Improving network robustness

Transcription

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

2 Table of Contents Motivation Sander Jurgens Improving network robustness 2 / 28

3 Graph theory Robustness Motivation Sander Jurgens Improving network robustness 3 / 28

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

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

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

7 Graph theory Robustness 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

8 Graph theory Robustness 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

9 Measure types Distance measures Sander Jurgens Improving network robustness 9 / 28

10 Measure types Distance measures Measure types Different types: Distance Centrality Connectivity Spectral Sander Jurgens Improving network robustness 10 / 28

11 Measure types Distance measures 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

12 Measure types Distance measures 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

13 Measure types Distance measures 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

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

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

16 Measure types Distance measures 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: D avg (v) = 1 n 1 d(u, v) 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 D avg 1 (u) = d(u, v). n n(n 1) u V u V v V Sander Jurgens Improving network robustness 12 / 28

17 Minimization problem NP-completeness Proof of complexity Sander Jurgens Improving network robustness 13 / 28

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

19 Minimization problem NP-completeness Proof of complexity 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

20 Minimization problem NP-completeness Proof of complexity Cover by 3-sets Name: Instance: Problem: Cover by 3-sets (3c) Given a set X with X = 3k 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 with C = k, such that every element of X occurs in at least one member of C. x 1 X:... x 3k c 1 C:... c l Figure: Graph representing the 3c problem. Sander Jurgens Improving network robustness 15 / 28

21 Minimization problem NP-completeness Proof of complexity Cover by 3-sets Name: Instance: Problem: Cover by 3-sets (3c) Given a set X with X = 3k 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 with C = k, such that every element of X occurs in at least one member of C. x 1 X:... x 3k c 1 C:... c l C Figure: Graph representing the 3c problem. Sander Jurgens Improving network robustness 15 / 28

22 Minimization problem NP-completeness Proof of complexity 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 ɛ G (a) 2 x 1 X:... x 3k C: c 1... c l b a Figure: Graph used in the proof of complexity for em. Sander Jurgens Improving network robustness 16 / 28

23 Minimization problem NP-completeness Proof of complexity 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 ɛ G (a) 2 x 1 X:... x 3k C: c 1... c l C b a Figure: Graph used in the proof of complexity for em. Sander Jurgens Improving network robustness 16 / 28

24 Basics Approach Clustering Improvement Sander Jurgens Improving network robustness 17 / 28

25 Basics Approach Clustering Improvement What is an approximation algorithm? Unlike heuristics, we want a provable solution quality Two types of results: Computed result ( ) Optimal result ( ) ratio ρ is an upper bound on the factor between the computed and optimal result: OPT f (x) ρ OPT ɛ (v) ɛ (v) ρ ɛ (v) Sander Jurgens Improving network robustness 18 / 28

26 Basics Approach Clustering Improvement Approach Problem Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph approach (for a given v V ): 1. Locate k vertices which are heads of clusters distant from v. 2. Let set F contain the k new edges between v and these heads. v Figure: Illustration of approach for k = 2. Sander Jurgens Improving network robustness 19 / 28

27 Basics Approach Clustering Improvement Approach Problem Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph approach (for a given v V ): 1. Locate k vertices which are heads of clusters distant from v. 2. Let set F contain the k new edges between v and these heads. v Figure: Illustration of approach for k = 2. Sander Jurgens Improving network robustness 19 / 28

28 Basics Approach Clustering Improvement Approach Problem Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph approach (for a given v V ): 1. Locate k vertices which are heads of clusters distant from v. 2. Let set F contain the k new edges between v and these heads. v Figure: Illustration of approach for k = 2. Sander Jurgens Improving network robustness 19 / 28

29 Basics Approach Clustering Improvement Clustering Locate vertices which are heads of clusters distant from a given v V Clustering to minimize objective function: max G (u, h) u V h H Cluster(G, d, v, k) (1) h 1 v (2) B 1 V (3) for i = 2 to k (4) Let u be the vertex furthest away from a head (5) h i u (6) B i {u} (7) foreach w (B 1 B i 1 ) (8) Let j be such that w B j (9) if d(w, h i ) d(w, h j ) (10) B j B j \ {w} (11) B i B i {w} (12) return (h 1,..., h k ) Sander Jurgens Improving network robustness 20 / 28

30 Basics Approach Clustering Improvement Improvement Let set F contain the new edges between v and the heads from the clustering ApproxSource(G, d, v, k) (1) H Cluster(G, d, v, k + 1) (2) F { (h 1, h i ) 1 < i k + 1 } (3) return F Sander Jurgens Improving network robustness 21 / 28

31 Basics Approach Clustering Improvement Improvement Let set F contain the new edges between v and the heads from the clustering. ApproxSource(G, d, v, k) (1) H Cluster(G, d, v, k + 1) (2) F { (h 1, h i ) 1 < i k + 1 } (3) return F Definition The additional edge weight ω is given by the maximum edge weight in set F, where F is the set of edges to be added. Sander Jurgens Improving network robustness 21 / 28

32 Basics Approach Clustering Improvement Improvement Let set F contain the new edges between v and the heads from the clustering. ApproxSource(G, d, v, k) (1) H Cluster(G, d, v, k + 1) (2) F { (h 1, h i ) 1 < i k + 1 } (3) return F Given R(B i ) D 2R 2ɛ (v), for 1 i k + 1, we have the following: } Our eccentricity: 2ɛ (v) + ω 2ɛ (v) + ω Optimal eccentricity: ɛ (v) ɛ = 2 + ω (v) ɛ (v) R(B1) h1 ω R(Bi) hi Figure: Graph illustrating the eccentricity bound. Sander Jurgens Improving network robustness 21 / 28

33 Basics Approach Clustering Improvement Improvement Similar results have been proven for other measures using the following algorithm ApproxGeneral(G, d, k, M) (1) F { }, m (2) foreach v V (3) F ApproxSource(G, d, v, k) (4) G (V, E F ) (5) if M G < m (6) F F (7) m M G (8) return F Sander Jurgens Improving network robustness 22 / 28

34 Results Future work Questions Sander Jurgens Improving network robustness 23 / 28

35 Results Future work Questions Results 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 24 / 28

36 Results Future work Questions Results Which graph measures are suited for measuring network robustness? Distance, Centrality and Spectral. How viable are distance-based measures in measuring the network robustness? Very viable, partially due to their correlation. What complexity class does robustness improvement, i.e. distance-based measure minimization, belong to? All minimization problems were proven to be NP-hard. Sander Jurgens Improving network robustness 25 / 28

37 Results Future work Questions Results How do we go about approximating the optimal robustness improvements? Connect given vertex to distant cluster heads. What approximation ratios can we achieve for robustness improvements? ratio Time complexity Eccentricity 2 + ω ɛ (v) O(kn) Radius 2 + ω R O(n 4 ) Diameter 2 + 2ω D O(n 4 ) SS-ASP 2 O(kn) ASP 4 O(n 4 ) Table: Results of the approximation algorithms. Sander Jurgens Improving network robustness 26 / 28

38 Results Future work Questions Future work Network properties: Expand to directed networks ratios: Improve approximation ratios Prove approximation ratio tightness Improvement technique: Introduction of potential targets Sander Jurgens Improving network robustness 27 / 28

39 Results Future work Questions Questions? Sander Jurgens Improving network robustness 28 / 28