1. RESILIENCE NOTIONS FOR SCALE-FREE NETWORKS GUNES ERCAL JOHN MATTA 1

2. THE STRUCTURE OF NETWORKS A graph, G = (V, E) represents a network. The degree of a node v in a network is the number of nodes that v is connected to. The distribution of node degrees in a network is clearly an important structural property of the network. Homogeneous degree distribution: all nodes have similar degrees Heterogeneous degree distribution: node degrees clearly variant 2

3. HIGH VARIANCE IN DEGREE DISTRIBUTION 3

4. MODELS FOR SCALE- FREE NETWORKS Two popular generative models: Preferential attachment: Dynamic model, “rich get richer” phenomenon Given parameters m, a, and b For node v arriving at time t, choose m neighbors of v with probability p(v, u) = probability that u is a neighbor of v p(v, u) = (degree(u) a +b)/N Where N = Σ (degree(x) a +b) Random scale-free: Assume that you have generated a degree distribution D that is scale-free (e.g. power-law) Randomly choose edges conditional upon D 4

5. ROBUSTNESS Characterizing the robustness of networks: under various forms of attack Nodes vs. Edges Targeted vs. Random for various generative models of such networks What is known so far: Lots of work on edge based resilience Theoretically: Spectral gap captures resilience Lots of work on general resilience for homogeneous nets Corollary of edge based resilience 5

6. CONDUCTANCE AS A MEASURE OF RESILIENCE 6

7. MORE ON CONDUCTANCE What does conductance say in the face of node attacks? 7

8. CONDUCTANCE 8 Two three-regular graphs with 10 nodes: High Conductance Low Conductance In homogeneous degree graphs, the property of having high conductance maps directly to being resilient against both node and edge attacks.

9. MORE ON CONDUCTANCE What does conductance say in the face of node attacks for heterogeneous degree graphs (e.g. scale-free graphs)? 9

10. CONDUCTANCE IN HETEROGENEOUS DEGREE GRAPHS 10 A highly heterogeneous degree graph with a high conductance An attack against the center node disconnects the entire graph. Conductance is not a good measure of this graph's resilience.

11. EDGE FAILURES VS NODE FAILURES Conductance captures resilience under a model of edge failures. This coincides with a measure of resilience under node failures when the graph has a homogeneous degree distribution Conductance no longer captures resilience under a model of node failures when the graph is highly heterogeneous, and in particular scale free What is needed is a measure of node-based resilience 11

12. A PROPOSED MEASURE OF NODE-BASED RESILIENCE 12

13. CALCULATIONS conductance 13

14. CALCULATIONS conductance 14

15. CALCULATIONS conductance 15

16. CONDUCTANCE VS S(G) 16 Conductance: s(G): 1 (high).2(low) 1 (high) 1 (high).2(low).1111 (low)

17. HOTNET 17 *As described in Fabrikant, Koutsoupias, Papadimitriou, Heuristically Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet

18. 18 *As described in C. Palmer and J. Steffan, Generating Network Topologies That Obey Power Laws PLOD