1. Optimal serverless networks attacks, complexity and some approximate algorithms Carlos Aguirre Maeso Escuela Politécnica Superior Universidad Autónoma de Madrid

2. Network attacks ● An attack is a set of objects of a network (nodes and/or edges) that are disabled removed (from the graph). ● The goal of a given attack is to produce the maximum possible damage in terms of connectivity.

3. Network attacks ● The connectivity (resistance) of the graph after the attack can be measured in different ways. Number of nodes that are disconnected from a given source node (server networks) (Aura et al.) Size of the biggest connected component (Albert- Barabasi). ● The efficiency (damage) of an attack algorithm for a given graph is the inverse of the resistance of the graph to the attack

4. Model of communication network ● We model a communication network as a cuadruple CN={V,E,c,s}:

5. Network attacks ● The cost function indicates how much costs to a possible enemy to disable the element ● The relevance function indicates the importance of the element, and therefore how bad is that other nodes of the network become disconnected with this node

6. Network attacks ● The cost of a given attack is the sum of the costs of the elements that are removed from the graph. ● The relevance of a set of nodes is the sum of the relevances of the nodes in the set.

7. Network attacks ● For a given graph, we define the core of the graph as the connected component with the highest importance. ● We define the damage produced over a communication network as the sum of the significances of the elements that do not belong to the core after the attack. ● The resistence of the network to the attack is defined as the size of the core.

8. Optimal network attacks ● Now, the problem op the optimal attack can be stablished in the following terms: ● Problem OPT_ATTACK: Given a serverless communication network CN and two fixed values C and D, does there exist an attack A such as C(A) = D ?

9. Optimal network attacks ● We look for attacks that produce the maximal damage with the minimum cost. ● The problem of finding such attack is NP- Complete (Aguirre et al.) even in the easier case of bidirectional links, unbreakable edges (nodes), identical cost for all nodes (edges) and indentical importance for all the nodes.

10. NP-Completness ● NP-Complete problems are problems that verify two conditions ● It is very hard to find a solution. ● But If we are presented a possible solution it is easy to check if this solution meets the conditions of the problem. ● This conditions makes this problems very suitable for approximate algorithms.

11. NP ● I n the OPT_ATTACK problem is very easy to check is a possible solution meets the conditions of the problem ● We count the cost of the attack O(|E|) ● We find the connected components after the attack O(|E|+|V|log|V|) ● We find the biggest connected component O(|V|) ● The damage now can be computed in O(|V|) ● To check if a possible solution meets the condition of the problem takes a time O(|E|+|V|log|V|).

12. Completness ● To show the completness of the problem, we stablish a polynomical reduction of this problem to other known complete problem. ● For the OPT_ATTACK problem we stablish a polynomical reduction with the problem of the bisection of a graph. ● As the bisection problem is complete, our problem is also complete.

14. Completness Lemma: The original graph G has a bisection of size B = 1/2(|V| 3 /4+|V|)+|V|/2 and C(A) <= 1/2(|V| 3 /4+|V|)+B. Sketch of the proof: For the only if part take as attack the edges of the bisection plus the edges of the paths from the extra node to one of the sides of G. For the if part show that is such attacks exists the numbers of nodes that are discconected from the extra node s cant not be higher that |V|/2 or lower than |V|/2.

15. Completness The same theorem can be stablished for attacks where only edges can be attacked. For a graph G consider the following graph G' where only the white nodes can be attacked and apply the previous theorem.

16. Approximate Algorithms for Attacks ● Random Walks, Brownian methods. ● Minimal Cuts (Shamir) ● Random Failure (Albert-Barabasi) ● Maximal node degree (Albert-Barabasi) ● Minimal All-Paths (Newman). ● Minimal Single Path (Aguirre et al.)

17. Random Attack (Failure) Failure j= 1 while j < n node=aleat(V) V = V - node j++ End Failure Failure has a computational complexity O(n)

18. Degree Based Attack AttackDegree j= 1 while j < n node=MostConected(V) V = V - node j++ End Attack ● AttackDegree has a computational complexity O(n 2 )

19. MinPath Based Attack AttackMinPath j= 1 while j < n For each u in V ind[u]=0 For each u,v in V C = minimal path from u to v for each node k in C ind[k]++ node= k such as the value ind[k] is maximum V = V - node j++ End AttackMinPath MinPath has a computational complexity O(n 3 ), this order can be reduced by selecting only a subset of the set of pairs of nodes in the graph

20. Graph topologies ● A given attack algorithm has different efficiencies in different kinds of graphs. ● This means that before selecting an attack algorithm is usually a good idea to know what kind of graph is going to be attacked and select the best attack algorithm for that graph. ● The graphs are classified using their intrinsic metrics

21. Metrics ● Esparsity (E/V) ● Node degree distribution ● Average degree ( ) ● Cluster Coefficient (C) ● Characteristic Path (L) ● Number of biconnected components (B)

22. 1830.7518.77HIERARCHIC NETW. 4280.05615.808POWER GRID 60.0043.89RANDOM 10.01863.409SCALE-FREE 10.1573.744MIXED 10.62614.2SMALL-WORLD 10.643125.438RING-LATTICE BCL

23. Network models ● Random Networks ● Regular Networks (Rings, grids) ● Small-World ● Scale Free ● Hierarchical networks ● Real Networks

24. Small-World

25. Ring

26. Scale-Free

27. Mix

28. Random

29. Hierarchic

30. Power Grid

31. Failure

32. Attack Degree

33. Attack Flow