Betweenness centrality on 1-dimensional periodic graphs Norie Fu, - - PowerPoint PPT Presentation

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betweenness centrality on 1 dimensional periodic graphs

Betweenness centrality on 1-dimensional periodic graphs Norie Fu, - - PowerPoint PPT Presentation

Nov 09, 2022 182 likes •505 views

Betweenness centrality on 1-dimensional periodic graphs Norie Fu, Vorapong Suppakitpaisarn June 10, 2014 1 / 12 Definition of periodic graphs Static graph G = ( V , E ) Periodic graph G = ( V , E ) finite and directed V = V Z

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Betweenness centrality on 1-dimensional periodic graphs

Norie Fu, Vorapong Suppakitpaisarn June 10, 2014

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Definition of periodic graphs

◮ Static graph G = (V, E)

◮ finite and directed ◮ allow contains self loops and

multiple edges

◮ a vector in Zd is associated

with each edge

◮ Periodic graph G = (V , E)

◮ V = V × Zd ◮ E = {((u, y), (v, y + z)) :

(uv, z) ∈ E, y ∈ Zd}.

1 1 2 2

u v

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Definition of periodic graphs

◮ Static graph G = (V, E)

◮ finite and directed ◮ allow contains self loops and

multiple edges

◮ a vector in Zd is associated

with each edge

◮ Periodic graph G = (V , E)

◮ V = V × Zd ◮ E = {((u, y), (v, y + z)) :

(uv, z) ∈ E, y ∈ Zd}.

1 1 2 2

u v

In this talk, G is 1-dimensional, undirected, and connected

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Applications of periodic graphs

Periodic graph is used as a model of

◮ Crystals ◮ System of uniform recurrence

equations [Karp, Miller, Winograd ’67]

◮ VLSI [Iwano, Steiglitz ’86]

Many algorithmic researches

◮ Planarity testing [Iwano, Steiglitz ’88] ◮ Connectivity testing [Cohen, Megiddo ’91] ◮ Shortest path problem [H¨

fting, Wanke ’95 (SODA)]

◮ Strongly polynomial-time algorithm on coherent and planar 2-dim.

periodic graphs [F. ’12 (ISAAC)]

◮ ...

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Betweenness centrality

◮ H = (U, F): a finite graph ◮ u, v, w ∈ U ◮ σH uv: # of the shortest paths from u to v on H ◮ σH uv(w): # of the shortest paths from u to v containing w on H

Definition

The betweenness centrality of w is g H(w) =

u,v∈U,

u=w=v

σH

uv(w)

σH

uv

.

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Betweenness centrality

◮ H = (U, F): a finite graph ◮ u, v, w ∈ U ◮ σH uv: # of the shortest paths from u to v on H ◮ σH uv(w): # of the shortest paths from u to v containing w on H

Definition

The betweenness centrality of w is g H(w) =

u,v∈U,

u=w=v

σH

uv(w)

σH

uv

.

◮ Defined by Freeman in 1977. ◮ Frequently used to deal with complex networks.

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Motivation

◮ Analysis of crystal structure

◮ average shortest path length of the finite subgraphs of periodic

graphs

◮ affects some properties of crystals [Ribeiro, Lind ’05] ◮ can significantly increase if nodes with high betweenness centrality

are removed [Cilia, Vuister, Lenaerts ’12]

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Definition of betweenness centrality on 1-dimensional periodic graphs

◮ x ∈ Z, Vx := {(w, z) : (w, z) ∈ V , z ≥ x} ◮ Gx: the subgraph of G induced by Vx. ◮ Gx(ν, D): the subgraph induced by Vx|D 0 (ν) = {λ : dG(ν, λ) < D}.

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Definition of betweenness centrality on 1-dimensional periodic graphs

◮ x ∈ Z, Vx := {(w, z) : (w, z) ∈ V , z ≥ x} ◮ Gx: the subgraph of G induced by Vx. ◮ Gx(ν, D): the subgraph induced by Vx|D 0 (ν) = {λ : dG(ν, λ) < D}.

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Definition of betweenness centrality on 1-dimensional periodic graphs

◮ x ∈ Z, Vx := {(w, z) : (w, z) ∈ V , z ≥ x} ◮ Gx: the subgraph of G induced by Vx. ◮ Gx(ν, D): the subgraph induced by Vx|D 0 (ν) = {λ : dG(ν, λ) < D}.

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Definition of betweenness centrality on 1-dimensional periodic graphs

◮ x ∈ Z, Vx := {(w, z) : (w, z) ∈ V , z ≥ x} ◮ Gx: the subgraph of G induced by Vx. ◮ Gx(ν, D): the subgraph induced by Vx|D 0 (ν) = {λ : dG(ν, λ) < D}.

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Definition of betweenness centrality on 1-dimensional periodic graphs

◮ x ∈ Z, Vx := {(w, z) : (w, z) ∈ V , z ≥ x} ◮ Gx: the subgraph of G induced by Vx. ◮ Gx(ν, D): the subgraph induced by Vx|D 0 (ν) = {λ : dG(ν, λ) < D}.

Definition

The half-infinite betweenness centrality of ν ∈ Vx with boundary x is hbcG

x (ν) = lim D→∞

1 |Vx|D

0 |g Gx(ν,D)(ν).

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VAP-free periodic graphs

Definition

A graph is VAP-free planar if it admits a planar drawing such that there is no vertex accumulation point in any finite bounded region.

VAP-free planar

[Iwano, Steiglitz. Networks, 18:205–222, 1988]

Not VAP-free planar

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Results

Theorem

For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbcG

x (ν) converges.

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Results

Theorem

For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbcG

x (ν) converges.

Corollary

There exists an algorithm to compute hbcG

x (ν). ◮ Note: hbcG x (ν) can be irrational. So more precisely, our algorithm

computes a value that is sufficiently close to hbcG

x (ν).

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Results

Theorem

For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbcG

x (ν) converges.

Corollary

There exists an algorithm to compute hbcG

x (ν). ◮ Note: hbcG x (ν) can be irrational. So more precisely, our algorithm

computes a value that is sufficiently close to hbcG

x (ν).

Theorem

If x is fixed and G is VAP-free planar, then the algorithm runs in polynomial time with respect to |V| and |E|.

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Our algorithm

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1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6

Our algorithm

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Our algorithm

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If , then shortest paths from to does not contain

Our algorithm

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Our algorithm

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Our algorithm

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period 0 period 1 period 2

Our algorithm

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# shortest paths from to: # shortest paths from to: period 0 period 1 period 2

Our algorithm

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# shortest paths from to: # shortest paths from to: period 0 period 1 period 2

Our algorithm

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Idea for the proof

◮ Computation of the function dGx((u, y), (w, y + b)) of b.

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1 1 1 1 1 1 1 1 1 1 1 1 1

min :

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1 1 1 1 1 1 1 1 1 1 1 1 1

min :

use the theory of unimodularity of 2-dim. VAP-free planar periodic graphs [F. '12]

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1 1 1 1 1 1 1 1 1 1 1 1

pt. val.
pt. val.
pt. val.
pt. val.
pt. val.

take min. value

1

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Concluding remarks

◮ We can do similar discussions on

◮ directed periodic graphs, ◮ periodic graphs with edge weights, and ◮ doubly infinite periodic graphs. ◮ Use the theory on Gr¨

bner bases and integer programming [Ho¸

sten, Thomas ’98]

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Concluding remarks

◮ We can do similar discussions on

◮ directed periodic graphs, ◮ periodic graphs with edge weights, and ◮ doubly infinite periodic graphs. ◮ Use the theory on Gr¨

bner bases and integer programming [Ho¸

sten, Thomas ’98]

◮ We need a new theory for 2-dimensional case.

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Concluding remarks

◮ We can do similar discussions on

◮ directed periodic graphs, ◮ periodic graphs with edge weights, and ◮ doubly infinite periodic graphs. ◮ Use the theory on Gr¨

bner bases and integer programming [Ho¸

sten, Thomas ’98]

◮ We need a new theory for 2-dimensional case.

Thank you for your attention.

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