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 d ◮ allow contains self loops and ◮ E = { (( u , y ) , ( v , y + z )) : ( uv , z ) ∈ E , y ∈ Z d } . multiple edges ◮ a vector in Z d is associated with each edge 2 u v 1 1 2 2 / 12
Definition of periodic graphs ◮ Static graph G = ( V , E ) ◮ Periodic graph G = ( V , E ) ◮ finite and directed ◮ V = V × Z d ◮ allow contains self loops and ◮ E = { (( u , y ) , ( v , y + z )) : ( uv , z ) ∈ E , y ∈ Z d } . multiple edges ◮ a vector in Z d is associated with each edge 2 u v 1 1 2 In this talk, G is 1-dimensional, undirected, and connected 2 / 12
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¨ ofting, Wanke ’95 (SODA)] ◮ Strongly polynomial-time algorithm on coherent and planar 2-dim. periodic graphs [F. ’12 (ISAAC)] ◮ ... 3 / 12
Betweenness centrality ◮ H = ( U , F ): a finite graph ◮ u , v , w ∈ U ◮ σ H uv : # of the shortest paths from u to v on H ◮ σ H uv ( w ): # of the shortest paths from u to v containing w on H Definition The betweenness centrality of w is σ H uv ( w ) � g H ( w ) = . σ H uv u , v ∈ U , u � = w � = v 4 / 12
Betweenness centrality ◮ H = ( U , F ): a finite graph ◮ u , v , w ∈ U ◮ σ H uv : # of the shortest paths from u to v on H ◮ σ H uv ( w ): # of the shortest paths from u to v containing w on H Definition The betweenness centrality of w is σ H uv ( w ) � g H ( w ) = . σ H uv u , v ∈ U , u � = w � = v ◮ Defined by Freeman in 1977. ◮ Frequently used to deal with complex networks. 4 / 12
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] 5 / 12
Definition of betweenness centrality on 1-dimensional periodic graphs ◮ x ∈ Z , V x := { ( w , z ) : ( w , z ) ∈ V , z ≥ x } ◮ G x : the subgraph of G induced by V x . ◮ G x ( ν, D ): the subgraph induced by V x | D 0 ( ν ) = { λ : d G ( ν, λ ) < D } . 6 / 12
Definition of betweenness centrality on 1-dimensional periodic graphs ◮ x ∈ Z , V x := { ( w , z ) : ( w , z ) ∈ V , z ≥ x } ◮ G x : the subgraph of G induced by V x . ◮ G x ( ν, D ): the subgraph induced by V x | D 0 ( ν ) = { λ : d G ( ν, λ ) < D } . 6 / 12
Definition of betweenness centrality on 1-dimensional periodic graphs ◮ x ∈ Z , V x := { ( w , z ) : ( w , z ) ∈ V , z ≥ x } ◮ G x : the subgraph of G induced by V x . ◮ G x ( ν, D ): the subgraph induced by V x | D 0 ( ν ) = { λ : d G ( ν, λ ) < D } . 6 / 12
Definition of betweenness centrality on 1-dimensional periodic graphs ◮ x ∈ Z , V x := { ( w , z ) : ( w , z ) ∈ V , z ≥ x } ◮ G x : the subgraph of G induced by V x . ◮ G x ( ν, D ): the subgraph induced by V x | D 0 ( ν ) = { λ : d G ( ν, λ ) < D } . 6 / 12
Definition of betweenness centrality on 1-dimensional periodic graphs ◮ x ∈ Z , V x := { ( w , z ) : ( w , z ) ∈ V , z ≥ x } ◮ G x : the subgraph of G induced by V x . ◮ G x ( ν, D ): the subgraph induced by V x | D 0 ( ν ) = { λ : d G ( ν, λ ) < D } . Definition The half-infinite betweenness centrality of ν ∈ V x with boundary x is 1 hbc G 0 | g G x ( ν, D ) ( ν ) . x ( ν ) = lim | V x | D D →∞ 6 / 12
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 7 / 12
Results Theorem For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbc G x ( ν ) converges. 8 / 12
Results Theorem For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbc G x ( ν ) converges. Corollary There exists an algorithm to compute hbc G x ( ν ) . ◮ Note: hbc G x ( ν ) can be irrational. So more precisely, our algorithm computes a value that is sufficiently close to hbc G x ( ν ). 8 / 12
Results Theorem For any connected, undirected and VAP-free 1-dimensional periodic graph G, hbc G x ( ν ) converges. Corollary There exists an algorithm to compute hbc G x ( ν ) . ◮ Note: hbc G x ( ν ) can be irrational. So more precisely, our algorithm computes a value that is sufficiently close to hbc G 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| . 8 / 12
Our algorithm 9 / 12
Our algorithm 2 3 4 5 6 1 2 3 1 3 4 4 5 5 6 6 2 3 2 5 6 1 4 9 / 12
Our algorithm 9 / 12
Our algorithm If , then shortest paths from to does not contain 9 / 12
Our algorithm 9 / 12
Our algorithm 9 / 12
Our algorithm period 0 period 1 period 2 9 / 12
Our algorithm # shortest # shortest paths from paths from to: to: period 0 period 1 period 2 9 / 12
Our algorithm # shortest # shortest paths from paths from to: to: period 0 period 1 period 2 9 / 12
Idea for the proof ◮ Computation of the function d G x (( u , y ) , ( w , y + b )) of b . 10 / 12
1 : min 1 1 1 1 1 1 1 1 1 1 1 1 11 / 12
use the theory of unimodularity of 2-dim. VAP-free planar periodic graphs [F. '12] 1 : min 1 1 1 1 1 1 1 1 1 1 1 1 11 / 12
opt. val. opt. val. take min. 1 value 1 1 opt. val. opt. val. opt. val. 1 1 1 1 1 1 1 1 1 1 11 / 12
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¨ obner bases and integer programming [Ho¸ sten, Thomas ’98] 12 / 12
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¨ obner bases and integer programming [Ho¸ sten, Thomas ’98] ◮ We need a new theory for 2-dimensional case. 12 / 12
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¨ obner bases and integer programming [Ho¸ sten, Thomas ’98] ◮ We need a new theory for 2-dimensional case. Thank you for your attention. 12 / 12
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