Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)

Adjacency lists

1. Birds eat the bread crumbs 2. They don’t random walk DFS/BFS Hansel & Gretel

Diffusion equation

Normal distribution Random walk

With bread crumbs one can find exit in time proportional to V+E DFS/BFS Hansel & Gretel

Breadth First Search

F A BCG DE H

F A BCG DE H Queue: A get 0 distance from A visit(A)

Breadth First Search F A BCG DE H Queue: 0 F 1 F discovered

Breadth First Search F A BCG DE H Queue: F 0 1 B 1 B discovered

Breadth First Search F A BCG DE H Queue: F B C 1 C discovered

Breadth First Search F A BCG DE H Queue: F B C G 1 G discovered

Breadth First Search F A BCG DE H Queue: F B C G get A finished

Breadth First Search F A BCG DE H Queue: B C G A already visited

Breadth First Search F A BCG DE H Queue: B C G D 2 D discovered

Breadth First Search F A BCG DE H Queue: B C G D E 2 E discovered

Breadth First Search F A BCG DE H Queue: B C G D E get F finished

Breadth First Search F A BCG DE H Queue: C G D E

Breadth First Search F A BCG DE H Queue: C G D E A already visited

Breadth First Search F A BCG DE H Queue: C G D E get B finished

Breadth First Search F A BCG DE H Queue: G D E A already visited

Breadth First Search F A BCG DE H Queue: G D E get C finished

Breadth First Search F A BCG DE H Queue: D E A already visited

Breadth First Search F A BCG DE H Queue: D E E already visited

Breadth First Search F A BCG DE H Queue: D E get G finished

Breadth First Search F A BCG DE H Queue: E E already visited

Breadth First Search F A BCG DE H Queue: E F already visited

Breadth First Search F A BCG DE H Queue: E get D finished

Breadth First Search F A BCG DE H Queue: D already visited

Breadth First Search F A BCG DE H Queue: F already visited

Breadth First Search F A BCG DE H Queue: G already visited

Breadth First Search F A BCG DE H Queue: H 3 H discovered

Breadth First Search F A BCG DE Queue: H get H 3 E finished

Breadth First Search F A BCG DE H Queue: E already visited

Breadth First Search F A BCG DE H Queue: STOP H finished

Breadth First Search F A BCG DE H distance from A

Breadth-First Search

b c a d a c d b v

Rod Steiger Martin Sheen Donald Pleasence #1 #2 #3 #876 Kevin Bacon Barabasi

Why Kevin Bacon? Measure the average distance between Kevin Bacon and all other actors. 876 Kevin Bacon Barabasi

Langston et al., A combinatorial approach to the analysis of differential gene expression data…. Minimum Dominating Set

size of dominating set

Expected size of dominating set Assume each node has at least d neighbors Naïve algorithm still n/2 in worst case Simple probabilistic algorithm:

1. For each vertex v, color v red with probability p

2. Color blue any non-dominated vertex

X= number of red nodes Y= number of blue nodes Size of dominating set = X+Y

Expected size of dominating set S =

Markov’s inequality proof j= k E|S|

Probability that is < 1/2 Run algorithm 10 times and keep smallest S with probability > 0.999

protein- protein interactions PROTEOME GENOME Citrate Cycle METABOLISM Bio- chemical reactions Barabasi

Tucker-Gera-Uetz

Local network motifs SIMMIMFFLFBL [Alon; Horak, Luscombe et al (2002), Genes & Dev, 16: 3017 ]

Barabasi

The New Science of Networks by Barabasi

Degree Distribution P(k) = probability a given node has exactly k neighbors P(k) = probability a given node has exactly k neighbors Random Network Random Network P(k) = Poisson ~ P(k) = Poisson ~ No hubs No hubs Scale free Network Scale free Network P(k) ~. P(k) ~. A few hubs A few hubs

Metabolic network Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, (2000) ArchaeaBacteriaEukaryotes Meta-P(k)

Barabasi & Albert, Science 286, 509 (1999) Actors Movies Web-pages Hyper-links Trans. stations Power lines Nodes: Links: Scale-free networks

Why scale-free topology in biological networks ?

Preferential attachment

Mean Field Theory γ = 3, with initial condition A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999) MFT

Clustering in protein interaction networks Goldberg and Roth, PNAS, 2003 high clustering = high quality of interaction

Scale-free model (1) GROWTH : A t every timestep we add a new node with m edges (connected to the nodes already present in the system). (2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity k i of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) P(k) ~k -3

Why scale-free topology in biological networks ?

Yeast protein network Nodes: proteins Links: physical interactions (binding) P. Uetz, et al. Nature, 2000; Ito et al., PNAS, 2001; …