Lecture 28 Network resilience

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Lecture 28Network resilience
Slides are modified from Lada Adamic
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Outline

• network resilience
• effects of node and edge removal
• example power grid
• example biological networks

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Network resilience

• Q If a given fraction of nodes or edges are
  removed
• how large are the connected components?
• what is the average distance between nodes in the
  components
• Related to percolation
• We say the network percolates when a giant
  component forms.

Source http//mathworld.wolfram.com/BondPercolati
on.html
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Bond percolation in Networks

• Edge removal
• bond percolation each edge is removed with
  probability (1-p)
• corresponds to random failure of links
• targeted attack causing the most damage to the
  network with the removal of the fewest edges
• strategies remove edges that are most likely to
  break apart the network or lengthen the average
  shortest path
• e.g. usually edges with high betweenness

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Percolation threshold in Erdos-Renyi Graphs
Percolation threshold the point at which the
giant component emerges As the average degree
increases to z 1, a giant component suddenly
appears Edge removal is the opposite process
As the average degree drops below 1 the network
becomes disconnected
av deg 3.96
av deg 0.99
av deg 1.18
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Quiz Q
In this network each node has average degree
4.64, if you removed 25 of the edges, by how
much would you reduce the giant component?
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Edge percolation
How many edges would you have to remove to break
up an Erdos Renyi random graph? e.g. each node
has an average degree of 4.64
50 nodes, 116 edges, average degree 4.64 after
25 edge removal - gt 76 edges, average degree
3.04 still well above percolation threshold
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Site percolation on lattices
Fill each square with probability p

• low p small isolated islands
• p critical giant component forms, occupying
  finite fraction of infinite lattice. Size of
  other components is power law distributed
• p above critical giant component rapidly spreads
  to span the lattice Size of other components is
  O(1)

• Interactive demonstration
• http//www.ladamic.com/netlearn/NetLogo501/Lattice
  Percolation.html

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Percolation on Complex Networks

• Percolation can be extended to networks of
  arbitrary topology
• We say the network percolates when a giant
  component forms

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Scale-free networks are resilient with respect to
random attack

• gnutella network
• 20 of nodes removed

574 nodes in giant component
427 nodes in giant component
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Targeted attacks are affective against scale-free
networks

• gnutella network,
• 22 most connected nodes removed (2.8 of the
  nodes)

301 nodes in giant component
574 nodes in giant component
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Quiz Q

• Why is removing high-degree nodes more effective?
• it removes more nodes
• it removes more edges
• it targets the periphery of the network

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random failures vs. attacks
Source Error and attack tolerance of complex
networks. Réka Albert, Hawoong Jeong and
Albert-László Barabási.
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Percolation Threshold scale-free networks

• What proportion of the nodes must be removed in
  order for the size (S) of the giant component to
  drop to 0?

• For scale free graphs there is always a giant
  component
• the network always percolates

Source Cohen et al., Resilience of the Internet
to Random Breakdowns
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Network resilience to targeted attacks

• Scale-free graphs are resilient to random
  attacks, but sensitive to targeted attacks.
• For random networks there is smaller difference
  between the two

random failure
targeted attack
Source Error and attack tolerance of complex
networks. Réka Albert, Hawoong Jeong and
Albert-László Barabási
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Real networks
Source Error and attack tolerance of complex
networks. Réka Albert, Hawoong Jeong and
Albert-László Barabási
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• the first few of nodes removed

Source Error and attack tolerance of complex
networks. Réka Albert, Hawoong Jeong and
Albert-László Barabási
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Skewness of power-law networks and effects and
targeted attack
of nodes removed, from highest to lowest degree
Source D. S. Callaway, M. E. J. Newman, S. H.
Strogatz, and D. J. Watts, Network robustness and
fragility Percolation on random graphs, Phys.
Rev. Lett., 85 (2000), pp. 54685471.
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Assortativity

• Social networks are assortative
• the gregarious people associate with other
  gregarious people
• the loners associate with other loners
• The Internet is disassortative

Disassortative hubs are in the periphery
Assortative hubs connect to hubs
Random
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Correlation profile of a network

• Detects preferences in linking of nodes to each
  other based on their connectivity
• Measure N(k0,k1) the number of edges between
  nodes with connectivities k0 and k1
• Compare it to Nr(k0,k1) the same property in a
  properly randomized network
• Very noise-tolerant with respect to both false
  positives and negatives

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Degree correlation profiles 2D
Internet
source Sergei Maslov
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Average degree of neighbors

• Pastor-Satorras and Vespignani 2D plot

probability of aquiring edges is dependent on
fitness degree Bianconi Barabasi
average degree of the nodes neighbors
degree of node
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Single number

• cor(deg(i),deg(j)) over all edges ij

rinternet -0.189
The Pearson correlation coefficient of nodes on
each side on an edge
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assortative mixing more generally

• Assortativity is not limited to degree-degree
  correlations other attributes
• social networks race, income, gender, age
• food webs herbivores, carnivores
• internet high level connectivity providers,
  ISPs, consumers
• Tendency of like individuals to associate
  homophily

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degree assortativity and resilience
will a network with positive or negative degree
assortativity be more resilient to attack?
assortative
disassortative
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Assortativity and resilience
assortative
disassortative
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Is it really that simple?

• Internet?
• terrorist/criminal networks?

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Power grid

• Electric power flows simultaneously through
  multiple paths in the network.
• For visualization of the power grid, check out
  NPRs interactive visualization
  http//www.npr.org/templates/story/story.php?story
  Id110997398

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Power grid

• Electric power does not travel just by the
  shortest route from source to sink, but also by
  parallel flow paths through other parts of the
  system.
• Where the network jogs around large geographical
  obstacles, such as the Rocky Mountains in the
  West or the Great Lakes in the East, loop flows
  around the obstacle are set up that can drive as
  much as 1 GW of power in a circle, taking up
  transmission line capacity without delivering
  power to consumers.

Source Eric J. Lerner, http//www.aip.org/tip/INP
HFA/vol-9/iss-5/p8.html
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Cascading failures

• Each node has a load and a capacity that says how
  much load it can tolerate.
• When a node is removed from the network its load
  is redistributed to the remaining nodes.
• If the load of a node exceeds its capacity, then
  the node fails

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Case study North American power grid
Modeling cascading failures in the North American
power grid R. Kinney, P. Crucitti, R. Albert, and
V. Latora, Eur. Phys. B, 2005

• Nodes generators, transmission substations,
  distribution substations
• Edges high-voltage transmission lines
• 14,099 substations
• NG 1,633 generators,
• ND 2,179 distribution substations
• NT the rest transmission substations
• 19,657 edges

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Degree distribution is exponential
Source Albert et al., Structural vulnerability
of the North American power grid
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Efficiency of a path

• efficiency e 0,1,
• 0 if no electricity flows between two endpoints,
• 1 if the transmission lines are working perfectly
• harmonic composition for a path

• path A, 2 edges, each with e0.5, epath 1/4
• path B, 3 edges, each with e0.5 epath 1/6
• path C, 2 edges, one with e0 the other with e1,
  epath 0
• simplifying assumption electricity flows along
  most efficient path

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Efficiency of the network

• Efficiency of the network
• average over the most efficient paths from each
  generator to each distribution station

eij is the efficiency of the most efficient path
between i and j
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capacity and node failure

• Assume capacity of each node is proportional to
  initial load

• L represents the weighted betweenness of a node
• Each neighbor of a node is impacted as follows

load exceeds capacity

• Load is distributed to other nodes/edges
• The greater a (reserve capacity), the less
  susceptible the network to cascading failures due
  to node failure

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power grid structural resilience

• efficiency is impacted the most if the node
  removed is the one with the highest load

highest load generator/transmission station
removed
Source Modeling cascading failures in the North
American power grid R. Kinney, P. Crucitti, R.
Albert, and V. Latora
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Quiz Q

• Approx. how much higher would the capacity of a
  node need to be relative to the initial load in
  order for the network to be efficient?
• (remember capacity C a L(0), the initial
  load).

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resilience power grids and cascading failures

• Vast system of electricity generation,
  transmission distribution is essentiallya
  single network
• Power flows throughall paths from source to
  sink(flow calculations areimportant for other
  networks,even social ones)
• All AC lines within an interconnect must be in
  sync
• If frequency varies too much (as line approaches
  capacity), a circuit breaker takes the generator
  out of the system
• Larger flows are sent to neighboring parts of the
  grid triggering a cascading failure

Source .wikipedia.org/wiki/FileUnitedStatesPower
Grid.jpg
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Cascading failures

• 158 p.m. The Eastlake, Ohio, First Energy
  generating plant shuts down (maintenance
  problems).
• 306 p.m. A First Energy 345-kV transmission line
  fails south of Cleveland, Ohio.
• 317 p.m. Voltage dips temporarily on the Ohio
  portion of the grid. Controllers take no action,
  but power shifted by the first failure onto
  another power line causes it to sag into a tree
  at 332 p.m., bringing it offline as well. While
  Mid West ISO and First Energy controllers try to
  understand the failures, they fail to inform
  system controllers in nearby states.
• 341 and 346 p.m. Two breakers connecting First
  Energys grid with American Electric Power are
  tripped.
• 405 p.m. A sustained power surge on some Ohio
  lines signals more trouble building.
• 40902 p.m. Voltage sags deeply as Ohio draws 2
  GW of power from Michigan.
• 41034 p.m. Many transmission lines trip out,
  first in Michigan and then in Ohio, blocking the
  eastward flow of power. Generators go down,
  creating a huge power deficit. In seconds, power
  surges out of the East, tripping East coast
  generators to protect them.

Source Eric J. Lerner, What's wrong with the
electric grid? http//www.aip.org/tip/INPHFA/vol-
9/iss-5/p8.html
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(dis) information cascades

• Rumor spreading
• Urban legends
• Word of mouth (movies, products)
• Web is self-correcting
• Satellite image hoax is first passed around, then
  exposed, hoax fact is blogged about, then written
  up on urbanlegends.about.com

Source undetermined
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Actual satellite images of the effect of the
blackout
20 hoursprior toblackout
7 hours after blackout
Source NOAA, U.S. Government
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Biological networks

• In biological systems nodes and edges can
  represent different things
• nodes
• protein, gene, chemical (metabolic networks)
• edges
• mass transfer, regulation
• Can construct bipartite or tripartite networks
• e.g. genes and proteins

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types of biological networks
genome
gene regulatory networks protein-gene
interactions
proteome
protein-protein interaction networks
metabolism
bio-chemical reactions
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protein-protein interaction networks

• Properties
• giant component exists
• longer path length than randomized
• higher incidence of short loops than randomized

Source Jeong et al, Lethality and centrality in
protein networks
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protein interaction networks

• Properties
• power law distribution with an exponential cutoff
• higher degree proteins are more likely to be
  essential

Source Jeong et al, Lethality and centrality in
protein networks
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resilience of protein interaction networks

• if removed
• lethal
• non-lethal
• slow growth
• unknown

Source Jeong et al, Lethality and centrality in
protein networks
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Implications

• Robustness
• resilient to random breakdowns
• mutations in hubs can be deadly
• gene duplication hypothesis
• new gene still has same output protein, but no
  selection pressure
• because the original gene is still present
• Some interactions can be added or dropped
• leads to scale free topology

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gene duplication

• When a gene is duplicated
• every gene that had a connection to it, now has
  connection to 2 genes
• preferential attachment at work

Source Barabasi Oltvai, Nature Reviews 2003
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Disease Network
source Goh et al. The human disease network
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Q do you expect disease genes to be the
essential genes?
- genetic origins of most diseases are shared
with other diseases - most disorders relate to a
few disease genes
source Goh et al. The human disease network
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Q where do you expect disease genes to be
positioned in the gene network
source Goh et al. The human disease network
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gene regulatory networks
translation regulation activating
inhibiting
slide after Reka Albert
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simple model of ON/OFF gene dynamics
Source Albert and Othmer, Journal of Theoretical
Biology 23(1), p. 1-18, 2003. doi10.1016/S0022-51
93(03)00035-3
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network interactions between segment polarity
genes
protein
mRNA
proteincomplex
translation activating inhibiting
Source Albert and Othmer, Journal of Theoretical
Biology 23(1), p. 1-18, 2003. doi10.1016/S0022-51
93(03)00035-3
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excellent agreement between model and observed
gene expression patterns

• test by observing the effect of gene mutation in
  specimen and in model

Source Albert and Othmer, Journal of Theoretical
Biology 23(1), p. 1-18, 2003. doi10.1016/S0022-51
93(03)00035-3
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predicting drosophila gene expression patterns
with a boolean model
initial state
predicted by model
Source Albert and Othmer, Journal of Theoretical
Biology 23(1), p. 1-18, 2003. doi10.1016/S0022-51
93(03)00035-3
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Metabolic networks

• metabolic reaction networks (tri-partite)

• metabolites (substrates or products)
• metabolite-enzyme complexes
• enzymes

Source Jeong et al., Nature 407, 651-654 (5
October 2000) doi10.1038/35036627
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Metabolic networks are scale-free

• In the bi-partite graph
• the probability that a given substrate
  participates in k reactions is k-a
• indegree a 2.2
• outdegree a 2.2

(a) A. fulgidus (Archae) (b) E. coli (Bacterium)
(c) C. elegans (Eukaryote), (d) averaged over 43
organisms
Source Jeong et al., Nature 407, 651-654 (5
October 2000) doi10.1038/35036627
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Is there more to biological networks than degree
distributions?

• No modularity
• Modularity
• Hierarchical modularity

Source E. Ravasz et al., Hierarchical
Organization of Modularity in Metabolic Networks
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clustering coefficients in different topologies
Source Barabasi Oltvai, Nature Reviews 2003
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How do we know that metabolic networks are
modular?

• clustering decreases with degree as
• C(k) k-1
• randomized networks (which preserve the power law
  degree distribution) have a clustering
  coefficient independent of degree

Source E. Ravasz et al., Hierarchical
Organization of Modularity in Metabolic Networks
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How do we know that metabolic networks are
modular?

• clustering coefficient is the same across
  metabolic networks in different species with the
  same substrate
• corresponding randomized scale free networkC(N)
  N-0.75 (simulation, no analytical result)

bacteria archaea (extreme-environment single cell
organisms) eukaryotes (plants, animals, fungi,
protists) scale free network of the same size
Source E. Ravasz et al., Hierarchical
organization in complex networks
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Constructing a hierarchically modular network

• RSMOB model
• Start from a fully connected cluster of nodes
• Create 4 identical replicas of the cluster,
  linking the outside nodes of the replicas to the
  center node of the original (N 25 nodes)
• This process can repeated indefinitely
• (initial number of nodes can be different than 5)

Source Ravasz and Barabasi, PRE 67 026112, 2003,
doi 10.1103/PhysRevE.67.026112
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Properties of the hierarchically modular model

• RSMOB model
• Power law exponent g 2.26
• in agreement with real world metabolic networks
• C 0.6, independent of network size
• also comparable with observed real-world values
• C(k) k-1,
• as in real world network
• How to test for hierarchically arranged modules
  in real world networks
• perform hierarchical clustering on the
  topological overlap map
• can be done with Pajek

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Discovering hierarchical structure using
topological overlap

• A Network consisting of nested modules
• B Topological overlap matrix

hierarchical clustering
Source E. Ravasz et al., Science 297, 1551 -1555
(2002)
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Modularity and the role of hubs

• Party hub
• interacts simultaneously within the same module
• Date hub
• sequential interactions
• connect different modules
• connect biological processes

Source Han et al, Nature 443, 88 (2004)
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Q which type of hub is more likely to be
essential?
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metabolic network of e. coli
Source Guimera Amaral, Functional cartography
of complex metabolic networks
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summing it up

• resilience depends on topology
• also depends on what happens when a node fails
• e.g. in power grid load is redistributed
• in protein interaction networks other proteins
  may start being produced or cease to do so
• in biological networks, more central nodes cannot
  be done without