Improving network robustness using distance-based graph measures - PowerPoint PPT Presentation

Improving network robustness using distance-based graph measures Sander Jurgens November 10, 2014 dr. K.A. Buchin dr. D.T.H. Worm Eindhoven University of Technology TNO ICT, Delft Department of Math and Computer Science Performance of Network and Systems

Table of Contents Motivation Graph measures Complexity Approximation Conclusion Sander Jurgens Improving network robustness 2 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion Motivation Sander Jurgens Improving network robustness 3 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion What is a graph? A network consisting of: ◮ Vertices ◮ Edges Additional properties: ◮ Connected ◮ Undirected ◮ Weighted Sander Jurgens Improving network robustness 4 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion What can graphs do? Model relations and processes Many domains: ◮ Communication ◮ Energy ◮ Transportation ◮ ... Sander Jurgens Improving network robustness 5 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion What can graphs do? Model relations and processes Many domains: ◮ Communication ◮ Energy ◮ Transportation ◮ ... Sander Jurgens Improving network robustness 6 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion What is robustness? Networks contain weaknesses Definition Robustness is the ability of a network to continue performing well when it is subject to attacks and random failures. Sander Jurgens Improving network robustness 7 / 28

Motivation Graph measures Graph theory Complexity Robustness Approximation Conclusion Research questions Research question How can we determine the robustness of a simple, connected, undirected, weighted graph, and use the resulting measures in the improvement process? Secondary research questions ◮ Which graph measures are suited for measuring network robustness? ◮ How viable are distance-based measures in measuring the network robustness? ◮ What complexity class does robustness improvement, i.e. distance-based measure minimization, belong to? ◮ How do we go about approximating the optimal robustness improvements? ◮ What approximation ratios can we achieve for robustness improvements? Sander Jurgens Improving network robustness 8 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Graph measures Sander Jurgens Improving network robustness 9 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Measure types Different types: ◮ Distance ◮ Centrality ◮ Connectivity ◮ Spectral Sander Jurgens Improving network robustness 10 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Measure types Different types: ◮ Distance ◮ Centrality ◮ Connectivity ◮ Spectral (a) Complete (b) Cycle (c) Star (d) Line (e) Empty Figure: Graph topologies in order of decreasing robustness. Sander Jurgens Improving network robustness 10 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Measure types Different types: ◮ Distance (viable, good correlation) ◮ Centrality (viable) ◮ Connectivity (not viable, no distinct values) ◮ Spectral (viable, but not intuitive) (a) Complete (b) Cycle (c) Star (d) Line (e) Empty Figure: Graph topologies in order of decreasing robustness. Sander Jurgens Improving network robustness 10 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Measure types Different types: ◮ Distance (viable, good correlation) ◮ Centrality (viable) ◮ Connectivity (not viable, no distinct values) ◮ Spectral (viable, but not intuitive) Distance The distance d ( u , v ) is the combined weight of all edges that make up the shortest path between vertex u ∈ V and vertex v ∈ V . d : V × V → R + 0 Sander Jurgens Improving network robustness 10 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Distance measures Eccentricity Eccentricity , for a vertex v ∈ V , represents the greatest distance between v and some other vertex in the graph: ǫ ( v ) = max u ∈ V d ( v , u ) . Sander Jurgens Improving network robustness 11 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Distance measures Eccentricity Eccentricity , for a vertex v ∈ V , represents the greatest distance between v and some other vertex in the graph: ǫ ( v ) = max u ∈ V d ( v , u ) . Radius The radius of a graph is the minimum eccentricity over all of its vertices: R = min v ∈ V ǫ ( v ) . Diameter The diameter of a graph is the maximum eccentricity over all of its vertices: D = max v ∈ V ǫ ( v ) . Sander Jurgens Improving network robustness 11 / 28

Motivation Graph measures Measure types Complexity Distance measures Approximation Conclusion Distance measures Single Source Average Shortest Path ( SS-ASP ) The single source average shortest path length of a vertex is the average over the shortest paths to all other vertices in the graph: 1 � D avg ( v ) = d ( u , v ) n − 1 u ∈ V Average Shortest Path ( ASP ) The average shortest path length of a graph is the average over the shortest paths between all combinations of vertices: D avg = 1 1 � � � D avg ( u ) = d ( u , v ) . n n ( n − 1) u ∈ V u ∈ V v ∈ V Sander Jurgens Improving network robustness 12 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion Complexity Sander Jurgens Improving network robustness 13 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion Minimization problem Eccentricity Eccentricity , for a vertex v ∈ V , represents the greatest distance between v and some other vertex in the graph: ǫ ( v ) = max u ∈ V d ( v , u ) . Problem Name : Eccentricity Minimization ( em ) Instance : An undirected graph G = ( V , E ), a distance function d : V × V → R + 0 , a vertex v ∈ V , and a bound k ∈ N . Problem : Finding a set F of non-existing edges, with | F | ≤ k , such that eccentricity for vertex v , ǫ G ′ ( v ), is minimized in supergraph G ′ = ( V , E ∪ F ). Sander Jurgens Improving network robustness 13 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion What is computational complexity? Classifying computational problems according to their difficulty Non-deterministic polynomial time (NP) For a given problem C : ◮ Any given solution to C can be verified quickly (in polynomial time) ◮ The time required to solve C increases quickly as the size of the problem grows To prove that a problem C is NP-complete we need to show that 1. a candidate solution to C is verifiable in polynomial time, and 2. every problem in NP is reducible to C in polynomial time. Sander Jurgens Improving network robustness 14 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion Cover by 3-sets Name : Cover by 3-sets ( 3c ) Instance : Given a set X with | X | = 3 k and a collection C of 3-element subsets of X such that each element of X occurs in at least one member of C . Finding an exact cover of X in C , i.e. a subcollection C ′ ⊂ C Problem : with | C ′ | = k , such that every element of X occurs in at least one member of C ′ . X: x 1 . . . x 3 k C: c 1 . . . c l Figure: Graph representing the 3c problem. Sander Jurgens Improving network robustness 15 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion Cover by 3-sets Name : Cover by 3-sets ( 3c ) Instance : Given a set X with | X | = 3 k and a collection C of 3-element subsets of X such that each element of X occurs in at least one member of C . Finding an exact cover of X in C , i.e. a subcollection C ′ ⊂ C Problem : with | C ′ | = k , such that every element of X occurs in at least one member of C ′ . X: x 1 . . . x 3 k C: c 1 . . . c l C ′ Figure: Graph representing the 3c problem. Sander Jurgens Improving network robustness 15 / 28

Motivation Graph measures Minimization problem Complexity NP-completeness Approximation Proof of complexity Conclusion Reduction C contains a subset C ′ ⊂ C , with | C ′ | = k , such that every element of X occurs in at least one element of C ′ ⇔ G has a supergraph G ′ , obtained by adding all k edges from set F , where ǫ G ′ ( a ) ≤ 2 X: . . . x 1 x 3 k C: . . . c 1 c l b a Figure: Graph used in the proof of complexity for em . Sander Jurgens Improving network robustness 16 / 28