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 Degree distributions A problem Degree distributions Degree distributions Assortativity 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?
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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 degree distribution Complex Networks hierarchical scaling Networks A problem A problem Degree distributions assortativity ∗ Degree distributions Assortativity network distances Assortativity Clustering homophily Clustering Motifs centrality Motifs Concurrency clustering Concurrency Branching ratios Branching ratios Network distances efficiency Network distances Interconnectedness motifs Interconnectedness interconnectedness Nutshell modularity 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 1 2 ∑ ∈� � 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 : 1 2∈� 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: