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Complex Networks Structure and Dynamics Ying-Cheng Lai Department of Mathematics and Statistics Department of Electrical Engineering Arizona 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 L C 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 k i of the node: i k i. 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 association Motter, 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 CL Actual configuration30244 * 59.90.533.16 Random configuration3024459.90.0022.5 * 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 networks Motter, Nishikawa, & Lai (2002) The range R( L ij ) of a link L ij connecting nodes i. and j is the length of the shortest path between i. and j in the absence of L ij. 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 i R( L ij ) = 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)]

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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 normalized efficiency N=5000 Newman, Strogatz, & Watts (2001)

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Load of a link L ij is the number of shortest paths passing through L ij. 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 - k i Short average path length in scale-free networks is mainly due to short-range links. Other scale-free models fraction of removed linksnormalized efficiency =6, N=5000 Dorogovtsev & Mendes (2000)

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Cascade-based attacks on complex networks Motter & 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: C j = (1 + ) l j (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 l j (n) > C j. 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 problem Motter, 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=10 6, 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.

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