 8 8 13 11 4 3 1 6 13 11 5 3 2 9 13 15 11 7 7 7 9 10 15 12 12 12 12 12 15 14 14 14 14 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 3 3 3 2 3 3 3 3 3 3 3 3 3 3 8 7 11 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 2 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 3 3 3 3 3 3 3 3 3 3 3 9 3 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 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

s(t) quantifies (locally) tree-like/clique-like structure of graphs

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SLIDE 5

convex expansion in networks

convex infrastructure and collaborations & non-convex food web

5 10 15 0.2 0.4 0.6 0.8 1

Western US power grid

% nodes s(t) # steps t

5 10 15 0.2 0.4 0.6 0.8 1

European highways

% nodes s(t) # steps t

5 10 15 0.2 0.4 0.6 0.8 1

US airports connections

% nodes s(t) # steps t

5 10 15 0.2 0.4 0.6 0.8 1

Networks coauthorships

% nodes s(t) # steps t

network edge rewiring Erdos−Renyi

5 10 15 0.2 0.4 0.6 0.8 1

Little Rock food web

% nodes s(t) # steps t

network edge rewiring Erdos−Renyi

5 10 15 0.2 0.4 0.6 0.8 1

Oregon Internet map

% nodes s(t) # steps t

random graphs fail to reproduce convexity in empirical networks random graphs convex for < O(ln n) & non-convex for > O(ln2 n) core-periphery networks have convex periphery & non-convex c-core

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SLIDE 6

global measure c-convexity

Xc = 1 −

n−1

t=1

c

max(∆s(t) − 1/n, 0)

Xc ≥ X RW

c

≥ X ER

c

Xc highlights tree-like/clique-like networks (cliques connected tree-like)

X1 X RW

1

X ER

1

X1.1 X RW

1.1

X ER

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

Xc measures global & regional (periphery) convexity in networks

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SLIDE 7

local measure of convexity

Lc = 1 + max{ t | s(t) < (t + c + 1)/n } L1 ≤ LER

1

≈ ln n/ ln k Lc highlights locally tree-like/clique-like networks & random graphs

Lt LER

t

L1 LER

1

ln n/ ln k 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

Lc measures local & absolute (tree/clique) convexity in networks

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SLIDE 8

convexity in graphs & networks

25 36 25 36 25 25 51 42 70 76 36 72 96 61 61 50 47 63 52 98 67 70 42 95 89 95 89 5 12 16 11 89 21 10 2 41 8 13 35 53 73 25 55 80 89 89 75 74 91 89 95 65 23 23 23 26 26 85 19 23 32 45 23 25 79 83 56 4 49 43 15 3 7 6 1 94 30 9 94 28 22 14 82 57 90 37 26 26 97 33 26 87 39 44 29 66 62 43 43 59 31 31 34 25 25 84 93 23 81 86 86 86 77 46 99 60 69 68 58 18 71 17 54 48 54 54 38 24 92 78 27 40 20 30 89 89 51 51 61 64 88 95 89 89 12

global convexity

tree/clique-like networks

5 5 5 5 5 5 5 5 5 5 5 5 31 5 5 60 5 5 5 5 18 5 5 5 8 5 35 5 77 5 5 5 5 5 5 5 5 5 15 5 5 5 5 5 5 55 5 5 5 5 5 19 5 64 8 5 5 5 5 5 5 5 5 5 84 68 5 5 35 89 89 89 60 5 83 5 5 5 5 5 9 5 5 81 15 5 29 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 18 13 5 34 8 7 49 5 20 45 48 5 98 74 71 5 5 5 5 5 5 5 5 5 5 42 5 5 14 46 5 2 5 5 8 5 85 5 5 5 5 61 75 16 51 44 56 21 78 65 66 27 43 62 96 38 54 99 5 5 4 33 39 5 5 87 5 5 28 32 3 5 36 1 52 80 5 6 22 17 73 5 5 5 5 5 30 5 5 5 58 5 5 5 5 5 5 5 8 5 5 5 5 8 72 77 5 5 63 5 76 5 5 86 39 47 5 5 5 5 5 5 82 5 5 5 5 5 5 55 5 5 93 5 5 5 58 5 5 69 5 11 5 88 5 28 8 8 5 5 5 8 5 5 8 5 5 5 5 30 40 30 30 5 5 5 5 95 5 5 5 5 5 5 5 77 73 5 5 87 26 5 41 50 58 5 5 5 5 5 8 5 5 5 70 5 59 8 5 5 5 5 40 5 5 5 25 23 23 86 5 8 8 58 5 46 53 13 5 8 5 5 94 12 37 79 91 92 97 90 5 67 24 5 5 57 10

regional convexity

core-periphery networks etc.

6 6 6 6 13 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 14 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 24 6 6 6 6 6 6 6 10 6 6 6 6 6 6 22 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 12 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 17 6 6 6 6 6 6 6 16 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 11 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 21 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 15 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 20 6 9 6 6 6 6 6 6 23 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 25 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 6 6 6 6 6 5 6 6 6 6 6 6 6 6 6 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 19 6 6 6 6 6 6 6 6 6 6 6 6 6

local convexity

random graphs

< ln n/ ln k

c-convexity = standard measures & c-core = k-cores

robustness, navigation, optimization, abstraction, comparison etc.

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SLIDE 9

to be continued. . .

arXiv:1608.03402v3

Marc & ˇ Subelj (2017) Convexity in complex networks, Network Science, pp. 27

Tilen Marc

University of Ljubljana tilen.marc@imfm.si http://www.imfm.si

Lovro ˇ Subelj

University of Ljubljana lovro.subelj@fri.uni-lj.si http://lovro.lpt.fri.uni-lj.si

SLIDE 10

convex skeletons of networks

Lovro ˇ Subelj

University of Ljubljana Faculty of Computer and Information Science

COSTNET ’17

SLIDE 11

convexity under randomization

Xs = s −

sn−1

t=1

c

max(s∆s(t) − 1/n, 0)

s = fraction of nodes in LCC Xs under degree-preserving/full randomization by edge rewiring Xs very sensitive to random perturbations of network structure

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SLIDE 12

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

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SLIDE 13

statistics of convex skeletons

C = 1 n

i

2ti ki(ki − 1) σ = 2 n(n − 1)

i<j

σij Xs = . . .

statistics of convex skeletons & spanning trees of networks

clustering C geodesics σ convexity Xs N CS ST N CS ST N CS ST Jazz musicians 0.62 0.81 0.00 9.71 1.97 1.00 0.12 0.84 1.00 Network scientists 0.74 0.75 0.00 2.66 1.47 1.00 0.85 0.95 1.00 Computer scientists 0.48 0.54 0.00 4.08 1.42 1.00 0.64 0.95 1.00 Plasmodium falciparum 0.02 0.07 0.00 3.71 1.77 1.00 0.43 0.95 1.00 Saccharomyces cerevisiae 0.07 0.10 0.00 2.58 1.19 1.00 0.68 0.88 1.00 Caenorhabditis elegans 0.06 0.12 0.00 6.79 3.03 1.00 0.56 0.85 1.00 AS (January 1, 1998) 0.18 0.21 0.00 3.87 2.32 1.00 0.66 0.91 1.00 AS (January 1, 1999) 0.18 0.27 0.00 3.54 2.05 1.00 0.49 0.95 1.00 AS (January 1, 2000) 0.20 0.25 0.00 4.81 3.07 1.00 0.59 0.90 1.00 Little Rock Lake 0.32 0.69 0.00 22.13 4.32 1.00 0.02 0.82 1.00 Florida Bay (wet) 0.33 0.79 0.00 9.17 1.37 1.00 0.03 0.92 1.00 Florida Bay (dry) 0.33 0.82 0.00 9.37 1.65 1.00 0.03 0.93 1.00

convex skeleton is generalization of spanning tree retaining clustering

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SLIDE 14

distributions of convex skeletons

distributions of convex skeletons & spanning trees of networks

10 10

1

10

2

10

−3

10

−2

10

−1

10

Network scientists

fraction of nodes pk node degree k

Full network Spanning tree Convex skeleton 10 10

1

10

2

10

3

10

−3

10

−2

10

−1

10

AS (January 1, 1999)

fraction of nodes pk node degree k

Full network Convex skeleton Power−law ∼k−2.17 10 10

1

10

2

10

3

10

−4

10

−3

10

−2

10

−1

10

Caenorhabditis elegans

fraction of nodes pk node degree k

Full network Convex skeleton Power−law ∼k−2.28 2 4 6 8 10 12 14 16 0.05 0.1 0.15 0.2 0.25

Network scientists

fraction of nodes pd node distance d

Full network Spanning tree Convex skeleton 2 4 6 8 10 12 14 16 0.1 0.2 0.3 0.4 0.5

AS (January 1, 1999)

fraction of nodes pd node distance d

Full network Spanning tree Convex skeleton 2 4 6 8 10 12 14 16 0.1 0.2 0.3 0.4 0.5

Caenorhabditis elegans

fraction of nodes pd node distance d

Full network Spanning tree Convex skeleton

convex skeletons retain distributions in contrast to spanning trees

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SLIDE 15

convex skeletons of coauthorships

convex skeleton ∼ network abstraction technique convex skeleton of Slovenian computer scientists coauthorships

computer theory ( ), information systems ( ), intelligent systems ( ), programming technologies ( ) & other ( )

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SLIDE 16

network backbones of coauthorships

convex skeleton ≫ high-betweenness & high-salience backbones properties of backbones of Slovenian computer scientists coauthorships

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

weight of coauthorship ties 〈w〉 backbone

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

modularity of field classification Q backbone

0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82

fraction of inter−field ties remainder

Full network Spanning tree Edge betweenness Salience skeleton Convex skeleton

convex skeletons enhance properties in contrast to other backbones

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SLIDE 17

convex skeletons of networks

spanning tree

tree w/o cliques

convex skeleton

tree w/ cliques

convex skeleton ≫ backbones & c-centrality = centralities

abstraction, sampling, visualization, modeling, dynamics etc.

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SLIDE 18

thank you!

arXiv:1709.00255v2

ˇ Subelj (2017) Convex skeletons of complex networks, pp. 19

Lovro ˇ Subelj

University of Ljubljana lovro.subelj@fri.uni-lj.si http://lovro.lpt.fri.uni-lj.si