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Complex Networks Structure and Dynamics Ying-Cheng Lai Department of Mathematics and Statistics...
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![Page 1: Complex Networks Structure and Dynamics Ying-Cheng Lai Department of Mathematics and Statistics Department of Electrical Engineering Arizona State University.](https://reader030.fdocuments.us/reader030/viewer/2022032521/56649d5c5503460f94a3afc6/html5/thumbnails/1.jpg)
Complex NetworksStructure and Dynamics
Ying-Cheng LaiDepartment of Mathematics and Statistics
Department of Electrical EngineeringArizona State University
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Collaborators Adilson E. Motter, now at Max-Planck
Institute for Physics of Complex Systems, Dresden, Germany
Takashi Nishikawa, now at Department of Mathematics, Southern Methodist University
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Complex Networks
Structures composed of a large number of elements linked together in an apparently fairly sophisticated fashion.
Examples:- Social networks- Internet and WWW (world-wide web)- Power grids- Brain and other neural networks
- Metabolic networks
Characteristics: - Large, sparse, and continuously evolving.
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Social networks
Contacts and Influences – Poll & Kochen (1958)
– How great is the chance that two people chosen at random from the population will have a friend in common?
– How far are people aware of the available lines of contact?
The Small-World Problem – Milgram (1967)
– How many intermediaries are needed to move a letter from person A to person B through a chain of acquaintances?– Letter-sending experiment: starting in Nebraska/Kansas,
with a target person in Boston.
“Six degrees of separation”
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Random graphs – Erdos & Renyi (1960)
Start with N nodes and for each pair of nodes, with probability p, add a link between them.
For large N, there is a giant connected component if the average connectivity (number of links per node) is larger than 1.
The average path length L in the giant component scales as L ln
N.
Minimal number of links one needs to follow to go from one node to another, on average.
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Small-world networks – Watts & Strogatz (1998)
Start with a regular lattice and for each link, with probability p, rewire one extreme of the link at random.
fraction p of the links is converted into shortcuts
LC
Clustering coefficient C is the probability that two nodes are connected to each other,
given that they are both connected to a common node.
p
regular sw random
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Scale-free networks – Barabasi & Albert (1999)
Growth: Start with few nodes and, at each time step, a new node with m links is added.
Preferential attachment: Each link connects with a node in the network according to a probability i proportional to the connectivity ki of the node: i ki .
The result is a network with an algebraic (scale-free) connectivity distribution: P(k) k -, where =3.
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Questions
Which are the generic structural properties of real-world networks?
What sort of dynamical processes govern the emergence of these properties?
How does individual behavior aggregate to collective behavior?
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Questions
Structure
Which are the generic structural properties of real-world networks?
Dynamics of the network
What sort of dynamical processes govern the emergence of these properties?
Dynamics on the network
How does individual behavior aggregate to collective behavior?
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Network of word associationMotter, de Moura, Lai, & Dasgupta (2002)
Words correspond to nodes of the network; a link exists between two words if they express similar concepts.
• Motivation: structure and evolution of language, cognitive science.
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Word association is a small-world network
N <k> C L
Actual configuration 30244* 59.9 0.53 3.16
Random configuration 30244 59.9 0.002 2.5
*Source: online Gutenberg Thesaurus dictionary
Featured in Nature Science Update, New Scientist, Wissenschaft-online, etc.
“Three degrees of separation for English words”
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Word association as a growing network
Preferential and random attachments [Liu et al (2002)]:
i (1- p) k i + p, 0 p 1 Scaling for the connectivity distribution:
P(k) [k + p/(1- p) ] - , = 3 + m-1 p/(1- p)
P(k): exponential for small k, algebraic for large k
= 3.5
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Small-world phenomenon in scale-free networksMotter, Nishikawa, & Lai (2002)
The range R(Lij) of a link Lij connecting nodes i. and j is the length of the shortest path between i. and j in the absence of Lij.
Watts-Strogatz model: short average path length is due to long-range links
(shortcuts).
Scale-free networks also present very short L.
Are long-range links responsible for the short average path length of scale-free networks?
j
iR(Lij) = 3
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Range-based attack
Short-range attack: links with shorter range are removed first.
Long-range attack: links with longer range are removed first.
Average of the inverse path length
Efficiency [Latora & Marchiori (2001)]
ijLNNE
1
)1(
2
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Range-based attack on scale-free networks
Results for semirandom scale-free networks: P(k) k -
The connectivity distribution is more heterogeneous for smaller .
fraction of removed links
norm
aliz
ed
effi
cien
cy
N=5000
Newman, Strogatz, & Watts (2001)
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Load of a link Lij is the number of shortest paths passing through Lij.
Links between highly connected nodes are more likely to have high load and small range.
Heterogeneity versus homogeneity
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Results for growing networks with aging: i i- ki
Short average path length in scale-free networks
is mainly due to short-range links.
Other scale-free models
fraction of removed links
norm
aliz
ed
effi
cien
cy
<k>=6, N=5000
Dorogovtsev & Mendes (2000)
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Cascade-based attacks on complex networksMotter & Lai (2002)
Statically: – L increases significantly in scale-free networks when highly
connected nodes are removed [Albert et al (2000)]; – the existence of a giant connected component does not
depend on the presence of these nodes [Broder et al (2000)]. Dynamically, if 1. the flow of a physical quantity, as characterized by load on
nodes, is important, and 2. the load can redistribute among other nodes when a node is
removed, intentional attacks may trigger a global cascade of overload
failures in heterogeneous networks.
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Flow: at each time step, one unit of the relevant quantity is exchanged between every pair of nodes along the shortest path.
Capacity is proportional to the initial load:
Cj = (1 + ) lj (0), ( j=1,2, … N, 0).
Cascade: a node fails whenever the updated load exceeds the capacity, i.e., node j is removed at step n if lj (n) > Cj.
Simple model for cascading failure
load on a node = number of shortest paths passing through that node
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Simulations
( =3, N 5000)
– random (squares) – connectivity (stars) – load (circles)
G: relative number of nodes in the largest connected component
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Simulations
( =3, N 5000) (Western U.S. power grid, N=4941)
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Simulations
Featured in Newsletter (Editorial), Equality: Better for network security;
NewsFactor, Cascading failures could crash the global Internet;
The Guardian, Electronic Pearl Harbor; etc.
Networks with heterogeneous distribution of load: “robust-yet-fragile”
( =3, N 5000) (Western U.S. power grid, N=4941)
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Revisiting the original small-world problemMotter, Nishikawa, & Lai (2003)
After talking to a strange for a few minutes, you and the stranger often realize that you are linked through a mutual friend or through a short chain of acquaintances.
discovery of short paths existence of short paths
We want to model this phenomenon and find a criterion for plausible models of social networks.
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Model for the identification of mutual acquaintances
People are naturally inclined to look for social connections that can identify them with a newly introduced person.
We assume that a person knows another person when this person knows the social coordinates of the other.
We also assume that when two people are introduced:
1. they exchange information defining their own social coordinates;
2. they exchange information defining the social coordinates of acquaintances that are socially close to the other
person.
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Network model
Hierarchy of social structure: individuals are organized into groups, which in turn belong to groups of groups
and so on [Watts, Dodds, & Newman (2002)].
The distance along the tree structure defines a social distance between individuals in a hierarchy.
The society is organized into different but correlated hierarchies.
The network is built by connecting with higher probability pairs of closer individuals.
Social coordinates set of positions a person occupies in the hierarchies.
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Trade-off between short paths and high correlations
Probability of discovering mutual acquaintances, acquaintances in the same social group, and acquaintances who know each other, after citing m=1, 2, and 20 acquaintances.
Scaling with system size: P N -1
N=106, n=250, H=2, g=100, b=10, =
: correlation between hierarchies : correlation between distribution of social ties and social distance
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Discovery versus existence
The probability of finding a short chain of acquaintances between two people does not scale with typical distances in the underlying network of social ties.
Random networks are usually “smaller” than small-world networks, and because of that they are sometimes called themselves small-world networks.
But a random society would not allow people to find easily that “It is a small world!”
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Word association is a small-world network, with a crossover from exponential to algebraic distribution of connectivity.
The short average path length observed in scale-free network is mainly due to short-range links.
Networks with skewed distribution of load may undergo cascades of overload failures.
The “small-world phenomenon” results from a trade-off between short paths and high correlations in the network of social ties.
Conclusions
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Word association is a small-world network, with a crossover from exponential to algebraic distribution of connectivity.
The short average path length observed in scale-free network is mainly due to short-range links.
Networks with skewed distribution of load may undergo cascades of overload failures.
The “small-world phenomenon” results from a trade-off between short paths and high correlations in the network of social ties.
Conclusions
Recent developments in complex networks offer a framework to approach new and old problems in various disciplines.