Complex Networks

1. Complex Networks Structure and Dynamics Ying-Cheng Lai Department of Mathematics and Statistics Department of Electrical Engineering Arizona State University

2. 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

3. 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.

4. 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”

5. 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  lnN. Minimal number of links one needs to follow to go from one node to another, on average.

6. 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 regular sw random Clusteringcoefficient C is the probability that two nodes are connected to each other, given that they are both connected to a common node. C L p

7. 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.

8. 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?

9. Questions • Structure Which are the generic structural properties of real-world networks? • Dynamics ofthe network What sort of dynamical processes govern the emergence of these properties? • Dynamics on the network How does individual behavior aggregate to collective behavior?

10. 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.

11. Word association is a small-world network *Source: online Gutenberg Thesaurus dictionary “Three degrees of separation for English words” Featured in Nature Science Update, New Scientist, Wissenschaft-online, etc.

12. Word association as a growing network • Preferentialandrandom 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

13. j R(Lij)=3 i Small-world phenomenon in scale-free networksMotter, Nishikawa, & Lai (2002) • The rangeR(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?

14. 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)]

15. normalized efficiency fraction of removed links N=5000 Range-based attack on scale-free networks • Results for semirandom scale-free networks: P(k)  k- • The connectivity distribution is more heterogeneous for smaller . Newman, Strogatz, & Watts (2001)

16. Heterogeneity versus homogeneity • 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.

17. normalized efficiency fraction of removed links <k>=6, N=5000 Other scale-free models • Results for growing networks with aging:  i  i- ki Short average path length in scale-free networks • is mainly due to short-range links. Dorogovtsev & Mendes (2000)

18. 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 giantconnected componentdoes 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.

19. Simple model for cascading failure • 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. load on a node = number of shortest paths passing through that node

20. G: relative number of nodes in the largest connected component – random (squares) – connectivity (stars) – load (circles) ( =3, N5000) Simulations

21. ( =3, N5000) (Western U.S. power grid, N=4941) Simulations

22. ( =3, N5000) (Western U.S. power grid, N=4941) Simulations Networks with heterogeneous distribution of load: “robust-yet-fragile” Featured in Newsletter (Editorial), Equality: Better for network security; NewsFactor, Cascading failures could crash the global Internet; The Guardian, Electronic Pearl Harbor; etc.

23. 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.

24. 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.

25. 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 individualsin 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.

26. 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. : correlation between hierarchies : correlation between distribution of social tiesandsocialdistance N=106, n=250, H=2, g=100, b=10, = Scaling with system size: P  N-1

27. 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!”

28. Conclusions • 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.

29. Conclusions • 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. Recent developments in complex networks offer a framework to approach new and old problems in various disciplines.