Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Measures and Metrics, Networks saverio . giallorenzo @gmail.com � 1
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo The Small-world Effect i i A renowned (and measurable) network phenomenon is the small-world e ff ect. � N 1 Informally, we have a small-world e ff ect when we can j find shorter-than-expected distances between pairs of nodes. The typical example to illustrate a small-world e ff ect is i i Milgram’s experiment, where people were asked to get a letter from an initial holder to a distant target person by passing it from acquaintance to acquaintance S h o r t through their social network. The letters that made it to c u t the target did so in a remarkably small number of � N 2 steps. j j saverio . giallorenzo @gmail.com � 2
� Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo The Small-world Effect i i Mathematically, let � be the length of the shortest path d ij through a network between nodes � and � ; then, the i j mean distance � for a node � corresponds to � N 1 ℓ i i ∑ j d ij j and the mean distance for the whole ℓ i = n ∑ ij d ij ∑ i ℓ i network corresponds to � (for i i ℓ = = n 2 n single-component networks). S Simplistically—as we will see more accurate measures h o r t c u t using random graphs —a family of networks shows small- � N 2 world e ff ects when � (i.e., when � is directly ℓ ∝ log n ℓ proportional to � by a constant � ). j j log n k saverio . giallorenzo @gmail.com � 3
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo The Small-world Effect i i Properties of small-world networks include: - many highly-clustered groups (e.g., cliques) where all � N 1 nodes are densely connected j - many hubs that serve as “mediators” to shorten the lengths between other edges i i - according to Barabási’s hypothesis, these networks are particularly robust to random perturbations (e.g., deletion of a random node rarely causes a sensible S h change of � ) —thanks to the low hub-to-leaf ratio. Vice o ℓ r t c u t versa, rare/selective deletions of hubs dramatically � N 2 increase � ℓ j j saverio . giallorenzo @gmail.com � 4
� saverio . giallorenzo @gmail.com Web Science • Measures and Metrics, Networks p 0 = 1/10 nodes that have degree � . E.g., in the network on the right Consider an undirected network and let � have that degree. That ratio is essentially the probability of a given node to we have: of edges attached to that node. Reminder: the degree of a node corresponds to the number Degree Distribution p 1 = 2/10 p 2 = 4/10 d p 3 = 2/10 p d p 4 = 1/10 be the fraction of p 5+ = 0/10 MA Digital Humanities and Digital Knowledge, UniBo � 5
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks Let us take the degrees of (a portion) of the Internet and plot the degree distribution —bottom-left. The figure shows that most of the nodes in the network have a low degree. However, there exists d 0.4 of nodes with degree � a significant “tail” of nodes with substantially higher degree (indeed it reaches a degree of 2000+). 0.2 p d Fraction � 0 0 5 10 15 20 Degree � d saverio . giallorenzo @gmail.com � 6
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks Let us take the degrees of (a portion) of the Internet and plot the degree distribution —bottom-right. The figure shows that most of the nodes in the network have a low degree. However, there exists d 0.4 of nodes with degree � a significant “tail” of nodes with substantially higher degree (indeed it reaches a degree of 2000+). 0.2 p d Fraction � 0 0 5 10 15 20 Degree � d saverio . giallorenzo @gmail.com � 7
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks More specifically, when 0 10 d plotted in a log-log scale, of nodes with degree � d 0.4 of nodes with degree � power-law distributions -2 tend to follow a straight-line 10 behaviour -4 10 g n i t t o l p g o l - 0.2 g o L -6 10 p d Fraction � p d Fraction � -8 10 1 10 100 1000 Degree � d 0 0 5 10 15 20 Degree � d saverio . giallorenzo @gmail.com � 8
� Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks Distributions of this kind are described by the formula � ln p d = − α ln d + c where � and � are constants that respectively modify the slope and α c normalise the curve of the distribution . 0 10 d Taking the exponential of both sides of the formula, we have of nodes with degree � p d = Cd − α C = e c (with � ). -2 10 Since the distribution is dependent on a power (with -4 exponent � ) of the degree � , it is called a α d 10 “ power law ” distribution. -6 10 p d Fraction � -8 10 1 10 100 1000 Degree � d saverio . giallorenzo @gmail.com � 9
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks Detecting power-laws by just visualising the distribution (particularly in log-log form) cannot be trusted. Indeed, in our example we see a “deceiving” non-monotonically decreasing (direct scale) and non-straight (log-log scale) distribution curve. 0 10 d 0.4 of nodes with degree � d of nodes with degree � -2 10 -4 10 0.2 p d p d Fraction � -6 Fraction � 10 -8 0 10 0 5 10 15 20 Degree � d Degree � d 1 10 100 1000 saverio . giallorenzo @gmail.com � 10
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks To detect power-law 1 behaviours, we can use the of nodes with degree � or greater cumulative distribution 0.1 function , which is defined by d ∞ ∑ the formula � , so P d = p d ′ � 0.01 d ′ � = d is the fraction of nodes that � 0.001 P d p d that have degree � or greater. d Fraction � 0.0001 1 10 100 1000 Degree � d saverio . giallorenzo @gmail.com � 11
� Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks Also here we are looking for a straight-line of nodes with degree � or greater 1 behaviour. However, while the curve lends itself to less-statistically-biased visual interpretations, 0.1 d we can get a precise measure of how close our distribution approximates a power-law by 0.01 calculating the value of � . α p d = Cd − α Indeed, if � then 0.001 p d ∞ ∞ d ′ � − α ≃ C ∫ C Fraction � d ′ � − α ∂ d ′ � = ∑ 0.0001 α − 1 d − ( α − 1) P d = C 1 10 100 1000 { d d ′ � = d Degree � d Assuming � α > 1 so that � becomes the exponent determining the distribution (on � ) as α = 1 + n ( ∑ α d − 1 d min − 1/2 ) d i ln Empirically, in power-law distributions � . 2 ≥ α ≥ 3 i saverio . giallorenzo @gmail.com � 12
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Power Laws and Scale-free Networks of nodes with degree � or greater 1 Networks whose degree distribution follows a power-law behaviour are usually called 0.1 scale-free networks . d The reason for the name comes from the 0.01 fact that power laws are scale-invariant, i.e., that scaling the argument, here � , by a d 0.001 constant factor just causes a multiplication of the original power-law relation by that p d Fraction � constant. 0.0001 1 10 100 1000 This is also why we look for straight-line Degree � d behaviours in log-log plots, which reduce the “noise” derived from constant multiplications. saverio . giallorenzo @gmail.com � 13
Web Science • Measures and Metrics, Networks MA Digital Humanities and Digital Knowledge, UniBo Properties of Scale-free Networks Scale-free networks are highly robust networks that can survive the failure of a sensible number of their nodes. d 0.4 of nodes with degree � E.g., if we removed nodes randomly from the Internet, the network would retain its characterising behaviours. If central hubs were to be removed (by choice or luck), we should repeat that operation many times to significantly change the behaviours (e.g., disrupt the connectivity) of the network. 0.2 p d Fraction � 0 0 5 10 15 20 Degree � d saverio . giallorenzo @gmail.com � 14
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