1. Lecture 28 Network resilience
Slides are modified from Lada Adamic

2. Outline network resilience effects of node and edge removal
example: power grid
example: biological networks

3. 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:

4. 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

5. Percolation threshold in Erdos-Renyi Graphs
average degree
size of giant component
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

6. 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?

7. 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 - > 76 edges, average degree 3.04
still well above percolation threshold

8. 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:

9. Percolation on Complex Networks
Percolation can be extended to networks of arbitrary topology
We say the network percolates when a giant component forms

10. 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

11. Targeted attacks are affective against scale-free networks
gnutella network,
22 most connected nodes removed (2.8% of the nodes)
574 nodes in giant component
301 nodes in giant component

12. Why is removing high-degree nodes more effective?
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

13. random failures vs. attacks
Source: Error and attack tolerance of complex networks. Réka Albert, Hawoong Jeong and Albert-László Barabási.

14. 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?
Percolation transition for networks with power-law
connectivity distribution. Plotted is the fraction of nodes that
remain in the spanning cluster after breakdown of a fraction p
of all nodes, P`pP`0, as a function of p, for a 3.5
(crosses) and a 2.5 (other symbols), as obtained from computer
simulations of up to N In the former case, it
can be seen that for p . pc 0.5 the spanning cluster disintegrates
and the network becomes fragmented. However, for
a 2.5 (the case of the Internet), the spanning cluster persists
up to nearly 100% breakdown. The different curves for K 25
(circles), 100 (squares), and 400 (triangles) illustrate the finite
size effect: the transition exists only for finite networks, while
the critical threshold pc approaches 100% as the networks grow
in size.
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

15. 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

16. Real networks
Source: Error and attack tolerance of complex networks. Réka Albert, Hawoong Jeong and Albert-László Barabási

17. the first few % of nodes removed
a, Comparison between the exponential (E) and scale-free (SF) network models, each containing N = 10,000 nodes and 20,000 links (that is, k = 4). The blue symbols correspond to the diameter of the exponential (triangles) and the scale-free (squares) networks when a fraction f of the nodes are removed randomly (error tolerance). Red symbols show the response of the exponential (diamonds) and the scale-free (circles) networks to attacks, when the most connected nodes are removed. We determined the f dependence of the diameter for different system sizes (N = 1,000; 5,000; 20,000) and found that the obtained curves, apart from a logarithmic size correction, overlap with those shown in a, indicating that the results are independent of the size of the system. We note that the diameter of the unperturbed ( f = 0) scale-free network is smaller than that of the exponential network, indicating that scale-free networks use the links available to them more efficiently, generating a more interconnected web. b, The changes in the diameter of the Internet under random failures (squares) or attacks (circles). We used the topological map of the Internet, containing 6,209 nodes and 12,200 links (k = 3.4), collected by the National Laboratory for Applied Network Research c, Error (squares) and attack (circles) survivability of the World-Wide Web, measured on a sample containing 325,729 nodes and 1,498,353 links3, such that k = 4.59.
Source: Error and attack tolerance of complex networks. Réka Albert, Hawoong Jeong and Albert-László Barabási

18. Skewness of power-law networks and effects and targeted attack
% of nodes removed, from highest to lowest degree
Size of the giant component S in graphs with
power-law degree distribution and all vertices with degree
greater than kmax unoccupied, for t 2.4 (circles), 2.7
(squares), and 3.0 (triangles). Points are simulation results for
systems with 107 vertices; solid lines are the exact solution.
Upper frame: S as a function of fraction of vertices unoccupied.
Lower frame: S as a function of the cutoff parameter kmax.
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. 5468–5471.

19. Social networks are assortative:
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

20. 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

21. Degree correlation profiles: 2D
Internet
source: Sergei Maslov

22. Average degree of neighbors
Pastor-Satorras and Vespignani: 2D plot
probability of aquiring edges is dependent on ‘fitness’ + degree
Bianconi & Barabasi
of the node’s neighbors
average degree
degree of node

23. rinternet = -0.189 Single number
cor(deg(i),deg(j)) over all edges {ij}
rinternet =
The Pearson correlation coefficient of nodes on each
side on an edge

24. 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’

25. degree assortativity and resilience
will a network with positive or negative degree assortativity be more resilient to attack?
assortative
disassortative

26. Assortativity and resilience
assortative
disassortative

27. Is it really that simple?
Internet?
terrorist/criminal networks?

28. Power grid
Electric power flows simultaneously through multiple paths in the network.
For visualization of the power grid, check out NPR’s interactive visualization:

29. 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,

30. 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

31. 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

32. Degree distribution is exponential
The probability that a substation has more than K transmission lines. The straight line
represents the exponential function (1). Inset: the fraction Fg(k) of generating substations among
substations with degree k.
Source: Albert et al., ‘Structural vulnerability of the North American power grid

33. 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 e=0.5, epath = 1/4
path B, 3 edges, each with e=0.5 epath = 1/6
path C, 2 edges, one with e=0 the other with e=1, epath = 0
simplifying assumption: electricity flows along most efficient path

34. 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

35. 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

36. power grid structural resilience
efficiency is impacted the most if the node removed is the one with the highest load
Global efficiency of the power grid after the removal
of random (triangles) or high-load (circles) generators (a) or
transmission substations (b). The unperturbed efficiency is
E(G0) = As the overload tolerance α of the substations
increases, the final efficiency approaches the unperturbed
value. The random disruption curves were obtained by averaging
over 10–100 individual removals. The load-based disruption
curve is obtained by removing the highest load generator and
transmission node, respectively.
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

37. 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).

38. resilience: power grids and cascading failures
Vast system of electricity generation, transmission & distribution is essentially a single network
Power flows through all paths from source to sink (flow calculations are important 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/File:UnitedStatesPowerGrid.jpg

39. Cascading failures
1:58 p.m. The Eastlake, Ohio, First Energy generating plant shuts down (maintenance problems).
3:06 p.m. A First Energy 345-kV transmission line fails south of Cleveland, Ohio.
3:17 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 3:32 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.
3:41 and 3:46 p.m. Two breakers connecting First Energy’s grid with American Electric Power are tripped.
4:05 p.m. A sustained power surge on some Ohio lines signals more trouble building.
4:09:02 p.m. Voltage sags deeply as Ohio draws 2 GW of power from Michigan.
4:10:34 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?”

40. (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

41. Actual satellite images of the effect of the blackout
20 hours prior to blackout
7 hours after blackout
Source: NOAA, U.S. Government

42. In biological systems nodes and edges can represent different things
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

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

44. 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’

45. 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’

46. resilience of protein interaction networks
if removed: lethal non-lethal slow growth unknown
Source: Jeong et al, ‘Lethality and centrality in protein networks’

47. gene duplication hypothesis
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

48. 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

49. Disease Network
source: Goh et al. The human disease network

50. 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

51. Q: where do you expect disease genes to be positioned in the gene network
source: Goh et al. The human disease network

52. gene regulatory networks
translation
regulation: activating
inhibiting
slide after Reka Albert

53. simple model of ON/OFF gene dynamics
Source: Albert and Othmer, Journal of Theoretical Biology 23(1), p. 1-18, doi: /S (03)

54. network interactions between segment polarity genes
protein
mRNA
protein complex
translation
activating
inhibiting
Source: Albert and Othmer, Journal of Theoretical Biology 23(1), p. 1-18, doi: /S (03)

55. test by observing the effect of gene mutation in specimen and in model
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, doi: /S (03)

56. 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, doi: /S (03)

57. metabolic reaction networks (tri-partite)
Metabolic networks
metabolic reaction networks (tri-partite)
metabolites (substrates or products)
metabolite-enzyme complexes
enzymes
Source: Jeong et al., Nature 407, (5 October 2000) | doi: /

58. 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, (5 October 2000) | doi: /

59. 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

60. clustering coefficients in different topologies
Source: Barabasi & Oltvai, Nature Reviews 2003

61. 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

62. 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 network: C(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

63. 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 , 2003, doi: /PhysRevE

64. 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

65. 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, (2002)

66. 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)

67. Q: which type of hub is more likely to be essential?

68. metabolic network of e. coli
Source: Guimera & Amaral, Functional cartography of complex metabolic networks

69. resilience depends on topology
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