1. Complex Networks Analysis
Information Systems Engineering
‎ (2012 A)
Properties of networks
Slides are taken in part from
Network Science class 2012 (

2. Basic properties 𝑛 – number of nodes 𝑚 – number of links
Trees have 𝑛−1 links
𝑘 →2
𝑛 – number of nodes
𝑚 – number of links
𝑚/𝑛 – density of the network
2𝑚 𝑛 =<𝑘> – average degree
𝑛=12
𝑚=19
2𝑚 𝑛 =3.16
Ben-Gurion University

3. Average degree?

4. Define 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑑𝑒𝑔𝑟𝑒𝑒
𝑑𝑒𝑔𝑟𝑒𝑒 𝑣 ≡𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣 ′ 𝑠 𝑙𝑖𝑛𝑘𝑠
Emanating or incoming links?
What about multigraphs?
Weighted networks? … ?
𝑑𝑒𝑔𝑟𝑒𝑒 𝑣 =𝑑𝑒𝑔𝑟𝑒𝑒(𝑢) 𝑤 𝑣 𝑢
Ben-Gurion University

5. Define 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑑𝑒𝑔𝑟𝑒𝑒
Local 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟ℎ𝑜𝑜𝑑 of a vertex 𝑣
Γ 𝑣 ≡ 𝑢 𝑢,𝑣 ∈𝐸∨ 𝑣,𝑢 ∈𝐸}
Γ in 𝑣 ≡ 𝑢 𝑢,𝑣 ∈𝐸}
Γ out 𝑣 ≡ 𝑢 𝑣,𝑢 ∈𝐸}
𝐷𝑒𝑔𝑟𝑒𝑒(𝑣)= Γ 𝑣
Γ 𝑣 will not include v even if there is a self loop
In rare cases we can allow Γ(𝑣) to be a fussy set and 𝐷𝑒𝑔𝑟𝑒𝑒 𝑣 to be a floating point number
Ben-Gurion University

6. Degree Distribution
𝑁 𝑘 ≡ the number of vertices having 𝑑𝑒𝑔𝑟𝑒𝑒 𝑣 =𝑘 for each 0<𝑘 <𝑣
Probability Distribution
𝑃(𝑑𝑒𝑔𝑟𝑒𝑒(𝑣)=𝑘)= 𝑁 𝑘 𝑛
Cumulative Distribution
𝑃(𝑑𝑒𝑔𝑟𝑒𝑒 𝑣 ≥𝑘)= 𝑖=𝑘 ∞ 𝑃(𝑖)
Every node has some degree
𝑖=0 ∞ 𝑃(𝑖) =1
=𝑃 𝑘 for short
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7. Example
What is the probability that a randomly selected node has degree < 3
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8. Degree Distribution of Random Graphs
Erdos-Renée
All edges have the same probability 𝑝 to exist
Degree distribution
𝑃 deg 𝑣 =𝑘 = 𝑛−1 𝑘 𝑝 𝑘 1−𝑝 𝑛−1−𝑘
𝑃 deg 𝑣 =𝑘 → 𝑛𝑝 𝑘 𝑒 −𝑛𝑝 𝑘!
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9. Probability Density Function (PDF)
WORLD WIDE WEB
Expected
Nodes: WWW documents Links: URL links
Over 3 billion documents
ROBOT: collects all URL’s found in a document and follows them recursively
Continuous:
Probability Density Function (PDF)
P(k) ~ k-
Found
R. Albert, H. Jeong, A-L Barabasi, Nature, (1999).
Network Science: Scale-Free Property 2012

10. Degree distribution of the WWW
Expected
P(k) ~ k-
Found
R. Albert, H. Jeong, A-L Barabasi, Nature, (1999).
Network Science: Scale-Free Property 2012

11. Nodes: WWW documents Links: URL links Exponential Network
What does the difference mean? Visual representation.
Nodes: WWW documents Links: URL links
Expected
Exponential Network
Over 3 billion documents
ROBOT: collects all URL’s found in a document and follows them recursively
P(k) ~ k-
Found
Scale-free Network
R. Albert, H. Jeong, A-L Barabasi, Nature, (1999).
Network Science: Scale-Free Property 2012

12. WORLD WIDE WEB
Network Science: Scale-Free Property 2012

13. The difference between a power law and an exponential distribution20 40 60 80
100
0.2
0.6 1
Above a certain x value, the power law is always higher than the exponential.
There appears to be little difference between them at the first look. Yet. There is some quite relevant different: the at high degrees the power law curve is always higher than the exponential.
Network Science: Scale-Free Property 2012

14. 10 The difference between a power law and an exponential distribution
This difference is particularly obvious if we plot them on a log vertical scale: for large x there are orders of magnitude differences between the two functions. 10 1 2 3 -4 -3 -2 -1
loglog
semilog
Network Science: Scale-Free Property 2012

15. Scale-free networks SCALE-FREE NETWORKS
Many real world networks have a similar architecture:
Scale-free networks
WWW, Internet (routers and domains), electronic circuits, computer software, movie actors, coauthorship networks, sexual web, instant messaging, web, citations, phone calls, metabolic, protein interaction, protein domains, brain function web, linguistic networks, comic book characters, international trade, bank system, encryption trust net, energy landscapes, earthquakes, astrophysical network…
Network Science: Scale-Free Property 2012

16. Universality
How generic is our finding of a power law degree distribution?
There appears to be little difference between them at the first look. Yet. There is some quite relevant different: the at high degrees the power law curve is always higher than the exponential.
Network Science: Scale-Free Property 2012

17. INTERNET BACKBONE Nodes: computers, routers Links: physical lines
(Faloutsos, Faloutsos and Faloutsos, 1999)
Network Science: Scale-Free Property 2012

18. Network Science: Scale-Free Property 2012

19. P(k) ~k- ( = 3) SCIENCE CITATION INDEX Nodes: papers
Links: citations
578... 25
H.E. Stanley,...
1736 PRL papers (1988)
P(k) ~k-
( = 3)
(S. Redner, 1998)
Network Science: Scale-Free Property 2012

20. SCIENCE COAUTHORSHIP Nodes: scientist (authors)
Links: joint publication
M: math
NS: neuroscience
(Newman, 2000, Barabasi et al 2001)
Network Science: Scale-Free Property 2012

21. ONLINE COMMUNITIES Nodes: online user Links: email contact
Pussokram.com online community;
512 days, 25,000 users.
Kiel University log files
112 days, N=59,912 nodes
Ebel, Mielsch, Bornholdtz, PRE 2002.
Holme, Edling, Liljeros, 2002.

22. ONLINE COMMUNITIES Twitter:
Nodes: online user
Links: contact
Twitter:
All distribtions show a fat-tail behavior: there are orders of magnitude spread in the degrees
Alan Mislove, Measurement and Analysis of Online Social Networks
Jake Hoffman, Yahoo,

23. Rual et al. Nature 2005; Stelze et al. Cell 2005
HUMAN INTERACTION NETWORK
2,800 Y2H interactions
4,100 binary LC interactions
(HPRD, MINT, BIND, DIP, MIPS)
Rual et al. Nature 2005; Stelze et al. Cell 2005
Network Science: Scale-Free Property 2012

24. CLEANING UP DEGREE DISTRIBUTIONS
Probability that a node has a degree bigger than x.
If the (noncumulative)
degree distribution decays with a slope >1,
the cumulative degree distribution
will decay with slope -1.
Does not apply for =1!
Probability that node has degree x.
Network Science: Scale-Free Property 2012

25. HUMAN INTERACTION DATA BY RUAL ET AL.
P(k) ~ k-g g  2
(linear scale)
Network Science: Scale-Free Property 2012

26. HUMAN INTERACTION DATA BY RUAL ET AL.
P(k) ~ (k+k0)-g k0  1.4, g  2.6.
(linear scale)
Network Science: Scale-Free Property 2012

27. Common Misconceptions
COMMON MISCONCEPTIONS
low-k saturation
Common Misconceptions
-if there is a low-k saturation, it is not scale-free
-if there is a high-k cutoff, it is not scale-free
Most real networks:
P(k) ~ (k+k0)-γexp(-k/k1)
High-k cutoff
Network Science: Scale-Free Property 2012

28. UNIVERSALITY AGAIN Critical phenomena:
Universality means that the exponents are the same for different systems… they are independent of details.
Networks:
The exponents vary from system to system.
Most are between 2 and 3
Universality:
the emergence of common features across different networks. Like the scale-free property.
Network Science: Scale-Free Property 2012

29. Practical implications (discussion)
Assume you have a giant social network (Facebook)
You need to crawl a small part of it (1%)
AND for every profile crawled you need to know the fraction of friends who have changed the default privacy settings…
What strategy would you choos to crawl the network?

30. Ultra small world behavior
SUMMARY OF THE BEHAVIOR OF SCALE-FREE NETWORKS
γcollab
γmetab
γintern
γsynonyms
γwin
γwout
γactor
γcita
γsex
γ=1
γ=2
γ=3
<k2> diverges
<k2> finite
<k> diverges
<k> finite
Ultra small world behavior
Small world
Regime full of anomalies…
The scale-free behavior is relevant
Behaves like a random network
Network Science: Scale-Free Property 2012

31. Properties of networks (cont.)
Ben-Gurion University

32. What is the difference? 𝑛=7, 𝑚=9, <𝑘>≈2.6
𝑃 𝑘 = 1 8 , 3 8 , 3 8 , 1 8
Contains a densely connected cluster with 4 triangles
𝑛=7,
𝑚=9,
<𝑘>≈2.6
𝑃 𝑘 = 1 8 , 3 8 , 3 8 , 1 8
Homogeneous with 0 triangles

33. Clustering Coefficient
Clusters are tightly connected components of the graph
𝐶𝐶 𝐺 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝐺 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑟𝑖𝑝𝑙𝑒𝑡𝑠 𝑖𝑛 𝐺
Clustering coefficient of a complete graph is 1
Local Clustering Coefficient of a vertex
𝐸 𝑣 ≡{ 𝑖,𝑗 ∈𝐸|i,j∈Γ 𝑣 } links between 𝑣’s neighbors
Undirected: 𝐶𝐶 𝑣 = 2⋅ 𝐸 𝑣 Γ 𝑣 ⋅ Γ 𝑣 −1
Directed: 𝐶𝐶 𝑣 = 𝐸 𝑣 Γ 𝑣 ⋅ Γ 𝑣 −1
Ben-Gurion University

34. CONNECTIVITY OF UNDIRECTED GRAPHS
Connected (undirected) graph: any two vertices can be joined by a path.
A disconnected graph is made up by two or more connected components. B B A
Largest Component:
Giant Component A C D F C D F F
The rest: Isolates F G G
Bridge: if we erase it, the graph becomes disconnected.
The phone is useless as a communication device if con not call any user with a valid number. Similarly, the Internet is useless as a communication system if you cannot send a message to any other user that has access to it. From a network perspective this means that that the technology behind the phone must be able to establish a path between your device and the device you are calling and that there needs to exist a path on the internet between any two computers with Internet access.
This is infact the key utility of most networks: they connect the system's components. This does not require a direct link between any two nodes, but only the existence of a path between them. These paths play an essential role in most network effects and the essence of a network is to guarantee connectedness. In this section we discuss the graph theoretic formulation of the connectedness problem.
Network Science: Graph Theory

35. CONNECTIVITY OF DIRECTED GRAPHS
Strongly connected directed graph: has a path from each node to
every other node and vice versa (e.g. AB path and BA path).
Weakly connected directed graph: it is connected if we disregard the
edge directions.
Strongly connected components can be identified, but not every node is part
of a nontrivial strongly connected component. B E A F B A D E C D C G F G
In-component: nodes that can reach the scc,
Out-component: nodes that can be reached from the scc.
Network Science: Graph Theory

36. It Is a Small World
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37. DISTANCES IN RANDOM GRAPHS
Random graphs tend to have a tree-like topology with almost constant node degrees.
nr. of first neighbors:
nr. of second neighbors:
nr. of neighbours at distance d:
estimate maximum distance:
Network Science: Scale-Free Property 2012

38. SIX DEGREES Milgram’s Six Degrees The first chain letters
The destination: Boston, Massachusetts
Starting Points: Omaha, Nebraska & Wichita, Kansas
SIX DEGREES
Travers and Milgram, Sociometry 32,425 (1969)

39. Percolation
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40. Random Graphs Erdos-Renée
All edges have the same probability 𝑝 to exist
Degree distribution
𝑃 deg 𝑣 =𝑘 = 𝑛−1 𝑘 𝑝 𝑘 1−𝑝 𝑛−1−𝑘
𝑃 deg 𝑣 =𝑘 → 𝑛𝑝 𝑘 𝑒 −𝑛𝑝 𝑘!
𝑛𝑝<1 – no connected components larger than 𝑂 log 𝑛
𝑛𝑝=1 – a largest component whose size is of order 𝑛2/3
𝑛𝑝>1 – a giant connected component, no other components is larger than 𝑂 log 𝑛
𝑝≫ ln 𝑛 𝑛 – connected graph
with high probability
Ben-Gurion University

41. ROBUSTNESS node failure
Could the network structure affect a system’s robustness?
node failure
How do we describe in quantitave terms the breakdown of a network under node or link removal?
~percolation theory~
Network Science: Robustness Cascades March 23, 2011

42. Critical point pc: above pc we have a spanning cluster.
PERCOLATION THEORY
p= the probability that a node is occupied
Increasing p
Critical point pc: above pc we have a spanning cluster.
Network Science: Robustness Cascades March 23, 2011

43. I: Subcritical <k> < 1 II: Critical <k> = 1 III:
Supercritical
<k> > 1
IV:
Connected
<k> > ln N
<k>
N=100
<k>=0.5
<k>=1
<k>=3
<k>=5

44. ROBUSTNESS node failure
Could the network structure contribute to robustness?
node failure
How do we describe in quantitave terms the breakdown of a network under node removal?
~percolation theory~
Network Science: Robustness Cascades March 23, 2011

45. Gian Component Persists
ROBUSTNESS: INVERSE PERCOLATION TRANSITION
Remove nodes
Remove nodes
Unperturbed network
Gian Component Persists
Network Collapses
P∞:probability that a node belongs to the giant component
f: fraction of removed nodes. fc f
Network Science: Robustness Cascades March 23, 2011

46. (Inverse Percolation phase transition)
Damage is modeled as an inverse percolation process
f= fraction of removed nodes S
Component
structure
Graph fc f
(Inverse Percolation phase transition)
Network Science: Robustness Cascades March 23, 2011

47. BOTTOM LINE: ROBUSTNESS OF REGULAR NETWORKS IS WELL UNDERSTOOD
f=0: all nodes are part of the giant component, i.e.
S=N, P∞=1 fc f
0<f<fc: the network is fragmented into many clusters with average size S~|p-pc|-γ
there is a giant component; the probability that a node belongs to it: P∞~(p-pc)β
f>fc: the network collapses, falling into many small clusters;
giant component disappears
Network Science: Robustness Cascades March 23, 2011

48. ROBUSTNESS: OF SCALE-FREE NETWORKS
The interest in the robustness problem has three origins:
Robustness of complex systems is an important problem in many areas
Many real networks are not regular, but have a scale-free topology
In scale-free networks the scenario described above is not valid
Albert, Jeong, Barabási, Nature (2000)
Network Science: Robustness Cascades March 23, 2011

49. Albert, Jeong, Barabási, Nature 406 378 (2000)
ROBUSTNESS OF SCALE-FREE NETWORKS
Scale-free networks do not appear to break apart under random failures.
Reason: the hubs.
The likelihood of removing a hub is small. 1 S f
Albert, Jeong, Barabási, Nature (2000)
Network Science: Robustness Cascades March 23, 2011

50. Achilles’ Heel of scale-free networks
Robust-SF
Achilles’ Heel of scale-free networks 1 S f
Attacks
Failures
  3 : fc=1
(R. Cohen et al PRL, 2000) fc
Albert, Jeong, Barabási, Nature (2000)
Network Science: Robustness Cascades March 23, 2011

51. Course Project Topics
Clustering coefficient via vertex cover / independent set
Net comparison via kroneker graphs
Structural communities vs fictional clusters
GAP compiler with network reduction optimizations
Property preserving network sampling
Centrality computations using GPGPU
Network robustness and damage cost