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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
using distance-based graph measures Sander Jurgens November 10, 2014
Eindhoven University of Technology TNO ICT, Delft Department of Math and Computer Science Performance of Network and Systems
Motivation Graph measures Complexity Approximation Conclusion
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Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
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Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
A network consisting of:
◮ Vertices ◮ Edges
Additional properties:
◮ Connected ◮ Undirected ◮ Weighted Sander Jurgens Improving network robustness 4 / 28
Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
Model relations and processes Many domains:
◮ Communication ◮ Energy ◮ Transportation ◮ ... Sander Jurgens Improving network robustness 5 / 28
Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
Model relations and processes Many domains:
◮ Communication ◮ Energy ◮ Transportation ◮ ... Sander Jurgens Improving network robustness 6 / 28
Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
Networks contain weaknesses
Robustness is the ability of a network to continue performing well when it is subject to attacks and random failures.
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Motivation Graph measures Complexity Approximation Conclusion Graph theory Robustness
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 Complexity Approximation Conclusion Measure types Distance measures
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
Different types:
◮ Distance ◮ Centrality ◮ Connectivity ◮ Spectral Sander Jurgens Improving network robustness 10 / 28
Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
Different types:
◮ Distance ◮ Centrality ◮ Connectivity ◮ Spectral
(a) Complete (b) Cycle (c) Star (d) Line (e) Empty Figure: Graph topologies in order of decreasing robustness.
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
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.
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
Different types:
◮ Distance (viable, good correlation) ◮ Centrality (viable) ◮ Connectivity (not viable, no distinct values) ◮ Spectral (viable, but not intuitive)
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+
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
Eccentricity, for a vertex v ∈ V , represents the greatest distance between v and some
ǫ(v) = max
u∈V d(v, u).
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
Eccentricity, for a vertex v ∈ V , represents the greatest distance between v and some
ǫ(v) = max
u∈V d(v, u).
The radius of a graph is the minimum eccentricity over all of its vertices: R = min
v∈V ǫ(v).
The diameter of a graph is the maximum eccentricity over all of its vertices: D = max
v∈V ǫ(v).
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Motivation Graph measures Complexity Approximation Conclusion Measure types Distance measures
The single source average shortest path length of a vertex is the average over the shortest paths to all other vertices in the graph: Davg(v) = 1 n − 1
d(u, v)
The average shortest path length of a graph is the average over the shortest paths between all combinations of vertices: Davg = 1 n
Davg(u) = 1 n(n − 1)
d(u, v).
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
Eccentricity, for a vertex v ∈ V , represents the greatest distance between v and some
ǫ(v) = max
u∈V d(v, u).
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, ǫG′(v), is minimized in supergraph G ′ = (V , E ∪ F).
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
Classifying computational problems according to their difficulty
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
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
Name: Cover by 3-sets (3c) Instance: 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. Problem: 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
Figure: Graph representing the 3c problem.
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
Name: Cover by 3-sets (3c) Instance: 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. Problem: 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
Figure: Graph representing the 3c problem.
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
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 ǫG′(a) ≤ 2
Figure: Graph used in the proof of complexity for em.
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Motivation Graph measures Complexity Approximation Conclusion Minimization problem NP-completeness Proof of complexity
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 ǫG′(a) ≤ 2
Figure: Graph used in the proof of complexity for em.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
Unlike heuristics, we want a provable solution quality Two types of results:
◮ Computed result (′) ◮ Optimal result (∗)
Approximation ratio ρ is an upper bound on the factor between the computed and
OPT ≤ f (x) ≤ ρ · OPT ǫ∗(v) ≤ ǫ′(v) ≤ ρ · ǫ∗(v)
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph Approximation approach (for a given v ∈ V ):
Figure: Illustration of approach for k = 2.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph Approximation approach (for a given v ∈ V ):
Figure: Illustration of approach for k = 2.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
Minimize the distance measure (e.g. eccentricity) by adding k edges to a given graph Approximation approach (for a given v ∈ V ):
Figure: Illustration of approach for k = 2.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering Improvement
Locate vertices which are heads of clusters distant from a given v ∈ V Clustering to minimize objective function: max
u∈V min h∈H dG (u, h)
Cluster(G, d, v, k) (1) h1 ← v (2) B1 ← V (3) for i = 2 to k (4) Let u be the vertex furthest away from a head (5) hi ← u (6) Bi ← {u} (7) foreach w ∈ (B1 ∪ · · · ∪ Bi−1) (8) Let j be such that w ∈ Bj (9) if d(w, hi) ≤ d(w, hj) (10) Bj ← Bj \ {w} (11) Bi ← Bi ∪ {w} (12) return (h1, . . . , hk)
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering 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 ← { (h1, hi) | 1 < i ≤ k + 1 } (3) return F
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering 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 ← { (h1, hi) | 1 < i ≤ k + 1 } (3) return F
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.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering 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 ← { (h1, hi) | 1 < i ≤ k + 1 } (3) return F Given R(Bi) ≤ D∗ ≤ 2R∗ ≤ 2ǫ∗(v), for 1 ≤ i ≤ k + 1, we have the following: Our eccentricity: 2ǫ∗(v) + ω Optimal eccentricity: ǫ∗(v)
ǫ∗(v) = 2 + ω ǫ∗(v)
R(B1) R(Bi) ω h1 hi
Figure: Graph illustrating the eccentricity bound.
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Motivation Graph measures Complexity Approximation Conclusion Basics Approach Clustering 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 MG′ < m (6) F ← F ′ (7) m ← MG′ (8) return F
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Motivation Graph measures Complexity Approximation Conclusion Results Future work Questions
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Motivation Graph measures Complexity Approximation Conclusion Results Future work 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 24 / 28
Motivation Graph measures Complexity Approximation Conclusion Results Future work Questions
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
Motivation Graph measures Complexity Approximation Conclusion Results Future work Questions
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? Approximation ratio Time complexity Eccentricity 2 +
ω ǫ∗(v)
O(kn) Radius 2 +
ω R∗
O(n4) Diameter 2 + 2ω
D∗
O(n4) SS-ASP 2 O(kn) ASP 4 O(n4)
Table: Results of the approximation algorithms.
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Motivation Graph measures Complexity Approximation Conclusion Results Future work Questions
Network properties:
◮ Expand to directed networks
Approximation ratios:
◮ Improve approximation ratios ◮ Prove approximation ratio tightness
Improvement technique:
◮ Introduction of potential targets Sander Jurgens Improving network robustness 27 / 28
Motivation Graph measures Complexity Approximation Conclusion Results Future work Questions
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