Sample solution to Homework 2

Aufgabe 2:A degree of funIn G, calculate exactly:1) the expected degree of a UIS node (i.e., the average node degree),2) the expected average degree of neighbors of a UIS node,3) the expected degree of a WIS node,4) the expected degree of end-nodes of a UIS edge.Verify all of the above by sampling. Finally, implement RW and report the observed average node degree.Summarize the exact and sampled expected degrees in a table. Show how the variance of each sampleevolves with sample size, using, for example, box plots.Solution:N = number of nodes, E = number of edges, d i = degree of node iUISnUISWISeUISFormula Form. result Sample result∑E[D UIS ] = N 1N · d i 4.726 4.734i=1 ( )∑E[D nUIS ] = N d i1∑1N d j· d i 13.210 13.198i=1E[D WIS ] = N ∑i=1E[D eUIS ] = N ∑i=1j=1d PN i· dj=1 d i 11.604 11.596jd i2E · d i 11.604 11.650RW – – 11.6051412degree statistics1086UISnUISWISeUISRW420 1000 2000 3000 4000 5000 6000 7000sample size3

Aufgabe 3:BFSPlot the mean degree among nodes sampled by: (i) BFS, and (ii) DFS, as a function of the fraction ofsampled nodes. Average the plot over 100 different seed nodes, selected by UIS.What can you say about the BFS and DFS exploration processes from these plots?Solution:20BFS and DFS traversalsBFSDFS15average degree105010000 20000 30000 40000 50000# of samples per sourceBoth BFS and DFS first explore high degree nodes and then low degree nodes. This does not seem todepend on the starting node at all.Approaching the real average degree requires the exploration of almost the entire network.4

Aufgabe 4:Graph sizeRecall the graph size estimation formulas in slides 23–24 of the second lecture in Chapter 2 (Nov 12).Apply the corresponding formula to nodes collected by:1) UIS WOR,2) UIS WR,3) WIS WR,4) RW (using all nodes, or “SimpleThinning” as in slide 32 of the above-mentioned lecture).For RW, plot the estimated graph size versus the thinning factor k.What do you observe?Solution:n = number of samples, c = number of collisionsMethod Formula Size estimateReal – N = 62 ′ 586UIS WOR – ˆN = ∞UIS WR ˆN =n 2WIS WRRW WR2cˆN =Ψ 1 Ψ −12cˆN =Ψ 1 Ψ −12cˆN = 64 ′ 002ˆN = 62 ′ 306ˆN = 24 ′ 963180000Network size from thinned RW160000estimated network size1400001200001000008000060000400002000000 10 20 30 40 50thinning factor5

Importing the graph✞1 def l o a d _ g r a p h ( fname ) :2 f r = open ( fname , ’r’ )3 G = {} # d i c t i o n a r y : node −> s e t of n e i g h b o r s4 f o r l i n e in f r :5 i f not l i n e . s t a r t s w i t h ( ’#’ ) :6 a , b = map ( i n t , l i n e . s p l i t ( ) )7 i f a not in G: G[ a ] = s e t ( )8 i f b not in G: G[ b ] = s e t ( )9 G[ a ] . add ( b )10 G[ b ] . add ( a )11 f r . c l o s e ( )12 return G1314 G = l o a d _ g r a p h ( "p2p-Gnutella31.txt" )15 p r i n t G[ 4 2 ] # Neighbors of node 42✡✝✆Alternatively, you can use the networkx Python module.✞1 import networkx as nx23 G = nx . r e a d _ e d g e l i s t ( "p2p-Gnutella31.txt" , d e l i m i t e r =’\t’ , n o d e t y p e = i n t )✡✝✆6