NONUNIFORM NETWORK MODELS

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1 NONUNIFORM NETWORK MODELS SATU VIRTANEN Formal Methods Forum, March 10th, 2003

2 OUTLINE Network modeling: targets and goals Properties of natural networks Traditional network models Nonuniform network models Properties of the models Finding clusters in graphs Further work

3 NATURAL NETWORK MODELS chemistry, biology: metabolism, food chains sociology, biology: collaborations, epidemics linguistics, cognitive science: semantic relations engineering: communication, traffic

4 UNDIRECTED GRAPHS G = (V, E), V = n, E = m Γ(v) = {u (v, u) E} deg(v) = Γ(v)

5 INTERNET MODELING Waxman 1988: nodes on a grid, connections randomly depending on the Euclidean distance [Wax88] Doar 1996: layered graphs (LAN AS...) [Doa96] Zegura et al. 1997: adjusted connection probabilities [BT02] Medina et al. 2000: BRITE-generator [MMB00] Jin et al. 2000: Inet-generator [JCJ00]

6 WWW MODELING a directed graph research motivation from search engines Broder et al. 2000: bow-tie diagram [BKM + 00] number of pages per website classification of websites growth processes to imitate the birth and interconnection of websites

7 COLLABORATION NETWORKS sforrest mgirvan praghavan jhopcroft cmoore pholme bkim cyoon abroder dsivakumar zneda zdezso

8 ERDŐS-RENYÍ MODEL Random graphs G = (V, E), where V = {v 1,..., v n }. Edges are added with one of the below processes: G n,p : each of the ( n 2) edges is included in E with probability p, considering each pair of vertices {v, w} independently G n,m : a total of m edges are randomly drawn from the set of all possible edges [ER76a, ER76b, Gil59]

9 SMALL WORLDS The characteristic path length L(G) of a graph G = (V, E) is the average length of the shortest path between two vertices in G. The clustering coefficient C(G) of a graph G = (V, E) is the average over C v for v V, where C v is the density of the subgraph induced by Γ(v). The proximity ratio µ of a graph G is µ = C/L C r /L r. [WS98, Wal99]

10 MEASURES FROM NATURAL GRAPHS Network V E L L r C C r µ Ref. C. elegans 202 1, edu 11, [Ada99] IMDb 225,226 6,869, [WS98] Internet AS 12,709 27, [BT02] Power grid 4,941 6, [WS98] Thesaurus 30, [MdMLD02]

11 WATTS-STROGATZ MODEL [WS98]

12 C AND L OF THE WS-MODEL C L Rewiring probability p

13 C AND L OF THE WS-MODEL SWS L ER L SWS C ER C Connection/rewiring probability p

14 KLEINBERG S REWIRED LATTICES [Kle00]

15 PROPERTIES OF THE LATTICE GRAPHS p q

16 CAVEMAN GRAPHS c a b [Wat99]

17 HIERARCHICAL CAVEMAN GRAPHS 47

18 DEGREE DISTRIBUTION A function P (k) that assigns for each k [0, n) the probability that an arbitrary vertex v V has exactly k neighbors, Pr [deg(v) = k ]. frequency degree 7 2 4

19 ERDŐS-RENYÍ MODEL Linear scale Log-log scale

20 WATTS-STROGATZ MODEL Linear scale Log-log scale

21 KLEINBERG LATTICE MODEL Linear scale Log-log scale

22 AN EXAMPLE OF A NATURAL GRAPH Linear scale Log-log scale

23 POWER LAWS A non-negative random variable X obeys a power-law distribution if for constants c, γ > 0. Pr [X x ] c x γ Sometimes also a positive and measurable function f(x) for which t > 0 f(tx) f(x) as x is included as a coefficient of the probability [FFF99]. Several different measures have been observed to obey a power-law distribution: for the Internet, these include vertex degrees, their ranks, the number of vertices within h hops, and even the eigenvalues.

24 MEASURED POWER-LAW EXPONENTS Network V E γ γ in γ out Ref. Citations 783,339 6,716,198 3 [Red98] IMDb 212,250 3,054, ± 0.1 [BAJ99] Internet AS 8,613 18, [BT02] Power grid 4,941 6,596 4 [BA99] Synonyms 182, , [RB03] 325, , ± 0.1 [BA99] 325,729 1,497, [RB03] WWW 200 million 1.5 billion [BKM + 00]

25 A LARGE COLLABORATION GRAPH Logarithm of degree frequency Degree distribution Fitted line Ignoring frequency of one Logarithm of degree k

26 ZIPFIAN DISTRIBUTIONS The log-log plot of the ranks of the values of the random variable versus their frequency follows a straight line [Zip49].

27 RANKS OF THE EXAMPLE GRAPH 4.5 Logarithms of rank frequency Degree distribution Fitted line Logarithm of degree k

28 MATCHING A GIVEN DEGREE DISTRIBUTION Given V = {v 1,..., v n } and degree distribution p. Attach to vertex v i exactly k i stubs, where k i is drawn randomly and independently from p. After assigning all stubs, choose two random stubs and merge them into an edge until no stubs remain. Also graphs that match a given degree sequence d can be fairly easily constructed. [MR95]

29 OBTAINING A random INSTANCE FROM G(d) A Markov chain starting with any realization G G(d): pick two edges (u, v) and (s, t) at random from G (with distinct endpoints) and replace by (u, s) and (v, t) if this produces a connected graph G. In the limit, every possible graph in G(d) is reached with equal probability, independently of the start position G G(d). Stop condition: Compute sorted adjacency lists for all vertices that have a unique degree for G and G, and count the number of positions in which these lists differ. The larger the count, the more different the graphs G and G are expected to be. This measure increases linearly before leveling off. [MGSZ02]

30 PREFERENTIAL ATTACHMENT New vertices connect more eagerly to high-degree vertices: Pr [(v, w) E t ] = deg(w) deg(u) u V [BA99]

31 Number of vertices with degree k BARABÁSI-ALBERT MODEL Degree k

32 BA MODEL WITH CLUSTERING Values of C and L ER L CBA L CBA C n = ER C Clustering probability p

33 A DETERMINISTIC MODEL G 1 = (V 1, E 1 ) consists of two vertices v and w and the edge (v, w). At each discrete time step t 0 of the process, per each (u, v) E t 1, a new vertex w is added together with edges (u, w) and (v, w). [DGM02]

34 THE PSEUDOFRACTAL GRAPH G t FOR t { 1, 0, 1, 2, 3} G 1 G 0 G 3 G 1 G 2

35 PROPERTIES OF THE MODEL: C, L, δ DGM L DGM C DGM d

36 PROPERTIES OF THE MODEL: DEGREE DISTRIBUTION 1e

37 EPIDEMIC SPREADING infection rate µ among the healthy infections cured at rate δ effective spreading rate λ = µδ epidemic threshold λ c ; below this rate the epidemic dies out object of study: how does the epidemic behave in different network topologies

38 ERROR AND ATTACK TOLERANCE complex natural systems tolarate error well why? very important in practice error tolarance: how many random vertices may be removed from the network without significantly affecting interconnectivity attack tolarance: how many especially chosen vertices may be removed... huge differences have been observed related to percolation studies

39 FINDING CLUSTERS IN GRAPHS important in much of data analysis usually done by splitting the entire network in smaller pieces by some rule infeasible for large and partially unknown networks using local search? a good cluster: many connections within, few to the outside of the cluster

40 IDENTIFYING THE CAVES Cluster order f1 visit count f2 visit count

41 References [Ada99] [BA99] [BAJ99] [BKM + 00] Lada Adamic. The small world web. In S. Abiteboul and A.-M. Vercoustre, editors, Proceedings of ECDL 99, volume 1696 of Lecture Notes in Computer Science, pages , Berlin, Germany, Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286: , Albert-László Barabási, Réka Albert, and Hawoong Jeong. Mean-field theory for scale-free random networks. Physica A, 272: , Andrei Broder, S. Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie

42 Stata, Andrew Tomkins, and Janet Wiener. Graph structure in the web. Computer Networks, 33(1 6): , June [BT02] [DGM02] [Doa96] Tian Bu and Don Towsley. On distinguishing between internet power law topology generators. In IEEE Infocom: The Twenty-first Annual Joint Conference of the IEEE Computer and Communications Societies, Los Alamitos, CA, USA, IEEE Computer Society Press. Sergei N. Dorogovtsev, A. V. Goltsev, and José Ferreira F. Mendes. Pseudofractal scale-free web. Physical Review E, 65(6):066122, June Matthew B. Doar. A better model for generating test networks. In GLOBECOM 96: IEEE Global

43 Telecommunications Conference, Piscataway, NJ, USA, IEEE. [ER76a] [ER76b] [FFF99] Pál Erdős and Alfréd Rényi. On random graphs i. In Selected papers of Alfréd Renyí, volume 2, pages Akadémiai Kiadó, Budapest, Hungary, First publication in Publ. Math. Debrecen Pál Erdős and Alfréd Rényi. On the evolution of random graphs. In Selected papers of Alfréd Renyí, volume 2, pages Akadémiai Kiadó, Budapest, Hungary, First publication in MTA Mat. Kut. Int. Közl Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos. On power-law relationships of the Internet topology. In Proceedings of the ACM SIGCOMM 99 Conference on Applications, Technologies,

44 Architectures, and Protocols for Computer Communication, pages , New York, NY, USA, ACM Press. [Gil59] [JCJ00] [Kle00] E. N. Gilbert. Random graphs. Annals of Mathematical Statistics, 30(4): , December Cheng Jin, Qian Chen, and Sugih Jamin. Inet: Internet topology generator. Technical Report CSE-TR443-00, Department of EECS, University of Michigan, Jon M. Kleinberg. The small-world phenomenon: an algorithm perspective. In STOC: Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing, pages , New York, NY, USA, ACM Press. [MdMLD02] Adilson E. Motter, Alessandro P. S. de Moura,

45 Ying-Cheng Lai, and Partha Dasgupta. Topology of the conceptual network of language. Physical Review E, 65(6):065102, June [MGSZ02] [MMB00] [MR95] Milena Mihail, Christos Gkantsidis, Amin Saberi, and Ellen Zegura. On the semantics of Internet topologies. Technical Report GIT-CC-02-07, College of Computing, Georgia Institute of Technology, January Alberto Medina, Ibrahim Matta, and John Byers. On the origin of power laws in Internet topologies. ACM Computer Communication Review, 30(2):18 28, April Mike Molloy and Bruce Reed. A critical point for random graphs with a given degree sequence. Random Structures and Algorithms, 6: , 1995.

46 [RB03] [Red98] [Wal99] [Wat99] [Wax88] Erzsébet Ravasz and Albert-Laszlo Barabási. Hierarchical organization in complex networks. Physical Review E, 67(2):026112, February Sidney Redner. How popular is your paper? an empirical study of the citation distribution. European Physical Journal B, 4(2): , Toby Walsh. Search in a small world. In IJCAI 99: Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, volume 2, pages , San Francisco, CA, USA, Morgan Kaufmann Publishers. Duncan J. Watts. Small Worlds. Princeton University Press, Princeton, NJ, USA, Bernard M. Waxman. Routing of multipoint connections.

47 IEEE Journal on Selected Areas in Communications, 6(9): , [WS98] [Zip49] Duncan J. Watts and Steven H. Strogatz. Collective dynamics of small world networks. Nature, 393: , June George Kingsley Zipf. Human behavior and the principle of least effort. Addison-Wesley Press, Cambridge, MA, USA, 1949.

The Establishment Game. Motivation

The Establishment Game. Motivation Motivation Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Motivation The network models so far neglect