PoCS | @pocsvox PoCS | @pocsvox Properties of Outline Properties of Complex Complex Properties of Complex Networks Networks Networks Principles of Complex Systems | @pocsvox Properties of Complex Networks CSYS/MATH 300, Fall, 2017 Properties of Properties of Complex A problem Complex Networks Networks A problem distributions A problem Degree distributions Degree distributions Assortativity Assortativity Prof. Peter Dodds | @peterdodds Clustering Clustering Motifs Clustering Motifs Concurrency Concurrency Dept. of Mathematics & Statistics | Vermont Complex Systems Center Branching ratios Motifs Branching ratios Network distances Network distances Vermont Advanced Computing Core | University of Vermont Interconnectedness Concurrency Interconnectedness Nutshell Branching ratios Nutshell References Network distances References Interconnectedness PoCS Principles of Complex Systems @pocsvox What's the Story? PoCS Nutshell PoCS Principles of Principles of Complex Systems Complex Systems @pocsvox @pocsvox What's the Story? What's the Story?

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. References

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PoCS | @pocsvox PoCS | @pocsvox These slides are brought to you by: Properties of A notable feature of large-scale networks: Properties of Complex Complex Networks  Graphical renderings are often just a big mess. Networks

Properties of Properties of Complex Complex Networks Networks A problem A problem Degree distributions Typical hairball Degree distributions Assortativity Assortativity Clustering  number of nodes = 500 Clustering Motifs Motifs Concurrency ⇐ Concurrency Branching ratios  number of edges = 1000 Branching ratios Network distances � Network distances Interconnectedness  average degree = 4 Interconnectedness Nutshell Nutshell References References ⟨⟩  And even when renderings somehow look good: “That is a very graphic analogy which aids PoCS understanding wonderfully while being, strictly Principles of Complex Systems @pocsvox What's the Story? speaking, wrong in every possible way” said Ponder [Stibbons] —Making Money, T. Pratchett.  We need to extract digestible, meaningful aspects...... 2 of 40 . . . 7 of 40

PoCS | @pocsvox PoCS | @pocsvox These slides are also brought to you by: Properties of Properties of Complex Complex Networks Some key aspects of real complex networks: Networks Special Guest Executive Producer: Pratchett Properties of  concurrency Properties of Complex  Complex Networks  hierarchical scaling Networks A problem  A problem Degree distributions assortativity ∗ Degree distributions Assortativity  network distances Assortativity Clustering  Clustering Motifs  Motifs Concurrency  clustering Concurrency Branching ratios Branching ratios Network distances  efficiency Network distances Interconnectedness  motifs Interconnectedness  interconnectedness Nutshell  Nutshell References  robustness References

 Plus coevolution of network structure PoCS and processes on networks. Principles of Complex Systems @pocsvox What's the Story? Degree distribution is the elephant in the room that we are now all very aware of...  On Instagram at pratchett_the_cat  ∗ ...... 3 of 40 . . . 8 of 40 PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Local socialness: Properties of Complex Complex Networks Networks 1. degree distribution 4. Clustering: Properties of Properties of  is the probability that a randomly selected Complex Complex Networks  Your friends tend to know Networks node has degree . � A problem A problem Degree distributions each other. Degree distributions  � = node degree = number of connections. Assortativity Assortativity Clustering  Two measures (explained Clustering Motifs Motifs  ex 1: Erdős-Rényi random networks have Poisson Concurrency on following slides): Concurrency Branching ratios [8] Branching ratios degree distributions: Network distances 1. Watts & Strogatz Network distances Insert question from assignment 7  Interconnectedness Interconnectedness Nutshell Nutshell

References References 1 2 12 ∑ ∈� � 1 � = ⟨ ⟩ −⟨⟩ [6] ⟨⟩ 2. Newman ( − 1)/2  ex 2: “Scale-free” networks:� = � ‘hubs’. ! #triangles  link cost controls skew. −� #triples  hubs may facilitate or impede� contagion.∝ ⇒ 2 3 × � = ...... 11 of 40 . . . 18 of 40

PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Properties of Complex  is the average fraction of Complex Networks pairs of neighbors who are Networks Example network: connected1 . Properties of � Properties of Complex  Fraction of pairs of Complex Networks Networks A problem neighbors who are A problem Note: Degree distributions Degree distributions Assortativity connected is Assortativity  Erdős-Rényi random networks are a mathematical Clustering Clustering Motifs Motifs construct. Concurrency Concurrency Branching ratios Branching ratios Network distances Network distances  ‘Scale-free’ networks are growing networks that Interconnectedness 1 2 Interconnectedness Calculation of : 12∈� form according to a plausible mechanism. Nutshell ∑ � Nutshell References where is node ’s degree, References  Randomness is out there, just not to the degree of 1 ( − 1)/2 � and is the set of ’s a completely random network. neighbors.  Averaging� over all nodes, we have:

∑12∈� � 1 2 1 � 1 � . . . =1 ( −1)/2 . . . � = ∈�∑ 12 = . . . 12 of 40 ∑ 1 2 � . . . 19 of 40 (−1)/2 ⟨ ⟩

PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Triples and triangles Properties of Complex Complex Networks Networks Example network:  Nodes , , and form a Properties of Properties of 2. Assortativity/3. Homophily: Complex triple around if is Complex Networks Networks 1 2 3  Social networks: Homophily  = birds of a feather A problem connected to and . A problem Degree distributions 1 1 Degree distributions  Assortativity  Assortativity e.g., degree is standard property for sorting: Clustering Nodes , , and form a Clustering Motifs 2 3 Motifs measure degree-degree correlations. Concurrency triangle if each pair of nodes is Concurrency [5] Branching ratios connected1 2 3 Branching ratios  Assortative network: similar degree nodes Network distances Triangles: Network distances Interconnectedness Interconnectedness connecting to each other.  The definition #triangles Nutshell #triples Nutshell

Often social: company directors, coauthors, actors. References measures the fraction3× of References 2  Disassortative network: high degree nodes closed triples � = connecting to low degree nodes.  The ‘3’ appears because for Often techological or biological: Internet, WWW, Triples: each triangle, we have 3 closed protein interactions, neural networks, food webs. triples.  Analysis (SNA): fraction of transitive triples...... 16 of 40 . . . 20 of 40 PoCS | @pocsvox PoCS | @pocsvox Clustering: Properties of Properties Properties of Complex Complex Sneaky counting for undirected, unweighted Networks Networks networks: 6. modularity and structure/community Properties of detection: Properties of  If the path – – exists then . Complex Complex Networks Networks A problem A problem  Otherwise, . Degree distributions Degree distributions ℓ ℓ � � = 1 Assortativity Assortativity  We want for good triples. Clustering Clustering ℓ Motifs Motifs � � = 0 Concurrency Concurrency  In general, a path of edges between nodes Branching ratios Branching ratios Network distances Network distances and travelling ≠ ℓ through nodes , ,… exists Interconnectedness Interconnectedness 1 � = 1. Nutshell Nutshell � 2 3 �−1  References References 12 23 34 �−2�−1 �−1� ⟺ � � � ⋯ � � triples Tr � � 1 2 2  # = (∑ ∑ [� ]ℓ − � ) 2 =1 ℓ=1 triangles Tr Clauset et al., 2006 [2]: NCAA football 3 1 ...... # = � . . . 21 of 40 . . . 25 of 40 6

PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Properties Properties of Complex Complex Networks Networks

Properties of 7. concurrency: Properties of Complex Complex Networks  Networks A problem transmission of a contagious element only occurs A problem Degree distributions Degree distributions  For sparse networks, tends to discount highly Assortativity during contact Assortativity Clustering Clustering connected nodes. Motifs  rather obvious but easily missed in a simple model Motifs 1 Concurrency Concurrency  is a useful and often� preferred variant Branching ratios  Branching ratios Network distances dynamic property—static networks are not Network distances  In general, . Interconnectedness enough Interconnectedness 2 Nutshell Nutshell  �  knowledge of previous contacts crucial is a global average of a local ratio. References References 1 2  is a ratio� of≠ two � global quantities.  beware cumulated network data 1 �  Kretzschmar and Morris, 1996 [4] 2 �  “Temporal networks” become a concrete area of study for Piranha Physicus in 2013.

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PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Properties Properties of Complex Complex Networks Networks 8. Horton-Strahler ratios: Properties of Properties of Complex  Complex Networks Metrics for branching networks: Networks 5. motifs: A problem  Method for ordering streams hierarchically A problem Degree distributions Degree distributions Assortativity  Number: Assortativity  small, recurring functional subnetworks Clustering Clustering Motifs  Segment length: Motifs  e.g., Feed Forward Loop: Concurrency  Area/Volume:� � �+1 Concurrency Branching ratios � = � /� Branching ratios Network distances �+1 � Network distances Interconnectedness � = ⟨ ⟩/⟨ ⟩ Interconnectedness � �+1 � Nutshell � = ⟨� ⟩/⟨� ⟩ Nutshell References References

Shen-Orr, Uri Alon, et al. [7]

...... 24 of 40 . . . 29 of 40 PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Properties Properties of Complex Complex Networks Networks Interconnected networks and robustness (two for 9. network distances: Properties of Properties of Complex one deal): Complex Networks Networks (a) shortest path length : A problem A problem Degree distributions “Catastrophic cascade of failures in interdependent Degree distributions Assortativity networks” [1]. Buldyrev et al., Nature 2010. Assortativity  Fewest number of steps between nodes and . Clustering Clustering Motifs abc Motifs � Concurrency Concurrency  (Also called the chemical distance between and Branching ratios Branching ratios Network distances Network distances .) Interconnectedness Interconnectedness Nutshell Nutshell References References (b) average path length :  Average shortest path length in whole network. Figure 1 | Modelling a blackout in Italy. Illustration of an iterative process of at the next step are marked in green. b, Additional nodes that were ⟨� ⟩ a cascade of failures using real-world data from a power network (located on disconnected from the Internet communication network giant component  Good algorithms exist for calculation. the map of Italy) and an Internet network (shifted above the map) that were are removed (red nodes above map). As a result the power stations implicated in an electrical blackout that occurred in Italy in September depending on them are removed from the power network (red nodes on 200320. The networks are drawn using the real geographical locations and map). Again, the nodes that will be disconnected from the giant cluster at the every Internet server is connected to the geographically nearest power next step are marked in green. c, Additional nodes that were disconnected  station. a, One power station is removed (red node on map) from the power from the giant component of the power network are removed (red nodes on Weighted links can be accommodated. network and as a result the Internet nodes depending on it are removed from map) as well as the nodes in the Internet network that depend on them (red the Internet network (red nodes above the map). The nodes that will be nodes above map). disconnected from the giant cluster (a cluster that spans the entire network)

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PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Nutshell: Properties of Complex Complex Networks Networks Overview Key Points: 9. network distances: Properties of Properties of Complex  Complex Networks The field of complex networks came into existence Networks  network diameter max: A problem A problem Degree distributions in the late 1990s. Degree distributions Maximum shortest path length between any two Assortativity Assortativity Clustering  Explosion of papers and interest since 1998/99. Clustering nodes. � Motifs Motifs Concurrency  Hardened up much thinking about complex Concurrency  closeness cl : Branching ratios Branching ratios Network distances systems. Network distances Interconnectedness Interconnectedness Average ‘distance’ between−1 � any−1 two nodes. 2 Nutshell  Specific focus on networks that are large-scale, Nutshell  Closeness� handles= [∑ disconnected� /( )] networks References sparse, natural or man-made, evolving and References ( ) dynamic, and (crucially) measurable.  only when all nodes are isolated. cl  Three main (blurred) categories:  Closeness� = ∞ perhaps compresses too much into one 1. Physical (e.g., river networks), �number= ∞ 2. Interactional (e.g., social networks), 3. Abstract (e.g., thesauri).

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PoCS | @pocsvox PoCS | @pocsvox Properties Properties of Properties of Complex Complex Networks Networks

Properties of Properties of Complex Complex 10. centrality: Networks Networks A problem A problem  Degree distributions Degree distributions Many such measures of a node’s ‘importance.’ Assortativity Assortativity Clustering Clustering  ex 1: Degree centrality: . Motifs Motifs Concurrency Concurrency  ex 2: Node ’s betweenness Branching ratios Branching ratios Network distances Network distances = fraction of shortest paths that pass through . Interconnectedness Interconnectedness Nutshell Nutshell  ex 3: Edge ’s betweenness References References = fraction of shortest paths that travel along .  ex 4: Recursiveℓ centrality: Hubs and Authorities (Jon Kleinberg [3]) ℓ

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Properties of [1] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, Complex Networks and S. Havlin. A problem Degree distributions Catastrophic cascade of failures in interdependent Assortativity Clustering networks. Motifs Concurrency Nature, 464:1025–1028, 2010. pdf  Branching ratios Network distances Interconnectedness

[2] A. Clauset, C. Moore, and M. E. J. Newman. Nutshell

Structural inference of hierarchies in networks, References 2006. pdf  [3] J. M. Kleinberg. Authoritative sources in a hyperlinked environment. Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, 1998. pdf  ...... 38 of 40

PoCS | @pocsvox References II Properties of Complex Networks [4] M. Kretzschmar and M. Morris. Measures of concurrency in networks and the Properties of Complex spread of infectious disease. Networks A problem Math. Biosci., 133:165–95, 1996. pdf  Degree distributions Assortativity Clustering [5] M. Newman. Motifs Concurrency Assortative mixing in networks. Branching ratios Network distances Phys. Rev. Lett., 89:208701, 2002. pdf  Interconnectedness Nutshell

[6] M. E. J. Newman. References The structure and function of complex networks. SIAM Rev., 45(2):167–256, 2003. pdf  [7] S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon. Network motifs in the transcriptional regulation network of Escherichia coli. Nature Genetics, 31:64–68, 2002. pdf 

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PoCS | @pocsvox References III Properties of Complex Networks

Properties of Complex Networks A problem Degree distributions Assortativity Clustering Motifs [8] D. J. Watts and S. J. Strogatz. Concurrency Branching ratios Collective dynamics of ‘small-world’ networks. Network distances Nature, 393:440–442, 1998. pdf  Interconnectedness Nutshell

References

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