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

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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

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HIGH VARIANCE IN DEGREE DISTRIBUTION 3

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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

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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

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CONDUCTANCE AS A MEASURE OF RESILIENCE 6

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

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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.

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

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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.

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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

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A PROPOSED MEASURE OF NODE-BASED RESILIENCE 12

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CALCULATIONS conductance 13

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CALCULATIONS conductance 14

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CALCULATIONS conductance 15

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

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

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

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