Lecture 28 Network resilience Slides are modified from Lada Adamic

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

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/BondPercolation.html

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

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

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?

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

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/LatticePercolation.html

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

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

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

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

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

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 106. 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

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

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

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 http://moat.nlanr.net/Routing/rawdata/. 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

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.

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

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

Degree correlation profiles: 2D Internet source: Sergei Maslov

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

rinternet = -0.189 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

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’

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

Assortativity and resilience assortative disassortative

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

Power grid Electric power flows simultaneously through multiple paths in the network. For visualization of the power grid, check out NPR’s interactive visualization: http://www.npr.org/templates/story/story.php?storyId=110997398

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/INPHFA/vol-9/iss-5/p8.html

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

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

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

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

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

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

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) = 0.04137. 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

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).

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

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?” http://www.aip.org/tip/INPHFA/vol-9/iss-5/p8.html

(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

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

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

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

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’

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’

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

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

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

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

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

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

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

simple model of ON/OFF gene dynamics Source: Albert and Othmer, Journal of Theoretical Biology 23(1), p. 1-18, 2003. doi:10.1016/S0022-5193(03)00035-3

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, 2003. doi:10.1016/S0022-5193(03)00035-3

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, 2003. doi:10.1016/S0022-5193(03)00035-3

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. doi:10.1016/S0022-5193(03)00035-3

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, 651-654 (5 October 2000) | doi:10.1038/35036627

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) | doi:10.1038/35036627

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

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

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

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

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

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

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)

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)

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

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

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