PoCS PoCS PoCS @pocsvox @pocsvox Properties @pocsvox Properties of Properties of Properties of Complex Some key aspects of real complex networks: Complex Complex Properties of Complex Networks Networks Networks Networks Last updated: 2019/08/26, 12:11:53  concurrency Properties of  distribution Properties of 2. /3. : Properties of Principles of Complex Systems | @pocsvox Complex  Complex Complex Networks hierarchical scaling Networks Networks  assortativity ∗  Social networks: Homophily  = birds of a feather CSYS/MATH 300, Fall, 2019 A problem  A problem A problem Degree distributions network distances Degree distributions  Degree distributions Assortativity  homophily Assortativity e.g., degree is standard property for sorting: Assortativity Clustering  Clustering measure degree-degree correlations. Clustering Motifs  clustering Motifs Motifs Dr. Nick Allgaier | @nallgaier Concurrency  efficiency Concurrency [5] Concurrency Branching ratios  Branching ratios  Assortative network: similar degree nodes Branching ratios Prof. Peter Sheridan Dodds | @peterdodds Network distances motifs Network distances Network distances Interconnectedness  interconnectedness Interconnectedness connecting to each other. Interconnectedness  Nutshell  robustness Nutshell Often social: company directors, coauthors, actors. Nutshell Dept. of Mathematics & Statistics | Vermont Complex Systems Center References References References Vermont Advanced Computing Core | University of Vermont  Disassortative network: high degree nodes  Plus coevolution of network structure connecting to low degree nodes. and processes on networks. Often techological or biological: Internet, WWW, Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. protein interactions, neural networks, food webs. is the elephant in the room that we are now all very aware of... ∗ ...... 1 of 39 . . . 6 of 39 . . . 14 of 39

PoCS PoCS PoCS Outline @pocsvox Properties @pocsvox Local socialness: @pocsvox Properties of Properties of Properties of Complex Complex Complex Networks Networks 4. Clustering: Networks Properties of Complex Networks 1. degree distribution A problem Properties of  is the probability that a randomly selected Properties of Properties of Complex 푘 Complex  Your friends tend to know Complex Degree distributions Networks node has degree . 푃 Networks Networks A problem A problem each other. A problem 푘 Assortativity Degree distributions  푃 = node degree = number of connections. Degree distributions Degree distributions Assortativity Assortativity  Two measures (explained Assortativity Clustering Clustering 푘 Clustering Clustering Motifs  ex 1: Erdős-Rényi random networks have Poisson Motifs on following slides): Motifs Motifs Concurrency Concurrency [8] Concurrency Branching ratios 푘degree distributions: Branching ratios 1. Watts & Strogatz Branching ratios Concurrency Network distances Network distances Network distances Interconnectedness Insert question from assignment 7  Interconnectedness Interconnectedness Branching ratios Nutshell Nutshell Nutshell

Network distances References References References 1 2 푖 푗1푗2 Interconnectedness ∑푗 푗 ∈풩 푎 푘 1 퐶 = ⟨ 푖 푖 ⟩ −⟨푘⟩ [6] 푖 푘 ⟨푘⟩ 2. Newman 푘 (푘 − 1)/2 Nutshell  ex 2: “Scale-free” networks:푃 = 푒 ‘hubs’. 푘! #triangles  link cost controls skew. −훾 푘 #triples References  hubs may facilitate or impede푃 contagion.∝ 푘 ⇒ 2 3 × 퐶 = ...... 2 of 39 . . . 9 of 39 . . . 16 of 39

PoCS PoCS PoCS A notable feature of large-scale networks: @pocsvox Properties @pocsvox @pocsvox Properties of Properties of  is the average fraction of Properties of Complex Complex Complex  Graphical renderings are often just a big mess. Networks Networks pairs of neighbors who are Networks 1 Example network: connected퐶 . Properties of Properties of  Fraction of pairs of Properties of Complex Complex Complex Networks Networks neighbors who are Networks Typical hairball A problem Note: A problem A problem Degree distributions Degree distributions connected is Degree distributions  number of nodes = 500 Assortativity  Erdős-Rényi random networks are a mathematical Assortativity Assortativity Clustering Clustering Clustering ⇐ Motifs construct. Motifs Motifs  number of edges = 1000 Concurrency Concurrency Concurrency 푁 Branching ratios Branching ratios Branching ratios Network distances  ‘Scale-free’ networks are growing networks that Network distances Network distances  1 2 푖 1 2 average degree = 4 Interconnectedness Interconnectedness Calculation of : 푗 푗 ∈풩 푗 푗 Interconnectedness 푚 form according to a plausible mechanism. ∑ 푎 Nutshell Nutshell Nutshell  where is푖 node푖 ’s degree, ⟨푘⟩ References Randomness is out there, just not to the degree of References 1 푘 (푘 − 1)/2 References  퐶 and is the set of ’s And even when renderings somehow look good: a completely random network. 푖 neighbors.푘 푖 “That is a very graphic analogy which aids 푖 understanding wonderfully while being, strictly  Averaging풩 over all nodes,푖 we speaking, wrong in every possible way” have: said Ponder [Stibbons] —Making Money, T. Pratchett. 푗 푗 푗1푗2∈풩푖 1 2  We need to extract digestible, meaningful aspects. 1 푛 ∑ 푎 1 푖 푖 ...... 푛 푖=1 푘 (푘 −1)/2 . . . 퐶 =푗 푗 ∈풩∑ 푗1푗2 = . . . 5 of 39 . . . 10 of 39 ∑ 1 2 푖 푎 . . . 17 of 39 푘푖(푘푖−1)/2 ⟨ ⟩푖 PoCS PoCS PoCS Triples and triangles @pocsvox Properties @pocsvox Properties @pocsvox Properties of Properties of Properties of Complex Complex Complex Example network: Networks Networks Networks  Nodes , , and form a 7. concurrency: triple around if is Properties of Properties of Properties of Complex Complex Complex 1 2 3  transmission of a contagious element only occurs connected푖 푖 to and푖 . Networks 5. motifs: Networks Networks 1 1 A problem A problem during contact A problem  Degree distributions Degree distributions Degree distributions Nodes , , and푖 푖 form a Assortativity  small, recurring functional subnetworks Assortativity Assortativity 2 3 triangle if each푖 pair of푖 nodes is Clustering Clustering  rather obvious but easily missed in a simple model Clustering Motifs  e.g., Feed Forward Loop: Motifs Motifs connected1 2 3 Concurrency Concurrency  dynamic property—static networks are not Concurrency Triangles: 푖 푖 푖 Branching ratios Branching ratios Branching ratios #triangles Network distances Network distances Network distances  Interconnectedness Interconnectedness enough Interconnectedness The definition #triples Nutshell Nutshell  Nutshell measures the fraction3× of knowledge of previous contacts crucial 2 References References References closed triples 퐶 =  beware cumulated network data  The ‘3’ appears because for Shen-Orr, Uri Alon, et al. [7]  Kretzschmar and Morris, 1996 [4] each triangle, we have 3 closed Triples:  “Temporal networks” become a concrete area of triples. study for Piranha Physicus in 2013.  Analysis (SNA): fraction of transitive triples.

...... 18 of 39 . . . 22 of 39 . . . 26 of 39

PoCS PoCS PoCS Clustering: @pocsvox Properties @pocsvox Properties @pocsvox Properties of Properties of Properties of Complex Complex Complex Sneaky counting for undirected, unweighted Networks Networks Networks networks: 6. modularity and structure/community 8. Horton-Strahler ratios: detection:  If the path – – exists then . Properties of Properties of  Properties of Complex Complex Metrics for branching networks: Complex Networks Networks  Method for ordering streams hierarchically Networks  Otherwise, . A problem A problem A problem 푖푗 푗ℓ 푖 푗 ℓ 푎 푎 = 1 Degree distributions Degree distributions  Number: Degree distributions  We want for good triples. Assortativity Assortativity Assortativity 푖푗 푗ℓ Clustering Clustering  Segment length: Clustering  푎 푎 = 0 Motifs Motifs  Area/Volume:푛 휔 휔+1 Motifs In general, a path of edges between nodes Concurrency Concurrency 푅 = 푁 /푁 Concurrency Branching ratios Branching ratios 푙 휔+1 휔 Branching ratios and travelling푖 ≠ ℓ through nodes , ,… exists Network distances Network distances 푅 = ⟨푙 ⟩/⟨푙 ⟩ Network distances 1 Interconnectedness Interconnectedness 푎 휔+1 휔 Interconnectedness 푛 = 1. 푖 푅 = ⟨푎 ⟩/⟨푎 ⟩ 푛 2 3 푛−1 Nutshell Nutshell Nutshell  푖 푖 푖 푖 References References References 푖1푖2 푖2푖3 푖3푖4 푖푛−2푖푛−1 푖푛−1푖푛 ⟺ 푎 푎 푎 ⋯ 푎 푎 triples Tr 푁 푁 1 2 2  # = (∑ ∑ [퐴 ]푖ℓ − 퐴 ) 2 푖=1 ℓ=1 triangles Tr Clauset et al., 2006 [2]: NCAA football 3 1 ...... # = 퐴 . . . 19 of 39 . . . 23 of 39 . . . 28 of 39 6 PoCS PoCS PoCS Properties @pocsvox Bipartite/multipartite affiliation structures: @pocsvox Properties @pocsvox Properties of Properties of Properties of Complex Complex Complex Networks  Many real-world Networks Networks networks have an 9. network distances: Properties of Properties of Properties of Complex underlying Complex Complex Networks multi-partite structure. Networks (a) shortest path length : Networks A problem A problem A problem  For sparse networks, tends to discount highly Degree distributions Degree distributions Degree distributions Assortativity Assortativity  Fewest number of steps between nodes and . Assortativity connected nodes. Clustering  Stories-tropes. Clustering 푖푗 Clustering 1 Motifs  Motifs  푑 Motifs  is a useful and often퐶 preferred variant Concurrency Boards and Concurrency (Also called the chemical distance between and Concurrency Branching ratios Branching ratios Branching ratios Network distances directors. Network distances .) Network distances  In general, . 푖 푗 Interconnectedness  Films-actors- Interconnectedness Interconnectedness 2 푖  퐶 is a global average of a local ratio. Nutshell directors. Nutshell Nutshell 1 2 References  References 푗 References  is a ratio퐶 of≠ two 퐶 global quantities. Classes-teachers- (b) average path length : 1 students. 퐶  Average shortest path length in whole network.  Upstairs- 푖푗 2 퐶 downstairs.  Good algorithms exist⟨푑 for⟩ calculation.  Unipartite networks  Weighted links can be accommodated. may be induced or co-exist...... 20 of 39 . . . 24 of 39 . . . 30 of 39 PoCS PoCS PoCS Properties @pocsvox Nutshell: @pocsvox References II @pocsvox Properties of Properties of Properties of Complex Complex Complex Networks Networks Networks Overview Key Points: [4] M. Kretzschmar and M. Morris. 9. network distances: Measures of concurrency in networks and the Properties of  The field of complex networks came into existence Properties of Properties of  Complex Complex spread of infectious disease. diameter max: Networks in the late 1990s. Networks Networks A problem A problem Math. Biosci., 133:165–95, 1996. pdf  A problem Maximum shortest path length between any two Degree distributions Degree distributions Degree distributions Assortativity  Explosion of papers and interest since 1998/99. Assortativity Assortativity nodes. 푑 Clustering Clustering [5] M. Newman. Clustering Motifs  Hardened up much thinking about complex Motifs Motifs Concurrency Concurrency Assortative mixing in networks. Concurrency  closeness cl : Branching ratios systems. Branching ratios Branching ratios Network distances Network distances Phys. Rev. Lett., 89:208701, 2002. pdf  Network distances Average ‘distance’ between−1 푛 any−1 two nodes. Interconnectedness Interconnectedness Interconnectedness 푖푗 푖푗 2  Specific focus on networks that are large-scale,  Closeness푑 handles= [∑ disconnected푑 /( )] networks Nutshell Nutshell Nutshell sparse, natural or man-made, evolving and [6] M. E. J. Newman. ( ) References References References dynamic, and (crucially) measurable. The structure and function of complex networks.  only when all nodes are isolated. SIAM Rev., 45(2):167–256, 2003. pdf  cl푖푗  Three main (blurred) categories:  Closeness푑 = ∞ perhaps compresses too much into one 1. Physical (e.g., river networks), [7] S. S. Shen-Orr, R. Milo, S. Mangan, and U. Alon. 푑number= ∞ 2. Interactional (e.g., social networks), Network motifs in the transcriptional regulation 3. Abstract (e.g., thesauri). network of Escherichia coli. Nature Genetics, 31:64–68, 2002. pdf 

...... 31 of 39 . . . 35 of 39 . . . 38 of 39

PoCS PoCS PoCS Properties @pocsvox @pocsvox References III @pocsvox Properties of Properties of Properties of Complex Complex Complex Networks Networks Networks

Properties of Properties of Properties of 10. centrality: Complex Complex Complex Networks Networks Networks  A problem A problem A problem Many such measures of a node’s ‘importance.’ Degree distributions Degree distributions Degree distributions Assortativity Assortativity Assortativity  ex 1: Degree centrality: . Clustering Clustering Clustering Motifs Motifs [8] D. J. Watts and S. J. Strogatz. Motifs  ex 2: Node ’s betweenness Concurrency Concurrency Concurrency 푖 Branching ratios Branching ratios Collective dynamics of ‘small-world’ networks. Branching ratios = fraction of shortest paths푘 that pass through . Network distances Network distances Network distances Interconnectedness Interconnectedness Nature, 393:440–442, 1998. pdf  Interconnectedness  ex 3: Edge 푖’s betweenness Nutshell Nutshell Nutshell = fraction of shortest paths that travel along .푖 References References References  ex 4: Recursiveℓ centrality: Hubs and Authorities (Jon Kleinberg [3]) ℓ

...... 32 of 39 . . . 37 of 39 . . . 39 of 39

PoCS scale-free-networks, PoCS Properties @pocsvox @pocsvox Properties of Properties of Complex References I Complex Networks Networks Interconnected networks and robustness (two for Properties of [1] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, Properties of one deal): Complex Complex Networks and S. Havlin. Networks “Catastrophic cascade of failures in interdependent A problem A problem [1] Degree distributions Catastrophic cascade of failures in interdependent Degree distributions networks” . Buldyrev et al., Nature 2010. Assortativity Assortativity Clustering Clustering abc networks. Motifs Motifs Concurrency Nature, 464:1025–1028, 2010. pdf  Concurrency Branching ratios Branching ratios Network distances Network distances Interconnectedness [2] A. Clauset, C. Moore, and M. E. J. Newman. Interconnectedness Nutshell Structural inference of hierarchies in networks, Nutshell References 2006. pdf  References [3] J. M. Kleinberg. 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 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 Authoritative sources in a hyperlinked 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 environment. 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) Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, 1998. pdf  ...... 34 of 39 . . . 37 of 39