Applications of Outline Applications of Random Networks Random Networks Applications of Random Networks Analysis of real Analysis of real networks networks How to build revisited How to build revisited Complex Networks, Course 295A, Spring, 2008 Motifs Motifs References References Analysis of real networks Prof. Peter Dodds How to build revisited Department of Mathematics & Statistics Motifs University of Vermont References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/17 Frame 2/17 More on building random networks Applications of Building random networks: Stubs Applications of Random Networks Random Networks Phase 1: Analysis of real Analysis of real networks networks ◮ Problem: How much of a real network’s structure is How to build revisited How to build revisited ◮ Idea: start with a soup of unconnected nodes with Motifs Motifs non-random? stubs (half-edges): References References ◮ Key elephant in the room: the degree distribution P k . ◮ First observe departure of P k from a Poisson distribution. ◮ Next: measure the departure of a real network with a degree frequency N k from a random network with the same degree frequency. ◮ Randomly select stubs ◮ Degree frequency N k = observed frequency of (not nodes!) and degrees for a real network. connect them. ◮ What we now need to do: Create an ensemble of ◮ Must have an even random networks with degree frequency N k and then number of stubs. compare. ◮ Initially allow self- and Frame 3/17 Frame 5/17 repeat connections.
Building random networks: First rewiring Applications of General random rewiring algorithm Applications of Random Networks Random Networks i 2 e 1 i Analysis of real Analysis of real 1 networks networks How to build revisited How to build revisited ◮ Randomly choose two edges. ◮ Motifs Motifs Phase 2: (Or choose problem edge and References References ◮ Now find any (A) self-loops and (B) repeat edges and a random edge) randomly rewire them. ◮ Check to make sure edges are disjoint. i 4 e i 3 2 (A) (B) i 2 i ◮ Being careful: we can’t change the degree of any 1 ◮ Rewire one end of each edge. node, so we can’t simply move links around. ◮ Node degrees do not change. ◮ Simplest solution: randomly rewire two edges at a e’ ◮ Works if e 1 is a self-loop or e’ time. 2 1 repeated loop. ◮ Same as finding on/off/on/off i 4 4-cycles. and rotating them. Frame 6/17 Frame 7/17 i 3 Sampling random networks Applications of Random sampling Applications of Random Networks Random Networks Analysis of real Analysis of real networks networks How to build revisited How to build revisited ◮ Problem with only joining up stubs is failure to Motifs Motifs References References Phase 2: randomly sample from all possible networks. ◮ Example from Milo et al. (2003) [1] : ◮ Use rewiring algorithm to remove all self and repeat (a) (b) loops. (c) 1 Phase 3: 0.5 go with the winners % frequency of occurrence ◮ Randomize network wiring by applying rewiring 0 1 algorithm liberally. 0.5 switching algorithm ◮ Rule of thumb: # Rewirings ≃ 10 × # edges [1] . 0 1 0.5 matching algorithm 1 configuration 90 configurations 0 Frame 8/17 Frame 9/17
Sampling random networks Applications of Network motifs Applications of Random Networks Random Networks Analysis of real Analysis of real networks networks How to build revisited How to build revisited ◮ Idea of motifs [2] introduced by Shen-Orr, Alon et al. Motifs Motifs References in 2002. References ◮ What if we have P k instead of N k ? ◮ Looked at gene expression within full context of ◮ Must now create nodes before start of the transcriptional regulation networks. construction algorithm. ◮ Specific example of Escherichia coli. ◮ Generate N nodes by sampling from degree ◮ Directed network with 577 interactions (edges) and distribution P k . 424 operons (nodes). ◮ Easy to do exactly numerically since k is discrete. ◮ Used network randomization to produce ensemble of ◮ Note: not all P k will always give nodes that can be alternate networks with same degree frequency N k . wired together. ◮ Looked for certain subnetworks (motifs) that appeared more or less often than expected Frame 10/17 Frame 12/17 Network motifs Applications of Network motifs Applications of Random Networks Random Networks Analysis of real Analysis of real networks networks single input module (SIM) c How to build revisited How to build revisited a feedforward loop Motifs Motifs a X X References References X X Y Y n Z n Z 1 Z 2 ... Z n b crp d araC argR araBAD argCBH argD argE argF argI ◮ Z only turns on in response to sustained activity in X . ◮ Turning of X rapidly turns of Z . e ◮ Analogy to elevator doors. ◮ Master switch. Frame 13/17 Frame 14/17
Network motifs Applications of Network motifs Applications of Random Networks Random Networks Analysis of real Analysis of real networks networks How to build revisited How to build revisited e dense overlapping regulons (DOR) Motifs Motifs References References X 1 X 2 X 3 ... X n X 1 X 2 X 3 X n ◮ Note: selection of motifs to test is reasonable but nevertheless ad-hoc. Z 1 Z 2 Z 3 Z 4 ... Z m ◮ For more, see work carried out by Wiggins et al. at Columbia. f nhaR oxyR rpoS rcsA ada hns crp lrp ihf fis alkA katG dps osmC ftsQAZ nhaA proP Frame 15/17 Frame 16/17 References I Applications of Random Networks Analysis of real networks How to build revisited Motifs References R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, and U. Alon. On the uniform generation of random graphs with prescribed degree sequences, 2003. pdf ( ⊞ ) S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon. Network motifs in the transcriptional regulation network of Escherichia coli . Nature Genetics , pages 64–68, 2002. pdf ( ⊞ ) Frame 17/17
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