RESILIENCE
NOTIONS FOR
SCALE-FREE
NETWORKS
GUNES ERCAL
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JOHN MATTA
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
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•
HIGH VARIANCE IN
DEGREE DISTRIBUTION
• Scale-Free degree distribution:
• High variance, heterogeneous degree distribution
• Heavy-tailed degree distribution
•
•
Most nodes have small degree, but…
For arbitrarily high degrees: non-negligibly many nodes
• Power Law:
•
•
1
Frequency of nodes with degree d = 𝑑α
for a constant α > 1.
Looks linear on a log-log scale
• Contrast with Erdős-Rényi random graphs:
3
• These have Gaussian degree distributions
MODELS FOR SCALEFREE 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
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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
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•
CONDUCTANCE AS A
MEASURE OF RESILIENCE
• Combinatorial measure of edge based resilience
|Cut(S, V−S)|
|S|
• Can think of Cut(S, V-S) as the “attacked edges” that
disconnect the vertex set
• If conductance is low:
• conductance = minimum{S non-majority subset of V}
•
•
There exists relatively few edges whose removal
disconnects two relatively large sets of vertices
Bad bottleneck
• Otherwise, there is no such set of bad edges
i.e. You need to attack proportionally many edges to
disconnect large sets from each other
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•
MORE ON CONDUCTANCE
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What does conductance say in
the face of node attacks?
CONDUCTANCE
Two three-regular graphs with 10 nodes:
High Conductance
Low Conductance
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In homogeneous degree graphs, the property of having high
conductance maps directly to being resilient against both
node and edge attacks.
MORE ON CONDUCTANCE
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What does conductance say in
the face of node attacks for
heterogeneous degree graphs (e.g.
scale-free graphs)?
CONDUCTANCE IN
HETEROGENEOUS DEGREE GRAPHS
A highly heterogeneous degree graph with a high conductance
𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑎𝑛𝑐𝑒 = 𝑚𝑖𝑛
|Cut(S, V−S)|
=1
|S|
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• An attack against the center node disconnects the entire graph.
• Conductance is not a good measure of this graph's resilience.
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
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• What is needed is a measure of node-based resilience
A PROPOSED MEASURE OF
NODE-BASED RESILIENCE
What we really wish to measure is the following function:
|𝑆|
𝑠 𝐺 = 𝑚𝑖𝑛{𝑜𝑣𝑒𝑟 𝑎𝑙𝑙 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑆 𝑜𝑓 𝑉}
|𝑉 − 𝑆 − 𝐶𝑚𝑎𝑥 | + 1
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where Cmax is the largest connected component that remains in
the graph G(V – S)
CALCULATIONS
|Cut(S, V−S)|
= 𝑚𝑖𝑛
|S|
4
=4=1
Cutting 4 edges
disconnects 4 nodes
s(G)
|𝑆|
= 𝑚𝑖𝑛
|𝑉 − 𝑆 − 𝐶𝑚𝑎𝑥 | + 1
1
= 𝑚𝑖𝑛
10 − 1 − 9 + 1
1
1
= =1
Disconnecting 1 node
leaves 9 nodes still
connected
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conductance
CALCULATIONS
= 𝑚𝑖𝑛
|Cut(S, V−S)|
|S|
1
= 5 = .2
Cutting 1 edges
disconnects 5 nodes
s(G)
|𝑆|
= 𝑚𝑖𝑛
|𝑉 − 𝑆 − 𝐶𝑚𝑎𝑥 | + 1
1
= 𝑚𝑖𝑛
10 − 1 − 5 + 1
1
5
= = .2
Disconnecting 1 node
leaves 5 nodes still
connected
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conductance
CALCULATIONS
|Cut(S, V−S)|
= 𝑚𝑖𝑛
|S|
1
=1=1
Cutting 1 edge
disconnects 1 node
s(G)
|𝑆|
= 𝑚𝑖𝑛
|𝑉 − 𝑆 − 𝐶𝑚𝑎𝑥 | + 1
1
=
10 − 1 − 1 + 1
1
9
= = .1111
Disconnecting 1 node
leaves a largest connected
component of only 1 node
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conductance
Conductance:
1 (high)
.2 (low)
1 (high)
s(G):
1 (high)
.2 (low)
.1111 (low)
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CONDUCTANCE
VS S(G)
HOTNET
conductance
1
8
= = .125
s(G)
=
1
25 −1 −8 + 1
=
1
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= .055235
*As described in Fabrikant, Koutsoupias, Papadimitriou, Heuristically
Optimized Tradeoffs: A New Paradigm for Power Laws in the Internet
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degree = 1.92
PLOD
Conductance: .5
s(G): .25
degree 2.88
1
cond = 2 = .5
1
*As described in C. Palmer and J. Steffan, Generating Network Topologies
That Obey Power Laws
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s(G) = 25 −1 − 21 + 1 = .25